Statistical Mechanics of Relativistic Aynon-like Systems

To study the manifestation of the Aharonov-Bohm effect in many-body systems we consider the statistical mechanics of the Gross-Neveu model on a ring (1+1 dimensions) and on a cylinder (2+1 dimensions) with a thin solenoid coinciding with the axis. For such systems with a non-trivial magnetic flux ($\theta$) many thermodynamical observables, such as the order parameter, induced current and virial coefficients, display periodic but non-analytic dependence on $\theta$. In the 2+1 dimensional case we further find that there is an interval of $\theta\in(1/3,2/3)$ (modulo integers) where parity is always spontaneously broken, independent of the circumference of the cylinder. We show that the mean-field character of the phase transitions is preserved to the leading order in $1/N$, by verifying the $\theta$-independence of all the critical exponents. The precise nature of the quasi-particle, locally fermion-like and globally anyon-like, is illuminated through the calculation of the equal-time commutator and the decomposition of the propagator into a sum over paths classified by winding numbers.