Janossy densities for chiral random matrix ensembles and their applications to two-color QCD

TL;DR

This paper develops a numerical method to compute individual low-lying eigenvalue distributions of chiral random matrix ensembles and applies it to analyze the Dirac spectra in two-color QCD, enabling precise determination of the chiral condensate.

Contribution

The paper introduces a compact formula for eigenvalue distributions and applies a Nyström-type quadrature method for accurate numerical evaluation in chiral ensembles.

Findings

Numerical evaluation of eigenvalue distributions for chiral ensembles.

Fitting of lattice QCD data to chiral symplectic ensemble predictions.

Precise determination of the chiral condensate in two-color QCD.

Abstract

We compute individual distributions of low-lying eigenvalues of massive chiral random matrix ensembles by the Nystr\"om-type quadrature method for evaluating the Fredholm determinant and Pfaffian that represent the analytic continuation of the Janossy densities (conditional gap probabilities). A compact formula for individual eigenvalue distributions suited for precise numerical evaluation by the Nystr\"om-type method is obtained in an explicit form, and the $k^{\rm{\small th}}$ smallest eigenvalue distributions are numerically evaluated for chiral unitary and symplectic ensembles in the microscopic limit. As an application of our result, the low-lying Dirac spectra of the SU(2) lattice gauge theory with $N_F=8$ staggered flavors are fitted to the numerical prediction from the chiral symplectic ensemble, leading to a precise determination of the chiral condensateof a two-color QCD-like system in the future.

Figures

Tables

Table 1. Table 1: Nodes and weights of the Gauss-Legendre rule

$M$	$x_i^{[-1,1]}$	$w_i^{[-1,1]}$
1	0	2
2	$\pm \sqrt{1/3}$	1
3	0	8/9
	$\pm \sqrt{3/5}$	5/9
4	$\pm \sqrt{\left(3-2\sqrt{6/5}\right)/7}$	$\frac{18+\sqrt{30}}{36}$
	$\pm \sqrt{\left(3+2\sqrt{6/5}\right)/7}$	$\frac{18+\sqrt{30}}{36}$
5	0	128/225
	$\pm \frac{1}{3}\sqrt{5-2\sqrt{10/7}}$	$\frac{322+13\sqrt{70}}{900}$
	$\pm \frac{1}{3}\sqrt{5+2\sqrt{10/7}}$	$\frac{322-13\sqrt{70}}{900}$

Table 2. Table 2: Fit result of Σ ^ ^ Σ \hat{\Sigma} , chiral condensate in the lattice unit, together with the corresponding value of μ 𝜇 \mu . The bare coupling constant is given through β = 4 / g 2 𝛽 4 superscript 𝑔 2 \beta=4/g^{2} . The most right column is the number of configurations we used in each of the fitting, which belong to the topological charge ν = 0 𝜈 0 \nu=0 sector. The reduced chi squared, χ 2 / d . o . f formulae-sequence superscript 𝜒 2 d o f \chi^{2}/\mathrm{d.o.f} , indicates the quality of the fitting.

lattice size	$\beta$	$\hat{\Sigma }$	$\mu$	${\chi }^2/d.o.f$	num($\nu =0$)
$8^4$	1.100	0.2176(27)	8.91(11)	0.30(17)	230
	1.200	0.1997(24)	8.18(10)	0.25(17)	260
	1.300	0.1651(14)	6.76(05)	0.45(20)	314
	1.350	0.1378(12)	5.65(05)	0.27(16)	467
	1.375	0.1238(13)	5.07(05)	0.52(20)	407
	1.400	0.0781(11)	3.20(04)	11.00(93)	843
	1.425	0.0215(01)	0.880(4)	2.85(47)	2338
${12}^4$	1.100	0.1903(23)	39.46(48)	0.66(25)	399
	1.300	0.1425(44)	29.54(91)	0.20(15)	63
	1.350	0.1263(23)	26.19(49)	0.37(20)	38
	1.375	0.1156(39)	23.98(81)	0.13(13)	45
	1.400	0.0831(14)	17.23(29)	0.50(23)	106
	1.425	0.0598(10)	12.41(21)	0.37(20)	206
	1.450	0.0209(04)	4.32(08)	6.74(73)	600
${16}^4$	1.350	0.1252(20)	82.0(1.3)	0.34(16)	105
	1.375	0.1064(34)	69.8(2.3)	0.22(16)	41
	1.400	0.0799(12)	52.35(80)	0.37(21)	155
	1.425	0.0521(05)	34.13(33)	0.48(20)	369
	1.450	0.0246(02)	16.14(17)	0.83(26)	561
	1.475	0.0083(01)	5.47(12)	1.32(30)	248

Table 3. Table 3: Number of independent Monte Carlo configurations used in Sec. 3 .

