Physica A: Statistical Mechanics and its Applications
Critical temperature in the two-layered Ising model
Introduction
Recently, one of the major research subjects in the statistical mechanics and solid-state physics is studying of surface and finite-size effects. The main interest in studying these phenomena comes from the fact that all real systems are of finite size and, thus, these effects certainly cause relevant corrections to thermodynamic quantities which correspond to the limiting case of infinite system size. This topic has already been described in some review articles 1, 2.
An important model studied in the context of finite-size effects is the layered Ising model. This model is composed of several coupled two-dimensional Ising layers. Since the two-dimensional layers are infinite the model has a critical point at finite temperature and its critical exponents are the same as in the single-layer case. Increasing the number of layers one can observe interesting crossover phenomena from two- to three-dimensional Ising universality class [3].
In this model an interesting situation appears when the inter-layer coupling K becomes infinitesimally small compared to the intra-layer coupling J. For such a case some scaling theories were constructed 4, 5, 6. They predict the values of the shift exponent φ which describes the deviation of the critical temperature Tc(K) from the critical temperature in the decoupled limit (K=0)In particular, these theories predict 4, 5that when the coupling J is the same in each layer, then φ=γ, where γ is the critical exponent describing divergence of susceptibility upon approaching the critical point. Extending these scaling arguments, Oitmaa and Enting [6]suggested that when the coupling J changes in each sublattice, then φ=γ/2. The second case is particularly interesting. Let us notice that in this case, in the decoupled limit the system is composed of subsystems which have different critical temperatures and the problem of coupling of such subsystems is of general theoretical interest.
These scaling predictions have only modest confirmation by other calculations. For the case of equal intra-layer couplings a certain qualitative agreement with scaling theories has been obtained using the high-temperature series expansion [6]and a certain variational method [7]. The case of unequal couplings has been approached, in our opinion, only using the high-temperature series expansion [6]and with this method only very rough estimations φ<1 were made.
In the present paper we calculate the critical temperature in the two-layered Ising model using the transfer-matrix mean-field approximation which has been introduced by Suzuki and Lipowski [8]. This method has been already applied to the layered Ising model [9], but in the case when the inter-layer coupling J is the same in each layer and mainly for K=J. In the present paper the main objective is to study the model in the case of the small inter-layer coupling K, which enables us to estimate the shift exponent φ and to confront our results against previous predictions. For equal intra-layer couplings our results generally support the scaling result φ=γ. However, for unequal intra-layer couplings our estimations of φ considerably deviate from those predicted by Oitmaa and Enting [6]. At present we cannot say in which point these scaling arguments are wrong but clearly they require reconsideration.
In Section 2we briefly describe the transfer-matrix mean-field approximation. Since we also study the case when intra-layer interactions are different, introduction of two effective fields is required and the method has to be generalized compared to the original formulation [8]. Our main numerical results and the approximate decimation scheme which gives φ consistent with numerical estimations are summarized in Section 3. Section 4contains a summary of our research.
Section snippets
Transfer-matrix mean-field approximation
In the present paper we calculate the critical temperature using the transfer-matrix mean-field approximation [8](TMMFA). This method has an interesting property, namely for a number of 2D solvable Ising models TMMFA gives exact critical temperatures. Moreover, even for some nonsolvable models this method leads to very precise estimations of critical temperature 8, 10, 11. This method has been applied to the n-layer Ising model (n=2,…,5) and also in this case precise estimations of
Results
We performed calculations in three different regimes which correspond to different ways of approaching the limit K→0. In the following we, put J1=1. Our method of calculation of Tc(K) is as follows: First, for a fixed K we solve Eq. (2.13)for L=1,…,4 and obtain Tc(L). Assuming that this series of critical temperatures has a formwe can easily extrapolate our results and obtain Tc(K).
One can ponder whether it is possible to make a reasonable extrapolation having such a short
Summary
Using the transfer-matrix mean-field approximation (TMMFA) we calculated the critical temperature in the two-layer Ising model. The obtained results were, in our opinion, very accurate and we were able to estimate the shift exponent φ. In the case of equal intra-layer couplings (J1=J2) our result φ=1.79 supports the scaling-theories 4, 5prediction that φ=γ. However, for the unequal intra-layer couplings our result clearly contradicts a prediction based on other [6]scaling arguments that φ=γ/2.
Acknowledgements
The research described in the present paper was initiated when the author stayed in the Graduate School of Information Sciences in Tohoku University. I would like to thank Prof. T. Horiguchi for interesting discussions and his warm hospitality. This research is supported by the research grant KBN 2 P302 091 07.
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