Bosonic partition functions at nonzero (imaginary) chemical potential

Published for SISSA by Springer Received: April 5, 2017 Accepted: June 21, 2017 Published: July 28, 2017 M. Kellerstein and J.J.M. Verbaarschot Department of Physics and Astronomy, Stony Brook University, Stony Brook, New York 11794, U.S.A. E-mail: , Abstract: 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. Keywords: Matrix Models, Spontaneous Symmetry Breaking, Random Systems ArXiv ePrint: 1610.02363 Open Access, c The Authors. Article funded by SCOAP3 . https://doi.org/10.1007/JHEP07(2017)144 JHEP07(2017)144 Bosonic partition functions at nonzero (imaginary) chemical potential Contents 1 Introduction 1 2 Random matrix theories 4 3 Phase quenched QCD 5 8 8 9 13 5 Bosonic partition function for real chemical potential 15 5.1 Heuristic derivation of the mean field result 15 5.2 The finite n massless bosonic partition function at nonzero chemical potential 17 5.3 Large n limit of the bosonic partition function 20 6 Conclusions 21 A Derivation of the fermionic partition function using superbosonization 21 B Massless one flavor bosonic partition function 24 C Bosonic partition function for n = 2 and n = 3 26 1 Introduction Universal random matrix behavior of QCD Dirac spectra can be understood in terms of chiral Lagrangians and is a direct consequence of spontaneous symmetry breaking in the 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 λV Σ (with λ the Dirac eigenvalue, V the space-time volume and Σ the chiral condensate) is kept fixed 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– JHEP07(2017)144 4 One flavor partition function at imaginary chemical potential 4.1 The fermionic partition function at nonzero (imaginary) chemical potential 4.2 The bosonic partition function 4.3 Limiting cases G(m, z) = d dz Z(m, z, z 0 ) (1.1) z 0 =z with Z(m, z, z 0 ) =  detNf (D + m) det(D + z) det(D + z 0 )  . (1.2) 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 defined 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 ρ(λ) ∼ V λα 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 note that the chiral condensate is obtained at fixed m in the thermodynamic limit, while random matrix behavior takes place on the scale of the average level spacing. Since the Mermin-Wagner-Coleman theorem requires a vanishing chiral condensate in two dimensions or less, we could satisfy the Banks-Casher relation if the lowlying eigenvalues scale as 1/V 1/(α+1) with α > 0. At the same time the noncompact chiral symmetry of the generating function could be broken spontaneously by these eigenvalues. Let us discuss what has been found in lattice simulations of the massless Nf -flavor Schwinger model. The average macroscopic spectral density is given by ρ(λ) ∼ V λα with –2– JHEP07(2017)144 It has been well established that lattice QCD Dirac spectra fluctuate 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 flavor 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 different from the agreement found in four dimensions. Yet this is not the case [4–8]. The picture that emerges from the two-flavor massless Schwinger model [4–6, 9], is that the low-lying eigenvalues are correlated according to chiral Random Matrix Theory while the chiral condensate defined 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 Π1 (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, z 0 ) as α = (Nf − 1)/(Nf + 2) [18, 19]. This results in a chiral condensate that vanishes as mα for m → 0. What transpires from lattice simulations [4–6] is that the chiral condensate vanishes as predicted while the rescaled low-lying Dirac eigenvalues, λk V 1/(α+1) fluctuate according to random matrix theory. The low-lying eigenvalues spontaneously break the symmetry of the generating function but because, they scale as 1/V 1/(α+1) with the volume, the chiral condensate remains zero. The generating function for the resolvent that reflects this behavior of the low-lying Dirac spectrum is of the form 0 Z 1/(α+1) TrM (U +U −1 ) (1.3) U ∈G/H with M = diag(m, · · · , m, z, z 0 ) and G → H the sp (...truncated)