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Physical Review

Diagrammatic Expansion for the Ising Model with Arbitrary Spin and Range of Interaction

Gerald Horwitz, Herbert B. Callen

Published Dec 15, 1961• 174 citations
Publisher

Abstract

A new and simple diagrammatic expansion is developed for the free energy of the Ising model with arbitrary spin and range of interaction. The contribution of each diagram is a product of (1) the reciprocal of the order of the symmetry group of the diagram, (2) a product of semi-invariants, with a factor ${M}_{n}$ for each spin in the diagram $n$ being the number of bonds joined to that spin) and (3) a sum of products $\ensuremath{\Sigma}{i,j,k,\ensuremath{\cdots}}^{}(2\ensuremath{\beta}{J}_{\mathrm{ij}})(2\ensuremath{\beta}{J}_{\mathrm{kl}})\ensuremath{\cdots}$, where $2\ensuremath{\beta}{J}_{\mathrm{ij}}$ corresponds to a bond between spins $i$ and $j$, and where the summation is carried out with no restrictions on the summation indices (i.e., no "excluded volume" corrections). The expansion variables are the spin deviation operators ${\ensuremath{\sigma}}_{i}\ensuremath{\equiv}\overline{S}\ensuremath{-}{S}_{\mathrm{iz}}$. The quantity $\overline{S}$ is chosen to eliminate a large class of diagrams; this choice also minimizes the corresponding free-energy contribution and implies $\overline{S}=〈{S}_{\mathrm{iz}}〉$. By further renormalization of the vertices all reducible (i.e., simply articulated) diagrams are eliminated. To zero order the molecular field solution is obtained. The next approximation consists of the summation of renormalized simple loop diagrams. The justification of this choice of diagrams rests, at low temperature, on the decrease of the value of the higher order semi-invariants, on the relationship of these loop diagrams to the random phase approximation, and on the agreement with the rigorous low-temperature series result. At the Curie temperature the same choice of diagrams is justified by a modification of Brout's $\frac{1}{z}$ criterion, so that the expansion can be viewed as an expansion in $\frac{1}{z}$, where $z$ is the effective number of spins interacting with a given spin. Finally the choice of loop diagrams is justified at high temperature by exact agreement to second order in $\frac{{T}_{c}}{T}$ (and by a very close agreement to fourth order) with the rigorous high-temperature series expansion. Thus the theory agrees with rigorous results in the low-temperature and paramagnetic regions and has a well-defined criterion of validity in the intermediate temperature region. For nearest-neighbor interactions and spin \textonehalf{} the specific heat is continuous through the Curie temperature, with an infinite slope on the low-temperature side of the transition.

