Magnetism: mathematical aspects

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Vieri Mastropietro and Daniel C. Mattis (2010), Scholarpedia, 5(7):10316. doi:10.4249/scholarpedia.10316 revision #91450 [link to/cite this article]
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Already as an infant, Albert Einstein (Nobel prize winner, 1921) wondered about the physics of magnetic “action at a distance”. His was not the only brilliant mind entranced by magnetic phenomena. Among other Nobelists who have made incidental or even major contributions to our understanding of this field we note (along with the year of their prize) the names of H. Lorentz and P. Zeeman (1902), P. Curie (1903), W. Heisenberg (1932), W. Pauli (1945), F. Bloch (1952), C.N. Yang (1957), H. Bethe (1967), L. Néel (1970) and P.W. Anderson (1977) (Levinovitz & Ringertz, Eds., 2001). The list grows longer if we include the ancillary topics of magnetic resonance, Hall effect, and superconductivity or developments in unrelated fields, such as the concept of the Goldstone mode in high-energy physics inspired by the spin waves of the theory of magnetism. It is not a coïncidence that in "statistical mechanics", which comprises the study of physics at finite temperature, contemporary concerns such as phase transitions and their critical exponents evolved out of the corresponding microscopic properties of magnetic materials at the Curie point. In short, much of the mathematics that was originally developed to unveil the sources of magnetism found subsequent applications in other branches of theoretical physics and – in the case of the Ising model – far afield in cryptology, epidemiology, economics, political science and even sociology!

In this article we show how theories of magnetism are classified according to their internal and external symmetries, spatial dimensionality and various other physical properties. But from the outset, it is important for the reader to understand that we are only seeking to describe the material origins of magnetic phenomena, insofar as they might originate in the many-body comportment of electrons in atoms, molecules and solids. This is quite distinct from studies of the resulting electromagnetic field, whether this last is treated in the classical version due to Maxwell, dating back to the mid-19th Century, or in the quantized (QED) version developed in the middle of the 20th‚ the more so when we turn to antiferromagnets in which the concomitant magnetic fields cancel already at a microscopic level (Mattis, 2006).

What we present below is a small survey of a few idealized models of magnetism culled from an incredibly large array; we discuss the motivation behind them together with the interesting mathematics that arises in the course of solving these many-body problems(Mattis, 2006; Chaps. 3-9).

Contents

Effects of spatial dimensionality and of various symmetries

Setting aside Dirac's hypothetical magnetic monopole (Dirac, 1931) and the fragile current loop of Ampère, the leading physical source (and the unit) of magnetism has to be the permanent magnetic dipole‚ such as the elementary "Bohr magneton" carried by each and every electron (Mattis, 2006; p31). It is sometimes useful to idealize magnetic materials as periodic arrays of unit cells each of which contains one or more atoms or molecules sporting one or more Bohr magnetons.

As example of this periodicity imagine a family of "hypercubic solids" in arbitrary dimensions, ranging from \(d= 0\) (just 1 cell), to \(N\) cells in \(d \ge 1\ ,\) and up to unphysically high dimensions \(d \gg 1\ .\) Each cell has \(2d\) nearest-neighbors and \(N\) is large. The lattices are called the linear chain (\(lc\)) in 1D, the square (or simple quadratic \(sq\)) in 2D, the simple cubic (\(sc\)) in 3D, etc. In the \(lc\ ,\) identical cells lie along a given axis at points \(R_n = an\) with \(n= 1,2,N\) their label and \(a\) the lattice parameter. Thus \(d=1\) applies to an hypothetical, ideal, polymer of great length. Cells of the \(sq\) lattice are located at \(R_{n,m} = a (n,m)\) and those of the \(sc\) lattice are at \(R_{n,m,l} = a(n,m,l)\ ,\) etc.

