Classical XY model

Summary

The classical XY model (sometimes also called classical rotor (rotator) model or O(2) model) is a lattice model of statistical mechanics. In general, the XY model can be seen as a specialization of Stanley's n-vector model for n = 2. Definition Given a D-dimensional lattice Λ, per each lattice site j ∈ Λ there is a two-dimensional, unit-length vector sj = (cos θj, sin θj) The spin configuration, s = (sj)j ∈ Λ is an assignment of the angle −π < θj ≤ π for each j ∈ Λ. Given a translation-invariant interaction Jij = J(i − j) and a point dependent external field \mathbf{h}_{j}=(h_j,0), the configuration energy is : H(\mathbf{s}) = - \sum_{i\neq j} J_{ij}; \mathbf{s}_i\cdot\mathbf{s}_j -\sum_j \mathbf{h}j\cdot \mathbf{s}j =- \sum{i\neq j} J{ij}; \cos(\theta_i-\theta_j) -\sum_j h_j\cos\theta_j The case in which Jij = 0 except for ij nearest neighbor is called nearest neighbor case. The configuration probability is giv

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Emergence of Concentration Effects in the Variational Analysis of the N-Clock Model

Matthias Ruf

We investigate the relationship between the N-clock model (also known as planar Potts model or DOUBLE-STRUCK CAPITAL ZN-model) and the XY model (at zero temperature) through a Gamma-convergence analysis of a suitable rescaling of the energy as both the number of particles and N diverge. We prove the existence of rates of divergence of N for which the continuum limits of the two models differ. With the aid of Cartesian currents we show that the asymptotics of the N-clock model in this regime features an energy that may concentrate on geometric objects of various dimensions. This energy prevails over the usual vortex-vortex interaction energy. (c) 2021 The Authors. Communications on Pure and Applied Mathematics published by Wiley Periodicals LLC.

WILEY2021

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We present a comprehensive discussion on lattice techniques for the simulation of scalar and gauge field dynamics in an expanding universe. After reviewing the continuum formulation of scalar and gauge field interactions in Minkowski and FLRW backgrounds, we introduce the basic tools for the discretization of field theories, including lattice gauge invariant techniques. Following, we discuss and classify numerical algorithms, ranging from methods of O(delta t(2)) accuracy like staggered leapfrog and Verlet integration, to Runge-Kutta methods up to O(delta t(4)) accuracy, and the Yoshida and Gauss-Legendre higher-order integrators, accurate up to O(delta t(10)) We adapt these methods for their use in classical lattice simulations of the non-linear dynamics of scalar and gauge fields in an expanding grid in 3+1 dimensions, including the case of 'self-consistent' expansion sourced by the volume average of the fields' energy and pressure densities. We present lattice formulations of canonical cases of: i) Interacting scalar fields, ii) Abelian U(1) gauge theories, and iii) Non-Abelian SU(2) gauge theories. In all three cases we provide symplectic integrators, with accuracy ranging from O(delta t(2)) up to O(delta t(10)) For each algorithm we provide the form of relevant observables, such as energy density components, field spectra and the Hubble constraint. We note that all our algorithms for gauge theories always respect the Gauss constraint to machine precision, including when 'self-consistent' expansion is considered. As a numerical example we analyze the post-inflationary dynamics of an oscillating inflaton charged under SU(2) x U(1). We note that the present manuscript is meant to be part of the theoretical basis for the code CosmoLattice, a multi-purpose MPI-based package for simulating the non-linear evolution of field theories in an expanding universe, publicly available at http://www.cosrnolattice.net.

2021

Combined preheating on the lattice with applications to Higgs inflation

Joël Repond, Javier Rubio

We use classical lattice simulations in 3+1 dimensions to study the interplay between the resonant production of particles during preheating and the subsequent decay of these into a set of secondary species. We choose to work in a simplified version of Higgs inflation in which the Higgs field non-minimally coupled to gravity plays the role of the inflaton. Our numerical results extend the analytical estimates in the literature beyond the linear regime and shed some light on the limitations of the analytical techniques. The inclusion of fast and inefficient decays postpones the onset of parametric resonance by depleting the particles produced at the bottom of the potential. In spite of this delay, fermions are shown to play an important role on the destruction of the inflaton field. The limitations of our approach and its applications to a realistic Higgs inflation scenario are also discussed.

Iop Publishing Ltd2016