Application of finite size scaling to Monte Carlo simulations

Abstract

A new application of finite size scaling to Monte Carlo simulations is introduced. Using this technique, critical behavior can be investigated at temperatures arbitrarily close to the critical point without large lattice sizes. Applying this method to the two-dimensional standard 0(3) model it is shown that for the correlation length asymptotic scaling holds for /3 > 2.25; the magnetic susceptibility converges to the asymptotic scaling very slowly. In the scaling region, it is observed that the specific heat decreases with /3, which implies no singular behavior of the specific heat for this model. PACS numbers: 02.70.-c, 12.38.Gc, 64.60.-i Monte Carlo simulations have been widely used for the studies of critical phenomena.The main idea behind these methods is to measure the thermodynamic values of a certain physical quantity in the scaling region and then to fit the data to a certain scaling function. The crucial diflftculty here is that, without an exact solution of the model, the scaling region cannot be determined precisely; in other words, it is impossible to predict at what range of the temperature the data can be ideally fitted to an appropriate scaling function. To avoid this ambiguity, measurements must be done arbitrarily close to the critical point. The cost is that the size of the lattice must be extremely large to obtain proper thermodynamic data. The series expansion method has the same difficulty: it is impossible to decide how many expansion terms are needed for the proper determination of critical behavior. Because of this difficulty, often, the less singular terms are included in the leading scaling function in order to fit the data obtained at temperatures not sufficiently close to the critical point. However, this procedure usually makes the fittings highly unstable.To overcome such difficulties, various methods such as finite size scaling (FSS) and the Monte Carlo renormalization group (MCRG) method have been developed. However, it turns out that the standard usage of these methods still requires large lattice sizes [1].Our technique is based on the observation that finite size eflfects of any thermodynamic quantities on a finite lattice of linear size L depend only on L/^oo? where (^oo is the bulk (thermodynamic) correlation length. More precisely, for a typical thermodynamic quantity P, Pooit)= fp(x{t)), Xit) ^oo (0 ' fl) where t, PL, P<X>! and fp are the reduced temperature (T -Tc)/Tc, the value of P on a lattice of linear size L, the thermodynamic value of P, and a function depending on P, respectively. Equation (1) was initially suggested by Brezin [2], and using the analyticity property of PL (t) it was shown [3] that if P has a power critical behavior, Poo{t) ~ t"'' with p>0, from Eq. (1),follows for small values of ^, where po? Pi r • • ^^^ constants depending on P. It is obvious that Eq. (2) implies the usual FSS at i = 0, PL(t = 0) ^ L^/^, and that it implies Tc{L)-Tc --l/I/^/'", considering Eq. (2) up to the second order of t [Tc{L) is the fictitious critical point on the finite la...

D-theory: field theory via dimensional reduction of discrete variables

“…For the classical 2-d O(3) model the universal function f has already been determined very precisely [18,17]. Since at low temperatures the quantum Heisenberg model reduces to a classical 2-d lattice O(3) model, we can use the same universal function to deduce infinite volume results from finite-volume correlation length data.…”

“…In the classical 2-d O(3) model this was possible using finite-size scaling methods [18,17]. To investigate the correlation length in the quantum Heisenberg model we use the same technique.…”

“…The continuous time loop cluster algorithm was used in a single cluster version, and an improved estimator has been implemented for the staggered correlation function. Correlation lengths are extracted using the second moment method [18,17]. In all cases at least 10 5 measurements have been performed.…”

Monte Carlo Methods in Statistical Mechanics: Foundations and New Algorithms

“…Typically one approaches the critical point with L ~ c~, where c ~ 2 -4, and then uses finite-size scaling [22,23] to extrapolate to the infinite-volume limit. Note Added 1996: Recently, radical advances have been made in applying finite-size scaling to Monte Carlo simulations (see [97,98] and especially [99,100,101,102,103]); the preceding two sentences can now be seen to be far too pessimistic. For reliable extrapolation to the infinite-volume limit, L and ~ must both be » I, but the ratio L/ ~ can in some cases be as small as 10-3 or even smaller (depending on the model and on the quality of the data).…”

Simulations on Infinite-Size Lattices

“…DMRG can achieve β = ∞ (mostly without dynamical information) for fairly small systems, or infinite size for large temperatures and finite ∆τ . The most powerful method to extrapolate to infinite system size is the Finite Size Scaling method of Kim [10] and Caracciolo et al [11], which allows extrapolation at correlation lengths far larger than the system size, but re- In summary, we have introduced a new method, as a small modification of existing cluster methods, to simulate both classical and quantum systems at infinite system size and/or at zero temperature, while obtaining all two-point functions and derived quantities. Larger distances (in space and/or imaginary time) are accessed for longer simulations.…”