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تسجيل مستخدم جديدDirac spectrum of one-flavor QCD at theta=0 and continuity of the chiral condensate
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We derive exact analytical expressions for the spectral density of the Dirac operator at fixed theta-angle in the microscopic domain of one-flavor QCD. These results are obtained by performing the sum over topological sectors using novel identities involving sums of products of Bessel functions. Because the fermion determinant is not positive definite for negative quark mass, the usual Banks-Casher relation is not valid and has to be replaced by a different mechanism first observed for QCD at nonzero chemical potential. Using the exact results for the spectral density we explain how this mechanism results in a chiral condensate that remains constant when the quark mass changes sign.
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In a sector of fixed topological charge, the chiral condensate has a discontinuity given by the Banks-Casher formula also in the case of one-flavor QCD. However, at fixed theta-angle, the chiral condensate remains constant when the quark mass crosses
zero. To reconcile these contradictory observations, we have evaluated the spectral density of one-flavor QCD at theta=0. For negative quark mass, it becomes a strongly oscillating function with a period that scales as the inverse space-time volume and an amplitude that increases exponentially with the space-time volume. As we have learned from QCD at nonzero chemical potential, if this is the case, an alternative to the Banks-Casher formula applies, and as we will demonstrate in this talk, for one-flavor QCD this results in a continuous chiral condensate. A special role is played by the topological zero modes which have to be taken into account exactly in order to get a finite chiral condensate in the thermodynamic limit.
We analyze the mass dependence of the chiral condensate for QCD at nonzero $theta$-angle and find that in general the discontinuity of the chiral condensate is not on the support of the Dirac spectrum. To understand this behavior we decompose the spe
ctral density and the chiral condensate into contributions from the zero modes, the quenched part, and a remainder which is sensitive to the fermion determinant and is referred to as the dynamical part. We obtain general formulas for the contributions of the zero modes. Expressions for the quenched part, valid for an arbitrary number of flavors, and for the dynamical part, valid for one and two flavors, are derived in the microscopic domain of QCD. We find that at nonzero $theta$-angle the quenched and dynamical part of the Dirac spectral density are strongly oscillating with an amplitude that increases exponentially with the volume $V$ and a period of order of $1/V$. The quenched part of the chiral condensate becomes exponentially large at $theta e0$, but this divergence is canceled by the contribution from the zero modes. The oscillatory behavior of the dynamical part of the density is essential for moving the discontinuity of the chiral condensate away from the support of the Dirac spectrum. As important by-products of this work we obtain analytical expressions for the microscopic spectral density of the Dirac operator at nonzero $theta$-angle for both one- and two-flavor QCD with nonzero quark masses.
In the $epsilon$-domain of QCD we have obtained exact analytical expressions for the eigenvalue density of the Dirac operator at fixed $theta e 0$ for both one and two flavors. These results made it possible to explain how the different contribution
s to the spectral density conspire to give a chiral condensate at fixed $theta$ that does not change sign when the quark mass (or one of the quark masses for two flavors) crosses the imaginary axis, while the chiral condensate at fixed topological charge does change sign. From QCD at nonzero density we have learnt that the discontinuity of the chiral condensate may move to a different location when the spectral density increases exponentially with the volume with oscillations on the order of the inverse volume. This is indeed what happens when the product of the quark masses becomes negative, but the situation is more subtle in this case: the contribution of the quenched part of the spectral density diverges in the thermodynamic limit at nonzero $theta$, but this divergence is canceled exactly by the contribution from the zero modes. We conclude that the zero modes are essential for the continuity of the chiral condensate and that their contribution has to be perfectly balanced against the contribution from the nonzero modes. Lattice simulations at nonzero $theta$-angle can only be trusted if this is indeed the case.
The chiral condensate in QCD at zero temperature does not depend on the quark chemical potential (up to one third the nucleon mass), whereas the spectral density of the Dirac operator shows a strong dependence on the chemical potential. The cancellat
ions which make this possible also occur on the microscopic scale, where they can be investigated by means of a random matrix model. We show that they can be understood in terms of orthogonality properties of orthogonal polynomials. In the strong non-Hermiticity limit they are related to integrability properties of the spectral density. As a by-product we find exact analytical expressions for the partially quenched chiral condensate in the microscopic domain at nonzero chemical potential.
We compute the chiral condensate in 2+1-flavor QCD through the spectrum of low-lying eigenmodes of Dirac operator. The number of eigenvalues of the Dirac operator is evaluated using a stochastic method with an eigenvalue filtering technique on the ba
ckground gauge configurations generated by lattice QCD simulations including the effects of dynamical up, down and strange quarks described by the Mobius domain-wall fermion formulation. The low-lying spectrum is related to the chiral condensate, which is one of the leading order low-energy constants in chiral effective theory, as dictated by the Banks-Casher relation. The spectrum shape and its dependence on the sea quark masses calculated in numerical simulations are consistent with the expectation from one-loop chiral perturbation theory. After taking the chiral limit as well as the continuum limit using the data at three lattice spacings ranging 0.080-0.045 fm, we obtain $Sigma^{1/3}$(2 GeV) = 270.0(4.9) MeV, with the error combining those from statistical and from various sources of systematic errors. Finite volume effect is confirmed to be under control by a direct comparison of the results from two different volumes at the lightest available sea quarks corresponding to 230 MeV pions.
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