$N$	$N_f=2$	$N_f=4$	$N_f=8$
250	2495000	1535950
1000	245000	495000	245000
2000	245000	245000	245000
4000			145000

Equations

$$\displaystyle Z_{N,\beta,\nu}(\{m_{a}\})=\int dW\mathrm{e}^{-\beta\,\mathrm{tr}(W^{\dagger}W)}\prod_{a=1}^{\alpha}\mathrm{det}\left(\begin{array}[]{cc}m_{a}&\mathrm{i}W\\ \mathrm{i}W^{\dagger}&m_{a}\end{array}\right),$$

$$Z_{N,\beta ,\nu }(\{m_a\})=\left(\prod _{a=1}^{\alpha }{m}_a^{\nu }\right){\int }_0^{\infty }\cdots {\int }_0^{\infty }\prod _{i=1}^N\left(d{x}_i{x}_i^{\frac{\beta (\nu +1)}{2}-1}{\mathrm{e}}^{-\beta x_i}\prod _{a=1}^{\alpha }(x_i+m_a^2)\right)\prod _{i>j}^N|x_i-x_j{|}^{\beta }.$$

$$R_{N,\beta ,\nu }^{(p)}({\lambda }_1,\dots ,{\lambda }_p;\{m_a\})=\left(2^p\prod _{j=1}^p|{\lambda }_j|\right){\sigma }_{N,\beta ,\nu }^{(p)}({\lambda }_1^2,\dots ,{\lambda }_p^2;\{m_a\}),$$

$${\sigma }_{N,\beta ,\nu }^{(p)}(x_1,\dots ,x_p;\{m_a\})=\frac{{\Sigma }_{N,\beta ,\nu }^{(p)}(x_1,\dots ,x_p;\{m_a\})}{{\Sigma }_{N,\beta ,\nu }^{(0)}(\{m_a\})},$$

$${\Sigma }_{N,\beta ,\nu }^{(p)}(x_1,\dots ,x_p;\{m_a\})$$

$$=\frac{1}{(N-p)!}{\int }_0^{\infty }d{x}_{p+1}\cdots {\int }_0^{\infty }d{x}_N\prod _{i=1}^N\left(x_i^{\frac{\beta (\nu +1)}{2}-1}{\mathrm{e}}^{-\beta x_i}\prod _{a=1}^{\alpha }(x_i+m_a^2)\right)\prod _{i>j}^N|x_i-x_j{|}^{\beta }.$$

$$\displaystyle z_{j}=\left\{\begin{array}[]{cl}-m_{j}^{2},&\quad j=1,\ldots,\alpha\\ x_{j-\alpha},&\quad j=\alpha+1,\ldots\alpha+p\end{array}\right..$$

$${\sigma }_{N,\beta =2,\nu }^{(p)}(x_1,\dots ,x_p;\{m_a\})=\frac{1}{N!{\Sigma }_{N,\beta =2,\nu }^{(0)}(\{m_a\})}\det [K(z_i,z_j){]}_{i,j=1}^{p+\alpha }.$$

$${\sigma }_{N,\beta =(1,4),\nu }^{(p)}(x_1,\dots ,x_p;\{m_a\})$$

$$=\frac{1}{N!{\Sigma }_{N,\beta =1,4,\nu }^{(0)}(\{m_a\})}\mathrm{Pf}\left(Z[K(z_i,z_j){]}_{i,j=1}^{p+\alpha }\right),$$

$$N\to \infty ,x_i,m_a\to 0,{\zeta }_i=\sqrt{8N{x}_i},\mathrm{and}{\mu }_a=\sqrt{8N}{m}_a:\mathrm{fixed}.$$

