256 12, Phase Transitions and Critical Phenomena:Classical Theories (d) What is the asymptotic expression of the curve of coexistence of phases in the immediate vicinity of the critical point? 13 (e) Use your results to obtain the critical exponents ß, ì, 8, and a. The Ising Model 5. Consider the Curie-Weiss equation for ferromagnetism, m = tanh(ßH + ßÀm) , Obtain an asymptotic expression for the isothermal susceptibility, X (T, H), at T = Tc for H -- O. Obtain asymptotic expressions for the spontaneous magnetization for T .:.: Tc (that is, for T -- 0) and T ;: Tc (that is, for t -- 0-). Most of the experiments in the neighborhood of critical points indicate that critical exponents assume the same universal values, far from the predic- tions of the "classical theories" (as represented by Landau's phenomenol- ogy, for example). We now recognize that the universal values of the critical exponents depend on a just few ingredients: (i) The dimension of physical systems. Usual three-dimensional systems are associated with a certain class of critical exponents. There are experimental realizations of two-dimensional systems, whose critical behavior is characterized by another class of distinct and well-defined critical exponents. .~~ I~J' ,:D '.~:1.__1 (ii) The dimension of the order parameter. For simple fluids and uniaxial ferromagnets, the order parameter is a scalar number. For an isotropic ,:'::::; i:r. ferromagnet, the critical parameter is a three-dimensional vector. !O ,10 I;: "~,.' (iii) The range of the microscopic interactions. For most systems of phys- ,~¡- ical interest, the microscopic interactions are of short range. We wil ;i"~- ',(/) see that statistical systems with long-range microscopic interactions cr.: .~ lead to the set of classical critical exponents. ~!-i i:~:~ r~J'" ::) Owing to the universal behavior of critical exponents, it is enough to ana- Lr,~¡ lyze very simple (but nontrivial) models in order to construct a microscopic I.,,~ ~~:: theory of the critical behavior. The Ising model, including short-range in- J..,- teractions between spin variables on the sites of a d-dimensional lattice, 1~1') plays the role of a prototypical system. The Ising spin Hamiltonian is given '''''')ij'~r.:..". I J:',." 258 13, The Ising Model 13. The Ising Model 259 by In one dimension, it is relatively easy to obtain an expression for this free N energy. We wil use the technique of the transfer matrices, which can also be written in higher dimensions, to obtain a solution for the Ising chain. 1í = -f2':aiaj - HLai, (13.1) (ij) i=l However, as shown by Ising in 1925, tils one-dimensional solution is quite deceptive, since the free energy is an analytic function of T and H (ex- where ai is a random variable assuming the values :f1 on the sites i = cept at the trivial point T = H = 0), which precludes the existence of a 1,2, ..., N of ad-dimensional hypercubic lattice. The first term, where the spontaneous magnetization (and of any phase transition). sum is over pairs of nearest-neighbor sites, represents the interaction ener- Several approximate techniques have been developed to solve the Ising gies introduced to bring about ~n ordered ferromagnetic state (if J ~ 0). model in two and three dimensions. Some of them are quite simple and The second term, involving the interaction between the applied field Hand useful, and may lead to reasonable qualitative results for the phase dia- the spin system, is of a purely paramagnetic character (as we have already grams (besides providing useful tools to investigate more complex model seen in previous chapters of this book). Since it was proposed by Lenz and systems). However, as pointed out before, phase transitions are associated solved in one dimension by Ernst Ising in 1925, the Ising model has gone with a nonanalytic behavior of the free energy in the thermodynamic limit. through a long history ¡see, for example, the paper by S. G. Brush, in Rev. As a consequence, we should be warned against any truncations or pertur- Mod. Phys. 39, 883 (1967)J. bative expansions around the critical point. Indeed, most of the approxi- The Ising model can represent the main features of distinct physical mate schemes can be written as a Landau expansion, leading to classical systems. In the usual magnetic interpretation, the Ising spin variables are critical exponents. taken as spin components (that may be pointing either up or down, along the direction of the applied field) of crystalline magnetic ions. We may also In a mathematical "tour de force," Lars Onsager, in 1944, obtained an consider a binary alloy of type AB. In this case, the spin variables indicate analytical solution for the Ising model on a square lattice, with nearest- whether a certain site on the crystalline lattice is occupied by an atom of neighbor interactions, in the absence of an external field. For