The addition of an adjoint Polyakov loop term to the action of a pure gauge theory at finite temperature leads to new phases of SU (N ) gauge theories. For SU (3), a new phase is found which breaks Z(3) symmetry in a novel way; for SU (4), the new phase exhibits spontaneous symmetry breaking of Z(4) to Z(2), representing a partially confined phase in which quarks are confined, but diquarks are not. The overall phase structure and thermodynamics is consistent with a theoretical model of the effective potential for the Polyakov loop based on perturbation theory.

We minimize the one-loop effective potential for SU (N ) gauge theories including fermions with finite mass in the fundamental (F), adjoint (Adj), symmetric (S), and antisymmetric (AS) representations. We calculate the phase diagram on S 1 × R 3 as a function of the length of the compact dimension, β, and the fermion mass, m, for various N and N f . We consider the effect of periodic boundary conditions [PBC(+)] on fermions as well as antiperiodic boundary conditions [ABC(-)]. With standard ABC(-) on fermions only the deconfined phase is found at one-loop for all representations considered. However, the use of PBC(+) produces a rich phase structure. These phases are distinguished by the eigenvalues of the Polyakov loop P . In the case of fundamental representation fermions [QCD(F,+)], a phase in which Re TrP is minimized (and negative) is favoured for all values of mβ. For N odd charge conjugation (C) symmetry is spontaneously broken in this phase due to O(1/N ) effects. Minimization of the effective potential for QCD(AS/S,+) results in a phase where |Im TrP | is maximized, resulting in C-breaking for all N and all values of mβ, however, the partition function is the same up to O(1/N ) corrections as when ABC are applied. Therefore, regarding orientifold planar equivalence, we argue that in the one-loop approximation C-breaking in QCD(AS/S,+) resulting from the application of PBC on fermions does not invalidate the large N equivalence with QCD(Adj,-). Similarly, with respect to orbifold planar equivalence, breaking of Z 2 interchange symmetry resulting from application of PBC to bifundamental (BF) representation fermions does not invalidate equivalence with QCD(Adj,-) in the one-loop perturbative limit because the partition functions of QCD(BF,-) and QCD(BF,+) are the same. Of particular interest as well is the case of adjoint fermions where for N f > 1 Majorana flavour confinement is obtained for sufficiently small mβ, and deconfinement for sufficiently large mβ. For N ≥ 3 these two phases are separated by one or more additional phases, some of which can be characterized as partially-confining phases.1 When PBC are applied to fermions the theory does not correspond to a finite temperature field theory [ABC are required in that case due to the trace over (physical) anticommuting fields in Z = Tr`e −βH´] . However, the formalism is the same so we do not refrain from using the language of thermodynamics, even though it only strictly applies in the case of ABC on fermions.

It has been suggested that for QCD at finite baryon density the distribution of the phase angle, i.e. the angle defined as the imaginary part of the logarithm of the fermion determinant, has a simple Gaussian form. This distribution provides the density in the density of states approach to the sign problem. We calculate this phase angle distribution using i) the hadron resonance gas model; and ii) a combined strong coupling and hopping parameter expansion in lattice gauge theory. While the former model leads only to a Gaussian distribution, in the latter expansion we discover terms which cause the phase angle distribution to deviate, by relative amounts proportional to powers of the inverse lattice volume, from a simple Gaussian form. We show that despite the tiny inverse-volume deviation of the phase angle distribution from a simple Gaussian form, such non-Gaussian terms can have a substantial impact on observables computed in the density of states/reweighting approach to the sign problem.Comment: 43 pages, 4 figure

Abstract:The phase structure of QCD-like gauge theories with fermions in various representations is an interesting but generally analytically intractable problem. One way to ensure weak coupling is to define the theory in a small finite volume, in this case S 3 × S 1 . Genuine phase transitions can then occur in the large N theory. Here, we use this technique to investigate SU(N ) gauge theory with a number N f of massive adjoint-valued Majorana fermions having non-thermal boundary conditions around S 1 . For N f = 1 we find a line of transitions that separate the weak-coupling analogues of the confined and de-confined phases for which the density of eigenvalues of the Wilson line transform from the uniform distribution to a gapped distribution. However, the situation for N f > 1 is much richer and a series of weak-coupling analogues of partially-confined phases appear which leave unbroken a Z p subgroup of the centre symmetry. In these Z p phases the eigenvalue density has p gaps and they are separated from the confining phase and from one-another by first order phase transitions. We show that for small enough mR (the mass of the fermions times the radius of the S 3 ) only the confined phase exists. The large N phase diagram is consistent with the finite N result and with other approaches based on R 3 × S 1 calculations and lattice simulations.

To leading order in perturbation theory, we solve QCD, defined on a small three sphere in the large N and N f limit, at finite chemical potential and map out the phase diagram in the (µ, T ) plane. The action of QCD is complex in the presence of a non-zero quark chemical potential which results in the sign problem for lattice simulations. In the large N theory, which at low temperatures becomes a conventional unitary matrix model with a complex action, we find that the dominant contribution to the functional integral comes from complexified gauge field configurations. For this reason the eigenvalues of the Polyakov line lie off the unit circle on a contour in the complex plane. We find at low temperatures that as µ passes one of the quark energy levels there is a third-order Gross-Witten transition from a confined to a deconfined phase and back again giving rise to a rich phase structure. We compare a range of physical observables in the large N theory to those calculated numerically in the theory with N = 3. In the latter case there are no genuine phase transitions in a finite volume but nevertheless the observables are remarkably similar to the large N theory.