Numerical transfer-matrix methods are applied to two-dimensional Ising spin systems, in the presence of a confining magnetic field which varies with distance |x| to a "trap center," proportionally to (|x|/ℓ)p, where p>0 . On a strip geometry, the competition between the "trap size" ℓ and the strip width L is analyzed in the context of a generalized finite-size scaling ansatz. In the low-field regime ℓ >> L, we use conformal-invariance concepts in conjunction with a linear-response approach to derive the appropriate (p-dependent) limit of the theory, which agrees very well with numerical results for magnetization profiles. For high fields ℓ ≲ L, correlation-length scaling data broadly confirm an existing picture of p-dependent characteristic exponents. Standard spin-1/2 and spin-1 Ising systems are considered, as well as the Blume-Capel model.
Numerical transfer-matrix methods are applied to two-dimensional Ising spin systems, in the presence of a confining magnetic field which varies with distance |x| to a "trap center," proportionally to (|x|/ℓ)p, where p>0 . On a strip geometry, the competition between the "trap size" ℓ and the strip width L is analyzed in the context of a generalized finite-size scaling ansatz. In the low-field regime ℓ >> L, we use conformal-invariance concepts in conjunction with a linear-response approach to derive the appropriate (p-dependent) limit of the theory, which agrees very well with numerical results for magnetization profiles. For high fields ℓ ≲ L, correlation-length scaling data broadly confirm an existing picture of p-dependent characteristic exponents. Standard spin-1/2 and spin-1 Ising systems are considered, as well as the Blume-Capel model.
We study the ferromagnetic Ising model on very long strips of random widths using transfer matrix techniques. The width distribution is Gaussian with mean L and rms deviation \ensuremath{\Delta}L, which may be constant for any L (type I) or proportional to L (type II). We calculate the initial susceptibility for strips with 4\ensuremath{\leqslant}L\ensuremath{\leqslant}12, obtaining estimates of the ratio of exponents (\ensuremath{\gamma}/\ensuremath{\nu})L and the pseudocritical temperature T∗(L,L-1). In strips of type I, those estimates satisfy finite-size scaling, with finite-size corrections increasing with \ensuremath{\Delta}L, and in strips of type II it is satisfied only for very large lengths L. We discuss the influence of nonuniform thicknesses on the magnetism of low-dimensional systems, comparing the finite-size corrections of this problem to those which fit the experimental data of magnetic thin films.
Finite-size scaling (FSS) of the five-dimensional (d=5) Ising model is investigated numerically. Because of the hyperscaling violation in d>4 , FSS of the d=5 Ising model no longer obeys the conventional scaling relation. Rather, it is expected that the FSS behavior depends on the geometry of the embedding space (boundary condition). In this paper, we consider a cylindrical geometry and explore its influence on the correlation length xi=L;{Omega}f(L;{y_{t};{*}},HL;{y_{h};{*}}) with system size L , reduced temperature , and magnetic field H ; the indices y_{t,h};{*} and Omega characterize FSS. For that purpose, we employed the transfer-matrix method with Novotny's technique, which enables us to treat an arbitrary (integral) number of spins, N=8,10,...,28 ; note that, conventionally, N is restricted in N(=L;{d-1})=16,81,256,... . As a result, we estimate the scaling indices as Omega=1.40(15) , y_{t};{*}=2.8(2) , and y_{h};{*}=4.3(1) . Additionally, postulating Omega=43 , we arrive at y_{t};{*}=2.67(10) and y_{h};{*}=4.0(2) . These indices differ from the naively expected ones Omega=1 , y_{t};{*}=2 and y_{h};{*}=3 . Rather, our data support the generic formulas Omega=(d-1)3 , y_{t};{*}=2(d-1)3 , and y_{h};{*}=d-1 , advocated for a cylindrical geometry in d4 .
