arXiv subjects

Shinsuke M. Nishigaki

Publications and source records attributed to Shinsuke M. Nishigaki.

At least 19 recordsLinked to original sources

arXiv2026

Integrability enabled computations relating to the fixed trace Laguerre ensemble

Studies of density matrices for random quantum states lead naturally to the fixed trace Laguerre ensemble in random matrix theory. Previous studies have uncovered explicit rational function formulas for moments of purity statistic (trace of the squared density matrix), and also a third order linear differential equation satisfied by the eigenvalue density. We further probe the origin of these results from the viewpoint of integrability, which is taken here to mean wider classes of recursions and differential equations, and give extensions. Prominent in our study are first order linear matrix differential equations. One application given is to the derivation of the third order scalar equation for the density. Another is to obtain the explicit rational function formula for the variance of the purity statistic in the $β$ generalised fixed trace Laguerre ensemble. In the original case ($β= 2$), the purity cumulants are expressed in terms of the large argument expansion of a particular $σ$-Painlevé IV transcendent. In a different but related direction, the exact computation of the two-point correlation for the fixed determinant circular unitary ensemble SU$(N)$ is given the Appendix.

math-ph↗

arXiv2025

Distributions of consecutive level spacings of circular unitary ensemble and their ratio: finite-size corrections and Riemann $ζ$ zeros

We compute the joint distribution of two consecutive eigenphase spacings and their ratio for Haar-distributed $\mathrm{U}(N)$ matrices (the circular unitary ensemble) using our framework for Jánossy densities in random matrix theory, formulated via the Tracy-Widom system of nonlinear PDEs. Our result shows that the leading finite-$N$ correction in the gap-ratio distribution relative to the universal sine-kernel limit is of $\mathcal{O}(N^{-4})$, reflecting a nontrivial cancellation of the $\mathcal{O}(N^{-2})$ part present in the joint distributions of consecutive spacings. This finding suggests the potential to extract subtle finite-size corrections from the energy spectra of quantum-chaotic systems and explains why the deviation of the gap-ratio distribution of the Riemann zeta zeros $\{1/2+iγ_n\}, γ_n\approx T\gg1$ from the sine-kernel prediction scales as $\left(\log(T/2π)\right)^{-3}$.

math-ph↗

arXiv2024

Distributions of consecutive level spacings of Gaussian unitary ensemble and their ratio: ab initio derivation

In recent studies of many-body localization in nonintegrable quantum systems, the distribution of the ratio of two consecutive energy level spacings, $r_n=(E_{n+1}-E_n)/(E_{n}-E_{n-1})$ or $\tilde{r}_n=\min(r_n,r_n^{-1})$, has been used as a measure to quantify the chaoticity, alternative to the more conventional distribution of the level spacings, $s_n=\barρ(E_n)(E_{n+1}-E_n)$, as the former makes unnecessary the unfolding required for the latter. Based on our previous work on the Tracy-Widom approach to the Janossy densities, we present analytic expressions for the joint probability distribution of two consecutive eigenvalue spacings and the distribution of their ratio for the Gaussian unitary ensemble (GUE) of random Hermitian $N\times N$ matrices at $N\to \infty$, in terms of a system of differential equations. As a showcase of the efficacy of our results for characterizing an approach to quantum chaoticity, we contrast them to arguably the most ideal of all quantum-chaotic spectra: the zeroes of the Riemann $ζ$ function on the critical line at increasing heights.

math-ph↗

arXiv2021

Tracy-Widom method for Janossy density and joint distribution of extremal eigenvalues of random matrices

