Multi-Grid Monte Carlo via $XY$ Embedding I. General Theory and Two-Dimensional $O(N)$-Symmetric Nonlinear $\sigma$-Models
Tereza Mendes, Andrea Pelissetto, Alan D. Sokal (New York, University)
TL;DR
This paper develops a multi-grid Monte Carlo method using XY embedding for 2D nonlinear sigma models, demonstrating reduced critical slowing-down and confirming theoretical predictions for correlation lengths.
Contribution
It introduces a novel MGMC variant based on XY embedding and analyzes its effectiveness on 2D $O(N)$ models, providing both theoretical insights and empirical results.
Findings
Critical exponents $z_{int,{\
static correlation length matches four-loop predictions
Critical slowing-down is significantly reduced but not eliminated
Abstract
We introduce a variant of the multi-grid Monte Carlo (MGMC) method, based on the embedding of an model into the target model, and we study its mathematical properties for a variety of nonlinear -models. We then apply the method to the two-dimensional -symmetric nonlinear -models (also called -vector models) with and study its dynamic critical behavior. Using lattices up to , we find dynamic critical exponents , , for , respectively (subjective 68\% confidence intervals). Thus, for these asymptotically free models, critical slowing-down is greatly reduced compared to local algorithms, but not completely eliminated; and the dynamic critical exponent does apparently vary with . We also analyze the static data for using a finite-size-scaling…
| 3-vector model: static data | |||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 32 | 1.500 | 143.20 ( | 0.80) | 6.34 ( | 0.04) | 10.08 ( | 0.06) | 3.82 ( | 0.03) | 0.603146 ( | 96) | 0.497267 ( | 61) |
| 64 | 1.450 | 114.61 ( | 0.56) | 4.03 ( | 0.01) | 8.49 ( | 0.06) | 2.41 ( | 0.05) | 0.582513 ( | 26) | 0.484598 ( | 16) |
| 64 | 1.500 | 177.31 ( | 0.81) | 5.33 ( | 0.01) | 11.17 ( | 0.06) | 3.33 ( | 0.03) | 0.601747 ( | 23) | 0.496433 ( | 15) |
| 64 | 1.550 | 265.44 ( | 1.24) | 7.40 ( | 0.02) | 14.06 ( | 0.07) | 4.66 ( | 0.03) | 0.619546 ( | 22) | 0.508044 ( | 15) |
| 64 | 1.600 | 397.54 ( | 1.58) | 11.10 ( | 0.04) | 18.11 ( | 0.07) | 6.62 ( | 0.03) | 0.636082 ( | 20) | 0.519465 ( | 14) |
| 64 | 1.650 | 545.23 ( | 2.26) | 16.85 ( | 0.07) | 22.09 ( | 0.10) | 8.74 ( | 0.04) | 0.651148 ( | 23) | 0.530473 ( | 17) |
| 64 | 1.700 | 702.27 ( | 2.15) | 25.02 ( | 0.10) | 26.34 ( | 0.10) | 10.89 ( | 0.04) | 0.664824 ( | 21) | 0.540996 ( | 16) |
| 64 | 1.725 | 768.21 ( | 2.06) | 29.55 ( | 0.11) | 27.91 ( | 0.09) | 11.77 ( | 0.04) | 0.671122 ( | 20) | 0.546022 ( | 16) |
| 64 | 1.750 | 834.56 ( | 1.81) | 34.51 ( | 0.12) | 29.64 ( | 0.08) | 12.65 ( | 0.04) | 0.677156 ( | 19) | 0.550953 ( | 16) |
| 64 | 1.775 | 895.44 ( | 2.11) | 40.06 ( | 0.16) | 31.11 ( | 0.10) | 13.53 ( | 0.05) | 0.682911 ( | 22) | 0.555767 ( | 19) |
| 64 | 1.800 | 956.71 ( | 1.87) | 46.13 ( | 0.17) | 32.69 ( | 0.10) | 14.40 ( | 0.05) | 0.688427 ( | 21) | 0.560468 ( | 19) |
| 64 | 1.850 | 1054.35 ( | 1.69) | 57.81 ( | 0.19) | 34.90 ( | 0.09) | 15.76 ( | 0.05) | 0.698672 ( | 20) | 0.569463 ( | 18) |
| 64 | 1.900 | 1147.05 ( | 2.09) | 70.87 ( | 0.29) | 37.09 ( | 0.12) | 17.08 ( | 0.07) | 0.708203 ( | 27) | 0.578129 ( | 26) |
| 64 | 1.950 | 1224.55 ( | 1.94) | 83.83 ( | 0.32) | 38.58 ( | 0.12) | 18.06 ( | 0.06) | 0.717119 ( | 25) | 0.586526 ( | 25) |
| 128 | 1.450 | 115.43 ( | 0.66) | 4.01 ( | 0.01) | 8.48 ( | 0.18) | 2.19 ( | 0.22) | 0.582493 ( | 16) | 0.484605 ( | 10) |
| 128 | 1.500 | 175.61 ( | 1.15) | 5.17 ( | 0.01) | 11.11 ( | 0.17) | 2.92 ( | 0.18) | 0.601597 ( | 16) | 0.496336 ( | 10) |
| 128 | 1.550 | 277.80 ( | 2.03) | 6.95 ( | 0.02) | 14.46 ( | 0.18) | 4.31 ( | 0.13) | 0.619374 ( | 16) | 0.507931 ( | 11) |
| 128 | 1.575 | 351.79 ( | 2.01) | 8.07 ( | 0.02) | 16.52 ( | 0.13) | 4.62 ( | 0.09) | 0.627728 ( | 11) | 0.513626 ( | 7) |
| 128 | 1.600 | 451.87 ( | 2.78) | 9.51 ( | 0.02) | 19.22 ( | 0.15) | 5.43 ( | 0.09) | 0.635722 ( | 10) | 0.519229 ( | 7) |
| 128 | 1.650 | 729.57 ( | 4.50) | 13.72 ( | 0.04) | 25.29 ( | 0.16) | 7.76 ( | 0.08) | 0.650668 ( | 10) | 0.530135 ( | 7) |
| 128 | 1.680 | 963.04 ( | 5.95) | 17.74 ( | 0.07) | 29.64 ( | 0.18) | 9.91 ( | 0.08) | 0.659010 ( | 9) | 0.536485 ( | 7) |
| 128 | 1.700 | 1159.82 ( | 4.95) | 21.44 ( | 0.07) | 33.23 ( | 0.14) | 11.64 ( | 0.06) | 0.664314 ( | 6) | 0.540623 ( | 5) |
| 128 | 1.720 | 1375.13 ( | 5.44) | 26.06 ( | 0.08) | 36.95 ( | 0.14) | 13.48 ( | 0.06) | 0.669437 ( | 6) | 0.544697 ( | 5) |
| 128 | 1.740 | 1580.42 ( | 5.76) | 31.49 ( | 0.11) | 40.16 ( | 0.15) | 15.28 ( | 0.06) | 0.674350 ( | 6) | 0.548676 ( | 5) |
| 128 | 1.750 | 1707.43 ( | 7.99) | 34.73 ( | 0.16) | 42.28 ( | 0.20) | 16.20 ( | 0.09) | 0.676748 ( | 8) | 0.550646 ( | 7) |
| 128 | 1.770 | 1933.84 ( | 7.98) | 41.79 ( | 0.19) | 45.79 ( | 0.20) | 17.98 ( | 0.08) | 0.681412 ( | 8) | 0.554521 ( | 7) |
| 128 | 1.800 | 2264.75 ( | 8.85) | 54.13 ( | 0.27) | 50.90 ( | 0.23) | 20.64 ( | 0.10) | 0.688095 ( | 9) | 0.560202 ( | 8) |
| 128 | 1.850 | 2774.23 ( | 10.10) | 79.97 ( | 0.42) | 58.39 ( | 0.27) | 24.89 ( | 0.11) | 0.698460 ( | 10) | 0.569294 ( | 9) |
| 128 | 1.900 | 3185.97 ( | 8.70) | 107.63 ( | 0.49) | 63.69 ( | 0.24) | 27.88 ( | 0.11) | 0.708021 ( | 10) | 0.577985 ( | 9) |
| 128 | 1.950 | 3580.66 ( | 6.58) | 140.95 ( | 0.53) | 69.23 ( | 0.20) | 31.07 ( | 0.10) | 0.716928 ( | 9) | 0.586342 ( | 9) |
| 128 | 2.000 | 3915.42 ( | 8.57) | 175.97 ( | 0.84) | 72.99 ( | 0.28) | 33.50 ( | 0.15) | 0.725183 ( | 12) | 0.594332 ( | 12) |
| 256 | 1.450 | 116.51 ( | 0.59) | 4.03 ( | 0.01) | 9.05 ( | 0.53) | 2.79 ( | 0.68) | 0.582516 ( | 8) | 0.484616 ( | 5) |
| 256 | 1.550 | 278.41 ( | 1.87) | 6.92 ( | 0.02) | 14.20 ( | 0.47) | 4.59 ( | 0.47) | 0.619387 ( | 8) | 0.507942 ( | 5) |
| 256 | 1.650 | 738.30 ( | 6.57) | 13.23 ( | 0.04) | 25.07 ( | 0.45) | 7.53 ( | 0.34) | 0.650642 ( | 7) | 0.530118 ( | 5) |
| 256 | 1.700 | 1261.53 ( | 9.26) | 19.02 ( | 0.05) | 34.27 ( | 0.34) | 9.57 ( | 0.22) | 0.664240 ( | 5) | 0.540568 ( | 4) |
| 256 | 1.750 | 2151.82 ( | 16.78) | 28.55 ( | 0.09) | 46.74 ( | 0.39) | 14.45 ( | 0.18) | 0.676629 ( | 4) | 0.550554 ( | 3) |
| 256 | 1.800 | 3550.48 ( | 22.89) | 45.81 ( | 0.19) | 62.41 ( | 0.39) | 21.53 ( | 0.16) | 0.687953 ( | 3) | 0.560088 ( | 3) |
| 256 | 1.850 | 5468.72 ( | 28.65) | 77.19 ( | 0.36) | 81.45 ( | 0.42) | 30.87 ( | 0.16) | 0.698351 ( | 3) | 0.569199 ( | 3) |
| 256 | 1.900 | 7422.24 ( | 34.60) | 124.83 ( | 0.67) | 98.82 ( | 0.51) | 39.95 ( | 0.20) | 0.707952 ( | 3) | 0.577921 ( | 3) |
| 256 | 2.000 | 10832.90 ( | 34.75) | 264.76 ( | 1.38) | 126.41 ( | 0.54) | 54.98 ( | 0.25) | 0.725141 ( | 4) | 0.594293 ( | 4) |
| 4-vector model: static data | |||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 32 | 2.000 | 92.57 ( | 0.58) | 5.48 ( | 0.03) | 7.57 ( | 0.05) | 2.76 ( | 0.03) | 0.577469 ( | 86) | 0.435869 ( | 65) |
| 32 | 2.100 | 134.79 ( | 0.72) | 8.39 ( | 0.05) | 9.65 ( | 0.05) | 3.89 ( | 0.03) | 0.602061 ( | 86) | 0.454645 ( | 69) |
| 32 | 2.200 | 180.64 ( | 0.74) | 12.75 ( | 0.07) | 11.75 ( | 0.05) | 5.07 ( | 0.03) | 0.623979 ( | 76) | 0.472560 ( | 66) |
| 64 | 2.100 | 160.69 ( | 1.02) | 7.04 ( | 0.02) | 10.40 ( | 0.08) | 3.26 ( | 0.05) | 0.600938 ( | 29) | 0.453774 ( | 23) |
| 64 | 2.150 | 205.83 ( | 1.09) | 8.61 ( | 0.02) | 12.13 ( | 0.07) | 4.10 ( | 0.04) | 0.612116 ( | 23) | 0.462739 ( | 19) |
| 64 | 2.200 | 257.32 ( | 1.33) | 10.57 ( | 0.04) | 13.71 ( | 0.07) | 4.83 ( | 0.04) | 0.622688 ( | 22) | 0.471510 ( | 19) |
| 64 | 2.250 | 325.27 ( | 1.59) | 13.34 ( | 0.05) | 15.89 ( | 0.08) | 5.89 ( | 0.04) | 0.632745 ( | 22) | 0.480107 ( | 19) |
| 64 | 2.300 | 394.34 ( | 1.36) | 16.82 ( | 0.06) | 17.81 ( | 0.06) | 6.98 ( | 0.03) | 0.642221 ( | 16) | 0.488456 ( | 15) |
| 64 | 2.350 | 471.57 ( | 1.85) | 21.36 ( | 0.09) | 19.83 ( | 0.08) | 8.10 ( | 0.04) | 0.651226 ( | 20) | 0.496598 ( | 19) |
| 64 | 2.400 | 553.20 ( | 1.45) | 27.09 ( | 0.09) | 21.97 ( | 0.06) | 9.33 ( | 0.03) | 0.659749 ( | 15) | 0.504527 ( | 14) |
| 128 | 2.250 | 346.48 ( | 3.88) | 11.91 ( | 0.05) | 16.20 ( | 0.27) | 4.91 ( | 0.17) | 0.632545 ( | 19) | 0.479943 ( | 17) |
| 128 | 2.350 | 577.66 ( | 6.79) | 17.74 ( | 0.10) | 21.70 ( | 0.29) | 6.80 ( | 0.16) | 0.650918 ( | 18) | 0.496337 ( | 16) |
| 128 | 2.450 | 939.66 ( | 7.63) | 28.32 ( | 0.16) | 28.74 ( | 0.24) | 10.35 ( | 0.12) | 0.667481 ( | 12) | 0.511894 ( | 11) |
| 128 | 2.550 | 1457.27 ( | 9.77) | 47.68 ( | 0.32) | 37.45 ( | 0.26) | 14.93 ( | 0.12) | 0.682445 ( | 11) | 0.526619 ( | 11) |
| 128 | 2.650 | 2004.60 ( | 9.45) | 77.35 ( | 0.47) | 45.51 ( | 0.24) | 19.54 ( | 0.12) | 0.695981 ( | 10) | 0.540488 ( | 11) |
| 8-vector model: static data | |||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 64 | 4.000 | 53.89 ( | 0.27) | 5.19 ( | 0.01) | 5.46 ( | 0.07) | 1.91 ( | 0.04) | 0.543613 ( | 29) | 0.347690 ( | 26) |
| 64 | 4.600 | 148.63 ( | 0.46) | 11.37 ( | 0.02) | 9.84 ( | 0.04) | 3.56 ( | 0.02) | 0.603866 ( | 13) | 0.405040 ( | 14) |
| 64 | 4.800 | 208.47 ( | 0.54) | 15.40 ( | 0.02) | 12.02 ( | 0.04) | 4.60 ( | 0.02) | 0.620987 ( | 10) | 0.422834 ( | 11) |
| 64 | 5.000 | 284.97 ( | 0.68) | 21.34 ( | 0.04) | 14.33 ( | 0.04) | 5.86 ( | 0.02) | 0.636812 ( | 10) | 0.439888 ( | 11) |
| 64 | 5.200 | 381.17 ( | 0.81) | 30.39 ( | 0.07) | 17.05 ( | 0.04) | 7.50 ( | 0.02) | 0.651454 ( | 9) | 0.456198 ( | 11) |
| 64 | 5.400 | 487.41 ( | 0.88) | 43.18 ( | 0.10) | 19.75 ( | 0.04) | 9.26 ( | 0.02) | 0.664983 ( | 9) | 0.471718 ( | 11) |
| 64 | 5.800 | 706.90 ( | 1.46) | 80.34 ( | 0.27) | 24.91 ( | 0.06) | 12.77 ( | 0.04) | 0.689116 ( | 15) | 0.500489 ( | 18) |
| 64 | 6.100 | 858.53 ( | 1.37) | 116.18 ( | 0.33) | 28.11 ( | 0.06) | 15.07 ( | 0.04) | 0.705067 ( | 14) | 0.520279 ( | 17) |
| 64 | 6.400 | 1005.06 ( | 1.27) | 159.23 ( | 0.38) | 31.30 ( | 0.06) | 17.35 ( | 0.04) | 0.719367 ( | 13) | 0.538548 ( | 17) |
| 64 | 6.600 | 1092.16 ( | 1.00) | 189.00 ( | 0.34) | 33.08 ( | 0.05) | 18.63 ( | 0.03) | 0.728185 ( | 10) | 0.550059 ( | 13) |
| 64 | 6.800 | 1177.94 ( | 0.97) | 221.60 ( | 0.36) | 34.86 ( | 0.05) | 19.94 ( | 0.03) | 0.736470 ( | 10) | 0.561049 ( | 13) |
| 64 | 7.100 | 1298.91 ( | 0.90) | 272.69 ( | 0.38) | 37.48 ( | 0.05) | 21.80 ( | 0.03) | 0.747930 ( | 9) | 0.576527 ( | 13) |
| 64 | 7.400 | 1405.80 ( | 0.90) | 323.59 ( | 0.43) | 39.62 ( | 0.05) | 23.35 ( | 0.03) | 0.758452 ( | 9) | 0.591024 ( | 12) |
| 64 | 7.800 | 1540.37 ( | 0.97) | 394.93 ( | 0.53) | 42.60 ( | 0.06) | 25.47 ( | 0.04) | 0.771157 ( | 10) | 0.608890 ( | 15) |
| 64 | 8.200 | 1660.89 ( | 0.96) | 466.20 ( | 0.58) | 45.26 ( | 0.07) | 27.36 ( | 0.04) | 0.782591 ( | 10) | 0.625310 ( | 14) |
| 64 | 8.700 | 1798.68 ( | 0.88) | 556.16 ( | 0.59) | 48.52 ( | 0.07) | 29.65 ( | 0.04) | 0.795351 ( | 9) | 0.644016 ( | 14) |
| 128 | 4.000 | 53.92 ( | 0.26) | 5.19 ( | 0.01) | 5.47 ( | 0.21) | 1.74 ( | 0.18) | 0.543583 ( | 14) | 0.347665 ( | 13) |
| 128 | 4.600 | 148.57 ( | 0.66) | 11.21 ( | 0.01) | 9.68 ( | 0.13) | 3.55 ( | 0.08) | 0.603836 ( | 9) | 0.405005 ( | 10) |
| 128 | 4.900 | 254.22 ( | 1.32) | 17.12 ( | 0.03) | 13.39 ( | 0.13) | 4.74 ( | 0.07) | 0.629018 ( | 9) | 0.431414 ( | 9) |
| 128 | 5.200 | 430.36 ( | 1.73) | 26.62 ( | 0.04) | 18.01 ( | 0.10) | 6.50 ( | 0.04) | 0.651306 ( | 6) | 0.456032 ( | 7) |
| 128 | 5.500 | 726.74 ( | 2.03) | 43.17 ( | 0.07) | 24.23 ( | 0.08) | 9.34 ( | 0.04) | 0.671168 ( | 4) | 0.478960 ( | 5) |
| 128 | 5.800 | 1175.09 ( | 2.89) | 74.24 ( | 0.17) | 31.96 ( | 0.08) | 13.73 ( | 0.04) | 0.688918 ( | 4) | 0.500251 ( | 4) |
| 128 | 5.930 | 1412.55 ( | 4.41) | 94.97 ( | 0.32) | 35.66 ( | 0.12) | 16.03 ( | 0.07) | 0.696030 ( | 5) | 0.508995 ( | 6) |
| 128 | 6.100 | 1731.01 ( | 4.59) | 128.76 ( | 0.43) | 40.07 ( | 0.12) | 18.94 ( | 0.07) | 0.704853 ( | 5) | 0.520012 ( | 6) |
| 128 | 6.400 | 2330.01 ( | 6.27) | 211.82 ( | 0.89) | 48.11 ( | 0.16) | 24.38 ( | 0.10) | 0.719216 ( | 7) | 0.538353 ( | 9) |
