Surprising Pfaffian factorizations in random matrix theory with Dyson index β = 2

The microscopic spectral density of the Wilson Dirac operator for two flavor lattice QCD is analyzed. The computation includes the leading order a 2 corrections of the chiral Lagrangian in the microscopic limit. The result is used to demonstrate how the Sharpe-Singleton first order scenario is realized in terms of the eigenvalues of the Wilson Dirac operator. We show that the Sharpe-Singleton scenario only takes place in the theory with dynamical fermions whereas the Aoki phase can be realized in the quenched as well as the unquenched theory. Moreover, we give constraints imposed by γ 5 -Hermiticity on the additional low energy constants of Wilson chiral perturbation theory.

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“…This Pfaffian form was first given in [46]. Here we give a proof in terms of chiral Lagrangians rather than random matrix theories.…”

Section: Appendix B Simplification Of the Partition Functionmentioning

confidence: 68%

“…In [46] it was shown that the four flavor partition function Z ν 4 can be expressed in terms of two flavor partition functions. A proof in terms of chiral Lagrangians is given in Appendix B.…”

Section: The Unquenched Spectrum Of D Wmentioning

confidence: 99%

Realization of the Sharpe-Singleton scenario

2012

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“…Even so, we can find one as follows. As it was shown in [61] each determinantal point process, especially the β = 2 random matrix ensembles, can also be written as a Pfaffian point process in a non-trivial way. They can then be solved in terms of skew-orthogonal polynomials as well, given again by (2.16).…”

Section: (B22)mentioning

confidence: 99%

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

Akemann¹,

Kieburg²,

Mielke³

et al. 2019

J. Stat. Mech.

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We consider a parameter dependent ensemble of two real random matrices with Gaussian distribution. It describes the transition between the symmetry class of the chiral Gaussian orthogonal ensemble (Cartan class B|DI) and the ensemble of antisymmetric Hermitian random matrices (Cartan class B|D). It enjoys the special feature that, depending on the matrix dimension N , it has exactly ν = 0 (1) zero-mode for N even (odd), throughout the symmetry transition. This "topological protection" is reminiscent of properties of topological insulators. We show that our ensemble represents a Pfaffian point process which is typical for such transition ensembles. On a technical level, our results follow from the applicability of the Harish-Chandra integral over the orthogonal group. The matrixvalued kernel determining all eigenvalue correlation functions is explicitly constructed in terms of skeworthogonal polynomials, depending on the topological index ν = 0, 1. These polynomials interpolate between Laguerre and even (odd) Hermite polynomials for ν = 0 (1), in terms of which the two limiting symmetry classes can be solved. Numerical simulations illustrate our analytical results for the spectral density and an expansion for the distribution of the smallest eigenvalue at finite N .

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“…Although these RMTs are more complicated than the chiral random matrix theory formulated in [17,18], in the case of Wilson fermions a complete analytical solution of the RMT has been achieved [9][10][11]13,[20][21][22][23]. Since the Wilson RMT shares the global symmetries of the Wilson-Dirac operator it will be equivalent to the corresponding (partially quenched) chiral Lagrangian in the microscopic domain (also known as the domain) [24][25][26][27][28][29].…”

Section: Introductionmentioning

confidence: 99%

“…Quite recently, there has been a breakthrough in deriving eigenvalue statistics of the infrared spectrum of the Hermitian [20] as well as the non-Hermitian [21][22][23] Wilson-Dirac operator. These results explain [13] why the Sharpe-Singleton scenario is only observed for the case of dynamical fermions [1,5,[30][31][32][33][34][35][36] and not in the quenched theory [37,38] while the Aoki phase has been seen in both cases.…”

Section: Introductionmentioning

confidence: 99%

Spectral properties of the Wilson-Dirac operator and random matrix theory

2013

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Random matrix theory has been successfully applied to lattice quantum chromodynamics. In particular, a great deal of progress has been made on the understanding, numerically as well as analytically, of the spectral properties of the Wilson-Dirac operator. In this paper, we study the infrared spectrum of the Wilson-Dirac operator via random matrix theory including the three leading order a 2 correction terms that appear in the corresponding chiral Lagrangian. A derivation of the joint probability density of the eigenvalues is presented. This result is used to calculate the density of the complex eigenvalues, the density of the real eigenvalues, and the distribution of the chiralities over the real eigenvalues. A detailed discussion of these quantities shows how each low-energy constant affects the spectrum. Especially we consider the limit of small and large (which is almost the mean field limit) lattice spacing. Comparisons with Monte Carlo simulations of the random matrix theory show a perfect agreement with the analytical predictions. Furthermore we present some quantities which can be easily used for comparison of lattice data and the analytical results.

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Surprising Pfaffian factorizations in random matrix theory with Dyson index β = 2

Cited by 7 publications

References 92 publications

Realization of the Sharpe-Singleton scenario

Realization of the Sharpe-Singleton scenario

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

Spectral properties of the Wilson-Dirac operator and random matrix theory

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