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Physical review. B, Condensed matter

Universal scaling form of the correlation length in Ising strips with periodic, free, fixed, and mixed boundary conditions

Theodore W. Burkhardt, Ihnsouk Guim

Published Feb 1, 1987• 23 citations
Publisher

Abstract

According to finite-size scaling theory, the correlation length in an Ising strip with infinite length and a width of L lattice constants has the form ${\ensuremath{\xi}}_{L}$(T)=Lf(c(T-${T}_{c}$)L) in the scaling regime L\ensuremath{\gg}1,\ensuremath{\Vert}T-${T}_{c}$\ensuremath{\Vert}/${T}_{c}$\ensuremath{\ll}1. Here f(x) is a universal scaling function, which does, however, depend on the boundary conditions at the edges of the strip. Utilizing the correspondence between the two-dimensional Ising model and the one-dimensional quantum Ising model in a transverse field, we calculate the explicit form of the function f(x) for spin-spin and energy-energy correlations in strips with periodic, free, fixed, and mixed free-fixed boundary conditions.

DOI
10.1103/physrevb.35.1799

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According to finite-size scaling theory, the correlation length in an Ising strip with infinite length and a width of L lattice constants has the form \ensuremathξL{\ensuremath{\xi}}_{L}\ensuremathξL​(T)=Lf(c(T-Tc{T}_{c}Tc​)L) in the scaling regime L\ensuremath{\gg}1,\ensuremath{\Vert}T-Tc{T}_{c}Tc​\ensuremath{\Vert}/Tc{T}_{c}Tc​\ensuremath{\ll}1. Here f(x) is a universal scaling function, which does, however, depend on the boundary conditions at the edges of the strip. Utilizing the correspondence between the two-dimensional Ising model and the one-dimensional quantum Ising model in a transverse field, we calculate the explicit form of the function f(x) for spin-spin and energy-energy correlations in strips with periodic, free, fixed, and mixed free-fixed boundary conditions.
We study the (analytic) finite-size corrections in the Ising model on the strip with free, fixed (++), and mixed boundary conditions. For fixed (++) boundary conditions, the spins are fixed to the same values on two sides of the strip. We find that subdominant finite-size corrections to scaling should be to the form a(p)/N(2p-1) for the free energy f(N) and b(p)/N(2p-1) for inverse correlation length ξ(N)(-1), with integer value of p. We investigate the set {a(p),b(p)} by exact evaluation and their changes upon varying anisotropy of coupling. We find that the amplitude ratios b(p)/a(p) remain constant upon varying coupling anisotropy. Such universal behavior is correctly reproduced by the conformal perturbative approach.
We consider strips of Ising spins at criticality. For strips of width N sites, subdominant (additive) finite-size corrections to scaling are assumed to be of the form a(k)/N(k) for the free energy, and b(k)/N(k) for inverse correlation length, with integer values of k. We investigate the set {a(k),b(k)} (k≥2) by exact evaluation and numerical transfer-matrix diagonalization techniques, and their changes upon varying anisotropy of couplings, spin quantum number S, and (finite) interaction range, in all cases for both periodic (PBCs) and free (FBCs) boundary conditions across the strip. We find that the coefficient ratios b(k)/a(k) remain constant upon varying coupling anisotropy for S=1/2 and first-neighbor couplings, for both PBCs and FBCs (albeit at distinct values in either case). Such apparently universal behavior is not maintained upon changes in S or interaction range.
The specific heat, energy, and free energy of an infinitely long, square-lattice ferromagnetic Ising strip consisting of nnn parallel layers with free boundary conditions and a surface magnetic field H1=kBTh1{H}_{1}={k}_{B}{\mathrm{Th}}_{1}H1​=kB​Th1​ imposed on the last layer, is analyzed for large nnn in the light of finite-size scaling theory. It is shown rigorously that the free energy (and, similarly, the energy and specific heat) can be written asymptotically in the scaling form $\ensuremath{\Delta}f(n,t,{h}_{1})\ensuremath{\approx}{n}^{\ensuremath{-}2}(\mathrm{ln}n){W}_{1}(nt,{n}^{\frac{1}{2}}{h}_{1})+{n}^{\ensuremath{-}2}{W}_{2}(nt,{n}^{\frac{1}{2}}{h}_{1})$, where $t=\frac{(T\ensuremath{-}{T}_{c})}{{T}_{c}}$. The scaling functions Wi(x,y){W}_{i}(x,y)Wi​(x,y) are computed in explicit closed form and shown to verify all the analyticity and asymptotic requirements anticipated by scaling theory. Furthermore, in the limit $n\ensuremath{\rightarrow}\ensuremath{\infty}$ at fixed $t\ensuremath{\ne}0$, the bulk and surface contributions to the thermodynamic properties are found to account for all except a correction of order ${e}^{\frac{\mathrm{pna}}{\ensuremath{\xi}(T)}}$, where aaa is the lattice spacing and $\ensuremath{\xi}(T)$ is the bulk correlation length; the value of the small rational constant ppp is interpreted in terms of interference effects between the two opposite boundaries (or "surfaces").
Anisotropy effects on the finite-size critical behavior of a two-dimensional Ising model on a general triangular lattice in an infinite-strip geometry with periodic, antiperiodic, and free boundary conditions (bc) in the finite direction are investigated. Exact results are obtained for the scaling functions of the finite-size contributions to the free energy density. With ξ(>) the largest and ξ(<) the smallest bulk correlation length at a given temperature near criticality, we find that the dependence of these functions on the ratio ξ(<)/ξ(>) and on the angle parametrizing the orientation of the correlation volume is of geometric nature. Since the scaling functions are independent of the particular microscopic realization of the anisotropy within the two-dimensional Ising model, our results provide a limited verification of universality. We explain our observations by considering finite-size scaling of free energy densities of general weakly anisotropic models on a d-dimensional film (i.e., in an L×∞(d-1) geometry) with bc in the finite direction that are invariant under a shear transformation relating the anisotropic and isotropic cases. This allows us to relate free energy scaling functions in the presence of an anisotropy to those of the corresponding isotropic system. We interpret our results as a simple and transparent case of anisotropic universality, where, compared to the isotropic case, scaling functions depend additionally on the shape and orientation of the correlation volume. We conjecture that this universality extends to cases where the geometry and/or the bc are not invariant under the shear transformation and argue in favor of validity of two-scale factor universality for weakly anisotropic systems.
The correlation length of boundary spins in the Ising model, defined on strips of triangular lattice with free boundary conditions, is determined with an efficient numerical procedure based on the star-triangle transformation. In the case of isotropic critical interactions, the extrapolated amplitude of the correlation length is in excellent agreement with the value 2/( pi eta /sub ///) predicted by conformal invariance. An analytical formula for the amplitude in strips with anisotropic interactions is proposed. Fixing the spins on one edge reduces the amplitude of the correlation length on the other edge by a factor 1 / 2 . The convergence of phenomenological renormalisation with free boundary conditions is studied.