TY - JOUR

T1 - The N-Clock Model

T2 - Variational Analysis for Fast and Slow Divergence Rates of N

AU - Cicalese, Marco

AU - Orlando, Gianluca

AU - Ruf, Matthias

N1 - Publisher Copyright:
© 2022, The Author(s).

PY - 2022/8

Y1 - 2022/8

N2 - We study a nearest neighbors ferromagnetic classical spin system on the square lattice in which the spin field is constrained to take values in a discretization of the unit circle consisting of N equi-spaced vectors, also known as the N-clock model. We find a fast rate of divergence of N with respect to the lattice spacing for which the N-clock model has the same discrete-to-continuum variational limit as the classical XY model (also known as planar rotator model), in particular concentrating energy on topological defects of dimension 0. We prove the existence of a slow rate of divergence of N at which the coarse-grain limit does not detect topological defects, but it is instead a BV-total variation. Finally, the two different types of limit behaviors are coupled in a critical regime for N, whose analysis requires the aid of Cartesian currents.

AB - We study a nearest neighbors ferromagnetic classical spin system on the square lattice in which the spin field is constrained to take values in a discretization of the unit circle consisting of N equi-spaced vectors, also known as the N-clock model. We find a fast rate of divergence of N with respect to the lattice spacing for which the N-clock model has the same discrete-to-continuum variational limit as the classical XY model (also known as planar rotator model), in particular concentrating energy on topological defects of dimension 0. We prove the existence of a slow rate of divergence of N at which the coarse-grain limit does not detect topological defects, but it is instead a BV-total variation. Finally, the two different types of limit behaviors are coupled in a critical regime for N, whose analysis requires the aid of Cartesian currents.

UR - http://www.scopus.com/inward/record.url?scp=85132861858&partnerID=8YFLogxK

U2 - 10.1007/s00205-022-01799-9

DO - 10.1007/s00205-022-01799-9

M3 - Article

AN - SCOPUS:85132861858

SN - 0003-9527

VL - 245

SP - 1135

EP - 1196

JO - Archive for Rational Mechanics and Analysis

JF - Archive for Rational Mechanics and Analysis

IS - 2

ER -