T -7 Te, the either type A or type B (neighbors of the same type contribute with an specific heat diverges according to a logarithmic asymptotic form, energy -J; neighbors of different types, contribute with +J). As another CH=O r- In IT - Tel, (13.4) example, take the :f1 spin variables to indicate either the presence (+ 1) or the absence (-1) of a molecule in a certain cell of a "lattice gas" (which is a with a well-defined critical temperature, kBTefJ = 21 In (1 + J2). There- useful model for the critical behavior of a fluid system). This multiplicity of fore, the free energy is not analytic at Te, and cannot be written as a interpretations is compatible with the ability of the Ising model to represent Landau expansion. The Onsager solution has been reproduced and con- the main features of the critical behavior of many different physical systems. firmed by different techniques on many planar lattices (with first-neighbor From the point of view of magnetism, the Ising Hamiltonian may be interactions). It represents a true milestone in the development of the mod- regarded as a kind of approximation for the Heisenberg Hamiltonian, asso- ern theories of critical phenomena, By the first time, it was shown that a ciated with a highly anisotropic spin-1/2 magnetic insulator. The energy J microscopic model leads to nonanalytic behavior within the framework of is interpreted as the quantum exchange parameter of electrostatic origin. :~i:. equilibrium statistical mechanics. The origins of this nonanalyticity were ;:i~o. In this chapter, we take advantage of the more intuitive language of this 1.,.~.J later explained, under much more general grounds, by the Yang and Lee ,!i.-- i:i.i: magnetic analogy to derive some properties of the Ising modeL. theory of phase transitions (see Chapter 7), including the remarkable "circle ¡;\::~ '(:i In order to solve the Ising problem, we have to obtain the canonical theorem" about the zeros of the partition function in the thermodynamic !iie) partition function 1:1(' , limit. In the 1950s, C. N. Yang checked a result of Onsager for the spon- 1:1... taneous magnetization of the Ising ferromagnet on the square lattice to !,t-:.~-- ZN=Z(T,H,N)= Lexp(-ß1í), (13.2) obtain the exponent ß = 1/8, in sharp contrast with the classical value. iië7-5 fUiì ,;c:r Nowadays, although there are no exact solutions in a field, we may be sure ¡iloLl where the sum is over all configurations of spin variables, and the Hamil- that, = 714 in two dimensions. All planar lattices, with short-range in- '~I:t~ it~il" tonian is given by equation (13.1). From tils partition function, we have teractions, lead to the same set of critical exponents (a = 0, ß = 1/8, :~:) the magnetic free energy per site, , = 7/4), which are far from the experimental values for three-dimensional ¡Lr..i systems, and far as well from the classical Landau results. 1.,.- ~r' The solution of the Ising model in three dimensions remains an open (and ¡~,.~:: 9 = 9 (T, H) =N-.oo lim (-.ß1 N inZN). (13.3) P') probably impossible) problem. However, we can use an argument due to F". i r) 260 13. The Ising Model 13.1 Exact solution in one dimension 261 Peierls to prove the existence of spontaneous magnetization at suffciently transfer matrix, low temperatures. Also, since the 1960s there have been many efforts to obtain quite long series expansions (at high as well as low temperatures) T ( +, -) ) _ ( exp (K + L) exp ( - K) ) for several thermodynamic quantities associated with the three-dimensional T= (T(+,+)T(-,+) T(-,-) - exp(-K) exp(K-L) , Ising modeL. From refined asymptotic analyses of these series, we obtain a (13.9) range of values for the critical exponents in agreement with experimental measurements (ß ~ 5/16, 'Y ~ 5/4, æ ~ 1/8). Also, more recent, and and use the matrix formalism to see that equation (13.7) for the canonical much more sophisticated, renormalization-group techniques lead to similar partition function is a trace of a product of N identical transfer matrices, results, In the table below, we give the values of some usual thermodynamic critical exponents. ZN = Tr(T)N. (13.10) Landau Ising (d = 2) Ising (d = 3) Experiments Furthermore, the transfer matrix (13.9) is symmetric, and can thus be diagonalized by a unitary transformation, ß 1/2 1/8 ~ 5/16 0.3 - 0.35 'Y 1 7/4 ~ 5/4 1.2 - 1.4 UTU-1 = D, with U-1 = ut, (13.11) 8 3 15 ~ 5 4,2 - 4.8 æ 0 o (log) ~ 1/8 ~O where D is a diagonal matrix. Therefore, the canonical partition function can be written in terms of the eigenvalues of the transfer matrix, 13.1 Exact solution in one dimension ZN = Tr (U-1DUt = Tr (D)N = Àt + À!J, (13.12) In one dimension (d = 1), the Ising Hamiltonian is written as where N N À1,2 = eK cosh L:l (e2K cosh2 L - 2 sinh (2K)) 1/2 , (13.13) H = -JLO"iO"i+l - HLO"i' i=1 i=1 (13.5) are given by the roots of the secular equation, det (T - ÀI) = O.