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

Abstract

Motivated by the statistical fluctuation of Dirac spectrum of QCD-like theories subjected to (pseudo)reality-violating perturbations and in the ε regime, we compute the smallest eigenvalue distribution and the level spacing distribution of chiral and nonchiral parametric random matrix ensembles of Dyson-Mehta-Pandey type. To this end we employ the Nyström-type 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.2 boundary condition or imaginary chemical potential) [19][20][21]. These cases are distinct from the previously mentioned case in that the Dirac eigenvalues stay real even after the inclusion of symmetry violations and their statistical behavior exhibits crossover, rather than develops into the complex plane. In terms of the effective σ-model description, these two cases are almost identical, save for the difference of the sign of trBQ †B Q term (µ 2 F 2 or (iµ) 2 F 2 , see Sec. IV.).Crossover between universality classes of Hermitian random matrix ensembles [22][23][24], namely GOE-GUE and GSE-GUE, is extensively studied in the context of disordered [25,26] and quantum-chaotic Hamiltonians [27][28][29] with its time-reversal invariance slightly broken by weak magnetic field or AB flux applied [30]. On the other hand, the chiral or superconducting variant of universality crossover appears to be a relatively unexplored field so far. Previous attempts in this area either focused on the level number variance and the spectral form factor (both of which are integral transforms of the two-level correlator and insensitive to chiralness) of two-color QCD with AB fluxes versus GOE-GUE crossover [19], or have fruited in a series of tours de force by Damgaard and collaborators [31,32] devoted to the analytical computation of the level density and individual small eigenvalue distributions for the spectral crossover within the chiral Gaussian unitary ensemble (chGUE) class, due to the imaginary isospin chemical potential. Despite that analytic results for the microscopic spectral correlation functions are known for some time for chGOE-chGUE and chGSE-chGUE crossover [33,34], 1 they are yet to encounter with physical application. To the best of our knowledge, the only example of the crossover involving different Hermitian chiral universality classes discussed in a physical setting is the CI-C transition ...

Preserving topology while breaking chirality: from chiral orthogonal to anti-symmetric Hermitian ensemble

“…Certainly, other, more sophisticated methods exist for a controlled approximation of the Fredholm expansion, see e.g. [33] for the distribution of the smallest eigenvalues of a random two-matrix model that describes the chGUE-chGSE transition. Because we deal with quantities at finite (and small) n we have not aimed at a better precision.…”

“…Let us highlight one peculiarity that is distinct from the other models, which is its corresponding symmetry group. Whereas most models, e.g., in [2,3,18,19,20,21,26,27,28,29,32,33,36], are usually invariant with respect to a unitary group in one or another limit, our model always satisfies an orthogonal symmetry, regardless of the value of a, including infinity. This difference is remarkable because of the group integral that has to be solved to obtain the joint probability density function (jpdf) of the eigenvalues.…”

Statistical properties of eigenvalues of the non-Hermitian Su-Schrieffer-Heeger model with random hopping terms

“…Non-Hermitian systems with randomness have been studied in the context of Anderson localization [43][44][45][46][47][48][49][50][51][52][53][54][55], lowenergy QCD [56][57][58][59][60][61][62][63], and more [64][65][66][67]. The spectral statistics of random Hermitian Hamiltonians usually exhibits universal behavior, depending only on symmetries of the system [68].…”

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

“…On actual implementation of the above method, the numerical integration over k scaled variables in the third step becomes resource-consuming. To circumvent such technical issue, we will consider Fredholm determinants and Pfaffians for the chiral random matrix ensembles with α mass parameters as the generating function of the joint distribution of the first k eigenvalues, and utilize the quadrature method [49] to evaluate them numerically [50][51][52][53]. In this section, we will derive a compact formula 4 of Fredholm determinants and Pfaffians which will be efficient for numerical computations.…”