Anomalies and persistent order in the chiral Gross-Neveu model

Abstract

We study the 2d chiral Gross-Neveu model at finite temperature T and chemical potential μ. The analysis is performed by relating the theory to a SU(N) × U(1) Wess-Zumino-Witten model with appropriate levels and global identifications necessary to keep track of the fermion spin structures. At μ = 0 we show that a certain ℤ2-valued ’t Hooft anomaly forbids the system to be trivially gapped when fermions are periodic along the thermal circle for any N and any T > 0. We also study the two-point function of a certain composite fermion operator which allows us to determine the remnants for T > 0 of the inhomogeneous chiral phase configuration found at T = 0 for any N and any μ. The inhomogeneous configuration decays exponentially at large distances for anti-periodic fermions while it persists for T > 0 and any μ for periodic fermions, as expected from anomaly considerations. A large N analysis confirms the above findings.

Detecting inhomogeneous chiral condensation from the bosonic two-point function in the (1 + 1)-dimensional Gross–Neveu model in the mean-field approximation*

“…( 28), were studied successfully in Refs. [39][40][41] at finite N on the lattice and another collaboration confirmed their finite-N results with analytic methods [35,42].…”

“…Several recent works are also tackling the problem of going beyond the mean-field approximation in order to incorporate bosonic fluctuations. While in the GN model the literature is not fully conclusive regarding whether the existence of (quasi)-long range order at T ̸ = 0 somewhere in the phase diagram [36][37][38], recent results in χGN model indicate that the chiral spiral solution is indeed destabilized by bosonic quantum fluctuations such that only remnants of the IP can be detected in bosonic correlation functions, which are oscillating but exponentially suppressed [35,[39][40][41][42]]. 3…”

“…For N → ∞ we can ignore its breaking and treat it as a symmetry of the model, while its role at finite N was recently discussed in Ref. [35,42].…”

Revisiting the spatially inhomogeneous condensates in the ( 1 + 1 ) -dimensional chiral Gross–Neveu model via the bosonic two-point function in

“…Within the meanfield (or infinite-N) approximation, the phase diagram of the (1 + 1)-dimensional χGN model in the (µ, T) plane is rather simple, compare figure 1. At the critical temperature T c = e γ /π, it undergoes a second-order phase transition from the symmetric phase (SP) without condensation to an IP where two bosonic fields σ and η condense and form a so-called chiral spiral [17,18,[20][21][22][23][24][25][26][27]. As first obtained in [27] using an ansatz calculation, this chiral spiral is the solution for the thermodynamic ground state in the whole (µ, T) plane.…”

“…Several recent works are also tackling the problem of going beyond the mean-field approximation in order to incorporate bosonic fluctuations. While in the GN model the literature is not fully conclusive regarding whether the existence of (quasi)-long range order at T = 0 somewhere in the phase diagram [37][38][39], recent results in χGN model indicate that the chiral spiral solution is indeed destabilized by bosonic quantum fluctuations such that only remnants of the IP can be detected in bosonic correlation functions, which are oscillating but exponentially suppressed [26,[40][41][42][43] 5 .…”