The Jánossy density for a determinantal point process is the probability density that an interval $I$ contains exactly $p$ points except for those at $k$ designated loci. The Jánossy density associated with an integrable kernel $\mathbf{K}\doteq (φ(x)ψ(y)-ψ(x)φ(y))/(x-y)$ is shown to be expressed as a Fredholm determinant $\mathrm{Det}(\mathbb{I}-\tilde{\mathbf{K}}|_I)$ of a transformed kernel $\tilde{\mathbf{K}}\doteq (\tildeφ(x)\tildeψ(y)-\tildeψ(x)\tildeφ(y))/(x-y)$. We observe that $\tilde{\mathbf{K}}$ satisfies Tracy and Widom's criteria if $\mathbf{K}$ does, because of the structure that the map $(φ, ψ)\mapsto (\tildeφ, \tildeψ)$ is a meromorphic $\mathrm{SL}(2,\mathbb{R})$ gauge transformation between covariantly constant sections. This observation enables application of the Tracy--Widom method to Jánossy densities, expressed in terms of a solution to a system of differential equations in the endpoints of the interval. Our approach does not explicitly refer to isomonodromic systems associated with Painlevé equations employed in the preceding works. As illustrative examples we compute Jánossy densities with $k=1, p=0$ for Airy and Bessel kernels, related to the joint distributions of the two largest eigenvalues of random Hermitian matrices and of the two smallest singular values of random complex matrices.

math-ph↗

arXiv2019

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

We compute individual distributions of low-lying eigenvalues of massive chiral random matrix ensembles by the Nyström-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 Nyström-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.

hep-lat↗

arXiv2018

Individual eigenvalue distributions of crossover chiral random matrices and low-energy constants of SU(2)$\times$U(1) lattice gauge theory

We compute individual distributions of low-lying eigenvalues of a chiral random matrix ensemble interpolating symplectic and unitary symmetry classes by the Nyström-type method of evaluating the Fredholm Pfaffian and resolvents of the quaternion kernel. The one-parameter family of these distributions are shown to fit excellently the Dirac spectra of SU(2) lattice gauge theory with a constant U(1) background or dynamically fluctuating U(1) gauge field, which weakly breaks the pseudo-reality of the unperturbed SU(2) Dirac operator. Observed linear dependence of the crossover parameter with the strength of U(1) perturbations leads to precise determination of the pseudo-scalar decay constant, as well as the chiral condensate in the effective chiral Lagrangian of AI class.

hep-lat↗

arXiv2016

Distribution of the k-th smallest Dirac operator eigenvalue : an update

Based on the exact relationship to random matrix theory, we present an alternative method of evaluating the probability distribution of the k-th smallest Dirac eigenvalue in the epsilon-regime of QCD and QCD-like theories. By utilizing the Nystrom-type discretization of Fredholm determinants and Pfaffians, practical trouble of evaluating multiple integrations is circumvented and technical restrictions on the parities of the number of flavors and of the topological charge present in our previous treatment for beta=1 and 4 cases [Phys. Rev. D 63, 045012 (2001)] are partly lifted. This method is also applied to the distributions of spacings between k-th nearest-neighboring levels in the mobility edges of Anderson Hamiltonian and Dirac operator in high-temperature QCD.

hep-lat↗

arXiv2015

Lee-Yang zero distribution of high temperature QCD and Roberge-Weiss phase transition

Canonical partition functions and Lee-Yang zeros of QCD at finite density and high temperature are studied. Recent lattice simulations have confirmed that the free energy of QCD is a quartic function of quark chemical potential at temperature slightly above pseudo-critical temperature $T_c$, as in the case with a gas of free massless fermions. We present analytic derivation of the canonical partition functions and Lee-Yang zeros for this type of free energy using the saddle point approximation. We also perform lattice QCD simulation in a canonical approach using the fugacity expansion of the fermion determinant, and carefully examine its reliability. By comparing the analytic and numerical results, we conclude that the canonical partition functions follow the Gaussian distribution of the baryon number, and the accumulation of Lee-Yang zeros of these canonical partition functions exhibit the first-order Roberge-Weiss phase transition. We discuss the validity and applicable range of the result and its implications both for theoretical and experimental studies.

hep-lat↗

arXiv2015

Individual eigenvalue distributions for chGSE-chGUE crossover and determination of low-energy constants in two-color QCD+QED

