Chiral symmetry breaking in confining theories

Abstract:We exposit the eigenvalue distribution of the lattice Dirac operator in Quantum Chromodynamics with two colors (i.e. two-color QCD). We explicitly calculate all the eigenvalues in the presence of finite quark chemical potential µ for a given gauge configuration on the finite-volume lattice. First, we elaborate the Banks-Casher relations in the complex plane extended for the diquark condensate as well as the chiral condensate to relate the eigenvalue spectral density to the physical observable. Next, we evaluate the condensates and clarify the characteristic spectral change corresponding to the phase transition. Assuming the strong coupling limit, we exhibit the numerical results for a random gauge configuration in two-color QCD implemented by the staggered fermion formalism and confirm that our results agree well with the known estimate quantitatively. We then exploit our method in the case of the Wilson fermion formalism with two flavors. Also we elucidate the possibility of the Aoki (parity-flavor broken) phase and conclude from the point of view of the spectral density that the artificial pion condensation is not induced by the density effect in strong-coupling two-color QCD.

Section: Chiral Condensatementioning

confidence: 99%

Characteristics of the eigenvalue distribution of the Dirac operator in dense two-color QCD

Fukushima¹

2008

“…The spectral density ρ(λ) of the massive Dirac operator D m has a well defined thermodynamic limit. It can be defined as [39] …”

Section: Jhep04(2007)090mentioning

confidence: 99%

“…breaking in infinite volume [17,18,39]. It is due to the long-range correlations left in the system at finite volume.…”

Section: Jhep04(2007)090mentioning

confidence: 99%

Spontaneous chiral symmetry breaking in QCD: a finite-size scaling study on the lattice

Giusti¹,

Necco²

2007

Spontaneous chiral symmetry breaking in QCD with massless quarks at infinite volume can be seen in a finite box by studying, for instance, the dependence of the chiral condensate from the volume and the quark mass. We perform a feasibility study of this program by computing the quark condensate on the lattice in the quenched approximation of QCD at small quark masses. We carry out simulations in various topological sectors of the theory at several volumes, quark masses and lattice spacings by employing fermions with an exact chiral symmetry, and we focus on observables which are infrared stable and free from mass-dependent ultraviolet divergences. The numerical calculation is carried out with an exact variance-reduction technique, which is designed to be particularly efficient when spontaneous symmetry breaking is at work in generating a few very small low-lying eigenvalues of the Dirac operator. The finite-size scaling behaviour of the condensate in the topological sectors considered agrees, within our statistical accuracy, with the expectations of the chiral effective theory. Close to the chiral limit we observe a detailed agreement with the first Leutwyler-Smilga sum rule. By comparing the mass, the volume and the topology dependence of our results with the predictions of the chiral effective theory, we extract the corresponding low-energy constant.

“…In the continuum theory, or if chiral symmetry is preserved by the lattice theory, the spectral density vanishes at α < m 2 . Moreover, for slightly larger values of α, the Banks-Casher relation [17] implies the asymptotic form…”

Section: Spectral Densitymentioning

confidence: 99%

“…, n) (A. 17) and the maximal eigenvalue ǫ 2 of the n × n residual matrix R ij = (ρ i , ρ j ). (A.18) A well-known lemma then asserts that there are n orthonormal eigenvectors of A with eigenvaluesα i such that |α i −α i | ≤ ǫ for all i = 1, .…”

Section: A5 Stopping Criterionmentioning

confidence: 99%

Stability of lattice QCD simulations and the thermodynamic limit

Debbio¹,

Giusti²,

Lüscher³

et al. 2006

149

253

We study the spectral gap of the Wilson-Dirac operator in two-flavour lattice QCD as a function of the lattice spacing a, the space-time volume V and the current-quark mass m. It turns out that the median of the probability distribution of the gap scales proportionally to m and that its width is practically equal to a/ √ V . In particular, numerical simulations are safe from accidental zero modes in the large-volume regime of QCD.