How Should We Choose the Boundary Conditions in a Simulation Which Could Detect Anyons in One and Two Dimensions?
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How Should We Choose the Boundary Conditions in a Simulation Which Could Detect Anyons in One and Two Dimensions? Riccardo Fantoni1 Received: 18 February 2020 / Accepted: 13 September 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract We discuss the problem of anyonic statistics in one and two spatial dimensions from the point of view of statistical physics. In particular, we want to understand how the choice of the Born–von Karman or the twisted periodic boundary conditions necessary in a Monte Carlo simulation to mimic the thermodynamic limit of the many body system influences the statistical nature of the particles. The particles can either be just bosons, when the configuration space is simply connected as for example for particles on a line. They can be bosons and fermions, when the configuration space is doubly connected as for example for particles in the tridimensional space or in a Riemannian surface of genus greater or equal to one (on the torus, etc.). They can be scalar anyons with arbitrary statistics, when the configuration space is infinitely connected as for particles on the plane or in the circle. They can be scalar anyons with fractional statistics, when the configuration space is the one of particles on a sphere. One can further have multi-components anyons with fractional statistics when the configuration space is doubly connected as for particles on a Riemannian surface of genus greater or equal to one. We determine an expression for the canonical partition function of hard core particles (including anyons) on various geometries. We then show how the choice of boundary condition (periodic or open) in one and two dimensions determines which particles can exist on the considered surface. Keywords Statistical physics · Fractional statistics · Anyons · Computer simulation · Periodic boundary conditions · Twisted boundary conditions
* Riccardo Fantoni [email protected] 1
Dipartimento di Fisica, Università di Trieste, strada Costiera 11, 34151 Grignano, Trieste, Italy
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Vol.:(0123456789)
Journal of Low Temperature Physics
1 Introduction For the statistical mechanics of a systems of many anyons, very partial results can be obtained, because the exact solution of a gas of anyons is not known. In fact, in contrast to the bosonic or fermionic case where the statistics is implemented by hand on the many body Hilbert space by constructing completely symmetric or antisymmetric products of single particle wave functions, for anyons the complicated boundary conditions for the interchange of any two particles require the knowledge of the complete many-body configurations. Only the two-body problem is exactly soluble for anyons, and hence only the two-body partition function can be computed exactly. Since the thermodynamic limit cannot be performed, one has to resort to approximate or alternative methods to study the statistical mechanics of anyons [1, 2]. For example, if the thermodynamic functions are analytic in the particle density, it is we
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