Ising model

A model [a1] defined by the following Hamiltonian (cf. Hamilton function) (i.e. energy functional of variables; in this case the "spins" on the sites of a regular lattice in a space of dimension )

(a1)

Here, are "exchange constants" , is a (normalized) magnetic field, involving an interpretation of the model to describe magnetic ordering in solids ( is "magnetization" , the Zeeman energy in (a1) is the energy gained due to application of the field).

Since its solution for in 1925 [a1], the model became a "fruitfly" for the development of both concepts and techniques in statistical thermodynamics. It appears also in other interpretations in lattice statistics: defining occupation variables , where lattice site is empty () if or occupied () if . This is the lattice gas model of a fluid. One can also interpret the cases as two chemical species , for describing ordering or unmixing of binary alloys , etc.

Statistical thermodynamics [a2] aims to compute average properties of systems with a large number of degrees of freedom (i.e., in the thermodynamic limit ). These averages at a temperature are obtained from the free energy (per spin) or the partition function ,

(a2)

Here, is the Boltzmann constant [a2], and the trace operation stands for a sum over all the states in the phase space of the system (which here is the set of states ). Magnetization per spin , susceptibility , entropy , etc. are then found as partial derivatives of [a2]:

(a3)

where stands for a canonical average of a quantity :

The Ising model is important since for it exhibits phase transitions. In the simplest case, if sites , are nearest neighbours on the lattice and zero elsewhere, a transition occurs for from a paramagnet () to a ferromagnet () at a critical temperature . In the disordered paramagnet , while in the ordered ferromagnet the spontaneous magnetization occurs:

(a4)

This is an example of spontaneous symmetry breaking: for does not single out a sign of (replacing all by leaves invariant). However, for and the equilibrium state of the system is two-fold degenerate (). This degeneracy is already obvious from the groundstate of (a1), for , found from the absolute minimum of as a functional of the : for this minimum occurs for either all or all .

Interestingly, for no such phase transition at occurs; rather [a1]. The problem (a1)–(a3) is solved exactly by transfer matrix methods [a3]. Rewriting (a1) as with the periodic boundary condition , one finds

(a5)

The -matrix is defined as

(a6)

Now is simply the trace of an -fold matrix product,

(a7)

where the property was used that the trace of a symmetric matrix is independent of the representation, and so one can evaluate the trace by first diagonalizing ,

(a8)

where the eigenvalues , are found from the vanishing of the determinant, , being the unit -matrix:

(a9)

In the limit the largest eigenvalue dominates, , and hence

(a10)

(a11)

Indeed, for there is no spontaneous magnetization, and for the susceptibility becomes .

It is remarkable that (a11) strongly contradicts the popular molecular field approximation (MFA). In the molecular field approximation one replaces in the interaction of every spin with its neighbours, , the spins by their averages, ; , the problem becomes a single-site Hamiltonian where is exposed to an effective field , which needs to be calculated self-consistently; carrying out the average over the two states one finds

(a12)

which yields and with a critical exponent , and a Curie–Weiss law for . Thus, the Ising model shows that the molecular field approximation in this case yields unreliable and misleading results!

For the Ising-model in , exact transfer matrix methods are applicable for ; they show that a phase transition at does exist [a3], [a4], [a5]. But the critical exponents , differ very much from their molecular field approximation values; namely, and . This is important, since the exponents and also follow from the Landau theory of phase transitions [a2], which only requires that can be expanded in a power series in , with the coefficient at the term changing sign at as , which are plausible assumptions on many grounds. The Ising model testifies that neither molecular field approximation nor Landau theory are correct. The Ising model then prompted the development of entirely new theoretical concepts, namely renormalization group analysis [a6], by which one can understand how non-mean-field critical behaviour arises. The Ising model also became a very useful testing ground for many numerical methods: e.g. systematic expansions of at low (in the variable ) or at high in the variable [a7], or Monte-Carlo methods [a8]. It also played a pivotal role for the concepts on surface effects on phase transitions, and for phase coexistence (domains of oppositely oriented magnetization, separated by walls). Such problems were described with a mathematical rigor that is seldomly found in the statistical thermodynamics of many-body systems. Rigorous work includes the existence of a spontaneous magnetization for ( "Peierls proof" ), inequalities between spin correlations, theorems on the zeros of the partition function, etc.; see [a9]. The Ising model is the yardstick against which each new approach is measured.

Finally, there are extensions of the Ising model. One direction is to make the more complicated rather than uniformly ferromagnetic (). E.g., if in one lattice direction between nearest neighbours but between next nearest neighbours, the resulting anisotropic next nearest neighbour Ising model (ANNNI model) is famous [a10] for its phase diagram with infinitely many phases and transitions; choosing the at random from a prescribed distribution, the resulting Ising spin glass [a11] is a prototype model of glasses and other disordered solids.

Another extension adds "time t" as a variable: by a transition probability per unit time one is led to a master equation for the probability that a state occurs at time . Such kinetic Ising models [a12] are most valuable to test concepts of non-equilibrium statistical mechanics, and provide the basis for simulations of unmixing in alloys ( "spinodal decomposition" ), etc. Finally, one can generalize the Ising model by replacing the spin by a more complex variable, e.g. in the Potts model [a13] each site may be in one of states where is integer (also, the limit is of interest; the so-called "percolation problem" [a14]). The techniques for the Ising model (transfer matrix, series expansions, renormalization, Monte Carlo, etc.) are valuable for all these related problems, too.

References

[a1]	E. Ising, "Beitrag zur Theorie des Ferromagnetismus" Z. Phys. , 31 (1925) pp. 253–258
[a2]	L.D. Landau, E.M. Lifshitz, "Statistical physics" , Pergamon (1958)
[a3]	R.J. Baxter, "Exactly solved models in statistical mechanics" , Acad. Press (1982)
[a4]	L. Onsager, "Crystal statistics I. A two-dimensional model with an order-disorder transition" Phys. Rev. , 65 (1944) pp. 117–149
[a5]	B.M. McCoy, T.T. Wu, "The two-dimensional Ising model" , Harvard Univ. Press (1973)
[a6]	M.E. Fisher, "The renormalization group in the theory of critical behavior" Rev. Mod. Phys. , 46 (1974) pp. 597–616
[a7]	"Phase Transitions and Critical Phenomena" C. Domb (ed.) M.S. Green (ed.) , 3 , Acad. Press (1974)
[a8]	"Monte Carlo methods in statistical physics" K. Binder (ed.) , Springer (1979)
[a9]	R.B. Griffiths, "Rigorous results and theorems" C. Domb (ed.) M.S. Green (ed.) , Phase Transitions and Critical Phenomena , 1 , Acad. Press (1972) pp. 7–109
[a10]	W. Selke, "The Annni model-theoretical analysis and experimental application" Phys. Rep. , 170 (1988) pp. 213–264
[a11]	K. Binder, A.P. Young, "Spin glasses: experimental facts, theoretical concepts, and open questions" Rev. Mod. Phys. , 58 (1986) pp. 801–976
[a12]	K. Kawasaki, "Kinetics of Ising models" C. Domb (ed.) M.S. Green (ed.) , Phase Transitions and Critical Phenomena , 2 , Acad. Press (1972) pp. 443–501
[a13]	F.Y. Wu, "The Potts model" Rev. Mod. Phys. , 54 (1982) pp. 235–268
[a14]	D. Stauffer, A. Aharony, "Introduction to percolation theory" , Taylor&Francis (1992)