Toward a statistical theory of one-dimensional systems.

E. T. Y. Hsieh

Fei Ye, P. A. Marchetti, Zhixun Su +1 more

Abstract The relation between braid and exclusion statistics is examined in one-dimensional systems, within the framework of Chern–Simons statistical transmutation in gauge invariant form with an appropriate dimensional reduction. If the matter action is anomalous, as for chiral fermions, a relation between braid and exclusion statistics can be established explicitly for both mutual and nonmutual cases. However, if it is not anomalous, the exclusion statistics of emergent low energy excitations is not necessarily connected to the braid statistics of the physical charged fields of the system. Finally, we also discuss the bosonization of one-dimensional anyonic systems through T-duality.

Mitsusada M. Sano

Bill Sutherland

We present a microscopic theory of exclusion statistics, based on an analogy with the exactly solved integrable one-dimensional systems. The resulting theory differs from the usual theory of mutual exclusion statistics by certain dimension-dependent features. As an example, we carry out a calculation of the ground-state properties, low-lying excitations, and thermodynamics of a system of spinless fermions in two dimensions, obeying pure exclusion statistics.

Alexander A. Ovchinnikov, V. Ya. Krivnov

Hans-Peter Eckle

Abstract The connection between statistical mechanics on the one side and quantum mechanics and quantum field theory on the other side is based on the analogy between thermal and quantum fluctuations. Formally, the connection is expressed through the mathematical equivalence between the partition function in statistical mechanics and the propagator in quantum field theory. This chapter explores the equivalence between statistical mechanics and quantum mechanics or quantum field theory in general terms using the Feynman path integral and with the example of the equivalence between the classical XY Heisenberg model and the sigma model of quantum field theory. Invoking the concepts of the partition function and the transfer matrix, an example demonstrates the passage from the quantum mechanics of a single degree of freedom, a zero-dimensional system, to the statistical mechanics of a one-dimensional system represented by classical variables.