It is found that the leading term of various correlation functions of particles confined to such lattices, when expanded in powers of \(1/d\ ,\) yield formulas identical to those obtained in the self-consistent "mean-field", aka "molecular field" approximation. "Critical properties", that is, the singular contributions to thermodynamic functions (specific heat, susceptibility, etc.) near a phase transition, can sometimes be extrapolated to 3D from \(d = 4\) using the renormalization group (RG) rigged for \(d = 4- \epsilon \)‚ upon expanding in powers of \(\epsilon \ .\) This has justified the study of leading large \(d\) approximations, even though physical limitations restrict magnetic systems as well as other materials to \(d \le 3\ .\)

To further define a physical model it is necessary to specify the contents of each cell and the interactions among neighboring cells. For many models it is possible to solve the multi-cell model in \(d\) dimensions (or at the very least, to analyze its thermodynamical properties) by a "transfer matrix approach" on a \(d -1\) dimensional lattice. More precisely, the partition function in d dimensions is related to the largest eigenvalue of a transfer operator on a lattice in \(d -1\) dimensions. The eigenvalue problem that needs be solved is usually trivial in \(d=1\) but is, with some exceptions, too cumbersome to carry out for \(d > 2\ .\)

On the other hand and at the opposite extreme, for \(d \ge 4\) or 5 or even greater (\(d \gg 1\)), the statistical mechanics and phase diagram of many models of magnetism become easily solvable and predictable in leading \(1/d\) approximation, just as is the case in quantum field theories.

Clearly, whether \(d\) is large or small is important. Just as single strand polymers differ from 3D solids even when the constituent atoms are identical, the properties of magnetic polymers differ from those of magnetic solids. Once \(d\) is specified, the dynamics of the individual magnetic moments (point-group symmetry) together with the symmetries of their interactions (bonds) are what determine the collective properties in the ground state and at finite \(T\ .\) These symmetries and dynamics fall into various classes or categories.

Typically, once the class or category of the model is given, it is in the three-dimensional world \(d = 3\) in which we live that it is most difficult to find precise or merely reliable mathematical solutions. That is what keeps theoretical physicists in business. As an example let us next examine arguably the simplest model of anisotropic nearest-neighbor interactions among quantized spins on lattices in various dimensions \(d\ ,\) the one named after E. Ising who first studied it in the 1920's (Mattis, 2006; Chap. 8).

The one-dimensional Ising model vs. other possibilities

Ising’s model was based in the “old” quantum theory in which spins, affixed to pounts on a space lattice, can only point “up” or “down.” This discrete algebra is denoted Z(2). In \(d=1\) the Hamiltonian is \(H = J \sum_n S_n S_{n+1}\) (where each \(S_n = \pm 1\)).

The Ising model is better described in terms of Pauli matrices \(\vec S = \frac{\hbar}{2} (\sigma_x, \sigma_y, \sigma_z)\) with \(J\) absorbing the factor \(\left ( \frac{\hbar}{2} \right )^2\ .\) Each spin at \(n\) is assumed to interact with nearest-neighbors at \(n \pm 1\) via a highly anisotropic \(3 X 3\) “exchange” matrix characterizing the nearest-neighbor bond, which then assumes the appearance\[-(\sigma_{x,n} \sigma_{y,n} \sigma_{z,n}). \begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & J \end{pmatrix}. \begin{pmatrix} \sigma_{x,n+1} \\ \sigma_{y,n+1} \\ \sigma_{z,n+1} \end{pmatrix} \ .\] The total interaction with an external or self-consistently generated magnetic field is given by \(-B \sum_n \sigma_{z,n}\) if it is “longitudinal” or \(-B \sum_n \sigma_{x,n}\) if “transverse.” Also \(-(\sigma_{x,n} \sigma_{y,n} \sigma_{z,n}). \begin{pmatrix} J & 0 & 0 \\ 0 & J & 0 \\ 0 & 0 & J \end{pmatrix}. \begin{pmatrix} \sigma_{x,n+1} \\ \sigma_{y,n+1} \\ \sigma_{z,n+1} \end{pmatrix} \) is the Hamiltonian of an individual nearest-neighbor bond in Heisenberg’s model, with \(-B \sum_n \sigma_{z,n}\) the interaction with any orientation external field (as here, by symmetry, there can be no distinction between parallel and transverse.)