DOI
10.1103/physrev.124.1757

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•Jan 1, 2010•0 citations
A new and simple diagrammatic expansion is developed for the free energy of the Ising model with arbitrary spin and range of interaction. The contribution of each diagram is a product of (1) the reciprocal of the order of the symmetry group of the diagram, (2) a product of semi-invariants, with a factor Mn{M}_{n}Mn​ for each spin in the diagram nnn being the number of bonds joined to that spin) and (3) a sum of products \ensuremathΣi,j,k,\ensuremath⋯(2\ensuremathβJij)(2\ensuremathβJkl)\ensuremath⋯\ensuremath{\Sigma}{i,j,k,\ensuremath{\cdots}}^{}(2\ensuremath{\beta}{J}_{\mathrm{ij}})(2\ensuremath{\beta}{J}_{\mathrm{kl}})\ensuremath{\cdots}\ensuremathΣi,j,k,\ensuremath⋯(2\ensuremathβJij​)(2\ensuremathβJkl​)\ensuremath⋯, where 2\ensuremathβJij2\ensuremath{\beta}{J}_{\mathrm{ij}}2\ensuremathβJij​ corresponds to a bond between spins iii and jjj, and where the summation is carried out with no restrictions on the summation indices (i.e., no "excluded volume" corrections). The expansion variables are the spin deviation operators \ensuremathσi\ensuremath≡S‾\ensuremath−Siz{\ensuremath{\sigma}}_{i}\ensuremath{\equiv}\overline{S}\ensuremath{-}{S}_{\mathrm{iz}}\ensuremathσi​\ensuremath≡S\ensuremath−Siz​. The quantity S‾\overline{S}S is chosen to eliminate a large class of diagrams; this choice also minimizes the corresponding free-energy contribution and implies S‾=〈Siz〉\overline{S}=〈{S}_{\mathrm{iz}}〉S=〈Siz​〉. By further renormalization of the vertices all reducible (i.e., simply articulated) diagrams are eliminated. To zero order the molecular field solution is obtained. The next approximation consists of the summation of renormalized simple loop diagrams. The justification of this choice of diagrams rests, at low temperature, on the decrease of the value of the higher order semi-invariants, on the relationship of these loop diagrams to the random phase approximation, and on the agreement with the rigorous low-temperature series result. At the Curie temperature the same choice of diagrams is justified by a modification of Brout's 1z\frac{1}{z}z1​ criterion, so that the expansion can be viewed as an expansion in 1z\frac{1}{z}z1​, where zzz is the effective number of spins interacting with a given spin. Finally the choice of loop diagrams is justified at high temperature by exact agreement to second order in TcT\frac{{T}_{c}}{T}TTc​​ (and by a very close agreement to fourth order) with the rigorous high-temperature series expansion. Thus the theory agrees with rigorous results in the low-temperature and paramagnetic regions and has a well-defined criterion of validity in the intermediate temperature region. For nearest-neighbor interactions and spin \textonehalf{} the specific heat is continuous through the Curie temperature, with an infinite slope on the low-temperature side of the transition.
The partition function (ZZZ) for the classical Ising model of cooperative phenomena in lattice of arbitrary dimensionality and with interactions of unspecified range is expressed as a vacuum-state expectation value of a product of two operators, each constructed from boson annihilation and creation operators. In the absence of external fields, ZZZ, and similarly the spin-pair correlation function $\ensuremath{\psi}(\mathrm{r})$, are expanded thereupon into a series of Feynman diagrams. In the case of $\ensuremath{\psi}(\mathrm{r})$, a formally exact diagram summation (1) shows how the spherical model may be recovered in low-order approximation, (2) suggests a way of introducing systematic corrections to this approximation, and (3) leads to a generalized criterion for suppression of antiferro-magnetic order in "nonfitting" lattices. A full topological-diagram reduction to restricted sets of "elementary" subdiagrams is carried out.
We consider a “generalized Heisenberg model”, with a Hamiltonian given by ℋ︁(D) = –JΣij S(D)j, where the quantities Si are isotropically-interacting D-dimensional classical spins and the summation is restricted to nearest-neighbor pairs of sites <i,j>. Thus when D = 1, 2, and 3, this model reduces to the S = 1/2 Ising, plane rotator, and classical Heisenberg models which are useful for describing phase transitions in, respectively, liquid-gas systems, Bose fluids, and magnetic materials. Unfortunately this Hamiltonian is exactly soluble only in the limit D → ∞, in which case the solution corresponds to the spherical model of Berlin and Kac. For finite D, we must consider approximation procedures in order to obtain any useful information. In this paper we develop a diagrammatic representation, valid for arbitrary D, for successive terms in the high-temperature expansions of the free energy and the spin-spin correlation function—from which we can obtain series expansions for all the thermodynamic properties of the system.
