Spectral universality of real chiral random matrix ensembles

We consider two quenched, chiral ensembles which are coupled in such a way that a combined chiral symmetry is preserved. The coupling also links the topology of the two systems such that the number of exact zero modes in the coupled system equals the sum of the number of zero modes in the two uncoupled systems counted with sign. The canceled modes that turn non-topological due to the coupling become near-zero modes at small coupling. We analyze the distribution of these would-be zero modes using effective field theory. The distribution is universal and, in the limit of small coupling, the would-be zero modes are distributed according to a finite size chiral Gaussian ensemble, where the width of the distribution scales as the inverse square root of the volume.

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“…and it follows from the definitions of the resolvent and eigenvalue density, (33) and (34), with [40][41][42]…”

Section: A Large C 2 -Approximation For Choementioning

confidence: 99%

Universal distribution of would-be topological zero modes in coupled chiral systems

Mielke

Splittorff

2017

Phys. Rev. D

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“…of the eigenvalues of the Euclidean Dirac operator / Dψ k = λ k ψ k in QCD [2,3]. The spectral density for small eigenvalues is in fact determined completely by the symmetries of the QCD partition function: The chiral symmetry breaking pattern, and the axial symmetry due to which all non-zero eigenvalues appear in pairs ±λ k .…”

Section: Random Matrix Theorymentioning

confidence: 99%

Finite-Volume Effects in Quantum Chromodynamics and Functional Renormalization Group Methods

Klein

2012

Few-Body Syst

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Quantum Chromodynamics (QCD) at finite temperature and density is currently a subject of great interest, due to both advances in experimental heavy-ion collisions and theoretical improvement of our understanding in recent years. Simulations of the theory on finite space-time lattices provide important theoretical advances. But in particular for investigating QCD phase transitions, finite quark masses and finite volumes in the simulations need to be taken into account. The investigation of finite-volume effects in QCD has a long tradition in the framework of chiral perturbation theory and random matrix theory and can provide useful tools for the analysis. With regard to a description of the chiral phase transition in QCD, it is very important to correctly take into account the effects of long-range fluctuations such as pions as the Goldstone bosons of spontaneous chiral symmetry breaking. It is natural to employ Renormalization Group methods for this purpose. Together with Hans-Jürgen Pirner, we have initiated work in this direction, and there has been an increasing interest in the investigation of such finite-volume effects. I will give an overview over recent developments.

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“…In a parallel development it was shown that the spectral density of the Dirac operator for a gauge theory near its zero eigenvalues should only depend on the symmetries in question [17][18][19]. Although the original work [17][18][19] used a Gaussian random matrix model, the results from the random matrix theory can be proven to be universal [20][21][22][23][24]. This implies that the spectral density of the Dirac operator near the origin can be extracted from random matrix theories which provide a description of common aspects of various quantum phenomena (for a review see [25]).…”

Section: Introductionmentioning

confidence: 99%

Character expansions for the orthogonal and symplectic groups

Balantekin

Cassak

2002

Journal of Mathematical Physics

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Formulas for the expansion of arbitrary invariant group functions in terms of the characters for the Sp(2N ), SO(2N + 1), and SO(2N ) groups are derived using a combinatorial method. The method is similar to one used by Balantekin to expand group functions over the characters of the U (N ) group. All three expansions have been checked for all N by using them to calculate the known expansions of the generating function of the homogeneous symmetric functions. An expansion of the exponential of the traces of group elements, appearing in the finite-volume gauge field partition functions, is worked out for the orthogonal and symplectic groups.

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Spectral universality of real chiral random matrix ensembles

Cited by 22 publications

References 76 publications

Universal distribution of would-be topological zero modes in coupled chiral systems

Universal distribution of would-be topological zero modes in coupled chiral systems

Finite-Volume Effects in Quantum Chromodynamics and Functional Renormalization Group Methods

Character expansions for the orthogonal and symplectic groups

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