Lattice Gauge Theory of Three Dimensional Thirring Model

TL;DR

This study uses lattice gauge theory simulations to investigate the three-dimensional Thirring model, revealing spontaneous chiral symmetry breaking for certain fermion flavors and suggesting a critical flavor number where behavior changes.

Contribution

It provides the first lattice gauge theory simulation evidence for chiral symmetry breaking in the three-dimensional Thirring model with multiple fermion flavors.

Findings

Chiral symmetry is spontaneously broken for N=2, 4, 6.

Evidence of a critical fermion flavor number N_cr between 2 and 6.

Finite volume effects are significant but do not obscure the main results.

Abstract

Three dimensional Thirring model with $N$ four-component Dirac fermions, reformulated as a lattice gauge theory, is studied by computer simulation. According to an $8^{3}$ data and preliminary $16^3$ data, chiral symmetry is found to be spontaneously broken for $N=2,\;4$ and 6. $N=2$ data exhibits long tail of the non-vanishing chiral condensate into weak coupling region, and $N=6$ case shows phase separation between the strong coupling region and the weak coupling region. Although the comparison between $8^3$ data and $16^3$ data shows large finite volume effects, an existence of the critical fermion flavor number $N_{{\rm cr}}$ $(2<N_{{\rm cr}}<6)$ for which the chiral behavior changes its character, is suggested by the current numerical data.

Equations

$$\mathcal{L}=\sum _a{\bar{\psi }}_a({\gamma }_{\mu }{\partial }_{\mu }+m){\psi }_a+\frac{G}{2N}(\sum _a{\bar{\psi }}_a{\gamma }_{\mu }{\psi }_a{)}^2.$$

$${\mathcal{L}}_{\mathrm{HLS}}=\sum _a{\bar{\psi }}_a({\gamma }_{\mu }{D}_{\mu }+m){\psi }_a+\frac{1}{2G}(A_{\mu }-\sqrt{N}{\partial }_{\mu }\varphi {)}^2,$$

$$N_{\mathrm{cr}}=\frac{128}{3{\pi }^2}\approx 4.32$$

$$g_{\rm cr}\propto\exp\Bigl{(}\frac{\pi^{2}}{16}N\Bigr{)},$$

$$\bigl{<}\bar{\psi}\psi\bigr{>}\approx A_{N}\exp\biggl{(}-\frac{B_{N}}{\sqrt{\frac{g}{g_{\rm cr}}-1}}\biggr{)}.$$

$${\mathcal{L}}_L=\sum _a{{\Phi }^{†}}_a(M^{†}M{)}^{-1}{\Phi }_a-N\beta \sum _{\mu }\cos (\varphi (x+\mu )-\varphi (x)+{\theta }_{\mu }(x)),$$

$$M_{x,y}=\frac{1}{2}\sum _{\mu }{\eta }_{\mu }(x)(e^{i{\theta }_{\mu }(x)}{\delta }_{y,x+\mu }-e^{-i{\theta }_{\mu }(x-\mu )}{\delta }_{y,x-\mu })+m{\delta }_{x,y}$$

$$\bigl{<}\bar{\psi}\psi\bigr{>}=\bigl{<}{\rm Tr}M^{-1}\bigr{>},$$

$$\bigl{<}\bar{\psi}\psi\bigr{>}\sim(\beta_{c}-\beta)^{\frac{1}{2}}.$$

Full text

Lattice Gauge Theory

of

Three Dimensional Thirring Model

Seyong Kim

*Center for Theoretical Physics, Seoul National University, Seoul 151-742, Korea

*skim$@$ctp.snu.ac.kr

Yoonbai Kim

*Department of Physics, Pusan National University, Pusan 609-735, Korea

yoonbai$@$top.phys.pusan.ac.kr

**Abstract

**

Three dimensional Thirring model with $N$ four-component Dirac fermions, reformulated as a lattice gauge theory, is studied by computer simulation. According to an $8^{3}$ data and preliminary $16^{3}$ data, chiral symmetry is found to be spontaneously broken for $N=2,\;4$ and 6. $N=2$ data exhibits long tail of the non-vanishing chiral condensate into weak coupling region, and $N=6$ case shows phase separation between the strong coupling region and the weak coupling region. Although the comparison between $8^{3}$ data and $16^{3}$ data shows large finite volume effects, an existence of the critical fermion flavor number $N_{{\rm cr}}$ $(2<N_{{\rm cr}}<6)$ for which the chiral behavior changes its character, is suggested by the current numerical data.

