Bosonic partition functions at nonzero (imaginary) chemical potential

Journal of High Energy Physics, Jul 2017

We consider bosonic random matrix partition functions at nonzero chemical potential and compare the chiral condensate, the baryon number density and the baryon number susceptibility to the result of the corresponding fermionic partition function. We find that as long as results are finite, the phase transition of the fermionic theory persists in the bosonic theory. However, in case that the bosonic partition function diverges and has to be regularized, the phase transition of the fermionic theory does not occur in the bosonic theory, and the bosonic theory is always in the broken phase.

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Bosonic partition functions at nonzero (imaginary) chemical potential

HJE Bosonic partition functions at nonzero (imaginary) chemical potential M. Kellerstein 0 J.J.M. Verbaarschot 0 Stony Brook 0 New York 0 U.S.A. 0 0 Department of Physics and Astronomy, Stony Brook University We consider bosonic random matrix partition functions at nonzero chemical potential and compare the chiral condensate, the baryon number density and the baryon number susceptibility to the result of the corresponding fermionic partition function. We nd that as long as results are nite, the phase transition of the fermionic theory persists in the bosonic theory. However, in case that the bosonic partition function diverges and has to be regularized, the phase transition of the fermionic theory does not occur in the bosonic theory, and the bosonic theory is always in the broken phase. Matrix Models; Spontaneous Symmetry Breaking; Random Systems - 5 Bosonic partition function for real chemical potential Heuristic derivation of the mean eld result The nite n massless bosonic partition function at nonzero chemical potential 17 Large n limit of the bosonic partition function 6 Conclusions A Derivation of the fermionic partition function using superbosonization B Massless one avor bosonic partition function C Bosonic partition function for n = 2 and n = 3 1 Introduction Random matrix theories Phase quenched QCD 2 3 4 4.1 4.2 4.3 5.1 5.2 5.3 One avor partition function at imaginary chemical potential The fermionic partition function at nonzero (imaginary) chemical potential The bosonic partition function Limiting cases presence of a mass gap so that at low energies the theory reduces to a system of weakly interacting Goldstone modes. Spontaneous symmetry breaking also occurs in random matrix theories in the limit of large matrices, and because they also have a mass gap, the low energy limit of the random matrix theory partition function reduces to an integral over \Goldstone modes". In the microscopic scaling domain, where V (with the Dirac eigenvalue, V the space-time volume and the chiral condensate) is kept xed in the thermodynamic limit, the generating function for Dirac spectra of QCD or QCD-like theories coincides with the one obtained from random matrix theories with the same global symmetries and is identical to the one obtained from the corresponding chiral Lagrangian. The reason is that, in all cases we know of, the global symmetries in QCD are broken spontaneously in the same way as in the corresponding random matrix theory. { 1 { It has been well established that lattice QCD Dirac spectra uctuate according to the corresponding random matrix theory in the microscopic domain (see [1{3]). Because this agreement is based on the spontaneous breaking of the avor symmetry, one would expect that, as a consequence of the Coleman-Mermin-Wagner theorem, the agreement with Random Matrix Theory in two dimensions is structurally di erent from the agreement found in four dimensions. Yet this is not the case [4{8]. The picture that emerges from the two- avor massless Schwinger model [4{6, 9], is that the low-lying eigenvalues are correlated according to chiral Random Matrix Theory while the chiral condensate de ned in the usual way vanishes. For two-dimensional QCD [7], a nonzero chiral condensate was found for U(Nc) theories, while for SU(Nc) theories the mass dependence of the chiral condensate is consistent with m(Nf 1)=(Nf +2), the same as for the Schwinger model. Since 1(U(Nc)) = Z, the former observation could be interpreted in terms of a KosterlitzThouless phase. We performed quenched lattice simulations of two-dimensional QCD at strong coupling [8] and found that the agreement of QCD Dirac spectra with random matrix theory is as good as in four dimensions for comparable statistics. The resolvent of the Dirac operator D for Nf quarks with mass m can be expressed in terms of the generating function Z(m; z; z0) as G(m; z) = Z(m; z; z0) d dz z0=z detNf (D + m) det(D + z) det(D + z0) : (1.1) (1.2) with Because of the inverse determinant, this generating function has a noncompact symmetry [10]. It has been argued that the Mermin-Wagner-Coleman theorem can be violated for noncompact continuous symmetries [11{14]. In particular, it has been shown that the SO(2,1) symmetry of a hyperbolic spin chain is spontaneously broken also in one and two dimensions. In essence, the reason is that a partition function with a noncompact symmetry can only be de ned if this symmetry is spontaneously broken to its compact subgroup SO(2). In a conformal invariant theory the spectral density of the Dirac operator also scales as ( ) V and this scenario might reconcile conformal behavior with universal random matrix statistics [15{17]. As is the case for the hyperbolic spin chain, we could have the scenario that the compact symmetry remains unbroken, so that we have a vanishing chiral condensate, while the noncompact symmetry is spontaneously broken resulting in universal random matrix behavior. It is important to no (...truncated)


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M. Kellerstein, J.J.M. Verbaarschot. Bosonic partition functions at nonzero (imaginary) chemical potential, Journal of High Energy Physics, 2017, pp. 144, Volume 2017, Issue 7, DOI: 10.1007/JHEP07(2017)144