We compute statistical distributions of individual low-lying eigenvalues of random matrix ensembles interpolating chiral Gaussian symplectic and unitary ensembles. To this aim we use the Nystrom-type discretization of Fredholm Pfaffians and resolvents of the dynamical Bessel kernel containing a single crossover parameter ρ. The ρ-dependent distributions of the four smallest eigenvalues are then used to fit the Dirac spectra of modulated SU(2) lattice gauge theory, in which the reality of the staggered SU(2) Dirac operator is weakly violated either by the U(1) gauge field or by a constant background flux. Combined use of individual eigenvalue distributions is effective in reducing statistical errors in ρ; its linear dependence on the imaginary chemical potential μ_I enables precise determination of the pseudo-scalar decay constant F of the SU(2) gauge theory from a small lattice. The U(1)-coupling dependence of an equivalent of F^2 μ_I^2 in the SU(2) x U(1) theory is also obtained.

hep-lat↗

arXiv2014

Tracy-Widom distribution as instanton sum of 2D IIA superstrings

We present an analytic expression of the nonperturbative free energy of a double-well supersymmetric matrix model in its double scaling limit, which corresponds to two-dimensional type IIA superstring theory on a nontrivial Ramond-Ramond background. To this end we draw upon the wisdom of random matrix theory developed by Tracy and Widom, which expresses the largest eigenvalue distribution of unitary ensembles in terms of a Painleve II transcendent. Regularity of the result at any value of the string coupling constant shows that the third-order phase transition between a supersymmetry-preserving phase and a supersymmetry-broken phase, previously found at the planar level, becomes a smooth crossover in the double scaling limit. Accordingly, the supersymmetry is always broken spontaneously as its order parameter stays nonzero for the whole region of the coupling constant. Coincidence of the result with the unitary one-matrix model suggests that one-dimensional type 0 string theories partially correspond to the type IIA superstring theory. Our formulation naturally allows for introduction of an instanton chemical potential, and reveals the presence of a novel phase transition, possibly interpreted as condensation of instantons.

hep-th↗

arXiv2013

Critical statistics at the mobility edge of QCD Dirac spectra

We examine statistical fluctuation of eigenvalues from the near-edge bulk of QCD Dirac spectra above the critical temperature. For completeness we start by reviewing on the spectral property of Anderson tight-binding Hamiltonians as described by nonlinear sigma models and random matrices, and on the scale-invariant intermediate spectral statistics at the mobility edge. By fitting the level spacing distributions, deformed random matrix ensembles which model multifractality of the wave functions typical of the Anderson localization transition, are shown to provide an excellent effective description for such a critical statistics. Next we carry over the above strategy for the Anderson Hamiltonians to the Dirac spectra. For the staggered Dirac operators of QCD with 2+1 flavors of dynamical quarks at the physical point and of SU(2) quenched gauge theory, we identify the precise location of the mobility edge as the scale-invariant fixed point of the level spacing distribution. The eigenvalues around the mobility edge are shown to obey critical statistics described by the aforementioned deformed random matrix ensembles of unitary and symplectic classes. The best-fitting deformation parameter for QCD at the physical point turns out to be consistent with the Anderson Hamiltonian in the unitary class. Finally, we propose a method of locating the mobility edge at the origin of QCD Dirac spectrum around the critical temperature, by the use of individual eigenvalue distributions of deformed chiral random matrices.

hep-lat↗

arXiv2012

Accuracy and range of validity of the Wigner surmise for mixed symmetry classes in random matrix theory

Schierenberg et al. [Phys. Rev. E 85, 061130 (2012)] recently applied the Wigner surmise, i.e., substitution of \infty \times \infty matrices by their 2 \times 2 counterparts for the computation of level spacing distributions, to random matrix ensembles in transition between two universality classes. I examine the accuracy and the range of validity of the surmise for the crossover between the Gaussian orthogonal and unitary ensembles by contrasting them with the large-N results that I evaluated using the Nystrom-type method for the Fredholm determinant. The surmised expression at the best-fitting parameter provides a good approximation for 0 \lesssim s \lesssim 2, i.e., the validity range of the original surmise.

math-ph↗

arXiv2012

Universality crossover between chiral random matrix ensembles and twisted SU(2) lattice Dirac spectra

