The ferromagnetic transition in the Ising model is the paradigmatic example of ergodicity breaking accompanied by symmetry breaking. It is routinely assumed that the thermodynamic limit is taken with free or periodic boundary conditions. More exotic symmetry-preserving boundary conditions, like cylindrical antiperiodic, are less frequently used for special tasks, such as the study of phase coexistence or the roughening of an interface. Here we show, instead, that when the thermodynamic limit is taken with these boundary conditions, a novel type of transition takes place below T_{c} (the usual Ising transition temperature) without breaking either ergodicity or symmetry. Then the low-temperature phase is characterized by a regime (condensation) of strong magnetization's fluctuations which replaces the usual ferromagnetic ordering. This is due to critical correlations perduring for all T below T_{c}. The argument is developed exactly in the d=1 case and numerically in the d=2 case.
The ferromagnetic transition in the Ising model is the paradigmatic example of ergodicity breaking accompanied by symmetry breaking. It is routinely assumed that the thermodynamic limit is taken with free or periodic boundary conditions. More exotic symmetry-preserving boundary conditions, like cylindrical antiperiodic, are less frequently used for special tasks, such as the study of phase coexistence or the roughening of an interface. Here we show, instead, that when the thermodynamic limit is taken with these boundary conditions, a novel type of transition takes place below T_{c} (the usual Ising transition temperature) without breaking either ergodicity or symmetry. Then the low-temperature phase is characterized by a regime (condensation) of strong magnetization's fluctuations which replaces the usual ferromagnetic ordering. This is due to critical correlations perduring for all T below T_{c}. The argument is developed exactly in the d=1 case and numerically in the d=2 case.
The infinite range Ising model is usually investigated in the weak coupling limit. Here the model is solved with ferromagnetic coupling at fixed and finite strength. Exact analytical expressions are found for the thermodynamic potentials as functions of enthalpy and external field. These results differ from the potentials for the weak coupling limit. The model shows a temperature driven first order phase transition from a paramagnetic phase at high temperatures into a low temperature phase from which thermal fluctuations are absent.
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.
We show that the spontaneous symmetry breaking can be defined also for finite systems based on the properly defined jump probability between the ground states in the 2d and 3d Ising models on a square and a cubic lattice respectively. Our analysis reveals the existence of an interval in the temperature (control parameter) space within which the spontaneous symmetry breaking takes place. The upper limit of this region is the pseudocritical point where the symmetric vacuum bifurcates in energetically degenerate non-symmetric vacua, initiating the spontaneous symmetry breaking process. The lower limit, identified as the temperature value at which the spontaneous symmetry breaking is completed, is characterized by maximal characteristic time for the decay of magnetization (order parameter) auto-correlations. We argue that this anomalous enhancement of auto-correlations is attributed to the transition from type I to on-off intermittency in the order parameter dynamics. Possible phenomenological implications of this behaviour are briefly discussed.
A review is given on some recent developments in the theory of the Ising model in a random field. This model is a good representation of a large number of impure materials. After a short repetition of earlier arguments, which prove the absence of ferromagnetic order in d≤2 space dimensions for uncorrelated random fields, we consider different random field correlations and in particular the generation of uncorrelated from anti-correlated random fields by thermal fluctuations. In discussing the phase transition, we consider the transition to be characterized by a divergent correlation length and compare the critical exponents obtained from various methods (real space RNG, Monte Carlo calculations, weighted mean field theory etc.). The ferromagnetic transition is believed to be preceded by a spin glass transition which manifests itself by replica symmetry breaking. In the discussion of dynamical properties, we concentrate mainly on the zero temperature depinning transition of a domain wall, which represents a critical point far from equilibrium with new scaling relations and critical exponents.
Phase transitions are defined, and the concepts of order parameter and spontaneously broken symmetry are discussed. Simple models for magnetic phase transitions are introduced, together with some experimental examples. Critical exponents and the notion of universality are defined, and the consequences of the scaling assumptions are derived. Phase transitions and order parameters It is a fact of everyday experience that matter in thermodynamic equilibrium exists in different macroscopic phases. Indeed, it is difficult to imagine life on Earth without all three phases of water. A typical sample of matter, for example, has the temperature–pressure phase diagram presented in Fig. 1.1: by changing either of the two parameters the system may be brought into a solid, liquid, or gas phase. The change of phase may be gradual or abrupt. In the latter case, the phase transition takes place at well defined values of the parameters that determine the phase boundary. Phase transitions are defined as points in the parameter space where the thermodynamic potential becomes non-analytic. Such a non-analyticity can arise only in the thermodynamic limit, when the size of the system is assumed to be infinite. In a finite system the partition function of any system is a finite sum of analytic functions of its parameters, and is therefore always analytic. A sharp phase transition is thus a mathematical idealization, albeit one that describes the reality extremely well. Macroscopic systems typically contain ∼ 10 23 degrees of freedom, and as such are very close to being in the thermodynamic limit. The phase boundaries in Fig. 1.1, for example, for this reason represent reproducible physical quantities.
Phase transitions in Ising models with tree surfaces are studied from various points of view, including a phenomenological Landau theory, high-temperature series expansions, and a scaling theory for thermodynamic quantities and correlation functions. In the presence of a surface a number of new critical exponents must be defined. These arise because of the existence of "surface" terms in the thermodynamic functions, and because of the anisotropy of space and lack of translational symmetry introduced by the surface. The need for these new critical exponents already appears in the phenomenological theory, which is discussed in detail and related to the microscopic mean-field approximation. The essential new parameter appearing in this theory is an extrapolation length $\ensuremath{\lambda}$ which enters the boundary condition on the magnetization at the surface. For magnetic systems this length is of the order of the interaction range, in contrast to superconductors, where it is usually much larger. In order to go beyond the mean-field theory, high-temperature series expansions are carried out for the Ising half-space, to tenth order in two dimensions and to eighth order in three dimensions. A scaling theory is developed both for thermodynamic functions and for spin correlations near the surface, and relations are found among the exponents of the half-space. Both the scaling theory and the numerical calculations are compared with the exact solution of the Ising half-plane (two dimensions) by McCoy and Wu, and agreement is found wherever the theory is applicable. In analogy to the bulk situation, the scaling theory is found to agree with mean-field theory in four dimensions. The prediction of the present work which is most easily accessible to experiment is the temperature dependence of the magnetization at the surface, with critical exponent estimated to be ${\ensuremath{\beta}}_{1}=\frac{2}{3}$. The mean-field result, ${\ensuremath{\beta}}_{1}=1$, seems to agree more closely with presently available experiment, and more work is needed to clarify the situation.