| 128 | 6.800 | 3080.42 ( | 5.92) | 359.97 ( | 1.23) | 56.98 ( | 0.15) | 30.75 ( | 0.10) | 0.736312 ( | 6) | 0.560839 ( | 8) |
| 128 | 7.100 | 3607.80 ( | 5.57) | 495.51 ( | 1.45) | 63.13 ( | 0.15) | 35.15 ( | 0.10) | 0.747807 ( | 6) | 0.576359 ( | 8) |
| 128 | 7.400 | 4093.51 ( | 5.05) | 644.66 ( | 1.58) | 68.54 ( | 0.15) | 39.01 ( | 0.09) | 0.758335 ( | 5) | 0.590861 ( | 8) |
| 128 | 7.800 | 4686.97 ( | 6.87) | 861.07 ( | 2.58) | 75.05 ( | 0.21) | 43.68 ( | 0.14) | 0.771073 ( | 7) | 0.608771 ( | 10) |
| 128 | 8.200 | 5227.49 ( | 6.30) | 1091.37 ( | 2.74) | 81.01 ( | 0.22) | 47.98 ( | 0.14) | 0.782502 ( | 7) | 0.625181 ( | 10) |
| 128 | 8.700 | 5857.80 ( | 5.58) | 1401.51 ( | 2.83) | 88.31 ( | 0.21) | 53.18 ( | 0.14) | 0.795294 ( | 7) | 0.643931 ( | 10) |
| 256 | 5.200 | 433.09 ( | 3.29) | 26.39 ( | 0.06) | 18.31 ( | 0.45) | 6.38 ( | 0.29) | 0.651300 ( | 6) | 0.456024 ( | 7) |
| 256 | 5.800 | 1291.13 ( | 7.12) | 66.30 ( | 0.13) | 33.48 ( | 0.27) | 11.91 ( | 0.13) | 0.688885 ( | 3) | 0.500210 ( | 4) |
| 256 | 6.100 | 2220.50 ( | 12.62) | 109.15 ( | 0.32) | 45.26 ( | 0.30) | 17.09 ( | 0.13) | 0.704805 ( | 3) | 0.519952 ( | 4) |
| 256 | 6.400 | 3650.42 ( | 15.65) | 188.23 ( | 0.67) | 59.60 ( | 0.27) | 24.87 ( | 0.14) | 0.719168 ( | 2) | 0.538291 ( | 3) |
| 256 | 6.600 | 4930.58 ( | 21.44) | 279.33 ( | 1.32) | 71.09 ( | 0.33) | 32.07 ( | 0.19) | 0.727985 ( | 2) | 0.549797 ( | 3) |
| 256 | 6.800 | 6250.21 ( | 19.41) | 399.68 ( | 1.58) | 81.14 ( | 0.28) | 38.54 ( | 0.16) | 0.736265 ( | 2) | 0.560776 ( | 3) |
| 256 | 7.100 | 8345.02 ( | 26.99) | 655.40 ( | 3.32) | 96.18 ( | 0.38) | 48.79 ( | 0.23) | 0.747763 ( | 3) | 0.576300 ( | 4) |
| 256 | 7.800 | 13095.48 ( | 22.82) | 1583.24 ( | 5.25) | 126.95 ( | 0.35) | 70.85 ( | 0.22) | 0.771032 ( | 3) | 0.608712 ( | 4) |
| 256 | 8.200 | 15497.34 ( | 29.21) | 2255.87 ( | 8.48) | 141.80 ( | 0.47) | 81.41 ( | 0.30) | 0.782477 ( | 4) | 0.625145 ( | 5) |
| 3-vector model: dynamic data | |||||||||||
| Sweeps | Discard | ||||||||||
| 32 | 1.500 | 105000 | 5000 | 7.16 ( | 0.30) | 3.43 ( | 0.10) | 2.45 ( | 0.06) | 2.00 ( | 0.04) |
| 64 | 1.450 | 305000 | 5000 | 6.54 ( | 0.15) | 1.43 ( | 0.02) | 2.12 ( | 0.03) | 1.75 ( | 0.02) |
| 64 | 1.500 | 405000 | 5000 | 8.92 ( | 0.21) | 2.04 ( | 0.02) | 2.24 ( | 0.03) | 1.85 ( | 0.02) |
| 64 | 1.550 | 405000 | 5000 | 11.42 ( | 0.30) | 3.42 ( | 0.05) | 2.38 ( | 0.03) | 1.94 ( | 0.02) |
| 64 | 1.600 | 485000 | 5000 | 14.35 ( | 0.38) | 5.42 ( | 0.09) | 2.38 ( | 0.03) | 1.96 ( | 0.02) |
| 64 | 1.650 | 305000 | 5000 | 14.69 ( | 0.50) | 6.27 ( | 0.14) | 2.29 ( | 0.03) | 1.93 ( | 0.02) |
| 64 | 1.700 | 300000 | 5000 | 12.96 ( | 0.42) | 6.15 ( | 0.14) | 1.98 ( | 0.03) | 1.74 ( | 0.02) |
| 64 | 1.725 | 300000 | 5000 | 12.13 ( | 0.38) | 5.99 ( | 0.13) | 1.94 ( | 0.02) | 1.74 ( | 0.02) |
| 64 | 1.750 | 300000 | 5000 | 10.12 ( | 0.29) | 5.46 ( | 0.12) | 1.86 ( | 0.02) | 1.68 ( | 0.02) |
| 64 | 1.775 | 200000 | 5000 | 9.20 ( | 0.31) | 5.27 ( | 0.13) | 1.71 ( | 0.02) | 1.57 ( | 0.02) |
| 64 | 1.800 | 200000 | 5000 | 7.85 ( | 0.24) | 5.07 ( | 0.13) | 1.66 ( | 0.02) | 1.57 ( | 0.02) |
| 64 | 1.850 | 200000 | 5000 | 6.76 ( | 0.20) | 4.55 ( | 0.11) | 1.59 ( | 0.02) | 1.50 ( | 0.02) |
| 64 | 1.900 | 100000 | 5000 | 5.52 ( | 0.21) | 4.27 ( | 0.14) | 1.54 ( | 0.03) | 1.47 ( | 0.03) |
| 64 | 1.950 | 100000 | 5000 | 4.95 ( | 0.18) | 4.03 ( | 0.13) | 1.48 ( | 0.03) | 1.43 ( | 0.03) |
| 128 | 1.450 | 200000 | 5000 | 4.99 ( | 0.12) | 1.14 ( | 0.01) | 2.12 ( | 0.03) | 1.75 ( | 0.03) |
| 128 | 1.500 | 200000 | 5000 | 6.75 ( | 0.19) | 1.33 ( | 0.02) | 2.25 ( | 0.04) | 1.86 ( | 0.03) |
| 128 | 1.550 | 200000 | 5000 | 9.03 ( | 0.30) | 1.64 ( | 0.02) | 2.25 ( | 0.04) | 1.88 ( | 0.03) |
| 128 | 1.575 | 400000 | 5000 | 11.48 ( | 0.30) | 1.97 ( | 0.02) | 2.24 ( | 0.03) | 1.87 ( | 0.02) |
| 128 | 1.600 | 400000 | 5000 | 14.29 ( | 0.42) | 2.38 ( | 0.03) | 2.24 ( | 0.03) | 1.88 ( | 0.02) |
| 128 | 1.650 | 400000 | 5000 | 17.93 ( | 0.59) | 3.99 ( | 0.06) | 2.21 ( | 0.03) | 1.87 ( | 0.02) |
| 128 | 1.680 | 400000 | 5000 | 21.34 ( | 0.77) | 5.99 ( | 0.11) | 2.10 ( | 0.02) | 1.80 ( | 0.02) |
| 128 | 1.700 | 800000 | 5000 | 24.08 ( | 0.65) | 7.38 ( | 0.11) | 2.06 ( | 0.02) | 1.78 ( | 0.01) |
| 128 | 1.720 | 800000 | 5000 | 24.80 ( | 0.68) | 8.44 ( | 0.13) | 2.04 ( | 0.02) | 1.79 ( | 0.01) |
| 128 | 1.740 | 800000 | 5000 | 24.73 ( | 0.68) | 9.14 ( | 0.15) | 1.93 ( | 0.01) | 1.73 ( | 0.01) |
| 128 | 1.750 | 400000 | 5000 | 23.06 ( | 0.86) | 9.32 ( | 0.22) | 1.95 ( | 0.02) | 1.74 ( | 0.02) |
| 128 | 1.770 | 400000 | 5000 | 21.99 ( | 0.80) | 9.07 ( | 0.21) | 1.90 ( | 0.02) | 1.71 ( | 0.02) |
| 128 | 1.800 | 300000 | 5000 | 19.79 ( | 0.79) | 9.11 ( | 0.25) | 1.81 ( | 0.02) | 1.65 ( | 0.02) |
| 128 | 1.850 | 200000 | 5000 | 17.86 ( | 0.84) | 8.27 ( | 0.26) | 1.66 ( | 0.02) | 1.58 ( | 0.02) |
| 128 | 1.900 | 200000 | 5000 | 14.10 ( | 0.59) | 7.39 ( | 0.22) | 1.60 ( | 0.02) | 1.55 ( | 0.02) |
| 128 | 1.950 | 200000 | 5000 | 9.09 ( | 0.30) | 6.16 ( | 0.17) | 1.52 ( | 0.02) | 1.47 ( | 0.02) |
| 128 | 2.000 | 100000 | 5000 | 7.71 ( | 0.34) | 5.56 ( | 0.21) | 1.44 ( | 0.03) | 1.41 ( | 0.03) |
| 256 | 1.450 | 200000 | 5000 | 3.83 ( | 0.08) | 1.05 ( | 0.01) | 2.06 ( | 0.03) | 1.74 ( | 0.03) |
| 256 | 1.550 | 200000 | 5000 | 6.83 ( | 0.20) | 1.36 ( | 0.02) | 2.27 ( | 0.04) | 1.90 ( | 0.03) |
| 256 | 1.650 | 200000 | 5000 | 12.87 ( | 0.51) | 2.06 ( | 0.03) | 2.08 ( | 0.03) | 1.79 ( | 0.03) |
| 256 | 1.700 | 400000 | 5000 | 19.52 ( | 0.67) | 2.80 ( | 0.04) | 2.06 ( | 0.02) | 1.78 ( | 0.02) |
| 256 | 1.750 | 400000 | 5000 | 26.65 ( | 1.07) | 4.74 ( | 0.08) | 1.87 ( | 0.02) | 1.66 ( | 0.02) |
| 256 | 1.800 | 600000 | 5000 | 36.89 ( | 1.42) | 9.90 ( | 0.20) | 1.81 ( | 0.02) | 1.66 ( | 0.01) |
| 256 | 1.850 | 600000 | 5000 | 39.86 ( | 1.60) | 13.09 ( | 0.30) | 1.71 ( | 0.01) | 1.59 ( | 0.01) |
| 256 | 1.900 | 400000 | 5000 | 34.30 ( | 1.57) | 13.57 ( | 0.39) | 1.61 ( | 0.02) | 1.54 ( | 0.01) |
| 256 | 2.000 | 200000 | 5000 | 19.26 ( | 0.94) | 9.64 ( | 0.33) | 1.47 ( | 0.02) | 1.43 ( | 0.02) |
| 4-vector model: dynamic data | |||||||||||
| Sweeps | Discard | ||||||||||
| 32 | 2.000 | 105000 | 5000 | 7.84 ( | 0.34) | 3.88 ( | 0.12) | 2.98 ( | 0.08) | 2.70 ( | 0.07) |
| 32 | 2.100 | 105000 | 5000 | 8.91 ( | 0.41) | 5.37 ( | 0.19) | 3.19 ( | 0.09) | 2.88 ( | 0.08) |
| 32 | 2.200 | 105000 | 5000 | 8.50 ( | 0.38) | 5.86 ( | 0.22) | 2.86 ( | 0.07) | 2.69 ( | 0.07) |
| 64 | 2.100 | 205000 | 5000 | 11.03 ( | 0.40) | 3.23 ( | 0.06) | 2.95 ( | 0.06) | 2.70 ( | 0.05) |
| 64 | 2.150 | 305000 | 5000 | 12.92 ( | 0.42) | 4.39 ( | 0.08) | 2.94 ( | 0.05) | 2.72 ( | 0.04) |
| 64 | 2.200 | 305000 | 5000 | 14.17 ( | 0.48) | 5.78 ( | 0.12) | 2.95 ( | 0.05) | 2.71 ( | 0.04) |
| 64 | 2.250 | 305000 | 5000 | 15.70 ( | 0.56) | 7.20 ( | 0.17) | 2.94 ( | 0.05) | 2.72 ( | 0.04) |
| 64 | 2.300 | 505000 | 5000 | 15.90 ( | 0.44) | 8.47 ( | 0.17) | 2.90 ( | 0.03) | 2.72 ( | 0.03) |
| 64 | 2.350 | 305000 | 5000 | 15.59 ( | 0.55) | 8.91 ( | 0.24) | 2.86 ( | 0.04) | 2.70 ( | 0.04) |
| 64 | 2.400 | 525000 | 5000 | 15.50 ( | 0.41) | 9.48 ( | 0.20) | 2.77 ( | 0.03) | 2.63 ( | 0.03) |
| 128 | 2.250 | 100000 | 5000 | 14.33 ( | 0.86) | 3.31 ( | 0.10) | 2.82 ( | 0.08) | 2.68 ( | 0.07) |
| 128 | 2.350 | 100000 | 5000 | 18.82 ( | 1.30) | 5.49 ( | 0.20) | 2.74 ( | 0.07) | 2.60 ( | 0.07) |
| 128 | 2.450 | 200000 | 5000 | 24.15 ( | 1.32) | 9.82 ( | 0.34) | 2.77 ( | 0.05) | 2.62 ( | 0.05) |
| 128 | 2.550 | 200000 | 5000 | 26.08 ( | 1.48) | 13.60 ( | 0.56) | 2.61 ( | 0.05) | 2.50 ( | 0.04) |
| 128 | 2.650 | 200000 | 5000 | 21.34 ( | 1.09) | 12.97 ( | 0.52) | 2.53 ( | 0.04) | 2.46 ( | 0.04) |
| 8-vector model: dynamic data | |||||||||||
| Sweeps | Discard | ||||||||||
| 64 | 4.000 | 200000 | 5000 | 10.60 ( | 0.38) | 3.79 ( | 0.08) | 6.03 ( | 0.16) | 5.89 ( | 0.16) |
| 64 | 4.600 | 800000 | 5000 | 20.57 ( | 0.51) | 9.39 ( | 0.16) | 6.33 ( | 0.09) | 6.21 ( | 0.08) |
| 64 | 4.800 | 1200000 | 5000 | 25.99 ( | 0.59) | 14.42 ( | 0.25) | 6.30 ( | 0.07) | 6.21 ( | 0.07) |
| 64 | 5.000 | 1200000 | 5000 | 27.99 ( | 0.66) | 18.92 ( | 0.37) | 6.35 ( | 0.07) | 6.25 ( | 0.07) |
| 64 | 5.200 | 1200000 | 5000 | 29.59 ( | 0.72) | 22.69 ( | 0.48) | 6.24 ( | 0.07) | 6.16 ( | 0.07) |
| 64 | 5.400 | 1200000 | 5000 | 29.62 ( | 0.72) | 24.13 ( | 0.53) | 6.26 ( | 0.07) | 6.19 ( | 0.07) |
| 64 | 5.800 | 400000 | 5000 | 25.42 ( | 1.00) | 22.92 ( | 0.86) | 6.12 ( | 0.12) | 6.10 ( | 0.12) |
| 64 | 6.100 | 400000 | 5000 | 22.39 ( | 0.83) | 20.78 ( | 0.74) | 5.96 ( | 0.11) | 5.97 ( | 0.11) |
| 64 | 6.400 | 400000 | 5000 | 20.49 ( | 0.72) | 19.66 ( | 0.68) | 5.98 ( | 0.11) | 5.98 ( | 0.11) |
| 64 | 6.600 | 600000 | 5000 | 19.65 ( | 0.55) | 18.88 ( | 0.52) | 5.86 ( | 0.09) | 5.85 ( | 0.09) |
| 64 | 6.800 | 600000 | 5000 | 19.29 ( | 0.54) | 18.54 ( | 0.51) | 5.74 ( | 0.09) | 5.73 ( | 0.09) |
| 64 | 7.100 | 600000 | 5000 | 17.69 ( | 0.47) | 17.29 ( | 0.46) | 5.92 ( | 0.09) | 5.91 ( | 0.09) |
| 64 | 7.400 | 600000 | 5000 | 18.73 ( | 0.51) | 18.26 ( | 0.50) | 5.72 ( | 0.09) | 5.72 ( | 0.09) |
| 64 | 7.800 | 400000 | 5000 | 15.87 ( | 0.49) | 15.65 ( | 0.48) | 5.87 ( | 0.11) | 5.87 ( | 0.11) |
| 64 | 8.200 | 400000 | 5000 | 16.55 ( | 0.52) | 16.29 ( | 0.51) | 5.67 ( | 0.11) | 5.68 ( | 0.11) |
| 64 | 8.700 | 400000 | 5000 | 15.52 ( | 0.48) | 15.31 ( | 0.47) | 5.66 ( | 0.10) | 5.66 ( | 0.10) |
| 128 | 4.000 | 200000 | 5000 | 8.96 ( | 0.30) | 3.15 ( | 0.06) | 5.88 ( | 0.16) | 5.69 ( | 0.15) |
| 128 | 4.600 | 400000 | 5000 | 16.71 ( | 0.53) | 5.24 ( | 0.09) | 6.36 ( | 0.12) | 6.23 ( | 0.12) |
| 128 | 4.900 | 400000 | 5000 | 24.29 ( | 0.93) | 7.74 ( | 0.17) | 6.23 ( | 0.12) | 6.13 ( | 0.12) |
| 128 | 5.200 | 800000 | 5000 | 33.07 ( | 1.04) | 12.81 ( | 0.25) | 6.14 ( | 0.08) | 6.07 ( | 0.08) |
| 128 | 5.500 | 1600000 | 5000 | 40.85 ( | 1.01) | 22.58 ( | 0.42) | 6.18 ( | 0.06) | 6.12 ( | 0.06) |
| 128 | 5.800 | 1600000 | 5000 | 47.01 ( | 1.25) | 33.91 ( | 0.77) | 6.13 ( | 0.06) | 6.10 ( | 0.06) |
| 128 | 5.930 | 800000 | 5000 | 46.46 ( | 1.74) | 35.31 ( | 1.15) | 6.23 ( | 0.09) | 6.20 ( | 0.08) |
| 128 | 6.100 | 800000 | 5000 | 44.74 ( | 1.64) | 36.74 ( | 1.22) | 5.89 ( | 0.08) | 5.87 ( | 0.08) |
| 128 | 6.400 | 400000 | 5000 | 38.53 ( | 1.86) | 33.70 ( | 1.53) | 6.06 ( | 0.12) | 6.03 ( | 0.12) |
| 128 | 6.800 | 400000 | 5000 | 33.82 ( | 1.53) | 31.20 ( | 1.36) | 5.97 ( | 0.11) | 5.96 ( | 0.11) |
| 128 | 7.100 | 400000 | 5000 | 31.04 ( | 1.35) | 29.30 ( | 1.24) | 5.88 ( | 0.11) | 5.87 ( | 0.11) |
| 128 | 7.400 | 400000 | 5000 | 27.46 ( | 1.12) | 26.54 ( | 1.07) | 5.77 ( | 0.11) | 5.78 ( | 0.11) |
| 128 | 7.800 | 200000 | 5000 | 26.19 ( | 1.49) | 25.61 ( | 1.44) | 5.70 ( | 0.15) | 5.69 ( | 0.15) |
| 128 | 8.200 | 200000 | 5000 | 23.24 ( | 1.24) | 22.75 ( | 1.20) | 5.79 ( | 0.15) | 5.79 ( | 0.15) |
| 128 | 8.700 | 200000 | 5000 | 20.75 ( | 1.05) | 20.43 ( | 1.02) | 5.82 ( | 0.16) | 5.82 ( | 0.16) |
| 256 | 5.200 | 200000 | 5000 | 23.83 ( | 1.29) | 7.76 ( | 0.24) | 6.19 ( | 0.17) | 6.15 ( | 0.17) |
| 256 | 5.800 | 600000 | 5000 | 45.04 ( | 1.92) | 16.72 ( | 0.43) | 6.16 ( | 0.10) | 6.11 ( | 0.10) |
| 256 | 6.100 | 600000 | 5000 | 58.70 ( | 2.86) | 29.10 ( | 1.00) | 6.01 ( | 0.09) | 5.98 ( | 0.09) |
| 256 | 6.400 | 800000 | 5000 | 63.40 ( | 2.77) | 43.93 ( | 1.60) | 6.08 ( | 0.08) | 6.07 ( | 0.08) |
| 256 | 6.600 | 600000 | 5000 | 65.95 ( | 3.40) | 51.56 ( | 2.35) | 5.74 ( | 0.09) | 5.74 ( | 0.09) |
| 256 | 6.800 | 800000 | 5000 | 64.12 ( | 2.82) | 51.32 ( | 2.02) | 5.86 ( | 0.08) | 5.87 ( | 0.08) |
| 256 | 7.100 | 400000 | 5000 | 56.07 ( | 3.27) | 49.35 ( | 2.70) | 5.91 ( | 0.11) | 5.91 ( | 0.11) |
| 256 | 7.800 | 400000 | 5000 | 40.33 ( | 2.00) | 38.33 ( | 1.85) | 5.76 ( | 0.11) | 5.75 ( | 0.11) |
| 256 | 8.200 | 200000 | 5000 | 35.39 ( | 2.34) | 34.03 ( | 2.20) | 5.89 ( | 0.16) | 5.92 ( | 0.16) |
