Berichte der Bunsengesellschaft für physikalische Chemie

Finite size dependence of the helicity modulus within the mean spherical model

Daniel M. Danchev

Published Mar 1, 1994• 0 citations

Abstract

Abstract The finite‐size behavior of the helicity modulus is analyzed within the three‐dimensional mean spherical model. In contrast to previous heuristical predictions, for the helicity modulus in the finite‐size critical region there are no “resonant” logarithmic corrections.

DOI: 10.1002/bbpc.19940980348

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Berichte der Bunsengesellschaft für physikalische Chemie

Finite size dependence of the helicity modulus within the mean spherical model

Daniel M. Danchev

Published Mar 1, 1994• 0 citations

Publisher

Abstract

DOI: 10.1002/bbpc.19940980348

Related papers

Finite-size dependence of the helicity modulus within the mean spherical model

Daniel M. Danchev

Journal of Statistical PhysicsOct 1, 199314 citations

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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.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.

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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.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.

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The spherical model is solved on a hypercubic lattice in $d$ dimensions, each bond of which is decorated with $l$ spins. The thermodynamic functions and the helicity modulus, analogous to a superfluid density, are calculated. We find that at least two spherical fields are required for the model to exhibit low-temperature properties that can approximate reasonably those of $O(n)$ models. The heuristic prediction that the critical temperature behaves as ${T}_{c}(l)\ensuremath{\sim}{(l+1)}^{\ensuremath{-}1}$ is checked for the model and found to hold quite accurately even for small $l(\ensuremath{\gtrsim}2)$. The helicity modulus and magnetization of the two-constraint spherical model are found to scale approximately with the critical temperature, but the relation between them is more complex than in the undecorated model. This relation is used to check heuristic arguments concerning the helicity modulus at low temperatures. We comment on the relevance to physical systems, in particular, the problem of boson condensation in a restricted geometry.The spherical model is solved on a hypercubic lattice in $d$ dimensions, each bond of which is decorated with $l$ spins. The thermodynamic functions and the helicity modulus, analogous to a superfluid density, are calculated. We find that at least two spherical fields are required for the model to exhibit low-temperature properties that can approximate reasonably those of $O(n)$ models. The heuristic prediction that the critical temperature behaves as ${T}_{c}(l)\ensuremath{\sim}{(l+1)}^{\ensuremath{-}1}$ is checked for the model and found to hold quite accurately even for small $l(\ensuremath{\gtrsim}2)$. The helicity modulus and magnetization of the two-constraint spherical model are found to scale approximately with the critical temperature, but the relation between them is more complex than in the undecorated model. This relation is used to check heuristic arguments concerning the helicity modulus at low temperatures. We comment on the relevance to physical systems, in particular, the problem of boson condensation in a restricted geometry.

Physical review. B, Condensed matterAug 1, 198212 citations

The critical scaling of the helicity modulus of the classical O(3)

3d

Heisenberg ferromagnet is studied directly. Monte Carlo methods that impose either an antiperiodic boundary condition or a finite twist of definite handedness across otherwise periodic boundaries in one lattice direction are used to measure scale-dependent enthalpy variations in a simple cubic lattice at the ferromagnetic critical temperature. Finite-size scaling is then used to determine the critical exponents

{v}_{E}

and

{v}_{F}

for helicity and, by evaluating three independent hyperscaling-linked pairs of $\ensuremath{\nu}$ and $\ensuremath{\alpha}$, to test hyperscaling for this model. It is observed that antiperiodic boundary conditions in particular constrain the lattice to have a nonzero topological charge, establishing a connection between topological charge and helicity in the model.

The spherical model is solved on a hypercubic lattice in

d

dimensions, each bond of which is decorated with

l

spins. The thermodynamic functions and the helicity modulus, analogous to a superfluid density, are calculated. We find that at least two spherical fields are required for the model to exhibit low-temperature properties that can approximate reasonably those of

O(n)

models. The heuristic prediction that the critical temperature behaves as ${T}_{c}(l)\ensuremath{\sim}{(l+1)}^{\ensuremath{-}1}$ is checked for the model and found to hold quite accurately even for small $l(\ensuremath{\gtrsim}2)$. The helicity modulus and magnetization of the two-constraint spherical model are found to scale approximately with the critical temperature, but the relation between them is more complex than in the undecorated model. This relation is used to check heuristic arguments concerning the helicity modulus at low temperatures. We comment on the relevance to physical systems, in particular, the problem of boson condensation in a restricted geometry.