We consider the rectangular Ising model on a half-plane of infinite extent and study some of the consequences connected with the presence of the boundary. Only the spins on the boundary row are allowed to interact with a magnetic field
H. The method of Pfaffians is employed to obtain exact expressions for the partition function. It is found that the free energy is the sum of two terms, one of which is independent of
H and proportional to the total number of lattice sites, while the other depends on
H and is proportional to the number of lattice sites on the boundary. This separation makes it possible to define various thermodynamic quantities associated with the boundary. In particular, the boundary magnetization is shown to be discontinuous, in the ferromagnetic case, at zero magnetic field for temperatures below the bulk critical temperature
Tc. This discontinuity, which is the spontaneous boundary magnetization, goes to zero as ${(1\ensuremath{-}\frac{T}{{T}_{c}})}^{\frac{1}{2}}$ as $T\ensuremath{\rightarrow}{T}_{c}\ensuremath{-}$. For
T=Tc, the discontinuity is of course absent, and the boundary magnetization behaves as $\ensuremath{-}\mathfrak{H}\mathrm{ln}\mathfrak{H}$ for small
H. The boundary susceptibility at zero magnetic field in the ferromagnetic case exhibits a logarithmic singularity at
T=Tc, both above and below transition. An interesting feature is that the ferromagnetic boundary magnetization, although discontinuous for
T<Tc, may be analytically continued beyond the point
H=0. We interpret this as a hystersis phenomenon which we study in detail by computing the probability distribution function for the average boundary spin. The correlation function for two spins, both on the boundary row, is also obtained exactly and its asymptotic behavior is given. Finally, we derive an expression for the magnetization in any row and explicitly evaluate it for the second row, i.e., the row next to the boundary.
We investigate low- and high-temperature properties of the ferromagnetic Ising model of infinite-spin. By using the cummulant expansion methods, we obtain low-temperature expansions of the free energy, the specific heat, the spontaneous magnetization, the zero-field susceptibility and so on, and high-temperature expansions of the free energy and the zero-field susceptibility. We compare the results with those obtained by Thompson for the system on the one-dimensional lattice, with those by using Monte Carlo simulations for the systems on the square lattice and on the simple cubic lattice and with those with the aid of the mean-field approximation.
The critical two-dimensional Ising model is studied with four types boundary conditions: free, fixed ferromagnetic, fixed antiferromagnetic, and fixed double antiferromagnetic. Using bond propagation algorithms with surface fields, we obtain the free energy, internal energy, and specific heat numerically on square lattices with a square shape and various combinations of the four types of boundary conditions. The calculations are carried out on the square lattices with size N×N and 30<N<1000. The numerical data are analyzed with finite-size scaling. The bulk, edge, and corner terms are extracted very accurately. The exact results are conjectured for the corner logarithmic term in the free energy, the edge logarithmic term in the internal energy, and the corner logarithmic term in the specific heat. The corner logarithmic terms in the free energy agree with the conformal field theory very well.
We consider a semi-infinite two-dimensional Ising model with nearest-neighbor coupling constants that deviate from the bulk coupling by $A{m}^{\ensuremath{-}y}$ for large
m,
m being the distance from the edge. The case
A<0 of couplings which are weaker near the surface has been discussed by Hilhorst and van Leeuwen. We report exact results for the boundary magnetization and boundary pair-correlation function when
A>0. At the bulk critical temperature there is a rich variety of critical behavior in the $A \ensuremath{-}y$ plane with both paramagnetic and ferromagnetic surface phases. Some of our results can be derived and generalized with simple scaling arguments.
The interface method is used to derive the order-disorder critical temperature of the two-dimensional Ising model on a rectangular lattice with ferromagnetic interactions in the x direction and antiferromagnetic interactions in the y direction. Employing this approximation in two different ways gives different critical temperatures
Tc for nonzero uniform magnetic fields H. However, both of the phase boundaries
Tc(H) reduce to the exact results available at H=0 and T=0. .AE
The properties of a simple fluid or magnet with strong one-axis anisotropy can be studied by means of the Ising model. Such a model, but in an confined geometry with identical boundary fields, is studied along various thermodynamic paths using the density-matrix renormalization group technique. It has been found that also in the presence of the capillary condensation the critical exponent β of the bulk system can be found. Introduction In the thermodynamic limit of the two-dimensional system the correlation length and some other quantities diverge at the critical point (T = Tc, H = 0) [1]. One of them is magnetization that vanishes above Tc in zero bulk field (the paramagnetic regime). Phase coexistence occurs for temperatures T T). It results in a jump of the magnetization m = –∂f/∂H at H = 0, whereas the A. Drzewinski, K. Szota 30 limit value mb = m (T = const, H → 0) is finite and named the spontaneous magnetization. In the other words, when the thermodynamic path for fixed but very small H (the dashed arrow at Fig. 1) is closer and closer to the H = 0 axis, then the magnetization resembles more and more the spontaneous magnetization (full dots in Fig. 2). Fig. 1. Numerically determined coexistence curves for finite systems in the (T, H) phase diagram. The thick solid line indicates the bulk coexistence line. The dashed arrow denotes the thermodynamic path for the infinite systems but others for finite systems Fig. 2. Spontaneous magnetization mb as a function of temperature along the thermodynamic paths presented in the Fig. 1 with a constant shift –10 away from the coexistence lines There is an interesting question. What happens when the system is confined in the presence of the boundary fields. Is it possible to apply the same procedure and, exploiting the magnetizations along the (curvilinear) coexistence lines, reconstruct the behaviour of the spontaneous magnetization? Bulk critical exponents in a presence of the capillary condensation 31 1. The model and the method Our model Hamiltonian for the infinitely long strip has the form: − − ∑ ∑ ∑ ∑ l k, l k, k L k, 1 k k l' , k' l k, σ H + σ h + σ h σ σ J = H ,1 1 where the first sum is over all nearest-neighbour pairs and we have measured the surface field h1 and the bulk field H in units of J. In order to find out the free energy f and the longitudinal correlation length ξ|| we have used the density-matrix renormalization group (DMRG) [3]. It was originally proposed by White as a new tool for diagonalization of quantum spin chains [4] and was later adapted to classical two-dimensional equilibrium statistical mechanics by Nishino [5]. The DMRG allows one to study much larger systems (up to L = 650 in the present paper) than is possible with standard exact diagonalization methods (typically L ~50) and provides data with remarkable accuracy. We estimate that the errors in the plots are smaller than the symbol size.