| CPU timings (Cray C-90, ms/iteration) | ||||
|---|---|---|---|---|
| -embedding | direct | |||
| 32 | 14/15 | 14.5/17 | 16/24 | 19/20 |
| 64 | 38/42 | 39/45 | 42/61 | 50/52 |
| 128 | 113/123 | 116/135 | 128/178 | 136/140 |
| 256 | 380/405 | 388/432 | 426/580 | 400/410 |
| FSS curve, model | ||||||||||
|---|---|---|---|---|---|---|---|---|---|---|
| DF | ||||||||||
| ( 64, 64, 64) | 10 | 123.06 | 26.08 | 25.99 | 24.67 | 20.73 | 20.72 | 18.37 | 18.33 | |
| 0.0% | 2.5% | 1.7% | 1.6% | 3.6% | 2.3% | 3.1% | 1.9% | |||
| ( 64,128, 64) | 10 | 56.82 | 8.82 | 7.89 | 7.82 | 7.51 | 7.28 | 4.70 | 4.70 | |
| 0.0% | 55.0% | 54.5% | 45.1% | 37.8% | 29.6% | 45.4% | 32.0% | |||
| ( 64,128,128) | 10 | 10.95 | 5.92 | 5.65 | 4.87 | 3.59 | 3.58 | 0.00 | 0.00 | |
| 14.1% | 43.2% | 34.2% | 30.1% | 30.9% | 16.7% | 97.7% | —– | |||
| (128,128, 64) | 10 | 45.04 | 8.66 | 6.76 | 6.55 | 5.93 | 5.84 | 1.73 | 1.59 | |
| 0.0% | 37.2% | 45.4% | 36.5% | 31.3% | 21.1% | 63.1% | 45.1% | |||
| (128,128,128) | 10 | 8.01 | 5.62 | 4.33 | 3.18 | 0.41 | 0.00 | 0.00 | 0.00 | |
| 15.6% | 23.0% | 22.8% | 20.4% | 52.0% | —– | —– | —– | |||
| FSS curve, model | ||||||||||
|---|---|---|---|---|---|---|---|---|---|---|
| DF | ||||||||||
| ( 64, 64, 64) | 10 | 576.45 | 57.82 | 56.79 | 44.84 | 37.59 | 37.58 | 32.73 | 32.57 | |
| 0.0% | 0.0% | 0.0% | 0.0% | 0.0% | 0.0% | 0.0% | 0.0% | |||
| ( 64,128, 64) | 10 | 311.04 | 21.20 | 13.29 | 11.09 | 10.97 | 10.54 | 6.83 | 6.77 | |
| 0.0% | 2.0% | 15.0% | 19.7% | 14.0% | 10.4% | 23.4% | 14.9% | |||
| ( 64,128,128) | 10 | 22.43 | 11.84 | 8.54 | 6.89 | 3.93 | 3.81 | 0.84 | 0.00 | |
| 0.2% | 6.6% | 12.9% | 14.2% | 26.9% | 14.9% | 35.9% | —– | |||
| (128,128, 64) | 10 | 252.40 | 18.95 | 13.24 | 10.01 | 9.82 | 9.68 | 3.86 | 3.19 | |
| 0.0% | 1.5% | 6.6% | 12.4% | 8.0% | 4.6% | 27.7% | 20.3% | |||
| (128,128,128) | 10 | 19.43 | 11.72 | 7.56 | 5.94 | 0.36 | 0.00 | 0.00 | 0.00 | |
| 0.2% | 2.0% | 5.6% | 5.1% | 54.8% | —– | —– | —– | |||
| FSS curve, model | ||||||||||
|---|---|---|---|---|---|---|---|---|---|---|
| DF | ||||||||||
| ( 64, 64, 64) | 10 | 909.07 | 96.54 | 35.33 | 30.35 | 30.20 | 29.92 | 25.12 | 23.80 | |
| 0.0% | 0.0% | 0.1% | 0.2% | 0.1% | 0.1% | 0.3% | 0.2% | |||
| ( 64,128, 64) | 10 | 618.69 | 57.94 | 23.47 | 10.31 | 9.56 | 9.55 | 4.97 | 4.03 | |
| 0.0% | 0.0% | 0.5% | 24.4% | 21.5% | 14.5% | 42.0% | 40.3% | |||
| ( 64,128,128) | 10 | 173.90 | 22.91 | 7.31 | 7.26 | 3.57 | 1.68 | 0.06 | 0.00 | |
| 0.0% | 0.1% | 19.9% | 12.3% | 31.2% | 43.2% | 80.5% | —– | |||
| (128,128, 64) | 10 | 401.75 | 27.17 | 15.38 | 9.45 | 8.70 | 8.64 | 2.36 | 1.53 | |
| 0.0% | 0.1% | 3.1% | 15.0% | 12.2% | 7.1% | 50.1% | 46.6% | |||
| (128,128,128) | 10 | 86.43 | 13.64 | 6.75 | 6.68 | 0.01 | 0.00 | 0.00 | 0.00 | |
| 0.0% | 0.9% | 8.0% | 3.5% | 91.8% | —– | —– | —– | |||
| FSS curve, model | ||||||||||
|---|---|---|---|---|---|---|---|---|---|---|
| DF | ||||||||||
| ( 64, 64, 64) | 10 | 97.29 | 83.32 | 38.71 | 38.67 | 38.67 | 36.42 | 31.28 | 30.90 | |
| 0.0% | 0.0% | 0.0% | 0.0% | 0.0% | 0.0% | 0.0% | 0.0% | |||
| ( 64,128, 64) | 10 | 48.65 | 47.81 | 16.19 | 9.71 | 9.22 | 9.21 | 3.66 | 3.60 | |
| 0.0% | 0.0% | 6.3% | 28.6% | 23.7% | 16.2% | 60.0% | 46.3% | |||
| ( 64,128,128) | 10 | 10.21 | 6.04 | 5.90 | 5.84 | 1.51 | 1.12 | 0.81 | 0.00 | |
| 17.7% | 41.8% | 31.6% | 21.1% | 67.9% | 57.1% | 36.7% | —– | |||
| (128,128, 64) | 10 | 46.85 | 43.03 | 15.19 | 9.37 | 8.60 | 8.58 | 2.59 | 2.45 | |
| 0.0% | 0.0% | 3.4% | 15.4% | 12.6% | 7.2% | 45.9% | 29.3% | |||
| (128,128,128) | 10 | 8.69 | 5.81 | 5.67 | 5.53 | 0.18 | 0.00 | 0.00 | 0.00 | |
| 12.2% | 21.3% | 12.9% | 6.3% | 66.9% | —– | —– | —– | |||
| 8-vector model | |||||||||
|---|---|---|---|---|---|---|---|---|---|
| ( 64, 128, 64 ) | 4.80 | 212.99 ( | 0.74) | 12.13 ( | 0.05) | 14.86 ( | 0.03) | 4.37 ( | 0.02) |
| ( 64, 64, 64 ) | 4.80 | 212.98 ( | 0.73) | 12.13 ( | 0.05) | 14.88 ( | 0.03) | 4.37 ( | 0.02) |
| ( 64, 128, 64 ) | 4.90 | 254.30 ( | 1.31) | 13.39 ( | 0.13) | 17.11 ( | 0.03) | 4.74 ( | 0.07) |
| ( 64, 64, 64 ) | 4.90 | 254.30 ( | 1.33) | 13.39 ( | 0.13) | 17.11 ( | 0.03) | 4.74 ( | 0.07) |
| ( 64, 128, 64 ) | 5.00 | 301.04 ( | 1.34) | 14.68 ( | 0.08) | 19.73 ( | 0.06) | 5.28 ( | 0.03) |
| ( 64, 64, 64 ) | 5.00 | 300.89 ( | 1.30) | 14.68 ( | 0.08) | 19.76 ( | 0.06) | 5.29 ( | 0.03) |
| ( 64, 128, 64 ) | 5.20 | 432.63 ( | 1.41) | 18.08 ( | 0.09) | 26.43 ( | 0.03) | 6.44 ( | 0.04) |
| ( 64, 64, 64 ) | 5.20 | 432.30 ( | 1.42) | 18.07 ( | 0.09) | 26.44 ( | 0.03) | 6.44 ( | 0.04) |
| ( 64, 128, 64 ) | 5.40 | 621.49 ( | 2.90) | 22.16 ( | 0.13) | 35.73 ( | 0.12) | 7.94 ( | 0.05) |
| ( 64, 64, 64 ) | 5.40 | 620.71 ( | 2.93) | 22.12 ( | 0.13) | 35.67 ( | 0.12) | 7.93 ( | 0.05) |
| ( 64, 128, 64 ) | 5.50 | 743.24 ( | 2.67) | 24.47 ( | 0.11) | 41.61 ( | 0.09) | 8.85 ( | 0.04) |
| ( 64, 64, 64 ) | 5.50 | 743.19 ( | 2.62) | 24.47 ( | 0.10) | 41.66 ( | 0.09) | 8.86 ( | 0.04) |
| ( 64, 128, 64 ) | 5.80 | 1290.93 ( | 4.98) | 33.36 ( | 0.17) | 66.11 ( | 0.12) | 11.94 ( | 0.07) |
| ( 64, 64, 64 ) | 5.80 | 1290.50 ( | 4.95) | 33.37 ( | 0.17) | 66.16 ( | 0.12) | 11.96 ( | 0.07) |
| ( 64, 128, 64 ) | 5.93 | 1650.44 ( | 9.02) | 38.34 ( | 0.26) | 81.27 ( | 0.34) | 13.69 ( | 0.10) |
| ( 64, 64, 64 ) | 5.93 | 1647.92 ( | 8.94) | 38.29 ( | 0.26) | 81.23 ( | 0.34) | 13.67 ( | 0.10) |
| ( 64, 128, 64 ) | 6.10 | 2245.25 ( | 9.97) | 45.36 ( | 0.24) | 106.26 ( | 0.28) | 16.31 ( | 0.10) |
| ( 64, 64, 64 ) | 6.10 | 2236.29 ( | 9.60) | 45.15 ( | 0.23) | 105.93 ( | 0.26) | 16.21 ( | 0.09) |
| ( 64, 128, 64 ) | 6.40 | 3917.72 ( | 20.15) | 61.67 ( | 0.38) | 172.30 ( | 0.60) | 22.27 ( | 0.14) |
| ( 64, 64, 64 ) | 6.40 | 3917.57 ( | 19.28) | 61.64 ( | 0.36) | 172.53 ( | 0.56) | 22.26 ( | 0.13) |
| ( 64, 128, 64 ) | 6.60 | 5748.46 ( | 37.56) | 76.37 ( | 0.61) | 239.31 ( | 1.30) | 27.38 ( | 0.23) |
| ( 64, 64, 64 ) | 6.60 | 5711.84 ( | 34.80) | 75.75 ( | 0.52) | 237.66 ( | 1.02) | 27.06 ( | 0.19) |
| ( 64, 128, 64 ) | 6.80 | 8181.57 ( | 48.89) | 92.09 ( | 0.62) | 327.82 ( | 1.33) | 33.14 ( | 0.21) |
| ( 64, 64, 64 ) | 6.80 | 8203.61 ( | 47.71) | 92.35 ( | 0.58) | 328.73 ( | 1.22) | 33.26 ( | 0.20) |
| ( 64, 128, 64 ) | 7.10 | 14186.62 ( | 129.08) | 124.39 ( | 1.10) | 535.73 ( | 2.94) | 44.82 ( | 0.34) |
| ( 64, 64, 64 ) | 7.10 | 14442.55 ( | 115.01) | 126.32 ( | 0.95) | 542.57 ( | 2.52) | 45.45 ( | 0.30) |
| ( 64, 128, 64 ) | 7.40 | 25107.76 ( | 545.38) | 169.58 ( | 2.78) | 884.22 ( | 10.56) | 60.68 ( | 0.78) |
| ( 64, 64, 64 ) | 7.40 | 25478.71 ( | 327.09) | 171.33 ( | 1.65) | 894.13 ( | 7.01) | 61.55 ( | 0.53) |
| ( 64, 128, 64 ) | 7.80 | 53454.83 ( | 949.80) | 255.26 ( | 3.62) | 1742.55 ( | 16.91) | 91.72 ( | 0.98) |
| ( 64, 64, 64 ) | 7.80 | 54631.29 ( | 676.83) | 259.06 ( | 2.50) | 1769.85 ( | 12.66) | 93.29 ( | 0.75) |
| ( 64, 128, 64 ) | 8.20 | 114828.43 ( | 2825.84) | 385.50 ( | 6.83) | 3459.45 ( | 49.17) | 138.89 ( | 1.97) |
| ( 64, 64, 64 ) | 8.20 | 117744.40 ( | 2150.12) | 391.93 ( | 4.97) | 3516.35 ( | 42.02) | 141.65 ( | 1.54) |
| ( 64, 128, 64 ) | 8.70 | 302299.08 ( | 10890.33) | 648.17 ( | 15.33) | 8300.80 ( | 208.42) | 231.74 ( | 4.57) |
| ( 64, 64, 64 ) | 8.70 | 306388.42 ( | 10315.52) | 654.85 ( | 13.69) | 8384.16 ( | 202.32) | 234.93 ( | 4.31) |
| model, comparison with direct method | |||||||||
| embed | direct MGMC | ||||||||
| RRL | RRL | RATIO | |||||||
| 32 | 2.00 | 12762 | 7.84 ( | 0.34) | 35984 | 2.64 ( | 0.07) | 2.97 ( | 0.15) |
| 32 | 2.10 | 11222 | 8.91 ( | 0.41) | 34671 | 2.74 ( | 0.07) | 3.25 ( | 0.17) |
| 32 | 2.20 | 11767 | 8.50 ( | 0.38) | 35714 | 2.66 ( | 0.07) | 3.19 ( | 0.17) |
| 64 | 2.10 | 18129 | 11.03 ( | 0.40) | 56686 | 3.44 ( | 0.07) | 3.21 ( | 0.13) |
| 64 | 2.15 | 23212 | 12.92 ( | 0.42) | 50257 | 3.88 ( | 0.09) | 3.33 ( | 0.13) |
| 64 | 2.20 | 21166 | 14.17 ( | 0.48) | 44419 | 4.39 ( | 0.10) | 3.23 ( | 0.13) |
| 64 | 2.30 | 31446 | 15.90 ( | 0.44) | 39000 | 5.00 ( | 0.13) | 3.18 ( | 0.12) |
| 64 | 2.35 | 19244 | 15.59 ( | 0.55) | 38844 | 5.02 ( | 0.13) | 3.11 ( | 0.14) |
| 64 | 2.40 | 33555 | 15.50 ( | 0.41) | 40372 | 4.83 ( | 0.12) | 3.21 ( | 0.12) |
| 128 | 2.25 | 6631 | 14.33 ( | 0.86) | 44930 | 4.34 ( | 0.10) | 3.30 ( | 0.21) |
| 128 | 2.35 | 5046 | 18.82 ( | 1.30) | 32392 | 6.02 ( | 0.17) | 3.13 ( | 0.23) |
| 128 | 2.45 | 8074 | 24.15 ( | 1.32) | 25624 | 7.61 ( | 0.24) | 3.17 ( | 0.20) |
| 128 | 2.55 | 7476 | 26.08 ( | 1.48) | 25064 | 7.78 ( | 0.24) | 3.35 ( | 0.22) |
| 128 | 2.65 | 9136 | 21.34 ( | 1.09) | 27045 | 7.21 ( | 0.22) | 2.96 ( | 0.18) |
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Multi-Grid Monte Carlo
via Embedding
I. General Theory and Two-Dimensional -Symmetric Nonlinear -Models
Tereza Mendes
*Department of Physics
New York University
4 Washington Place
New York, NY 10003 USA
Andrea Pelissetto
Dipartimento di Fisica
Università degli Studi di Pisa
Pisa 56100, ITALIA
Alan D. Sokal
*Department of Physics
New York University
4 Washington Place
New York, NY 10003 USA
Abstract
We introduce a variant of the multi-grid Monte Carlo (MGMC) method, based on the embedding of an model into the target model, and we study its mathematical properties for a variety of nonlinear -models. We then apply the method to the two-dimensional -symmetric nonlinear -models (also called -vector models) with and study its dynamic critical behavior. Using lattices up to , we find dynamic critical exponents , , for , respectively (subjective 68% confidence intervals). Thus, for these asymptotically free models, critical slowing-down is greatly reduced compared to local algorithms, but not completely eliminated; and the dynamic critical exponent does apparently vary with . We also analyze the static data for using a finite-size-scaling extrapolation method. The correlation length agrees with the four-loop asymptotic-freedom prediction to within over the interval 12\mathrel{\hbox to0.0pt{\lower 3.0pt\hbox{\mathchar 536\relax}\hss}\raise 2.0pt\hbox{\mathchar 316\relax}}\xi\mathrel{\hbox to0.0pt{\lower 3.0pt\hbox{\mathchar 536\relax}\hss}\raise 2.0pt\hbox{\mathchar 316\relax}}650.
1 Introduction
By now it is widely recognized [1, 2, 3, 4] that better simulation algorithms, with strongly reduced critical slowing-down, are needed for high-precision Monte Carlo studies of statistical-mechanical systems near critical points and of quantum field theories (such as QCD) near the continuum limit. One promising class of such algorithms is multi-grid Monte Carlo (MGMC) [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24]: this is a collective-mode approach that introduces block updates (of fixed shape but variable amplitude) on all length scales. The basic ingredients of the method are111 See [6] for details. :
-
Interpolation operator: This is a rule specifying the shape of the block update. The interpolations most commonly used are piecewise-constant (square-wave updates) and piecewise-linear (pyramidal-wave updates).
-
Cycle control parameter : This is an integer number that determines the way in which the different block sizes are visited. In general, blocks of linear size are updated times per iteration. Thus, in the W-cycle () more emphasis is placed on large length scales than in the V-cycle ().
-
Basic (smoothing) iteration: This is the local Monte Carlo update that is performed on each level. Typically one chooses to use heat-bath updating if the distribution can be sampled in some simple way, and Metropolis updating otherwise.
-
Implementation: The computations can be implemented either in the recursive multi-grid style using explicit coarse-grid fields [25, 26, 5, 6, 7, 8, 9], or in the unigrid style using block updates acting directly on the fine-grid fields [27, 11, 12, 13, 14, 15]. For a -dimensional system of linear size , the computational labor per iteration is
[TABLE]
