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)