$$V\to \infty ,m_a\to 0,{\mu }_a=\Sigma V{m}_a:\mathrm{fixed},$$

$$\displaystyle R_{\beta=2,\nu}^{(p)}(\zeta_{1},\ldots,\zeta_{p};\{\mu_{a}\})=\frac{1}{Z_{\beta=2,\nu}(\{\mu_{a}\})}\det\left(\begin{array}[]{cc}\left[K_{--}(\mu_{a},\mu_{b})\right]_{a,b=1,\ldots,\alpha}&\left[K_{-+}(\zeta_{i},\mu_{b})\right]_{\begin{subarray}{c}i=1,\ldots,p\\ b=1,\ldots,\alpha\end{subarray}}\\ \left[K_{+-}(\mu_{a},\zeta_{j})\right]_{\begin{subarray}{c}a=1,\ldots,\alpha\\ j=1,\ldots,p\end{subarray}}&\left[K_{++}(\zeta_{i},\zeta_{j})\right]_{i,j=1,\ldots,p}\end{array}\right),$$

$$K_{++}(\zeta ,{\zeta }^{\prime })=\frac{\sqrt{\zeta {\zeta }^{\prime }}}{{\zeta }^{\prime 2}-{\zeta }^2}[J_{\nu }(\zeta ){\zeta }^{\prime }{J}_{\nu +1}({\zeta }^{\prime })-J_{\nu }({\zeta }^{\prime })\zeta J_{\nu +1}(\zeta )],$$

$$K_{++}(\zeta ,\zeta )=\frac{\zeta }{2}[J_{\nu }(\zeta {)}^2+J_{\nu +1}(\zeta {)}^2],$$

$$K_{+-}(\zeta ,{\mu }^{\prime })=\frac{-\sqrt{\zeta {\mu }^{\prime }}}{-{\mu }^{\prime 2}-{\zeta }^2}[J_{\nu }(\zeta )(-{\mu }^{\prime })I_{\nu +1}({\mu }^{\prime })-I_{\nu }({\mu }^{\prime })\zeta J_{\nu +1}(\zeta )],$$

$$K_{-+}(\mu ,{\zeta }^{\prime })=\frac{-\sqrt{\mu {\zeta }^{\prime }}}{{\zeta }^{\prime 2}+{\mu }^2}[I_{\nu }(\mu ){\zeta }^{\prime }{J}_{\nu +1}({\zeta }^{\prime })-J_{\nu }({\zeta }^{\prime })(-\mu )I_{\nu +1}(\mu )],$$

$$K_{--}(\mu ,{\mu }^{\prime })=\frac{\sqrt{\mu {\mu }^{\prime }}}{{\mu }^{\prime 2}-{\mu }^2}[I_{\nu }(\mu ){\mu }^{\prime }{I}_{\nu +1}({\mu }^{\prime })-I_{\nu }({\mu }^{\prime })\mu I_{\nu +1}(\mu )],$$

$$K_{--}(\mu ,\mu )=\frac{\mu }{2}[I_{\nu }(\mu {)}^2-I_{\nu +1}(\mu {)}^2],Z_{\beta =2,\nu }({\mu }_1,\dots ,{\mu }_{\alpha })=\det \left({\left[K_{--}({\mu }_a,{\mu }_b)\right]}_{a,b=1,\dots ,\alpha }\right),$$

$$I_{\nu }(x)={\mathrm{i}}^{-\nu }{J}_{\nu }(\mathrm{i}x)=\sum _{m=0}^{\infty }\frac{1}{m!\Gamma (m+\nu +1)}{\left(\frac{x}{2}\right)}^{2m+\nu }.$$

$$R_{\beta =4,\nu }^{(p)}({\zeta }_1,\dots ,{\zeta }_p;\{{\mu }_a\})=\frac{1}{Z_{\beta =4,\nu }(\{{\mu }_a\})}\mathrm{Pf}[Z{K}_{ij}].$$

$$P_{N+\alpha ,\beta ,\nu }(x_1,\dots ,x_{N+\alpha })=\frac{1}{N!C_{N,\beta ,\nu }^{(0)}(\{m_a\})}\prod _{i=1}^{N+\alpha }{x}_i^{\frac{\beta (\nu +1)}{2}-1}{\mathrm{e}}^{-\beta x_i}\prod _{i>j}^{N+\alpha }|x_i-x_j{|}^{\beta }.$$