PACS number(s): 11.15.Ha, 11.30.Qc, 11.30.Rd

Thirring model, appeared in the (1+1) dimensional world around forty years ago, has shed light on understanding various intriguing issues of field theories, e.g., exactly-soluble models, bosonization, conformal fixed point, etc. Recently, the very model with $N$ four-component Dirac fermions has attracted attention in the (2+1) dimensional world with regard to renormalizability of four-fermion models in the context of $1/N$-expansion [1]-[3], dynamical mass generation for the fermion in the analysis of Schwinger-Dyson equation [4]-[9] and so on. In the listed works, usually $1/N$ expansion technique is employed since interesting features of the model are inherently non-perturbative. In this note, using a Monte Carlo simulation technique, we consider the Thirring model on a three-dimensional(3D) cubic lattice, and ask whether the vector-type four fermion interaction of the model breaks chiral symmetry spontaneously.

The Lagrangian for the 3D Thirring model in Euclidean space time is given by

[TABLE]

Typical way of treating this model is to introduce an auxiliary vector field $A_{\mu}$ [1, 3, 4, 5, 7]. On a lattice, this leads to an auxiliary vector field formulation [10]. An alternative approach which is attractive to us, is to regard $A_{\mu}$ as a $U(1)$ gauge field. Indeed a reformulation of the Thirring model as a gauge theory can be achieved through a hidden local symmetry [11], [2, 8, 9]:

[TABLE]

where $\psi_{a}$ is a 4-component Dirac spinor, $\phi$ is a fictitious Nambu-Goldstone boson field (or equivalently a Stückelberg field), and $D_{\mu}=\partial_{\mu}-\frac{i}{\sqrt{N}}A_{\mu}$.

Noticing that the unitary gauge-fixed form of the above $U(1)$ gauge theory coincides exactly with the original Thirring model, we expect benefits from gauge-invariant formulation. One such benefit is that, in the limit of zero bare fermion mass, one can avoid complications like the dynamical generation of parity-violating mass [12] or Chern-Simons term [13]. The other is that we have a freedom to choose the most appropriate gauge for each particular situations of interest, e.g., analysis of the Schwinger-Dyson equation in a rigorous fashion [8] and a construction of a gauge-invariant effective potential for the chiral order parameter [9].

Since the model we are considering is free from parity violation by an exact argument in the gauge theory formulation, we can focus on the spontaneous breakdown of chiral symmetry (actually a flavor symmetry breaking of $U(2N)\rightarrow U(N)\times U(N)$). There are various interesting aspects to be addressed once the above chiral symmetry breaking occurs, which were raised by analytic methods. First, a rather systematic approach of Schwinger-Dyson equations in the gauge theory formulation [8] suggests a finite, critical number of the four-component fermion species,

[TABLE]

which is obtained in the infinite limit of four-fermion coupling $g$ ($g$ is a dimensionless coupling, $G$ rescaled by the ultraviolet cutoff $\Lambda,g\equiv G\Lambda$). In contrast, the auxiliary field formulation with the “Landau gauge fixing” implies otherwise.111In fact this Landau gauge fixing amounts to an $R_{\xi}$ gauge in the hidden local symmetry formulation if we concede that the auxiliary field is a gauge field. A crude constant mass approximation to this formulation seems to predict a chiral symmetry breaking for all $N$ [5]

[TABLE]

which also tells us the ratio of the values of the critical coupling $g_{\rm cr}$ for different $N$’s. Until now, there is no known equivalent relationship between $g_{\rm cr}$ and $N$ in the gauge theory formulation. However, if such a relation exists, it should be different from the above equation since there exists a critical number of fermion flavor $N_{\rm cr}$ in the gauge theory formulation [14]. Second, it may be interesting if we can read the chiral condensate \bigl{<}\bar{\psi}\psi\bigr{>} as a function of the coupling $g$ for a given $N$. In the gauge theory formulation [9], a critical scaling behavior was suggested to follow

[TABLE]

Of course, on a finite lattice, the behavior of the chiral condensate will be modified by finite volume effect and may be difficult to compare with this formula away from $g_{\rm cr}$.