We studied the ferromagnetic Ising model on two-dimensional finite systems with non-integer characteristic lengths. First we considered very long strips with a finite number I of complete rows and one partially filled row with probability x , which is the two-dimensional version of a layer-by-layer growth. At fixed temperature, when the characteristic length L=I+x increases, the free energy per spin, the specific heat and the magnetic susceptibility oscillate, attaining relative extremes at integer L . The oscillations in the free energy are interpreted as surface corrections related to the oscillations in the mean coordination number. Finite-size scaling relations are not satisfied with continuous L , but still hold for fixed x and variable I , where the differences of mean widths are integers. For fixed x , the fits of the free energy give the conformal anomaly with very good accuracy. We also studied strips with discretized Gaussian distributions of widths, with non-integer means L and rms deviation . In these structures, the thermodynamic quantities vary monotonically with continuous L , but some methods for calculating critical exponents do not work properly when generalized to continuous L . We also obtain with good accuracy in these systems. We discuss the possible implications of our results to real systems behaviour.
The equilibrium ensemble approach to disordered systems is used to investigate the critical behavior of the two-dimensional Ising model in the presence of quenched random site dilution. The numerical transfer matrix technique in semi-infinite strips of finite width, together with phenomenological renormalization and conformal invariance, is particularly suited to putting the equilibrium ensemble approach to work. A method by which to extract with great precision the critical temperature of the model is proposed and applied. A more systematic finite-size scaling analysis than in previous numerical studies has been performed. A parallel investigation, along the lines of the two main scenarios currently under discussion, namely, the logarithmic corrections scenario (with critical exponents fixed in the Ising universality class) versus the weak universality scenario (critical exponents varying with the degree of disorder), is carried out. In interpreting our data, maximum care is constantly taken to be open to both possibilities. A critical discussion shows that an unambiguous discrimination between the two scenarios is still not possible on the basis of the available finite-size data.
Transfer-matrix methods are used to calculate spin-spin correlation functions (G), Helmholtz free energies (f) and magnetizations (m) in the two-dimensional random-field Ising model close to the zero-field bulk critical temperature T(c 0), on long strips of width L=3-18 sites, for binary field distributions. Analysis of the probability distributions of G for varying spin-spin distances R shows that describing the decay of their averaged values by effective correlation lengths is a valid procedure only for not very large R. Connections between field and correlation function distributions at high temperatures are established, yielding approximate analytical expressions for the latter, which are used for computation of the corresponding structure factor. It is shown that, for fixed R/L, the fractional widths of correlation-function distributions saturate asymptotically with L-2.2. Considering an added uniform applied field h, a connection between f(h), m(h), the Gibbs free energy g(m) and the distribution function for the uniform magnetization in a zero uniform field, P0(m), is derived and first illustrated for pure systems, and then applied for nonzero random field. From finite-size scaling and crossover arguments, coupled with numerical data, it is found that the width of P0(m) varies against (nonvanishing, but small) random-field intensity H0 as H(-3/7)(0).
We consider long, finite-width strips of Ising spins with randomly distributed couplings. Frustration is introduced by allowing both ferromagnetic and antiferromagnetic interactions. Free energy and spin-spin correlation functions are calculated by transfer-matrix methods. Numerical derivatives and finite-size scaling concepts allow estimates of the usual critical exponents $\ensuremath{\gamma}/\ensuremath{\nu},$ $\ensuremath{\alpha}/\ensuremath{\nu},$ and $\ensuremath{\nu}$ to be obtained, whenever a second-order transition is present. Low-temperature ordering persists for suitably small concentrations of frustrated bonds, with a transition governed by pure-Ising exponents. Contrary to the unfrustrated case, subdominant terms do not fit a simple, logarithmic-enhancement form. Our analysis also suggests a vertical critical line at and below the Nishimori point. Approaching this point along either the temperature axis or the Nishimori line, one finds nondiverging specific heats. A percolationlike ratio $\ensuremath{\gamma}/\ensuremath{\nu}$ is found upon analysis of the uniform susceptibility at the Nishimori point. Our data are also consistent with frustration inducing a breakdown of the relationship between correlation-length amplitude and critical exponents, predicted by conformal invariance for pure systems.