$${\int }_{-\infty }^{\infty }\cdots {\int }_{-\infty }^{\infty }d{x}_1\cdots d{x}_{N+\alpha }{P}_{N+\alpha ,\beta ,\nu }(x_1,\dots ,x_{N+\alpha })\prod _{a=1}^{\alpha }{\chi }_{\{-m_a^2\}}(x_a)\prod _{i=\alpha +1}^{N+\alpha }{\chi }_{[0,\infty ]}(x_i)=1,$$

$$\displaystyle\chi_{I}(x)=\left\{\begin{array}[]{cc}1&(x\in I)\\ 0&(x\not\in I)\end{array}\right..$$

$${\chi }_{\{y\}}(x)=\delta (x-y).$$

$$E(k;I;\{m_a\})=$$

$$\times \prod _{a=1}^{\alpha }{\chi }_{\{-m_a^2\}}(x_a)\prod _{j=\alpha +1}^{\alpha +k}{\chi }_I(x_j)\prod _{l=\alpha +k+1}^{N+\alpha }(1-{\chi }_I(x_l)).$$

$$F_k(s)=1-\sum _{\ell =0}^{k-1}E(\ell ;[0,s];\{m_a\}),p_k(s)=\frac{\partial }{\partial s}{F}_k(s).$$

$$\tau (z;I;\{m_a\})$$

$$\displaystyle=\left\langle\sum_{i=1}^{N+\alpha}\Bigl{(}\prod_{a=1}^{\alpha}\delta(x_{i}+m_{a}^{2})\prod_{j(\neq i)}(1-z\chi_{I}(x_{j})\,)\Bigr{)}\right\rangle$$

$$=\frac{(N+\alpha )!}{\alpha !N!}{\int }_{-\infty }^{\infty }\cdots {\int }_{-\infty }^{\infty }d{x}_1\cdots d{x}_{N+\alpha }{P}_{N+\alpha ,\beta ,\nu }(x_1,\dots ,x_{N+\alpha })$$

$$\times \prod _{a=1}^{\alpha }{\chi }_{\{-m_a^2\}}(x_a)\prod _{j=\alpha +1}^{N+\alpha }(1-z{\chi }_I(x_j))$$

$$\tau (z;I;\sqrt{-y})$$

$$=(N+1)\left(\int -z{\int }_{I}d{x}_2\right)\cdots \left(\int -z{\int }_{I}d{x}_{N+1}\right)P_{N+1,\beta =2,\nu }(y,x_2,\dots ,x_{N+1})$$

$$=(N+1)\int d{x}_2\cdots d{x}_{N+1}{P}_{N+1,\beta =2,\nu }(y,x_2,\cdots ,x_{N+1})$$

$$-(N+1)Nz{\int }_{I}d{x}_2\int d{x}_3\cdots d{x}_{N+1}{P}_{N+1,\beta =2,\nu }(y,x_2,x_3,\cdots ,x_{N+1})$$

$$+(N+1)\frac{N(N-1)}{2!}{z}^2{\int }_{I}d{x}_{2}d{x}_3\int d{x}_4\cdots d{x}_{N+1}{P}_{N+1,\beta =2,\nu }(y,x_2,x_3,x_4,\cdots ,x_{N+1})$$

$$-\cdots$$

$$={\sigma }_{N,\beta =2,\nu }^{(0)}(\sqrt{-y})-z{\int }_{I}d{x}_2{\sigma }_{N,\beta =2,\nu }^{(1)}(x_2;\sqrt{-y})$$

$$+\frac{z^2}{2!}{\int }_{I}d{x}_{2}d{x}_3{\sigma }_{N,\beta =2,\nu }^{(2)}(x_2,x_3;\sqrt{-y})$$

$$-\frac{z^3}{3!}{\int }_{I}d{x}_{2}d{x}_{3}d{x}_4{\sigma }_{N,\beta =2,\nu }^{(3)}(x_2,x_3,x_4;\sqrt{-y})+\cdots .$$

Full text