for the recursive multi-grid approach, and
[TABLE]
for the unigrid approach. Thus, the unigrid implementation is marginally more expensive for a V-cycle, but prohibitively more expensive for a W-cycle.
The efficiency of the MGMC method can be analyzed rigorously in the case of the Gaussian (free-field) model, for which it can be proven [5, 6, 28] that critical slowing-down is completely eliminated. That is, the autocorrelation time is bounded as the correlation length and the lattice size tend to infinity, so that the dynamic critical exponent is zero.222 See [4] for a pedagogical discussion of the various autocorrelation times and their associated dynamic critical exponents.
More precisely, the algorithm with piecewise-linear interpolation exhibits for both the V-cycle and the W-cycle, while the algorithm with piecewise-constant interpolation has only for the W-cycle (the piecewise-constant V-cycle has ).
One is therefore motivated to apply MGMC to “nearly Gaussian” systems; one might hope that critical slowing-down would likewise be completely eliminated (possibly modulo a logarithm) or at least greatly reduced compared to the of local algorithms. In particular, we are interested in applying MGMC to asymptotically free two-dimensional -models, such as the -vector models [also called -invariant -models] for , the and principal chiral models, and the and -models.333 In [6, Section IX.C] we conjectured that that for asymptotically free theories — which are “almost Gaussian” except at very long distances — the critical slowing-down should be completely eliminated except for a possible logarithm. However, when we made a numerical study of MGMC in the two-dimensional 4-vector model [8], we found to our surprise that the dynamic critical exponent is not zero: (subjective 68% confidence interval). We then produced [8, Section 4.1] a heuristic explanation of this fact, based on the logarithmically decaying deviations from Gaussianness in an asymptotically free theory. Shortly thereafter, Richard Brower (private communication) pointed out to us that if our explanation is correct, it follows that for a one-dimensional -model — in which the deviations from Gaussianness decay much more rapidly, namely as a power law — the critical slowing-down should be completely eliminated, i.e. we should have (with no logarithm). We hardly doubted that this would be the case, but we decided to make an empirical test to settle the matter once and for all. We have recently [9] reported numerical evidence suggesting that the critical slowing-down is not in fact completely eliminated in the one-dimensional -symmetric -model, but is rather logarithmic: . It goes without saying that we have no idea why this is the case.
In view of the rigorous results for the Gaussian case, we want to investigate two questions:
-
Is for all asymptotically free two-dimensional -models? If not, does vary from one asymptotically free model to another?
-
Is the algorithm with piecewise-constant interpolation and a W-cycle as efficient (at least in order of magnitude) as the one with piecewise-linear interpolation and a V-cycle, i.e. is equal to for these models?
The key design choice in a MGMC algorithm is that of the interpolation operator; indeed, this choice determines most of the remaining ingredients. If one chooses a “smooth” interpolation such as piecewise-linear, then it is usually impossible to implement true recursive MGMC, as there is no simple form for the induced coarse-grid Hamiltonians.444 In particular, this occurs for the nonlinear -models. The trouble is that concepts like “an update of half the magnitude …” have no natural meaning in a nonlinear manifold; and whatever precise definition one chooses, one is left with a horribly nonlinear expression for the coarse-grid Hamiltonian.
Therefore one is obliged to use the unigrid style, a V-cycle, and Metropolis updating. We call this the “German” approach [10, 11, 12, 13, 14, 15]. On the other hand, if one chooses a “crude” interpolation such as piecewise-constant, then one is obliged to use a W-cycle in order to have a chance at ; but piecewise-constant interpolation usually gives rise to a simple coarse-grid Hamiltonian (typically a slight generalization of the fine-grid Hamiltonian555 For example, consider a model in which the fields are unitary matrices, and in which the block update is left-multiplication where is a fine-grid site, is a coarse-grid site, is the coarse-grid field, and the fine-grid block corresponding to coarse-grid site . Then, if the Hamiltonian on grid is
the induced Hamiltonian on grid will be
where the coarse-grid couplings { are given in terms of fine-grid couplings by
(Remark: In this paper grid 0 is the finest grid, and grid maps to blocks of linear size . This notation is opposite to that used in [6, 7, 8]; it now seems to us to be more natural.) ), so that one can use the recursive multi-grid style and, at least in principle, heat-bath updating. This is the approach taken by our group [5, 6, 7, 8, 9].
The “German” version has the advantage of being easy to implement for diverse models, but its use of Metropolis updates introduces several free parameters that have to be adjusted666 For each block size , one must choose the Metropolis hit size and the number of Metropolis hits per site. As shown by Grabenstein and Pinn [19, 20, 22], for smooth interpolations in Gaussian and “near-Gaussian” models (such as -models) it is natural to take and ; this is expected to ensure an acceptance fraction roughly independent of . So it suffices to choose and . , making it more difficult to test systematically. “Our” version has no free parameters, but its implementation is cumbersome and model-dependent, in the sense that the program (and in particular the heat-bath subroutine) has to be drastically rewritten for each distinct model. For example, among the -vector models, the only ones that can be handled conveniently by our “direct” MGMC method are [7] and [8], by exploiting the isomorphism with the and groups, respectively.777 For general -vector models, our direct MGMC method is still applicable, but one must first “lift” the theory from the manifold to the group , as described in [6, Section IV] and [8, Section 2]. One then has the problem of writing an efficient heat-bath subroutine for generating random variables with probability density , where is a general real matrix. For this seems to be quite messy.
With this problem in mind, we have developed a new implementation of MGMC that combines some of the advantages of both methods; in particular, it can be used conveniently for a large class of -models with very little modification of the program.888 We devised this approach after extensive discussions with Martin Hasenbusch and Steffen Meyer at the Lattice ’92 conference in Amsterdam. In particular, the idea of embedding is made explicit in their work: see equations (5)/(6) in [14] and equations (5)–(9) in [15].
The idea is to embed angular variables into the given -model, and then update the resulting induced model by our standard (piecewise-constant, W-cycle, heat-bath, recursive) MGMC method. We do not claim that this approach is superior in practice to the “German” method — that remains to be determined — but we do think that it is well suited for the systematic study of the dynamic critical behavior of MGMC algorithms.
In the simplest cases, the induced Hamiltonian takes the form
[TABLE]
where the induced couplings depend on the current configuration of the original model. Such a “generalized Hamiltonian” is easily simulated by MGMC; indeed, the coarse-grid Hamiltonians in -model MGMC are inevitably of the form (1.3), even when the fine-grid Hamiltonian is the standard model , [6, 7]. So one may just as easily start from (1.3) already on the finest grid.
For future reference, let us call the Hamiltonian (1.3) ferromagnetic if and for all bonds ; let us call it unfrustrated if there exists a configuration that simultaneously minimizes the bond energy on all bonds .
In models with isotensor or adjoint-representation couplings, the induced model will in some cases have also “isospin-2” terms:
[TABLE]
We shall call such a Hamiltonian ferromagnetic if and for all bonds ; we call it unfrustrated if there exists a configuration that simultaneously minimizes the bond energy on all bonds .
Already in a previous paper, we noted in passing [29, pp. 516–517] the possibility of -embedding MGMC; we argued, however, that it is no better than direct MGMC, because the updates in -embedding MGMC are simply a subset of those available in direct MGMC. This is true, but it is also true that -embedding MGMC is likely to be no worse than direct MGMC (except for a constant factor), at least for asymptotically free -models: this is because, in such a model, the spin waves along different axes in internal-spin space are weakly coupled.999 It would be very interesting to study direct versus -embedding MGMC also in non-asymptotically-free -models, such as the three-dimensional -vector model (e.g. with ). Of course, the dynamic critical behavior of MGMC in such a model is an open question even in the case .
And -embedding MGMC can be much more convenient to implement than direct MGMC.
One might wonder why we have bothered studying MGMC algorithms for two-dimensional -vector models, when Wolff’s [30] cluster algorithm already succeeds in completely eliminating the critical slowing-down for these models [30, 31, 32, 29, 33]. The reason is that the success of generalized Wolff-type embedding algorithms appears to be limited to -vector models (and possibly also models) [29], while MGMC can be expected to behave qualitatively similarly for all asymptotically free -models.101010 Additional difficulties might arise in models that have important discrete excitations (i.e. vortices) in addition to spin waves: for example, the model, the chiral model, and the adjoint chiral model. These difficulties would be analogous to the mediocre performance of MGMC in the two-dimensional model in the high-temperature (vortex) phase [7, 16].
Therefore, we want to understand systematically the dynamic critical behavior of MGMC in asymptotically free -models. We have started with the -vector models because of their simplicity, but our eye is ultimately on more general -models, such as those taking values in [14, 34, 35] or [14, 15].
The plan of this paper is as follows: In Section 2 we introduce embeddings for a variety of nonlinear -models, and we analyze their mathematical properties. In Section 3 we report our numerical data on the -embedding MGMC algorithm for the two-dimensional -vector models with . In Section 4 we analyze our static data for the case, using a finite-size-scaling extrapolation method [36, 37, 38, 39, 40] and comparing the results to the four-loop asymptotic-freedom predictions [41]. In Section 5 we analyze our dynamic data for , using traditional finite-size-scaling plots to extract the dynamic critical exponents . In Section 6 we discuss our findings.
A brief preliminary version of this work appeared in [34]. In a subsequent paper [35] we will report results for -embedding MGMC applied to the two-dimensional principal chiral model.
2 Embedding
In this section we introduce embeddings for a variety of nonlinear -models, and we analyze their mathematical properties. In particular, for each model and each choice of embedding, we want to answer the following two questions:
- (a)
Is the induced Hamiltonian of the simple form (1.3) [as opposed to the “mixed isospin-1/isospin-2” form (1.4)]? This question is relevant because it is much easier to write an MGMC program (and in particular a heat-bath subroutine) for (1.3) than for (1.4).
