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Journal of Physics C Solid State Physics

Finite-size effects of the helicity modulus in the two-dimensional classical XY model

Johannes E. Van Himbergen

Published Oct 10, 1984• 11 citations
Publisher

Abstract

A systematic numerical study of the relevant finite-size effects of the helicity modulus for the two-dimensional classical XY model demonstrates the usefulness of this quantity as a means of obtaining information about Kosterlitz-Thouless type phase transitions. In this way, one does not have to compute correlation functions on finite lattices, and fit or compare the data thus obtained to theoretical scaling predictions, which only pertain to extremely large lattices. The approach may have many applications to systems of current interest that exhibit XY-like phase transitions.

DOI
10.1088/0022-3719/17/28/019

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A systematic numerical study of the relevant finite-size effects of the helicity modulus for the two-dimensional classical XY model demonstrates the usefulness of this quantity as a means of obtaining information about Kosterlitz-Thouless type phase transitions. In this way, one does not have to compute correlation functions on finite lattices, and fit or compare the data thus obtained to theoretical scaling predictions, which only pertain to extremely large lattices. The approach may have many applications to systems of current interest that exhibit XY-like phase transitions.
We study the classical XY (plane rotator) model at the Kosterlitz–Thouless phase transition. We simulate the model using the single-cluster algorithm on square lattices of a linear size up to L = 2048. We derive the finite-size behaviour of the second moment correlation length over the lattice size ξ 2nd / L at the transition temperature. This new prediction and the analogous one for the helicity modulus ϒ are confronted with our Monte Carlo data. This way β KT = 1.1199 is confirmed as inverse transition temperature. Finally, we address the puzzle of logarithmic corrections of the magnetic susceptibility χ at the transition temperature.
Using Monte Carlo methods, we compute the finite-size scaling function of the helicity modulus \ensuremath{\Upsilon} of the two-dimensional O(3) model and compare it to the low-temperature expansion prediction. From this, we estimate the range of validity for the leading terms of the low-temperature expansion of the finite-size scaling function and for the low-temperature expansion of the correlation length. Our results strongly suggest that a Kosterlitz-Thouless transition at a temperature T>0 is extremely unlikely in this model.
The two-dimensional S=1/2S=1/2S=1/2 XY model is investigated with an extensive quantum Monte Carlo simulation. The helicity modulus is precisely estimated through a continuous-time loop algorithm for systems up to 128×128128 \times 128128×128 near and below the critical temperature. The critical temperature is estimated as TKT=0.3427(2)JT_{\rm KT} = 0.3427(2)JTKT​=0.3427(2)J. The obtained estimates for the helicity modulus are well fitted by a scaling form derived from the Kosterlitz renormalization group equation. The validity of the Kosterlitz-Thouless theory for this model is confirmed.
The helicity modulus of the S=1/2 XY model is precisely estimated through a world-line quantum Monte Carlo method enhanced by a cluster update algorithm. The obtained estimates for various system sizes and temperatures are well fitted by a scaling form with L replaced by log (L/L0{\mathrm{L}}_{0}L0​), which is inferred from the solution of the Kosterlitz renormalization group equation. The validity of the Kosterlitz-Thouless theory for this model is confirmed.
A new logarithmic term in the finite-size helicity modulus has been predicted recently for the three-dimensional classical XY-model universality class under nonperiodic boundary conditions. These theoretical ideas have important implications for the analysis of superfluid-density measurements in confined geometries. We present numerical results from Monte Carlo simulations of the helicity modulus for the three-dimensional classical XY model on a finite simple-cubic lattice, with pinned-surface-spin boundary conditions. Our data are consistent with the presence of an additional logarithmic term and an estimate for the amplitude is given.
We study the phase transition of thin films in the three-dimensional XY universality class. To this end, we perform a Monte Carlo study of the improved two-component ϕ 4 model, the improved dynamically diluted XY model and the standard XY model on the simple cubic lattice. We study films of a thickness up to L 0 = 32 lattice spacings. In the short direction of the lattice, free boundary conditions are employed. Using a finite size scaling (FSS) method, proposed recently, we determine the transition temperature with high accuracy. The effectively two-dimensional finite size scaling behaviour of the Binder cumulant U 4 , the second-moment correlation length over the lattice size ξ 2nd / L , the ratio of the partition functions with anti-periodic and periodic boundary conditions Z a / Z p and the helicity modulus ϒ clearly confirm the Kosterlitz–Thouless nature of the transition. We analyse the scaling of the transition temperature with the thickness L 0 of the film. We compute the universal ratio of the thickness of the film L 0 and the transverse correlation length ξ T in the three-dimensional thermodynamic limit at the Kosterlitz–Thouless transition temperature of a film of thickness L 0 : [ L 0,KT /ξ T ] * = 1.595(7). This results can be compared with experimental results on thin films of 4 He near the λ-transition.