Analytic studies in the framework of the Schwinger-Dyson equations combined with an $1/N$ expansion raised the questions in the above and it is interesting to investigate these questions by a lattice study. In choosing an appropriate action for numerical simulation, one can use an auxiliary vector field formulation without gauge symmetry or employ a gauge field formulation based on a hidden local symmetry. Recently simulation results of the auxiliary field formulation case was reported in Ref.[10]. Here we adopt the gauge theory formulation and take advantage of various benefits coming from the gauge theory formulation. A discretized version of the 3D action, ${\cal L}_{L}$ with a staggered fermion formulation for the fermion part, is

[TABLE]

where $\Phi$ is the pseudo fermion and $\theta_{\mu}$ is the gauge field,

[TABLE]

with staggered fermion phase factor, $\eta_{\mu}(x)$, and $\beta=1/g$, after dropping the constant piece. We include a finite bare fermion mass term and investigate the massless limit by simulating successively smaller fermion mass, $m$, since our numerical method prohibits simulating the chiral limit directly. The continuum limit of this lattice model is free from parity-violating mass or Chern-Simons term [15]. The gauge theory formulation of 3D Thirring model in Eq.(6) looks like 3-dimensional quantum electrodynamics (QED) + scalar QED with kinetic term of the gauge field and the Higgs degrees of freedom truncated. Note that the link variable $\exp(i\theta_{\mu})$ can not be used for the vector-fermion interaction term, $\bar{\psi}A_{\mu}\gamma_{\mu}\psi$, in the auxiliary field formulation. This was not encountered in lattice QED3 case because the gauge symmetry fixes uniquely the interaction between the fermion and the gauge field and it is the same as Eq.(7). Since this theory is non-asymptotically free, such differences may lead to different continuum limits. In addition, since in our gauge field formulation the gauge kinetic term will be induced by quantum effects, we can not say whether it will follow compact or non-compact form yet. Such an effect seems to be realized indeed by some kind of discontinuous behavior in the actual simulations for the strong coupling region [10].

Thus far, we have completed simulations for $N=2,\;4$ and 6 systems on a $8^{3}$ lattice volume with fermion mass, $m=0.05,\;0.025$, and 0.0125 by use of hybrid Monte Carlo algorithm (HMC). Preliminary data on $16^{3}$ lattice for $N=2$ and 6 will be used for finite volume effect study. As usual, by defining fermion fields on even sites only, one can avoid an additional fermion doubling from $M^{\dagger}M$ and can simulate even $N$ system with this HMC algorithm [17]. For odd $N$’s, we need to use other numerical algorithm such as hybrid molecular dynamics method [16]. Typically, we have accumulated $\tau\sim 10,000$ time unit after discarding initial $\tau=100$ time unit as the equilibration time with choosing appropriate $\Delta\tau$($=0.025,0.02,0.0125$) which gives reasonable acceptance rate (50% to 90%). The Metropolis step interval is set to $\tau=0.5$. As an order parameter for the chiral symmetry, we calculate the chiral condensate

[TABLE]

where we use a single noise vector to estimate the condensate. In addition, we also calculate internal energy \bigl{<}E_{\beta}\bigr{>} defined by \bigl{<}N\beta(3-\sum_{\mu}\cos{(\phi(x+\mu)-\phi(x)+\theta_{\mu}(x))})\bigr{>}, and specific heat \bigl{<}{E_{\beta}}^{2}\bigr{>}-\bigl{<}E_{\beta}\bigr{>}^{2} for each $N$. Mass spectra for the fermion, $\pi$(pseudo scalar channel) and $\rho$(vector channel) are also calculated. However, due to a short extent in the time direction, we were not able to fit the propagators reliably. Thus, in the following discussion, we will rely mainly on the chiral condensate measurement.