- (b)
Is the induced Hamiltonian unfrustrated? This question may be relevant to the dynamic critical behavior of the resulting algorithm (at any rate we want to find out whether it is relevant).
After making some general comments on embedding algorithms (Section 2.1), we introduce the models and the embeddings and compute the induced Hamiltonians (Section 2.2). Next we ask whether or not these induced Hamiltonians are frustrated (Section 2.3). We also discuss some alternative embeddings (Section 2.4). Finally, we summarize our results (Section 2.5).
2.1 Generalities on Embedding Algorithms
The general idea of embedding algorithms [4, 29] is the following: “Foliate” the configuration space of the original model into “leaves” isomorphic to the configuration space of some “embedded” model (which may be an Ising model, model, etc.). The Monte Carlo process then moves around the current leaf, using any legitimate Monte Carlo algorithm for simulating the conditional probability distribution of the original model restricted to that leaf; this conditional probability distribution (which of course depends on the current configuration of the original model) defines the induced Hamiltonian for the embedded model.111111 This same structure arises in the coarse-grid updates of MGMC algorithms; we have termed it “partial resampling” [1, 5, 6].
Of course, one must combine this procedure with other moves, or with different foliations, in order to make the algorithm ergodic.
Clearly, the updates available in an embedding algorithm with a specified foliation are a subset of all the updates available in a general Monte Carlo algorithm for the original model. So embedding never enlarges the space of possible algorithms; if anything, it restricts it. And the smaller the “leaves” of the foliation, the slower the algorithm is likely to move through the configuration space. The advantage of embedding is, rather, that the embedded model may be much simpler than the original model. As a result, it may be easier to invent a good algorithm for updating the embedded model than for updating the original model; or even if good algorithms for both models can be invented, the former algorithm may be easier to implement in practice. Thus, embedding may even lead to a loss in efficiency compared to alternative algorithms; but if this loss is not too great, it may be outweighed by the simplicity of programming and by the flexibility of the method (which is important in case we want to apply the method systematically to several similar models).
The dynamic critical behavior of an embedding algorithm is determined by the combined effect of two completely distinct issues:
- i)
How well the chosen embedding succeeds in “encoding” the important large-scale collective modes of the original model into the embedded variables.
- ii)
How well the chosen algorithm for updating the embedded model succeeds in its task.
Thus, if the physically relevant large-scale collective motions of the original model cannot be obtained by motions within a leaf, then the embedding algorithm will have severe critical slowing-down no matter what method is used to update the embedded variables. On the other hand, if the embedding algorithm with a particular choice of updating method for the embedded model shows severe critical slowing-down, this does not necessarily mean that the embedding per se works badly: the poor performance might be due to slow decorrelation in the inner-loop subroutine for updating the embedded model, and could possibly be remedied by switching to a better inner-loop algorithm.
To study question (i) independently of question (ii), it is conceptually useful to consider the idealized embedding algorithm [29], in which a new configuration on the current leaf is chosen, independent of the old configuration, with probabilities given by the conditional probability distribution restricted to that leaf. Of course, such an algorithm is not practical, but that is not its role. Rather, it serves as a standard of comparison (and presumed lower bound on the autocorrelation time) for all algorithms based on the given embedding. To approximate in practice the idealized embedding algorithm, we can update the embedded model by hits of some chosen algorithm, and extrapolate to .121212 Preferably we would perform simulations for successively increasing values of until the autocorrelation time is constant within error bars. However, this may not always be feasible.
To be sure, this test procedure can be very time-consuming; we have implemented it in our work on generalized Wolff-type embedding algorithms [29], for which the embedded variables are Ising spins.
Here, by contrast, the embedded variables are spins, and we suspect [29, p. 517] that the idealized embedding algorithm will have no critical slowing-down (because of the weak coupling between spin waves along different internal-spin axes in an asymptotically free -model). This prediction should, of course, be carefully tested; but in this initial study we have for simplicity restricted attention to the “practical” algorithm having . We want to know whether the -embedding MGMC algorithm lies in the same dynamic universality class as the direct MGMC algorithm for the same model; and we want to know whether (and how) the dynamic critical exponents vary from one asymptotically free model to another.
There is one situation in which one might worry that the practical () -embedding MGMC algorithm could behave poorly even though the idealized () algorithm would behave well: this is when the induced Hamiltonian is strongly frustrated, so that it has multiple ground states (or approximate ground states) separated by large energy barriers. In such a situation, the MGMC moves, which are basically spin waves, would probably be insufficient to surmount these energy barriers and to find the distant ground states.131313 A similar problem occurs in MGMC for the ordinary two-dimensional model just above the Kosterlitz-Thouless critical temperature: the MGMC moves are inefficient at creating and destroying vortices [7, 16].
We have therefore taken care to investigate, for each of our embeddings, whether the induced Hamiltonian is frustrated or not. (This is a natural theoretical question in any case: one wants to know the properties of the induced Hamiltonian. It is also relevant for analytical applications of embedding, such as the proof of correlation inequalities [42, 43, 44].) It turns out that in the case of -vector models the induced Hamiltonian is unfrustrated, while for chiral models it is generically frustrated. Nevertheless, -embedding MGMC performs well (dynamic critical exponents in the range 0.4–0.7) in both cases [34, 35]. We think this is because the induced Hamiltonian in the case is only weakly frustrated as ; we intend to study this point in more detail in a subsequent paper devoted specifically to chiral models [35].
Let us now define our notation for nonlinear -models. The original model will have field variables (“spins”) living on lattice sites and taking values in a compact manifold (“target space”) .141414 We shall use the notation for generic discussions. When discussing specific models we shall use standard notations: \mbox{\boldmath\sigma}_{x} for real unit vectors, for complex unit vectors, for matrices, etc.
The Hamiltonian of this model will have a global symmetry group , i.e. for all and all field configurations . We shall assume that is a compact Lie group with Lie algebra g, and that acts transitively on .
It is worth mentioning a useful heuristic principle for finding candidates for “good” embedding algorithms [29, p. 517]; this principle includes as special cases both the Wolff-type embedding of Ising spins and the multi-grid-type embedding of spin waves. The idea is to take an exact global symmetry of the model (global reflection in the case of Wolff, global rotation in the case of multi-grid) and to apply it in an inhomogeneous way (constant on clusters in the case of Wolff, constant on blocks in the case of multi-grid). In other words, one picks a subgroup (possibly all of ) and then embeds a model taking values in , via the update . Because is an exact global symmetry of the original model, the energy cost of such an update is zero whenever the field is constant (independent of ). Therefore, if is “almost constant” (e.g. constant on a large block), then the energy cost is only a “surface term”.151515 The principle of looking for collective-mode updates whose energy cost is only a “surface term” is also made explicit by Grabenstein and Pinn [22, 23].
Let us mention, finally, that embedding algorithms can be written in two alternate (but physically equivalent) parametrizations. On the one hand, the embedded variables can be considered as updates to the current values of the original spins; this is the interpretation taken in the formula . On the other hand, in some models the embedded variables can be taken to be a subset of the new values of the original spins: for example, in the -vector model the embedded variable can be taken to be the angle of certain components of with respect to some fixed basis [see (2.30) ff. for details]. The first parametrization always exists, while the second one exists only in some cases (e.g. in -vector models but apparently not in models). The advantage of the second parametrization, when it does exist, is that certain properties of the embedded Hamiltonian (e.g. nonfrustration) are more manifest. When both parametrizations exist, they are related by a gauge transformation with gauge group .
2.2 Embedding: Some Examples
2.2.1 Embedded Variables as Updates
The basic idea of our algorithm is to embed angular variables into the given -model; we then update the resulting induced model by our standard (piecewise-constant, W-cycle, heat-bath, recursive) MGMC method.
Consider, therefore, a -model with target manifold and global symmetry group ; we assume that is a connected compact Lie group with Lie algebra g, and that acts transitively on . The idea behind embedding is to choose randomly a subgroup , and to apply a “rotation” in this subgroup to the original spin variable . Thus, the angular variables are updates to the original variables . More precisely, we define the updated variable by
[TABLE]
where is a random element of , and is a fixed nonzero element (to be specified later) of the Lie algebra g. The embedded model consisting of the spins is then simulated using the induced Hamiltonian
[TABLE]
with initial condition (i.e. ) for all . At each iteration of the algorithm, a new random matrix is chosen.
In general the induced Hamiltonian (2.2) can be extremely complicated (and thus impractical to simulate by true recursive MGMC). However, if the original Hamiltonian is sufficiently “nice” and one makes a clever choice of the generator , then in some cases the induced Hamiltonian can be reasonably simple. Thus, we shall always choose the matrix to have all its eigenvalues in the set . This implies that
[TABLE]
where is the identity matrix.
Let us now illustrate the method in five models:
The -vector model: , . The original variables \mbox{\boldmath\sigma}_{x} are unit vectors in , and the original Hamiltonian is
[TABLE]
- 2)
The principal chiral model: , . The original variables of this model are matrices, and the original Hamiltonian is
[TABLE]
[Similar methods apply to other principal chiral models, e.g. .]
- 3)
The mixed isovector/isotensor model: , . This model generalizes the -vector model: the original variables \mbox{\boldmath\sigma}_{x} are again unit vectors in , but the Hamiltonian is now
[TABLE]
[If , this is the -vector model. If , this is the pure isotensor model, which has a local gauge group; it can therefore be thought of as taking values in .161616 Note also the special case and : since via the correspondence
where \mbox{\boldmath\sigma}_{x}\in S^{3} and , we have \mathop{\rm tr}\nolimits(V_{x}^{\rm T}V_{x^{\prime}})=4(\mbox{\boldmath\sigma}_{x}\cdot\mbox{\boldmath\sigma}_{x^{\prime}})^{2}-1, so that the model (with this choice of lattice action) is equivalent to the principal chiral model.
]
- 4)
The complex mixed isovector/isotensor model: , . The original variables are unit vectors in , and the original Hamiltonian is
[TABLE]
[If , this is -vector model written in complex notation. If , this is the pure complex isotensor model, which has a global symmetry group and a local gauge group; it can therefore be thought of as taking values in .171717 Note also the special case and : since via the correspondence \mbox{\boldmath\sigma}_{x}={\bf z}_{x}^{*}\mbox{\boldmath\tau}{\bf z}_{x} where are the Pauli matrices, we have |{\bf z}_{x}^{*}\cdot{\bf z}_{x^{\prime}}|^{2}=(1+\mbox{\boldmath\sigma}_{x}\cdot\mbox{\boldmath\sigma}_{x^{\prime}})/2, so that the model (with this choice of lattice action) is equivalent to the standard 3-vector model.
]
- 5)
The mixed fundamental/adjoint principal chiral model: , . Again the original variables of this model are matrices, and the Hamiltonian is
[TABLE]
where of course
[TABLE]
[If , this is the usual chiral model. If , this is the pure adjoint chiral model, which has a global symmetry group and a local gauge group; it can therefore be thought of as taking values in .181818Note also the special case : since via the correspondence U_{x}=\sigma_{x}^{(0)}I+i\,\vec{\mbox{\sigma}}_{x}\cdot\vec{\mbox{\tau}} where \mbox{\boldmath\sigma}_{x}\equiv(\sigma_{x}^{(0)},\vec{\mbox{\sigma}}_{x})\in S^{3} and \vec{\mbox{\tau}} are the Pauli matrices, we have \mathop{\rm tr}\nolimits(U_{x}^{\dagger}U_{x^{\prime}})=2\,\mbox{\boldmath\sigma}_{x}\cdot\mbox{\boldmath\sigma}_{x^{\prime}}, so that the mixed fundamental/adjoint model is equivalent to the mixed isovector/isotensor model.]
For models 1 and 3, the symmetry group is , and the update is
[TABLE]
where is a random matrix and is a fixed antisymmetric real matrix. As mentioned above, we want to have eigenvalues in . It follows that has eigenvalues , eigenvalues , and eigenvalues 0, for some integer satisfying . Here we shall choose ; the subgroup generated by is then rotation in a single plane in . (Other choices of will be discussed in Section 2.4.1 below.) Without loss of generality (because of the random choice of ) we can take
[TABLE]
We then have the following induced Hamiltonians:
Model 1 (-vector model): The induced Hamiltonian is
[TABLE]
where the induced couplings are given by
[TABLE]
(and for simplicity we have dropped the superscript old on the ’s). For the update with the explicit choice (2.11) for , we have
[TABLE]
Note that these couplings are in general disordered and non-ferromagnetic. However, we shall show below that they are unfrustrated: that is, they are -gauge-equivalent to a ferromagnetic set of couplings.
Model 3 (mixed isovector/isotensor model): The induced Hamiltonian is
[TABLE]
with the induced couplings given by
[TABLE]
where we have defined
[TABLE]
For the update with the explicit choice (2.11) for , we get
[TABLE]
Note that for we have in general , so that even in the pure isotensor model.191919
However, if is even we may choose the alternate embedding with : this yields and hence . With this embedding, the pure isotensor model () has . For further discussion, see Section 2.4.1.
For models 2 and 5, the symmetry group is , but we shall exploit here only the left-multiplication subgroup. Thus, we shall take as our update
[TABLE]
where is a random matrix and is a fixed traceless Hermitian matrix with eigenvalues in . It follows that has eigenvalues , eigenvalues , and eigenvalues 0, where . As before, we shall choose (see Section 2.4.1 for other choices). Without loss of generality we can take
[TABLE]
We then have the following induced Hamiltonians:
Model 2 ( principal chiral model): The induced Hamiltonian is again of the form (2.12), with couplings
[TABLE]
For the update with the explicit choice (2.20) for , we have
[TABLE]
Note that these couplings are in general disordered and non-ferromagnetic; moreover, as will be seen in the next subsection, they are also generically frustrated, in contrast to what happens for the -vector models.
Model 5 (mixed fundamental/adjoint chiral model): The induced Hamiltonian is again of the form (2.15), with couplings
[TABLE]
where we have defined
[TABLE]
Note that for we have and hence . On the other hand, for we have in general , so that even in the pure adjoint model.202020 However, if is even we may choose the alternate embedding with : this yields and hence . With this embedding, the pure adjoint model () has .
Model 4 (complex mixed isovector/isotensor model): The symmetry group is , and we take the update to be
[TABLE]
where is a random matrix and is a fixed Hermitian matrix with eigenvalues in .212121 Note that is not required to be traceless, even in the pure model (). Of course, in this latter model the subgroup of is a local gauge group, so only the traceless part of [i.e. ] plays an interesting dynamical role. In particular, two update matrices and that differ by a multiple of the identity give rise to physically equivalent embeddings (differing only by a gauge transformation) and identical induced Hamiltonians.
It follows that has eigenvalues , eigenvalues , and eigenvalues 0, for suitable integers and . Here we shall choose , (see also [15]); without loss of generality we can take
[TABLE]
The evaluation of the embedded Hamiltonian is analogous to model 3; it is of the form (2.15), with couplings
[TABLE]
where we have defined
[TABLE]
Note that because of the particular choice (2.26) of , we have and hence , which implies that . So, perhaps surprisingly, the embedded Hamiltonian is of the simple form (2.12) in spite of the isotensor couplings in (2.7). This happens whenever all the complex rotations are in the same direction, i.e. either (hence ) or (hence ). In these cases we have
[TABLE]
without double-angle terms.222222There is another interesting possibility: If we take , we have and hence . With this embedding, the pure isotensor model () has . Of course, this embedding is gauge-equivalent to one with , and replaced by (see footnote 21). For further discussion, see Section 2.4.1.
2.2.2 Alternate Interpretation
For the two models taking values in — viz. the -vector and mixed isovector/isotensor models — there is a simpler and more natural way of specifying the embedding of angular variables. The embedding is clearly given by choosing randomly a two-dimensional subspace and then rotating only the component of \mbox{\boldmath\sigma}_{x} in . [From (2.10)/(2.11) we see that is the image of the 1–2 plane under the rotation .] To be precise, let us decompose each vector \mbox{\boldmath\sigma}_{x} into its parts parallel and perpendicular to the plane :
[TABLE]
We then fix \{\mbox{\boldmath\sigma}_{x}^{\perp}\} at their current values, and consider only \{\mbox{\boldmath\sigma}_{x}^{\parallel}\} as dynamical variables. Substituting (2.30) into the original -vector Hamiltonian (2.4), we get
[TABLE]
The angular variables can then be defined in either of two convenient ways:
Let be the angles of \mbox{\boldmath\sigma}_{x}^{\parallel} with respect to the some fixed basis vectors in the plane . In this case \mbox{\boldmath\sigma}_{x}^{\parallel}\cdot\mbox{\boldmath\sigma}_{x^{\prime}}^{\parallel}=|\mbox{\boldmath\sigma}_{x}^{\parallel}|\,|\mbox{\boldmath\sigma}_{x^{\prime}}^{\parallel}|\,\cos(\theta_{x}-\theta_{x^{\prime}}), so that the embedding has the form (2.12), with couplings
[TABLE]
and the embedded model is simulated starting from the present configuration for the angles (call it ). These couplings are clearly ferromagnetic (provided that ). This is the embedding implementation that is used in our MGMC program.
- 2)
Alternatively, we can take to be the angles of \mbox{\boldmath\sigma}_{x}^{\parallel} with respect to its present orientation (given by in the previous coordinates). In this case the couplings are given by
[TABLE]
and the embedded model is simulated starting from . This is the approach taken in (2.10)–(2.14) above; and of course (2.33) is identical to (2.14).
