Long range disorder and Anderson transition in systems with chiral symmetry

García-García, Antonio M.; Takahashi, Kazutaka

doi:10.1016/j.nuclphysb.2004.08.035

Cited by 16 publications

(16 citation statements)

References 54 publications

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“…This seems to suggest that Anderson localization and the chiral phase transition cannot be so intimately related. However, in disordered systems with chiral symmetry (or any other additional discrete symmetry) the spectral density is already sensitive to the strength of disorder [30,31] so it may still play the role of an order parameter for the transition. Additionally the fluctuations of the order parameter are related to density-density correlations which are sensitive to localization effects even in systems with no chiral symmetry.…”

Section: Additional Discussion On Localization and Chiral Restoramentioning

confidence: 99%

Chiral phase transition in lattice QCD as a metal-insulator transition

2007

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We investigate the lattice QCD Dirac operator with staggered fermions at temperatures around the chiral phase transition. We present evidence of a metal-insulator transition in the low lying modes of the Dirac operator around the same temperature as the chiral phase transition. This strongly suggests the phenomenon of Anderson localization (localization by destructive quantum interference) drives the QCD vacuum to the chirally symmetric phase in a way similar to a metalinsulator transition in a disordered conductor. We also discuss how Anderson localization affects the usual phenomenological treatment of phase transitions a la Ginzburg-Landau.PACS numbers: 72.15. Rn, 71.30.+h, 05.45.Df, One of the most important features of the infrared limit of the strong interactions is the spontaneous breaking of the approximate chiral symmetry. The order parameter associated with this spontaneous chiral symmetry breaking (SχSB) is the chiral condensate, ψ ψ , which in the absence of SχSB would vanish as the quark mass goes to zero. In nature the lightest quarks are not massless so a nonzero condensate is expected even in a free theory. However the small quark mass can only account for a small percentage of the chiral condensate, the rest has its origin in the strong non-perturbative color interactions of QCD. Although lattice simulations have already provided overwhelming evidence that SχSB is a feature of QCD it is still highly desirable to understand its origin in more simple terms.Simplified models of QCD where gauge configurations are given by instantons have played a leading role in the description of the SχSB [1,2,3]. Instantons [4], originally introduced in QCD by t'Hooft [5] to solve the so called U (1) problem, are classical solutions of the Euclidean Yang-Mills equations of motion. Their relation with the SχSB stems from the fact that the QCD Dirac operator has an exact zero eigenvalue in the field of an instanton. In the QCD vacuum, these zero modes coming from different instantons mix together to form a band around zero. It turns out the spectral density, ρ(λ), of the Dirac operator in this region is directly related to the chiral condensate through the Banks-Casher relation [6],where V is the space-time volume. Based on this result it was shown that the instanton contribution was capable of producing a nonzero chiral condensate with a value close to the phenomenological one [2, 3] (see [7] for a detailed review). For sufficiently high energies the non-abelian gauge interaction of QCD is weak (asymptotic freedom) and chiral symmetry is restored. This poses an interesting question: How is the chiral symmetry restored as we go from low to high energies or, equivalently, from low to high temperatures? A standard approach to the chiral phase transition is to invoke universality arguments [8], namely, it is assumed that the transition is controlled by symmetries rather than by the dynamical details of QCD. The chiral phase transition is then studied by looking at the most general renormalizable Ginzburg-Landau Lagrang...

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Section: Additional Discussion On Localization and Chiral Restoramentioning

confidence: 99%

Chiral phase transition in lattice QCD as a metal-insulator transition

2007

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“…A practical problem of using the microscopic level density ρ(x) = K chi a (x, x) for fitting the Dirac spectrum is that ρ(x) becomes rather structureless at a finite deformation parameter a (see Fig.8 below) [49]. On the other hand, the level number variance (an integral transform of the twolevel correlation function −K chi a (x, x ) 2 ) for large x is sensitive to the parameter a, but one would need extremely large lattices for the window at the origin containing dozens of eigenvalues to be uniformly fitted with a single parameter a.…”

Section: Pos(lattice 2013)018mentioning

confidence: 99%

Critical statistics at the mobility edge of QCD Dirac spectra

Nishigaki¹,

Giordano²,

Kovács³

et al. 2014

Proceedings of 31st International Symposium on Lattice Field Theory LATTICE 2013 — PoS(LATTICE 2013)

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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 σ 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 scaleinvariant 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.

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“…Such an example can be found in Ref. [16]. The generalized random matrix model was used there and it was found that the quantity ⌬ is different from our result.…”

Section: Discussionmentioning

confidence: 56%

“…Using a chiral symmetric random matrix model we derived the nonlinear models (16) and (19). We demonstrated that they are equivalent to related chiral symmetric models.…”

Section: Discussionmentioning

confidence: 99%

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Nonlinearσmodel approach for level correlations in chiral disordered systems

Takahashi

2004

Phys. Rev. E

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We study level correlations of disordered systems with chiral unitary symmetry (AIII symmetry). We use a random matrix model with a finite correlation length to derive a supersymmetric nonlinear sigma model. The result is compared with existing results based on other models. Using the methods of Sov. Phys. JETP 60, 656 (1994)] and Phys. Rev. Lett. 75, 902 (1995)], we calculate the density of states and two-level correlation function. The result is expressed using the spectral determinant as in traditional nonchiral systems. We discuss the renormalization of the mean level spacing which is not present in the traditional systems.

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Long range disorder and Anderson transition in systems with chiral symmetry

Cited by 16 publications

References 54 publications

Chiral phase transition in lattice QCD as a metal-insulator transition

Chiral phase transition in lattice QCD as a metal-insulator transition

Critical statistics at the mobility edge of QCD Dirac spectra

Nonlinearσmodel approach for level correlations in chiral disordered systems

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