Qualitative nature of our numerical simulation result can be summarized in Fig. 1. The figure is the $m\rightarrow 0$ extrapolated chiral condensate on a $8^{3}$ lattice volume for $N=2,\;4$, and 6 respectively as a function of $\beta$, where we assumed a naive linear behavior in $m$ for the chiral condensate ($\sim c_{0}+c_{1}m$). The figure is similar to QED3 case; in [17], $N=2$ shows smooth change of the chiral condensate and a long tail into weak coupling regime, $N=3$ shows a complex behavior of the condensate near the critical point, and $N=5$ shows an abrupt change of the condensate. This supports the existence of the critical number of fermion flavors for chiral symmetry breaking in QED3 [18]. Likewise, if we believe that the naive linear extrapolation of the condensate is a correct procedure in the gauge theory formulation of 3D Thirring model, there appears to be a critical $N$ around which the phase transition changes its character. A complicated second order phase transition which connects the strong coupling region to the weak coupling region smoothly ($N=2$) turns into a phase transition which has a phase separation ($N=6$).

\bigl{<}\bar{\psi}\psi\bigr{>}$$\beta   -0.1  0  0.1  0.2  0.3  0.4  0.5  0.6  0.7  0  0.2  0.4  0.6  0.8  1  1.2  1.4  1.6  1.8     \Diamond$$\Diamond$$\Diamond$$\Diamond$$\Diamond$$\Diamond$$\Diamond$$\Diamond$$\Diamond$$\Diamond$$\Diamond$$\Diamond                                    +$$+$$+$$+$$+$$+$$+$$+$$+$$+$$+$$+$$+$$+                            \Box$$\Box$$\Box$$\Box$$\Box$$\Box$$\Box$$\Box$$\Box$$\Box$$\Box$$\Box$$\Box$$\Box$$\Box

Figure 1: The chiral condensate, \bigl{<}\bar{\psi}\psi\bigr{>}_{m\rightarrow 0}, vs. $\beta$ for $N=2$ ($\Diamond$ points), $N=4$ ($+$ points), and $N=6$ ($\Box$ points) on a $8^{3}$ lattice.

\beta^{2}\bigl{<}\bar{\psi}\psi\bigr{>}$$\beta  0  0.05  0.1  0.15  0.2  0.25  0.3  0.8  1  1.2  1.4  1.6  1.8     \Diamond$$\Diamond$$\Diamond$$\Diamond$$\Diamond$$\Diamond$$\Diamond$$\Diamond$$\Diamond$$\Diamond

Figure 2: \beta^{2}\bigl{<}\bar{\psi}\psi\bigr{>}_{m\rightarrow 0} vs. $\beta$ for $N=2$ on a $8^{3}$ lattice.

To elucidate these behaviors more clearly, in Fig. 2, we search for a continuum scaling window for $N=2$. Like QED3 $N=0$ and $1$ case, the chiral condensate exhibits a long plateau in weak coupling region. On the other hand, $N=4$ shows two slopes and hints flat region.

\bigl{<}\bar{\psi}\psi\bigr{>}^{2}$$\beta   -0.05  0  0.05  0.1  0.15  0.2  0.25  0.3  0.35  0.4  0  0.2  0.4  0.6  0.8  1  1.2  1.4  1.6  1.8     \Diamond$$\Diamond$$\Diamond$$\Diamond$$\Diamond$$\Diamond$$\Diamond$$\Diamond$$\Diamond$$\Diamond$$\Diamond$$\Diamond                                    +$$+$$+$$+$$+$$+$$+$$+$$+$$+$$+$$+$$+$$+                            \Box$$\Box$$\Box$$\Box$$\Box$$\Box$$\Box$$\Box$$\Box$$\Box$$\Box$$\Box$$\Box$$\Box$$\Box

Figure 3: \bigl{<}\bar{\psi}\psi\bigr{>}^{2}_{m\rightarrow 0} vs. $\beta$ for $N=2$($\Diamond$), $N=4$($+$), and $N=6$($\Box$) on a $8^{3}$ lattice.