The sets of couplings (2.32) and (2.33) are related by a gauge transformation: namely, the angle in the “first” variables is equal to in the “latter” variables, for all . Therefore, the couplings (2.14)/(2.33), though in general non-ferromagnetic, are nevertheless unfrustrated, since they are -gauge-equivalent to the ferromagnetic couplings (2.32).
For the mixed isovector/isotensor model we can use the same decomposition (2.30), but the embedded Hamiltonian becomes somewhat more complicated. In interpretation #2, the induced Hamiltonian has already been given in (2.15)–(2.18). In interpretation #1, we get
[TABLE]
The couplings are clearly ferromagnetic if .
Remark. A similar representation can be employed for the complex mixed isovector/isotensor model with the choice (2.26) of . The embedding is specified by choosing a random complex direction (in fact ) and then rotating only the component of along . That is, we decompose with
[TABLE]
The analogue of interpretation #1 is obtained by defining
[TABLE]
The induced Hamiltonian is of the form (2.12) (“single-angle” couplings only), with
[TABLE]
However, these couplings are not in general ferromagnetic, even if , since need not be real. In fact, we shall show in the next section that they are in general frustrated, except in the pure model (, ). In interpretation #2 we have the expression (2.27), with
[TABLE]
and hence
[TABLE]
2.3 Frustration
Although our original models (2.4)–(2.8) are ferromagnetic and translation-invariant, we have seen that the corresponding induced Hamiltonians are generically non-ferromagnetic and non-translation-invariant: they contain both and terms, with coefficients of arbitrary sign. This raises the question of whether the induced Hamiltonians may be frustrated. Recall that a Hamiltonian232323 More precisely, frustration is not a property of a Hamiltonian per se, but of a specific decomposition of that Hamiltonian into “elementary bonds”. Here we shall always consider the elementary terms in the Hamiltonian to be the complete energy associated with a single bond .
is said to be unfrustrated if there exists a configuration of the field variables that simultaneously minimizes the energy on each bond.
One case is easy: any ferromagnetic Hamiltonian is obviously unfrustrated, since any configuration (independent of ) simultaneously minimizes the energy on each bond. Furthermore, any Hamiltonian that is -gauge-equivalent to a ferromagnetic Hamiltonian is also unfrustrated. Thus, the induced Hamiltonian for the -vector model is unfrustrated: it is either ferromagnetic (interpretation #1) or gauge-equivalent to a ferromagnetic one (interpretation #2). This is also true for the mixed isovector/isotensor model with because of (2.34).242424For a more complete discussion of the mixed case, see Section 2.4.1.
For the principal chiral models, however, the situation is less clear, as no manifestly ferromagnetic representation appears to be available. Here we shall show that no such representation can exist: the induced Hamiltonians are in general frustrated for all . We shall also locate the underlying cause of this frustration.
Remark. When we say that the induced Hamiltonians are “in general frustrated”, what we mean is that there exist configurations of the original model (which we shall present explicitly) for which the induced Hamiltonians are frustrated. We conjecture that in fact a much stronger statement is true, namely that the induced Hamiltonians are generically frustrated, in the sense that frustration occurs for all configurations except those lying on a lower-dimensional submanifold of the configuration space. We expect that this can be proven by suitable modifications of our examples, but we have not attempted to do so.
To test whether the generalized model (2.12) is frustrated, we consider the product around a plaquette of the quantity . If this product is real and positive for all plaquettes, then the model is unfrustrated.252525 In periodic boundary conditions this statement is not quite correct: one should also insist that this product be real and positive for “Polyakov loops” that wind around the lattice by periodicity.
Conversely, if for some plaquette the product is not real and positive, then the model is frustrated. This is because we can rewrite (2.12) as
[TABLE]
where and ; to minimize the energy on bond we need
[TABLE]
(provided that ). This can be done simultaneously for all bonds if and only if for each plaquette we have
[TABLE]
or, equivalently,
[TABLE]
Let us now use the criterion (2.43) to test for frustration in our various models:
Model 2 (SU(N) principal chiral model): We shall prove frustration for all , by showing that (2.43) does not in general hold for the embedding (2.19)–(2.22). Without loss of generality we can take . We begin with the simplest case .
In the case of matrices, it is easy to show that for our choice of we get from (2.21) the simple form
[TABLE]
Then we want to show that around a plaquette with sites the product
[TABLE]
is not necessarily real and positive. It suffices to take and general and :
[TABLE]
where . We get for the product in (2.45)
[TABLE]
Although the first term is real and positive, the second is free to have an imaginary part (for example, take , , , ), and therefore the couplings are in general frustrated.
This example can be immediately extended to for : it suffices to take matrices which are block-diagonal, in which the uppermost block is as above, and the lowermost block is an arbitrary matrix. Thus, for all our algorithm leads in general to a frustrated induced model.
The presence of frustration even in the case may seem surprising, as there is an isomorphism relating the chiral model to the 4-vector model, and we know that the embedding (2.10)/(2.11) for -vector models is unfrustrated. The explanation is that the embedding (2.19) does not map into the 4-vector embedding (2.10)/(2.11), but rather into a different embedding! Let us work this out in detail, using the isomorphism
[TABLE]
between and ; here is the identity matrix, \vec{\mbox{\sigma}} represents the Pauli matrices, and we use the symbol \vec{\mbox{s}} to denote the -vector . The Hamiltonian (2.5) for the chiral model corresponds to the Hamiltonian (2.4) for the 4-vector model, with the identification .
Let us now work out the update at a single site (for simplicity we suppress the subscript from the notation). We have
[TABLE]
for some unit vector \vec{\mbox{a}}\in\hbox{\specialroman R}^{3}. Substituting (2.48) and (2.49) in (2.19), we get
[TABLE]
and hence
[TABLE]
where we have used
[TABLE]
For example, if we choose \vec{\mbox{a}}=(1,0,0), we have
[TABLE]
and we can see that this corresponds to two coupled rotations for the -vector : a counterclockwise rotation on the plane defined by the 0-direction and \vec{\mbox{a}}, and a clockwise rotation on the plane orthogonal to this one. By contrast, our standard 4-vector update (2.10), with given by the choice (2.11), corresponds to a rotation in one plane only. Clearly, the update (2.53) corresponds to the 4-vector update (2.10) with given instead by the choice
[TABLE]
The induced Hamiltonian is then (2.12) with the couplings
[TABLE]
where and are the initial angles of \mbox{\boldmath\sigma}_{x}^{\parallel}\equiv(s_{0},s_{1}) and \mbox{\boldmath\sigma}_{x}^{\perp}\equiv(s_{2},s_{3}), respectively. These couplings are clearly frustrated, since although there exist gauge transformations that will make the “parallel” or the “perpendicular” part of the couplings become ferromagnetic separately, we cannot in general make them simultaneously ferromagnetic (unless by chance we had for all , or some similar degenerate situation). For example, consider a plaquette with angles () given by , respectively. Then the product of
[TABLE]
around the plaquette will have a complex part proportional to
[TABLE]
which can clearly be nonzero.
Remark. The 4-vector update with both rotations clockwise, namely
[TABLE]
corresponds to the right update
[TABLE]
Model 5 (mixed fundamental/adjoint SU(N) chiral model): Let us now show, using (2.23), that frustration occurs also for the mixed fundamental/adjoint chiral model whenever (and are not both zero). The point is that for matrices we have and both real; it follows that
[TABLE]
Hence, if we write the induced Hamiltonian in terms of “frustration angles”,
[TABLE]
we have (mod ) provided that . It follows that the presence or absence of frustration can be determined solely by looking at . And as we have already seen, for certain configurations this product has a nonzero imaginary part.
To extend this proof to with , we take (when ) the matrices to be block-diagonal, in which the uppermost block is as above, and the lowermost block is the identity matrix. We then have each , so that the only change in the foregoing argument is that, in (LABEL:SU2_ab), the prefactor is replaced by .
Model 4 (complex mixed isovector/isotensor model): Likewise, we can show that the complex mixed isovector/isotensor model with the choice (2.26) of leads to a frustrated induced model in the following two cases:
- (i)
For all , provided that . Without loss of generality we can take , hence
[TABLE]
Consider the product of around a plaquette. We have, from (2.39),
[TABLE]
Now consider, on the plaquette ABCD, the configuration
[TABLE]
We get
[TABLE]
which shows that the embedding is frustrated if we take .
- (ii)
For all , whenever . Consider, on the plaquette the configuration
[TABLE]
Then, using (2.63) we get
[TABLE]
This has a nonzero imaginary part whenever , provided we choose so that the first three factors are nonvanishing; therefore the embedding is frustrated.
The remaining cases are unfrustrated:
- (iii)
If , this is equivalent to the -vector model with our standard update.
- (iv)
If and , this is the model, which is equivalent to the 3-vector model (see footnote 17); and it is easily seen that the update corresponds to our standard update for this latter model. The nonfrustration can also be seen directly from (2.63), once one notes that and are both one-dimensional.
2.4 Alternative Embeddings
2.4.1 Embeddings with
Thus far we have been concentrating on the embeddings in which the matrix has only one eigenvalue equal to , i.e. the cases . Here we shall examine the cases with ; in particular we shall investigate the occurrence of frustration.
Note first that we must always have (or in the case of the complex mixed isovector/isotensor model). For each model, therefore, we shall begin by investigating the minimal cases (or ). Once frustration has been proven for such a case, the proof can be immediately extended to all higher values of for the same fixed (and ): For the vector models, it suffices to extend the chosen configuration with zeros in the last (or ) components; then and are unchanged, while , so that the couplings are unchanged (even if ). For the matrix models, we can take (when ) the matrices to be block-diagonal, with the uppermost block being as previously constructed, and the lowermost block being the identity matrix; then and are unchanged, while , so that the only change in the reasoning is to replace by in the formulae for and .
Let us now consider separately the various models:
Model 1 (-vector model): For we have shown that the embedding is unfrustrated. For the minimal case is : we have already discussed this case in the preceding subsection, and have shown that the induced Hamiltonian (2.55) is in general frustrated because the couplings are a sum of contributions written in terms of unrelated angles. The same argument applies for general and : the update consists of plane rotations and leads to couplings that are a sum of contributions written in terms of unrelated angles. Finally, as explained above, the proof of frustration immediately extends to . Hence all cases with lead to frustration.
Model 2 ( principal chiral model): For we have shown frustration for all . Next consider and . If we look at configurations that are block-diagonal (in the same basis in which is block-diagonal) in which each of the blocks is an matrix, we get couplings that are the sum of terms each of which is as in (2.55). In other words, the couplings are a sum of contributions written in terms of unrelated angles, so they are in general frustrated. Finally, as explained above, the proof immediately extends to . Hence all cases lead to frustration.
Model 3 (mixed isovector/isotensor model): For we have shown that the couplings are unfrustrated if . For general , the couplings (2.16) lead to
[TABLE]
Hence, if we write the induced Hamiltonian in terms of “frustration angles” as in (2.61), we have (mod ) provided that . Hence the presence or absence of frustration can be determined solely by looking at . This product consists of two factors: the product of around the plaquette, which we have already studied in connection with the pure -vector model; and the product of , which is always real but may in some cases be negative. The cases with are simple, since we have and hence , so that the problem is reduced to that of frustration for the pure -vector model: frustration occurs whenever . The proof extends immediately to , as explained before.
The only cases left to investigate are the mixed models with for and . Now, for we have already seen that is real and positive, so that the model will be unfrustrated if and only if is positive. Let us treat separately the cases and :
Case (a): (pure model). The criterion for nonfrustration is
[TABLE]
Two distinct situations then arise:
- (i)
: Here the vectors \mbox{\boldmath\sigma}_{x}^{\perp} are one-dimensional, so that the product (2.69) is manifestly nonnegative (each \mbox{\boldmath\sigma}_{x}^{\perp} occurs exactly twice in the product). Therefore, the model is unfrustrated.
- (ii)
: For these cases one can easily find configurations with \prod\mbox{\boldmath\sigma}_{x}^{\perp}\cdot\mbox{\boldmath\sigma}_{x^{\prime}}^{\perp}<0: take, for example, on a plaquette the configuration
[TABLE]
with all . Then, we have frustration whenever the product
[TABLE]
is negative, e.g. for , and . We thus prove that all models with are frustrated when .
Case (b): . For any and any values , we can find configurations such that is negative. For example, take a configuration such that ,,, are all colinear, with ,, and relative to some fixed orientation of this line. Then, whenever and
[TABLE]
we have .
In summary, the mixed isovector/isotensor models are frustrated except for two cases with : (a) for any , and (b) the pure model.
Remark. As mentioned in footnote 16, the case and (namely the model) is equivalent to the principal chiral model. It is interesting to write the update in language. Recall that a 4-vector s=(s_{0},\vec{\mbox{s}})\in S^{3} maps into a matrix via
[TABLE]
Now, one case of the update for the model is given by (2.51). Using (2.73), we see that this translates into
[TABLE]
where \vec{\mbox{J}} are the generators of the algebra, i.e. . This is the standard left-update for the chiral model [analogous to (2.19) for the chiral model] except that the angle is doubled. This explains why the induced Hamiltonian has only “isospin-2” terms (see footnote 19).
Model 4 (complex mixed isovector/isotensor model): Let us start by considering the embeddings with , so that . (The cases with , are trivially equivalent.) All such embeddings have and
[TABLE]
where (resp. ) denotes the first (resp. last ) components of . We can prove frustration in three cases:
- (i)
If and . It suffices to choose configurations with . Then the couplings are those of a update for the -vector model, hence frustrated.
- (ii)
If and . The proof is identical to that of case (ii) in Section 2.3, if in (2.66) we insert zeros between the first and second coordinates.
- (iii)
If , and . This is a pure model, and adding a multiple of the identity to corresponds to a gauge transformation. So the case , is gauge-equivalent to , , which was proven to be frustrated for in case (ii) of Section 2.3.
The only remaining case with is , : this is a mere gauge transformation of the model, and the induced Hamiltonian is zero.
For cases with we no longer have , and a more careful analysis is needed. We have been able to obtain a result only for the pure model (). Consider first the minimal case : here we have and hence , so that . The criterion for frustration is then given in terms of
[TABLE]
where (resp. ) denotes the first (resp. the last ) components of . For this is manifestly unfrustrated, as ; but for all cases with and/or it is easy to devise frustrated configurations, by the same method as was used for the models with .262626 Alternate proof: The model with is gauge-equivalent to one with , and replaced by (see footnotes 21 and 22). Hence this is unfrustrated for and frustrated for , as proven in Section 2.3. Finally, the proof of frustration extends to , as explained earlier.
The remaining cases of the models are , with . Let (resp. ) denote the first (resp. second) component of , and let denote the last components.
- (a)
Suppose we choose a configuration with . From (2.28) we then have , so that and
[TABLE]
For it is easy to devise frustrated configurations, by the same method as was used for the models with .
- (b)
For we can construct a frustrated configuration as follows: consider on a plaquette the configuration
[TABLE]
with real and . Then for the links and we have from (2.27)/(2.28) (noting that is the diagonal matrix ) that and , so that the unique minimum of the Hamiltonian for these links occurs at . For the links and we have that and
[TABLE]
Therefore, this configuration is frustrated unless .
So the models with are frustrated for all .
Model 5 (mixed fundamental/adjoint chiral model): For all mixed models are frustrated. Therefore, for all mixed models will be likewise frustrated, since for we can take configurations that are block-diagonal formed by matrices; and we can extend the proof to as before.
2.4.2 Two-Sided Update for
In view of the results in Sections 2.3 and 2.4.1 concerning the frustration in the case, one might ask: Can we find a different update for the chiral model that would yield unfrustrated couplings for the induced model? One possibility is to make a simultaneous rotation on both the left and the right:
[TABLE]
where are random matrices and is a fixed traceless Hermitian matrix with eigenvalues in . In general has eigenvalues , eigenvalues , and eigenvalues 0, for some integer satisfying .
Inserting (2.80) into the chiral-model Hamiltonian (2.5) and using (2.3), we find that the induced Hamiltonian is of the “isovector/isotensor” form (2.15), with couplings
[TABLE]
where
[TABLE]
One interesting case is that in which is even and : then has only eigenvalues (and not also 0), so that
[TABLE]
It follows that , so that the induced Hamiltonian becomes again, after a relabeling , a “pure isovector” Hamiltonian (2.12).
For (and hence ) we can rewrite the update (2.80) in 4-vector notation. With given by (2.20), let RTR^{-1}=\vec{\mbox{a}}\cdot\vec{\mbox{\sigma}} and STS^{-1}=\vec{\mbox{b}}\cdot\vec{\mbox{\sigma}}. We have
[TABLE]
With the special choice \vec{\mbox{a}}=\vec{\mbox{b}} (which can be obtained from among other choices), we get
[TABLE]
which corresponds to a single rotation of angle in the plane defined by the 0-direction and \vec{\mbox{a}}. Likewise, the choice \vec{\mbox{a}}=-\vec{\mbox{b}} yields
[TABLE]
which corresponds to a single rotation of angle in the plane perpendicular to the 0-direction and to \vec{\mbox{a}}. With more general choices of and , we obtain rotations of angle in an arbitrary two-dimensional subspace . This (after a relabeling ) is our standard 4-vector update, and is therefore unfrustrated.