First, a new method proposed by Baker and Rushbrooke is used to study the simple ferromagnetic Ising model at and below the Curie temperature. Of course, the properties of the Ising model are already well known, so that the main aim here is to assess the potential and reliability of the new method, since it has wide applicability to other models which have not been otherwise studied. Between 8 and 16 coefficients of exact high-temperature expansions for fixed values of the magnetization are derived for various two- and three-dimensional lattices. A Pad\'e-approximant analysis of these expansions at the critical isotherm and magnetic phase boundary enables us to estimate the critical exponents $\ensuremath{\beta}$, ${\ensuremath{\gamma}}^{\ensuremath{'}}$, and $\ensuremath{\delta}$, and plot the spontaneous magnetization. The results are in good agreement with previous calculations. Secondly, an analysis of the exact series expansions provides no support for the conjecture that the phase boundary is a line of essential singularities. However, the same expansions strongly suggest the existence of a "spinodal" curve, whose properties are in reasonable agreement with the predictions of various heuristic arguments (based essentially upon analyticity at the phase boundary and one-phase homogeneity in the critical region). Finally, structure and a mild extension of the proven analyticity of the free energy are used to show the $\ensuremath{\Delta}\ensuremath{\le}{\ensuremath{\Delta}}^{\ensuremath{'}}$, $\ensuremath{\gamma}\ensuremath{\le}{\ensuremath{\gamma}}^{\ensuremath{'}}$.
The partition function of the Ising model of ferromagnetism is examined in the limit of high density in the anticipation that in the limit of infinite density one recovers the Weiss molecular field. The formal parameter of expansion is 1z\frac{1}{z}z1​ where zzz is the number of spins in the range of the exchange potential (not restricted to nearest neighbor interactions). In the absence of long-range order, only ring diagrams in the cluster expansion contribute. These give a divergence in the specific heat at $k{T}_{c}=\ensuremath{\Sigma}{j\ensuremath{\ne}i}^{}{v}_{\mathrm{ij}}$ where vij{v}_{\mathrm{ij}}vij​ is the exchange potential. This is the molecular field value for the Curie point Tc{T}_{c}Tc​. In the presence of a magnetic field the partition function is evaluated for fixed magnetic moment MMM in the same approximation, MMM being determined by minimization. This results in a susceptibility differing from the molecular field theory and hence an inconsistency in the theory.The inconsistency is traced back to the observation that the acceptance of ring diagrams is equivalent to the gaussian model of Kac and Berlin which violates the sum rule ${{\ensuremath{\Sigma}}_{i=1}}^{N}{{\ensuremath{\mu}}_{i}}^{2}=N$. Here ${\ensuremath{\mu}}_{i}$ is the "spin" per particle and NNN is the total number of particles. This condition is reinstated by insuring the sum rule. The result leads to the spherical model. Thus, a consistent high density approximation to the Ising model is the spherical model. Below the Curie point or for fixed magnetic field, MMM is again held fixed and only the Fourier components of the spin density with nonvanishing wave vector are "sphericalized." The result leads to a physically acceptable model which becomes the molecular field theory at low temperatures or high fields and deviates in O(1z)O(\frac{1}{z})O(z1​) in general. Formally, the results are simply expressed in terms of a temperature dependent Weiss field. These results differ from the ordinary spherical model which is physically unacceptable below the Curie point. However, a molecular field modification of the spherical model due to M. Lax yields the same result when properly interpreted.It is shown that the above results are also valid (to the same approximation) in the quantum mechanical Heisenberg model, for temperatures above the Curie point.
A study is made of an Ising model on the hexagonal closed-packed lattice, with ferromagnetic interactions D between nearest-neighbor spins located in adjacent layers, and antiferromagnetic interactions J between nearest-neighbor spins located in the same layer. The ground states of the model are studied for different values of the parameter κ = —J/D. For κ < 1/2 the ground states are ferromagnetic and for κ > 1/2 the ground state spin configurations consist of stacked identical layers, such that each layer is obtained by stacking rows of alternating spins. At the point (κ — 1/2,t = 0), where t = T/D, there exists a multitude of degenerate ground state spin configurations which are not stable for κ ≠ 1/2. Mean-field theory and low temperature expansions are used to study the phase diagram at low temperatures. Mean-field theory predicts that (κ= 1/2, t = 0) is a multiphase point where an infinite sequence of modulated phases coincide. In the vicinity of the multiphase point, the mean-field phase diagram is found to be similar to the mean-field phase diagram of the three-dimensional ANNNI model near its multiphase point. Low temperature expansions are performed to second order in x, where x = e [sup -2/t], around the phase boundary between the ferromagnetic and the modulated phases. In contrast to standard low temperature expansions, the complete contribution, to order x², is obtained by grouping the contributions from excitations which contribute to arbitrarily high orders in x. The phase boundary between the ferromagnetic and the modulated phases is found to coincide, to order x², with the line onto which Domany mapped a kinetic Ising model on the honeycomb lattice. This strongly suggests that the Domany line is a phase boundary in three-dimensions. Mean-field theory shows that this Ising model contains a continuous minimum-energy surface. A renormalization group method which applies to models which contain continuous minimum-energy surfaces is used to analyze the phase transition between the paramagnetic and the modulated phases. The calculation is performed using a Landau-Ginzburg-Wilson Hamiltonian whose minimum-energy surface consists of a hexagon and which contains fourth-order invariants due to the lattice. The calculation shows that the Hamiltonian does not contain a stable fixed point. This suggests that the paramagnetic-modulated phase transition of this Ising model is a fluctuation-induced first-order transition.