In Fig. 3, we plot \bigl{<}\bar{\psi}\psi\bigr{>}^{2}_{m\rightarrow 0} vs. $\beta$, in order to see whether the critical behavior follows a mean field theory,

[TABLE]

It appears that $N=2$ data does not agree with a mean field behavior and $N=6$ data seems to follow a mean field expectation. For $N=4$ data, it is very hard to say whether the system behaves in one way or the other. From internal energy density and specific heat peak, we obtain $\beta_{c}=0.9(1)$ for $N=2$, $0.35(5)$ for $N=4$, and $0.18(1)$ for $N=6$. These values are roughly consistent with those obtained from Fig. 1, 2, and 3.

According to the QED3 case, large finite volume effect is expected [17, 19]. In order to purchase this effect, simulations on larger lattices and smaller $m$ values are needed. Thus, we are investigating the same system on a $16^{3}$ lattice with the masses $m=0.05,0.025$ and $0.0125$. In Table 1, we compare \bigl{<}\bar{\psi}\psi\bigr{>}_{m\rightarrow 0} for $N=2$ and in Table 2 compare that for $N=6$ on $8^{4}$ and $16^{3}$ volume. The first column is the chiral condensate for $8^{3}$ using a linear extrapolation of all range of $m$ data, the second column is that for $16^{3}$ using a linear extrapolation of smaller $m$ data ($m=0.025,\;0.0125$), and the third column is that for $16^{3}$ using a quadratic fit to existing data. Although the dependence of the chiral condensate on both the size of lattices and the extrapolation method are considerable quantitatively (particularly in the very weak coupling region of $N=2$ case), we found that the overall qualitative nature of the chiral condensate does not seem to be changed: In all three cases, $N=6$ data are compatible with mean-field behavior but $N=2$ data is not.

[TABLE]

Table 1: Comparison of the chiral condensate for $N=2$

[TABLE]

Table 2: Comparison of the chiral condensate for $N=6$

In conclusion, we have found spontaneous chiral symmetry breaking for $N=2,\;4$ and 6 in 3D Thirring model by use of Monte Carlo simulation on a $8^{3}$ and a $16^{3}$ lattice. However, the chiral condensate in the $m\rightarrow 0$ limit suggests that the nature of the phase transition in $N=2$ is different from that in $N=6$. $N=2$ case has a long tail of non-vanishing chiral condensate in weak coupling region and $N=6$ case has a phase separation between the strong coupling region and the weak coupling region. It suggests a possibility that there exists $N_{\rm cr}$ where the chiral phase transition becomes a mean field type transition. This situation is quite similar to QED3. In contrast to the lattice simulation result of the auxiliary vector field formulation, our result seems to be more consistent with the gauge theory formulation of 3D Thirring model in continuum.

For a future work, we think that further detailed study on finite volume effects is needed although we have discussed our preliminary data on a $16^{3}$ lattice. Also, two further questions remain open: One is whether lattice 3D Thirring model is similar to 3D lattice QED. The other is whether the induced gauge field kinetic term takes the form of compact QED-like or noncompact QED-like. The latter issue will be determined by the importance of topological sector. At this moment, we do not know what the induced term looks like.

We would like to thank Ph. de Forcrand, D.K. Hong, J.B. Kogut, P. Maris, D.K. Sinclair, H.S. Song and K. Yamawaki for various comments and discussions. S. Ohta and the Computing Center of the Institute of Physical and Chemical Research (RIKEN) are acknowledged for their support and allowing us to access their Fujitsu VPP-500. This work was supported by the KOSEF(through CTP, Brain Pool Program and 95-0702-04-01-3) and the Korean Ministry of Education (BSRI-95-2413).