In retrospect we can see what is going on here. We’ve already seen that the left update (2.19) for corresponds in the 4-vector language to an update with given by (2.54). Likewise, a right update for corresponds in the 4-vector language to an update with given by (2.58). Now, a simultaneous left and right update for corresponds to the sum of these two matrices: we thus get twice the usual entry in the upper block, and zero in the lower block. This explains why we get a single rotation with a doubled angle.
Unfortunately, the update (2.80)/(2.83) for is generically frustrated. Let us choose the configuration so that the matrices are block-diagonal with matrices in each block. Then, the expressions for the couplings will be just sums of -type couplings, which can each be written in terms of -vectors as in (2.33) [since the update in each subspace corresponds to a single plane rotation in 4-vector language]. We have therefore independent angles, so that the couplings are frustrated whenever and .
This example can be immediately extended to the cases with : it suffices to choose the matrices to be block-diagonal, with the uppermost block as above, and the lowermost block being an arbitrary matrix. (For such a special configuration one has , even though this does not in general happen for .)
The only case left to investigate is , . In this case frustration can be proven by making use of configurations taking values in the group’s center:
[TABLE]
where and is the identity matrix. Then the couplings (2.21) become
[TABLE]
The are thus real numbers, and we will have frustration if their product around a plaquette is negative. But this occurs whenever three of the four sites have different values of , for example .
In conclusion, the two-sided embedding leads to a frustrated Hamiltonian in all cases except the exceptional case , .
2.5 Summary
In conclusion, we have a simple induced Hamiltonian in the following cases:
- (i)
-vector model, any
- (ii)
principal chiral model, any
- (iii)
complex mixed isovector/isotensor model, or
- (iv)
model, even,
- (v)
pure adjoint principal chiral model, even,
- (vi)
model,
- (vii)
principal chiral model, even, two-sided update
In the first three cases the induced Hamiltonian has only isospin-1 terms, while in the latter four cases it has only isospin-2 terms.
Furthermore, we have an unfrustrated induced Hamiltonian for the mixed isovector/isotensor models with and , as well as for models equivalent to one of these. We also have an unfrustrated Hamiltonian for the model with . All other cases appear to lead in general to frustrated Hamiltonians.
3 Numerical Results
In this section we present our numerical results for the two-dimensional -vector models (2.4) with .
3.1 Observables to be Measured
We wish to study various correlation functions of the isovector field \mbox{\boldmath\sigma}_{x} and the symmetric traceless isotensor field
[TABLE]
Our interest in the isotensor sector arose initially from our work on mixed isovector/isotensor models [45, 46, 44]; but even in the pure -vector model it is of some interest to measure isotensor observables. We thus define the isovector and isotensor 2-point correlation functions
[TABLE]
Note that invariance (which cannot be spontaneously broken in dimension ) determines the 1-point and 2-point correlation functions of and in terms of and :
[TABLE]
All our numerical work will be done on an lattice with periodic boundary conditions. We are interested in the following quantities:
- •
The isovector and isotensor energies272727 Actually (resp. ) is (resp. ) times the mean energy per link; we have chosen this normalization in order to have , where for a totally ordered state. Several other normalizations are in use in the literature.
[TABLE]
where stands for any nearest neighbor of the origin.
- •
The isovector and isotensor magnetic susceptibilities
[TABLE]
where .
- •
The isovector and isotensor correlation functions at the smallest nonzero momentum:
[TABLE]
where .
- •
The isovector and isotensor second-moment correlation lengths
[TABLE]
- •
The isovector and isotensor exponential correlation lengths
[TABLE]
and the corresponding mass gaps . [These quantities make sense only if the lattice is essentially infinite (i.e. ) in at least one direction. We will not measure any exponential correlation lengths in this work; but we will use as a theoretical standard of comparison.]
All these quantities except can be expressed as expectations involving the following observables:
[TABLE]
Thus,
[TABLE]
where ; here is the number of sites in the lattice.
3.2 Autocorrelation Functions and Autocorrelation Times
Let us now define the quantities — autocorrelation functions and autocorrelation times — that characterize the Monte Carlo dynamics. Let be an observable (i.e. a function of the field configuration ). We are interested in the evolution of in Monte Carlo time, and more particularly in the rate at which the system “loses memory” of the past. We define, therefore, the *unnormalized autocorrelation function282828 In the mathematics and statistics literature, this is called the autocovariance function. *
[TABLE]
where expectations are taken in equilibrium. The corresponding normalized autocorrelation function is
[TABLE]
We then define the integrated autocorrelation time
[TABLE]
[The factor of is purely a matter of convention; it is inserted so that if with .] Finally, the exponential autocorrelation time for the observable is defined as
[TABLE]
and the exponential autocorrelation time (“slowest mode”) for the system as a whole is defined as
[TABLE]
Note that whenever the observable is not orthogonal to the slowest mode of the system.
The integrated autocorrelation time controls the statistical error in Monte Carlo measurements of . More precisely, the sample mean
[TABLE]
has variance
[TABLE]
Thus, the variance of is a factor larger than it would be if the were statistically independent. Stated differently, the number of “effectively independent samples” in a run of length is roughly . The autocorrelation time (for interesting observables ) is therefore a “figure of (de)merit” of a Monte Carlo algorithm.
The integrated autocorrelation time can be estimated by standard procedures of statistical time-series analysis [47, 48]. These procedures also give statistically valid error bars on and . For more details, see [49, Appendix C]. In this paper we have used a self-consistent truncation window of width , where is a constant. We have here used ; this choice is reasonable whenever the autocorrelation function decays roughly exponentially.
3.3 Summary of our Runs
We performed runs on the -vector model with , using the -embedding MGMC algorithm described in Section 2.2, on lattices of size . We updated the induced model (2.12) using our standard -model MGMC program [7] with (W-cycle) and (one heat-bath pre-sweep and one heat-bath post-sweep). In all cases the coarsest grid is taken to be . All runs used an ordered initial configuration (“cold start”). The results of these computations are shown in Tables 1–3 (static data) and 4–6 (dynamic data).
For the case we made only a few runs, in order to compare -embedding MGMC to our previous detailed study of the direct MGMC algorithm for this model [8]. For and , by contrast, we made a reasonably extensive set of runs, enough to permit an ab initio dynamic finite-size-scaling analysis. Run lengths were in all cases between and .
Our static data for are in good agreement with data from previous simulations of these models [39, 44, 50, 8, 51]. To check this quantitatively, we made comparisons at each pair for which both “old” and “new” data are available, and summed the resulting values. The results are:
- •
, comparison with [39, 44, 50]:
degrees of freedom: 30
for isovector susceptibility = 38.73, confidence level = 13%
for isovector correlation length = 34.29, confidence level = 27%
for isovector energy = 36.18, confidence level = 20%
for isotensor susceptibility = 40.22, confidence level = 10%
for isotensor correlation length = 38.23, confidence level = 14%
for isotensor energy = 37.02, confidence level = 18%
- •
, comparison with [8] (isovector sector only)292929 We note that the energy defined in [8] is a factor of larger than the one employed here. :
degrees of freedom: 14
for isovector susceptibility = 18.75, confidence level = 17%
for isovector correlation length = 21.74, confidence level = 8%
for isovector energy = 13.70, confidence level = 47%
- •
, comparison with [51] (isovector sector only):
degrees of freedom: 4
for isovector susceptibility = 1.57, confidence level = 81%
for isovector correlation length = 2.58, confidence level = 63%
for isovector energy = 6.53, confidence level = 16%
These confidence levels are all individually acceptable, but taken together they are slightly lower than would be expected a priori. This may be due to a very slight (e.g. ) underestimation of the error bars in either the “old” or the “new” data (or both).
3.4 Computational Work
In Table 7 we show the CPU time per iteration for our -embedding MGMC program with (W-cycle) and , running on one processor of a Cray C-90, for and . We also show, for comparison, the corresponding CPU time for the direct MGMC algorithm, using our old program [8] recompiled for the Cray C-90. The first timing is for Monte Carlo iterations alone; the second timing includes measurement of observables (isovector only in the old program, isovector and isotensor in the new program). We see that:
(a) For each model these timing data grow sublinearly in the volume, in contrast to the theoretical prediction (1.1), because the vectorization is more effective on the larger lattices.303030 Note that the heat-bath subroutine uses von Neumann rejection to generate the desired random variables [7, Appendix A]. The algorithm is vectorized by gathering all the sites of one sublattice (red or black) into a single Cray vector, making one trial of the rejection algorithm, scattering the “successful” outputs, gathering and recompressing the “failures”, and repeating until all sites are successful. Therefore, although the original vector length in this subroutine is , the vector lengths after several rejection steps are much smaller. It is thus advantageous to make the original vector length as large as possible. Note also that our old direct MGMC program [8] used vectors of length instead of . As a result, the ratio time(old program)/time(new program for ) is slightly higher on the small lattices than on the large lattices.
However, the ratio is increasing with , and does appear to be approaching the theoretical value of 4 as .
(b) At each , the CPU time for Monte Carlo iterations grows roughly as with . This is because the bulk of the CPU time is spent in updating the induced model, which is independent of ; while a small fraction of the time is spent computing the induced Hamiltonian and updating the -vector spins, which are operations of order .
(c) At each , the CPU time for measurement of observables (i.e. the difference between the two timings in Table 7) grows roughly as with of order 1. This is because the measurement of isovector (resp. isotensor) observables takes a CPU time of order (resp. ). Our old program [8], by contrast, measured only isovector observables; the time spent in measurement is a factor of smaller.
The running speed on the Cray C-90 for our -embedding MGMC program at was about 329 MFlops for the case, 322 MFlops for , and 317 MFlops for . Our old direct MGMC program runs on the C-90 at 270 MFlops.
The total CPU time for the runs reported here was about 2000 Cray C-90 hours, of which about 2/3 were devoted to and about 1/3 to . The CPU time for was comparatively small.
The statistical efficiency of a Monte Carlo algorithm is inversely proportional to the integrated autocorrelation time for the observable(s) of interest, measured in CPU units. We can compare our largest values of for the cases at lattice size : we get Cray C-90 seconds, respectively, for (times for Monte Carlo iterations only). Thus, the CPU time to generate one “effectively independent” configuration grows (not surprisingly) with , perhaps slightly less than linearly. Note, finally, that for the direct MGMC method is about three times as efficient as the -embedding method: while the CPU time per iteration is roughly the same for the two programs (see Table 7), the autocorrelation time is smaller by a factor of for the direct algorithm (see Section 5).
4 Finite-Size-Scaling Analysis: Static Quantities
4.1 Preliminaries
We shall analyze our static data using a finite-size-scaling extrapolation method due originally to Lüscher, Weisz and Wolff [36] and elaborated recently by our group [38, 39, 40] (see also [37]). An exhaustive study of the case has been made elsewhere, using much more precise and extensive data [39, 44, 50]; and our data for are too sparse to support this type of analysis. Therefore, our analysis here of the static quantities will be restricted to the case .313131 A preliminary version of this analysis was reported in [52].
We wish to test the asymptotic-freedom predictions
[TABLE]
as , where
[TABLE]
is the fundamental mass scale323232 In (4.4), the exponential and power of are universal; the remaining factor is special to the standard nearest-neighbor action (2.4), and comes from a one-loop lattice calculation [53, 54]. ; here , and are universal (albeit non-perturbative) quantities characteristic of the continuum theory (and thus depending only on ), while the , and are nonuniversal constants that can be computed in weak-coupling perturbation theory on the lattice at loops; and denotes any one of . It is worth emphasizing that the same coefficients occur in all four correlation lengths: this is because the ratios of these correlation lengths take their continuum-limit values plus corrections that are powers of the mass , hence exponentially small in .
When analyzing the susceptibilities, it is convenient to study instead the ratios
[TABLE]
The advantage of this formulation in the case of is that one additional term of perturbation theory is available (i.e. but not or ).
For the standard nearest-neighbor action (2.4), the perturbative coefficients , , , , , and have been computed [55, 56, 57, 41]:
[TABLE]
where
[TABLE]
From these expressions it is straightforward to obtain
[TABLE]
Perturbation theory predicts trivially — or rather, assumes — that the lowest mass in the isotensor channel is the scattering state of two isovector particles, i.e. there are no isotensor bound states:
[TABLE]
The non-perturbative universal quantity for the standard continuum -model has been computed exactly by Hasenfratz, Maggiore and Niedermayer (HMN) [58, 59, 60] using the thermodynamic Bethe Ansatz: it is
[TABLE]
For the rest, we know only their values at large [61, 62, 41, 63]
[TABLE]
where is Euler’s constant and
[TABLE]
so that and . From (4.18)/(4.19) combined with (4.16), we get
[TABLE]
We can also write asymptotic scaling in terms of an “improved expansion parameter” [64, 65, 51, 66, 57, 44] based on the energy E_{V}=\langle\mbox{\boldmath\sigma}_{0}\cdot\mbox{\boldmath\sigma}_{1}\rangle. First we invert the perturbative expansion [67, 57]
[TABLE]
and substitute into (4.1); this gives a prediction for as a function of :
[TABLE]
where
[TABLE]
In a similar way we can obtain “energy-improved” expansions for and and the ratios and . Note that for the basic observables and , which grow exponentially as , from the three-loop energy (4.24) we can get at best a three-loop “energy-improved” prediction, no matter how many terms in (4.1)–(4.3) are computed; but for the ratios , in which the exponential cancels out, we can obtain a four-loop “energy-improved” prediction using our knowledge of and .
Our extrapolation method [38] is based on the finite-size-scaling Ansatz
[TABLE]
where is any long-distance observable, is a fixed scale factor (usually ), is a suitably defined finite-volume correlation length, is the linear lattice size, is a scaling function characteristic of the universality class, and is a correction-to-scaling exponent. Here we will use in the role of ; for the observables we will use the four “basic observables” , , , as well as certain combinations of them such as and .
In an asymptotically free model, the functions can be computed in perturbation theory, yielding the following expansions [44] in powers of , where :
[TABLE]
and also
[TABLE]
where , , and
[TABLE]
In the case of the limit it is possible to derive exact expressions for the finite-size-scaling functions [63]. We will compare these curves with our empirically obtained finite-size-scaling functions for .
Our extrapolation method [38] proceeds as follows: We make Monte Carlo runs at numerous pairs and . We plot versus , using those points satisfying both some value and some value . If all these points fall with good accuracy on a single curve — thus verifying the Ansatz (4.28) for , — we choose a smooth fitting function . Then, using the functions and , we extrapolate the pair successively from .
We have chosen to use functions of the form
[TABLE]
(Other forms of fitting functions can be used instead.) This form is partially motivated by theory, which tells us that in some cases exponentially fast as [44]. Typically a fit of order is sufficient; the required order depends on the range of values covered by the data and on the shape of the curve. Empirically, we increase until the of the fit becomes essentially constant. The resulting value provides a check on the systematic errors arising from corrections to scaling and/or from the inadequacies of the form (4.32).
4.2 Data Analysis for
We now apply the method described in the previous subsection to our data for the case. As mentioned before, since the primary purpose of this paper was to study the MGMC embedding algorithm, and since MGMC is in any case less efficient than Wolff’s cluster algorithm [30] for the special case of the -vector models, we have made only a modest effort to produce static data, concentrating on the case. Moreover, not all of our runs can be used to construct the extrapolation curve333333 Although all runs with can be extrapolated using the curve. (only 20 pairs), since at the beginning of this project we were not yet making use of the extrapolation method in our data analyses. Nevertheless, we get interesting results, which confirm the asymptotic-freedom prediction with the correct nonperturbative constant.
Our data cover the range , and we found tentatively that a sixth-order fit is indicated: see Tables 8–11 for the four basic observables .
Next we took and sought to choose to avoid any detectable systematic error from corrections to scaling.343434 A choice would have been equivalent for our set of data for obtaining the finite-size-scaling functions, since although we have points with that can be extrapolated using these curves, we do not have any pairs of points falling in this interval.
We began with the conservative choice and plotted the deviations from the fitting curves for the four basic observables: see Figure 1.
Note that in each case there are three points (at ) that deviate upwards from the fitting curve by 1–5 standard deviations. We do not understand the cause of these deviations. They could be due to corrections to scaling, but this explanation seems unlikely to us because: (a) for the corrections to scaling are negative and smaller in magnitude than those found here [39, 44, 50]; and (b) at the corrections to scaling are extremely small [63]. Possibly there is some instability in the fitting procedure because of the small number of pairs at our disposal (only eight). So, not knowing whether to include or exclude these three “deviant” data points, we try both possibilities and treat the discrepancy between the two results as an added systematic error. More precisely, we divided the axis into three intervals — , and — and allowed different values for each.
See Tables 8–11 for the results of the tests. The best values are obtained, as expected, for the choice , which eliminates the three “deviant” data points as well as one other point, but leaves intact the rest. But since we don’t really have good cause to eliminate these points, we shall take to be our preferred fit. In any case, we shall carry out the extrapolations using both of these fits, and compare the results. In all cases we take .
The finite-size-scaling curve for coming from our preferred fit is shown in Figure 2(a) (solid curve). We also show, for comparison, the perturbative prediction (LABEL:xiV_FSS_PT) through orders and (dashed curves). In Figure 2(b) we show the same solid curve and, for comparison, the corresponding curves for (Monte Carlo) [39, 50] and (exact) [63]. In Figures 3–5 we show the analogous curves for the observables . In all cases, the agreement with perturbation theory is excellent for x\mathrel{\hbox to0.0pt{\lower 3.0pt\hbox{\mathchar 536\relax}\hss}\raise 2.0pt\hbox{\mathchar 318\relax}}0.5 (isovector sector) and x\mathrel{\hbox to0.0pt{\lower 3.0pt\hbox{\mathchar 536\relax}\hss}\raise 2.0pt\hbox{\mathchar 318\relax}}0.6 (isotensor sector). The finite-size-scaling curves fall, as expected, between the and curves, and rather closer to the latter.