Motivated by the statistical fluctuation of Dirac spectrum of QCD-like theories subjected to (pseudo)reality-violating perturbations and in the epsilon-regime, we compute the smallest eigenvalue distribution and the level spacing distribution of chiral and non-chiral parametric random matrix ensembles of Dyson-Mehta-Pandey type. To this end we employ the Nystrom method to numerically evaluate the Fredholm Pfaffian of the integral kernel for the chG(O,S)E-chGUE and G(O,S)E-GUE crossover. We confirm the validity and universality of our results by comparing them with several lattice models, namely fundamental and adjoint staggered Dirac spectra of SU(2) quenched lattice gauge theory under the twisted boundary condition (imaginary chemical potential) or perturbed by phase noise. Both in the zero-virtuality region and in the spectral bulk, excellent one-parameter fitting is achieved already on a small 4^4 lattice. Anticipated scaling of the fitting parameter with the twisting phase, mean level spacing, and the system size allows for precise determination of the pion decay (diffusion) constant F in the low-energy effective Lagrangian.

hep-lat↗

arXiv2012

Level spacings of parametric chiral random matrices and two-color QCD with twisted boundary condition

We evaluate level spacing and smallest eigenvalue distributions of chiral random matrix ensembles transiting from symplectic or orthogonal to unitary symmetry classes with a crossover parameter rho. As expected from the effective sigma model description, these results can be fitted perfectly to the fundamental or adjoint staggered Dirac spectrum of SU(2) quenched lattice gauge theory under the imaginary chemical potential (twisting) mu. The linear dependence of the parameter rho on mu determines the pion decay constant F as its proportionality constant.

hep-lat↗

arXiv2003

3D two-color QCD at finite temperature and baryon density

We study the phase diagram for two-color QCD in three-dimensional spacetime, as a function of temperature and baryon chemical potential, using the low-energy effective Lagrangian approach. We show one-loop renormalizability at zero temperature, and then use the one-loop effective Lagrangian at finite temperature and chemical potential to show that at low temperature there is a critical line separating the normal and diquark phase, with this critical line ending at a tricritical point. This phase structure is qualitatively similar to that found recently by Splittorff et al for two-color QCD in four-dimensional spacetime, although the details are quite different, due to the different symmetries and the different loop and infrared properties of three-dimensional spacetime.

hep-ph↗

arXiv2003

Two-color QCD in 3D at finite baryon density

We study the low energy phase structure of SU(2) gauge theories in three-dimensional spacetime, at finite baryon density. The pseudoreality of representations of SU(2) permits an analytic study of a real baryon chemical potential, and the restriction to 3D results in a different global symmetry breaking pattern from the corresponding 4D model studied previously by Kogut et al. We find a second-order phase transition separating the normal phase and the baryon superconducting phase. The chemical potential dependence of condensates and baryon density are computed. We find that the phase structure and the excitation spectrum are essentially the same as in 4D, despite the different symmetry groups, indicating a universality that is rooted in the properties of Riemannian symmetric spaces.

hep-ph↗

arXiv2002

Correlation functions of the BC Calogero-Sutherland model

The BC-type Calogero-Sutherland model (CSM) is an integrable extension of the ordinary A-type CSM that possesses a reflection symmetry point. The BC-CSM is related to the chiral classes of random matrix ensembles (RMEs) in exactly the same way as the A-CSM is related to the Dyson classes. We first develop the fermionic replica sigma-model formalism suitable to treat all chiral RMEs. By exploiting ''generalized color-flavor transformation'' we then extend the method to find the exact asymptotics of the BC-CSM density profile. Consistency of our result with the c=1 Gaussian conformal field theory description is verified. The emerging Friedel oscillations structure and sum rules are discussed in details. We also compute the distribution of the particle nearest to the reflection point.

cond-mat.str-el↗

arXiv2002

Fundamental Matter and the Deconfining Phase Transition in 2+1 D

We analyze the effect of heavy fundamentally charged particles on the finite temperature deconfining phase transition in the 2+1 dimensional Georgi-Glashow model. We show that in the presence of fundamental matter the transition turns into a crossover. The near critical theory is mapped onto the 2 dimensional Ising model in an external magnetic field. Using this mapping we determine the width of the crossover region as well as the specific heat as a function of the fundamental mass.

hep-th↗