Next we checked for consistency between the extrapolated values of each observable coming from different lattice sizes at the same . In most cases these are consistent at better than the 5% level; but in two cases () they are inconsistent at worse than the 0.1% level (we don’t know why). If we exclude these two values and sum over the rest, we get (15 degrees of freedom), respectively, for the observables . This corresponds to confidence levels of .
In Table 12 we show the extrapolated values of the four basic observables from our preferred fit and also from the fit . The discrepancies between these values (if larger than the statistical errors) can serve as a rough estimate of the remaining systematic errors due to corrections to scaling. The statistical errors in the extrapolated values from our preferred fit range from about 0.2–0.5% when \xi_{V}\mathrel{\hbox to0.0pt{\lower 3.0pt\hbox{\mathchar 536\relax}\hss}\raise 2.0pt\hbox{\mathchar 316\relax}}20 to about 2–3% when . The systematic errors are of the same order as the statistical errors, or smaller. The statistical errors at different are strongly positively correlated, but we did not evaluate this correlation quantitatively.
In Figure 6 (points , , +––\tiny\vdash$$\tiny\dashv) we plot divided by the two-loop, three-loop and four-loop predictions for given in equations (4.1)/(4.7)/(4.8)/(4.17). This curve is theoretically predicted to converge to a constant as , and the limiting value is . We also compare to the prediction (4.25) using the “improved expansion parameter” (points and ). [For in (4.25) we use the value measured on the largest lattice; the statistical errors and finite-size corrections on are less than , and therefore induce a negligible error (less than 0.5%) on the predicted .] We show for comparison the expected limiting value 0.9996 coming from the expansion (4.18) through order , evaluated at . Both the four-loop and the “improved” 3-loop prediction are in excellent agreement with the data, showing discrepancies of less than 1%.
In Figure 7 we proceed analogously for , by plotting (points , , +––\tiny\vdash$$\tiny\dashv) divided by the two-loop, three-loop and four-loop predictions for given in equations (4.1)/(4.7)/(4.8)/(4.16)/(4.17). We also compare to the prediction (4.25) using the “improved expansion parameter” (points and ). The four-loop and improved three-loop data are relatively flat, and suggest a limiting value . This can be compared with the expected limiting value 0.6937 coming from the expansion (4.18) through order , evaluated at . Clearly the order- corrections are still significant (of order 3–4%), which is not surprising since the order- correction was 15%.
For the susceptibility we proceed in two different ways, using either directly or else using the ratio . The advantage of the latter approach is that one additional term of perturbation theory is available. Thus, in Figure 8(a) we plot divided by the theoretical prediction (4.2) with the prefactor omitted; the limit of this curve thus gives an estimate of . Here we have two-loop, three-loop and four-loop predictions (points , , +––\tiny\vdash$$\tiny\dashv) as well as “improved” two-loop and three-loop predictions ( and ). Similarly, we could plot divided by the theoretical prediction (4.2)/(4.1) with the prefactors and omitted; the limit would then give an estimate of . However, in order to make the vertical scale of this graph more directly comparable to that of Figure 8(a), we have multiplied the quantity being plotted by the factor . Note that this does not in any way alter the logic of the analysis, as is an explicit number defined in (4.17). The resulting curve is plotted in Figure 8(b); its limit gives an estimate of . In this case we have two-loop, three-loop, four-loop and five-loop predictions (, , +––\tiny\vdash$$\tiny\dashv, \times$$\tiny\times$$\tiny\times$$\tiny\times$$\tiny\times) as well as “improved” two-loop, three-loop and four-loop predictions (, , \small\Box$$\rm\backslash$$\rm/$$\rm/$$\rm\backslash). To convert this to an estimate for itself, we need to multiply by
[TABLE]
The first factor on the right side is theoretically expected to be , and the second factor is theoretically expected to be 1; moreover, our data for itself (Figure 6) are consistent with this prediction (of course they are incapable of distinguishing 0.9992 from 1). For all practical purposes the factor (4.33) is 1, and Figures 8(a) and 8(b) can be compared directly. The two approaches give consistent results, but the one based on seems to work better. We get the value
[TABLE]
This can be compared with the predicted value coming from the expansion (4.20) through order , evaluated at .353535 Our preliminary report [52] mistakenly wrote 5.30 instead of 5.40. We apologize for the arithmetic error! Clearly the order- corrections are still significant (of order 3%), which is not surprising since the order- correction was 14%.
We proceed analogously for , using either directly or else using the ratio . In Figure 9(a) we plot divided by the theoretical prediction (4.3) with the prefactor omitted; the limit of this curve gives an estimate of . Here we have two-loop, three-loop and four-loop predictions (points , , +––\tiny\vdash$$\tiny\dashv) as well as “improved” two-loop and three-loop predictions ( and ). In Figure 9(b) we plot divided by the theoretical prediction (4.3)/(4.1) for this same quantity, with the prefactors and omitted, and the whole thing finally multiplied by ; the limit of this curve gives an estimate of . As before, this is for all practical purposes equal to . Here we have two-loop, three-loop and four-loop predictions (, , +––\tiny\vdash$$\tiny\dashv) as well as “improved” two-loop, three-loop and four-loop predictions (, , \small\Box$$\rm\backslash$$\rm/$$\rm/$$\rm\backslash), namely we do not have an additional term of conventional perturbation theory, but we do have an additional term in the “improved” perturbation theory. The two approaches give consistent results, and here the one based on does not seem to work better. We get the value
[TABLE]
This can be compared with the predicted value coming from the expansion (4.21) through order , evaluated at . Clearly the order- corrections are still significant (of order 31%) which is surprising since the order- correction was 0.4%.
5 Finite-Size-Scaling Analysis: Dynamic Quantities
5.1 Integrated Autocorrelation Times
Of all the observables we studied, the slowest mode (by far) is the squared isovector magnetization : this quantity measures the relative rotations of the spins in different parts of the lattice, and is the prototypical -invariant “long-wavelength observable”. For all three values of , the autocorrelation time has the same qualitative behavior: as a function of it first rises to a peak and then falls; the location of this peak shifts towards as increases; and the height of this peak grows as increases. A similar but less pronounced peak is observed in . By contrast, the integrated autocorrelation times of the energies, and , are uniformly small — less than 2.5, 3.2, 6.4 for , respectively — and vary only weakly with and . This is because the energies are primarily “short-wavelength observables”, and have only weak overlap with the modes responsible for critical slowing-down.
Let us now make these considerations quantitative, by applying finite-size scaling to the dynamic quantities and . We use the Ansatz
[TABLE]
for and . Here is an unknown scaling function, and is supposed to be finite and nonzero.363636 It is of course equivalent to use the Ansatz
\tau_{int,A}(\beta,L)\;\approx\;L^{z_{int,A}}\,h_{A}\Bigl{(}\xi(\beta,L)/L\Bigr{)}\;,
and indeed the two Ansätze are related by . However, to determine whether is nonzero, it is more convenient to inspect a graph of than one of .
We determine by plotting versus and adjusting until the points fall as closely as possible onto a single curve (with priority to the larger values). We emphasize that the dynamic critical exponent is in general different from the exponent associated with the exponential autocorrelation time [1, 68, 4].
Using the procedure just described, we find
[TABLE]
(subjective 68% confidence limits). In Figures 10 and 11 we show the “best” finite-size-scaling plots for each case. Note that the corrections to scaling are quite strong: the point deviates considerably from the asymptotic scaling curve, and even for the deviation is noticeable. However, it is reasonable to hope that these corrections to scaling will have decayed to a small value by ; if so, our estimates of for the W-cycle — which attempt to place the and points on top of one another — will be afflicted by only a small systematic error. In any case, the error bars in (5.2) take account of this potential systematic error. Similarly, for the isotensor sector we have
[TABLE]
(subjective 68% confidence limits). In Figures 12 and 13 we show the “best” finite-size-scaling plots for each case. It thus appears that, for , is strictly smaller than . This behavior is by no means surprising; indeed, the exponents are expected to vary from one observable to another [1, 4]. On the other hand, for we have within statistical error. We have no explanation for the very different behavior in the two cases. Perhaps is strictly smaller than for all , but the difference between the two exponents goes rapidly to zero as .
For the case, our data are insufficient to support a good finite-size-scaling analysis. Instead, let us compare our values of to the corresponding values for the direct MGMC algorithm (taken from [8]): see Table 13. We find that the ratio of the two autocorrelation times is compatible with a constant (independent of and ):
[TABLE]
with (13 DF, level = 92%), using all the data; or with (10 DF, level = 94%), if we use only the data for and . Therefore, the two algorithms have the same dynamic critical exponent, which was found in [8] to be
[TABLE]
and the same dynamic finite-size-scaling functions; in other words, they belong (as expected) to the same dynamic universality class. It could also have been expected that the ratio of autocorrelation times would be : there are distinct internal-spin directions in an -vector model, and the -embedding algorithm handles them one at a time, while the direct algorithm handles them all at once.
The dynamic critical exponent for W-cycle MGMC thus appears to be decreasing as increases. (However, our evidence for this is at present on the borderline of statistical significance.) It is tempting to speculate that (-vector) tends to zero as ; this would be compatible with the vague idea that the model is “essentially Gaussian”. However, it is far from clear that the model is Gaussian in the sense relevant for MGMC.
5.2 Autocorrelation Functions and Exponential Autocorrelation Times
Finally, we want to test the more detailed dynamic finite-size-scaling Ansatz
[TABLE]
where is an unknown exponent and is an unknown scaling function. If , then (5.6) can equivalently be written as
[TABLE]
In this latter situation373737 Contrary to much belief, need not equal . Indeed, if , we have . See [1, 4] for further discussion. , and have the same dynamic critical exponent , and we furthermore have
[TABLE]
where
[TABLE]
In Figure 14 we test the Ansatz (5.7) for and , by plotting versus , and mapping different ranges of to different symbols. (For visual clarity we have also “thinned out” the data points; and to reduce the effects of statistical noise, we have included data only from those runs that are longer than .) The data support the Ansatz (5.7) reasonably well, with each range of defining roughly a single curve (until that curve falls into the statistical noise). The curve for small is close to straight (i.e. close to a pure exponential), while the curves for larger are increasingly convex.383838 The rescaled horizontal axis in Figure 14 ensures that the total area under each curve is 1. Therefore, the more convex curves must be below the straight curve for small , but above it for large .
This means that the ratio is close to 1 for small , and less than 1 for larger . Indeed, our crude estimates of , based on linear fits (taking proper account of correlations) to the approximately linear regime in Figure 14, suggest that is 0.90–1 for \xi(\beta,L)/L\mathrel{\hbox to0.0pt{\lower 3.0pt\hbox{\mathchar 536\relax}\hss}\raise 2.0pt\hbox{\mathchar 316\relax}}0.1, and decreases to for . It is conceivable that tends to 1 as ; if true, this would mean that truly becomes the “slowest mode” in the limit , .
In Figure 15 we show the analogous plot for . Again the Ansatz (5.7) seems to be well verified, but here the dependence on is much weaker (in truth, it is virtually invisible in our data). We estimate that is 0.98 for \xi(\beta,L)/L\mathrel{\hbox to0.0pt{\lower 3.0pt\hbox{\mathchar 536\relax}\hss}\raise 2.0pt\hbox{\mathchar 316\relax}}0.2, and decreases to for . It is conceivable that this dependence on disappears entirely in the limit , i.e. we might have for all in the limit , .
6 Discussion
6.1 Dynamic Critical Behavior of Piecewise-Constant MGMC
Let us now summarize our conclusions about the dynamic critical behavior of W-cycle MGMC with piecewise-constant interpolation:
-
The direct and -embedding MGMC algorithms for the 4-vector model fall, as expected, into the same dynamic universality class. Indeed, the ratio of their autocorrelation times is approximately 1:3, as expected from the fact that there are distinct internal-spin directions in an -vector model.
-
The dynamic critical exponent for W-cycle MGMC with piecewise-constant interpolation is clearly nonzero for two-dimensional asymptotically free models: we have
[TABLE]
(subjective 68% confidence intervals). Furthermore, this exponent apparently varies from one asymptotically free model to another (but the evidence for this is on the borderline of statistical significance, especially in the -vector models). Finally, it is conceivable that (-vector) tends to zero as ; testing this will require measurements at larger values of .
In [8, Section 4.1] we produced a heuristic explanation of why , based on the logarithmically decaying deviations from Gaussianness in an asymptotically free theory. This explanation does not, unfortunately, give a quantitative prediction for ; it merely makes the weak prediction that . Moreover, our study of MGMC in the one-dimensional 4-vector model [9] has cast some doubt on the exactness of this explanation (it seems to be off by at least a logarithm). Clearly, the dynamic critical behavior of MGMC is far from being understood.
6.2 Piecewise-Constant vs. Smooth Interpolation
As noted in the Introduction, Mack and collaborators [11, 12, 13, 14, 15] have advocated the use of a smooth interpolation (e.g. piecewise-linear or better) in place of piecewise-constant. The key question is this: Does the MGMC algorithm have a different dynamic critical exponent depending on whether smooth or piecewise-constant interpolation is used?
In the Gaussian case, the answer is “yes” for a V-cycle but “no” for a W-cycle (see the Introduction). For non-Gaussian asymptotically free models, we conjecture that the answer is the same as in the Gaussian case, but to date no full-scale test has been made. Some time ago, Hasenbusch, Meyer and Mack [13] reported preliminary data for the 3-vector model suggesting that for a V-cycle with smooth interpolation. (These authors did not study a W-cycle, as this would be extremely costly in their unigrid approach, but it is safe to assume that .) This is considerably lower than our estimate for the piecewise-constant W-cycle — suggesting that the smooth interpolation might indeed lead to a smaller dynamic critical exponent than piecewise-constant interpolation, even for a W-cycle. However, the estimate of [13] is based on runs at only four pairs, which moreover have different values of , so it is very difficult to perform a correct finite-size-scaling analysis. It would be very useful to have data at additional pairs, and with higher statistics.
More recently, Hasenbusch and Meyer [14, 15] have reported data for the and -models, using a V-cycle with piecewise-linear interpolation. For the data are consistent with the dynamic finite-size-scaling Ansatz (5.1) with . For , however, the data are very erratic, and do not appear to be consistent with (5.1). More data would be very useful here too; both the dynamic critical exponent and the dynamic finite-size-scaling functions could be compared with those for the piecewise-constant W-cycle [34, 35].
6.3 Comparison with Other Collective-Mode Algorithms
We have previously [8, Section 4.3] compared the behavior of MGMC with that of Fourier acceleration [69, 70, 71, 72]. To our knowledge, no new data on the dynamic critical behavior of Fourier acceleration has appeared since that discussion, so we have nothing new to add.
A very different approach to collective-mode Monte Carlo is represented by Wolff’s [30] cluster algorithm for -vector models.393939 This algorithm was also proposed independently by Hasenbusch [73].
This algorithm is based on embedding a field of Ising variables into the -vector model, and then simulating the induced Ising model by the Swendsen-Wang [74] algorithm (or its single-cluster variant [30]). It can be argued heuristically [31, 29] that this algorithm is effective at creating long-wavelength spin waves; indeed, extensive numerical tests show that critical slowing-down is completely eliminated in the two-dimensional -vector models for [30, 31, 32, 29, 33].404040 For the complete elimination of critical slowing-down requires also that the algorithm be effective at creating vortex-antivortex pairs; the mechanism for this is explained in [31].
The principal difference between MGMC and Fourier acceleration on the one hand, and algorithms of Swendsen-Wang and Wolff type on the other, is how proposals are made for updating the long-wavelength modes. MGMC and Fourier acceleration propose additive updates of fixed shape (square, triangular or sinusoidal) and variable amplitude, while cluster algorithms like Swendsen-Wang and Wolff allow the system, in some sense, to choose its own collective modes.
At present, the Wolff algorithm is clearly the best algorithm for simulating -vector models (at least in two dimensions); neither MGMC nor Fourier acceleration appears to be a serious competitor. However, this favorable situation for Wolff-type algorithms probably does not extend to other -models: indeed, there are strong reasons to believe [29] that a generalized Wolff-type embedding algorithm can have dynamic critical exponent only if the target manifold is a sphere (as in the -vector model), a product of spheres, or the quotient of such a space by a discrete group (e.g. real projective space ).414141 Moreover, the practical Wolff-type embedding algorithm (with one Swendsen-Wang hit on the induced Ising model per iteration) appears to behave rather poorly in the models, with [75, 33].
By contrast, the MGMC algorithm (especially in its -embedding form) has a straightforward extension to -models taking values in an arbitrary homogeneous space , and it appears to work well quite generally [e.g. for ]. This is a strong motivation for pursuing the study of MGMC algorithms.
Acknowledgments
We want to thank Martin Hasenbusch and Steffen Meyer for the discussions that initiated this project, and for many helpful comments thereafter. We also wish to thank Gustavo Mana for many discussions. Most of the computations reported here were carried out on the Cray C-90 at the Pittsburgh Supercomputing Center (PSC). This work was supported in part by the U.S. National Science Foundation grants DMS-9200719 and PHY-9520978 (A.D.S.), by the Istituto Nazionale di Fisica Nucleare (A.P.), and by PSC grants PHY890035P and MCA94P032P.
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