Finally, the interaction \(-(\sigma_{x,n} \sigma_{y,n} \sigma_{z,n}). \begin{pmatrix} J & 0 & 0 \\ 0 & J & 0 \\ 0 & 0 & 0 \end{pmatrix}. \begin{pmatrix} \sigma_{x,n+1} \\ \sigma_{y,n+1} \\ \sigma_{z,n+1} \end{pmatrix} \) describes the “X-Y” model in which we again need distinguish “in-plane” from “out-of-plane” external fields. These three models are discussed separately below.

The most general bilinear Hamiltonian for a bond connecting two sites is given by an Hermitean \(3 X 3\) matrix \(J_{\beta}^{\alpha}\ .\) This generalization has 9 independent parameters, with the Ising version in its most anisotropic limit, all the way to the totally isotropic Heisenberg model, \(J_{\beta}^{\alpha} = J \delta_{\beta}^{\alpha}\ ,\) representing isotropic limit, given that it is explicitly invariant under arbitrary spatial rotations. ( \(\delta_{\beta}^{\alpha}=1\) if \(\alpha = \beta\)and 0 otherwise is Kronecker’s delta.) Additionally, quartic forms for two-site interactions and interactions that involve three sites, have sometimes been considered in the literature in connection with various physical applications, but they are not discussed further here.

Ground states, elementary excitations and Tc

The spin \(1/2\) Ising \(lc\) of \(N\) sites connected by \(A-1\) (ferromagnetic) bonds \(J > 0\) was described above. The combined ground state (lowest energy) solution of the Hamilton or Schrödinger equation \(( H = E )\) is \(E_0 = -J(N-1)\) for either of two ground state configurations: all spins “up” (+1) or all “down” (–1). Either configuration manifests perfect long-range order (LRO).

The next higher energy levels are associated with a single “domain wall” defined as follows: the first \(q\) spins are all parallel, say “up,” followed by spins numbered \(q+1,\ldots, N\ ,\) that are all “down.” Because the \(q^{th}\) bond is promoted from energy \(-J\) to energy \(+ J\ ,\) the energetic cost of this defect is \(+2J\ .\) There are \(N-1\) possible values of \(q\ ,\) that is, \(N-1\) distinct positions on which to place the break. A second break can occur at \(q + r\) (\(r \neq 0)\) where \(- q < r < N-q\ ;\) a third one at some third distinct place, etc. Because a bond can only be broken once a sort of “exclusion principle” prevents any two breaks from sharing the same bond. Thus it appears that the domain walls are fermions, unusual only in the sense that each such fermions adds a constant amount \(2J\) to the total energy – regardless how many others are present. The corresponding Boltzmann factor of each is \(e^{- \frac{2J}{k_B T}}\ .\) It follows that the entropy \(\mathcal{S}\) ( \(\mathcal{S}\) = Boltzmann’s constant natural logarithm of the number of allowed configurations) is the sum of \(k_B \log 2\) (recall: initially there were two configurations) and of \((N-1)k_B \log (1+e^{-2J/k_B T})\ .\) The free energy \(F = E_0 - T\mathcal{S}\) becomes:

\[\tag{1} F = E_0 - (N-1)k_B T \log (1+e^{-2J/k_B T})-k_B T \log 2\]


The last term can be neglected in the large \(N\) limit. From this one can infer that at temperature \(T\) the thermodynamic average of the number \(\mathcal{N}\) of domain walls is,

\[\tag{2} \mathcal{N}=(N-1) \times \frac{1}{e^{-2J/k_B T}+1}\]


in which the second factor is the “Fermi-Dirac distribution function” (being the average number of fermions that lives on each of the \(N-1\) bonds in thermal equilibrium at temperature \(T\ .\)) Both \(F\) and \(\mathcal{N}\) are analytic in \(T\) except in the limit \(T \to 0\ ,\) signaling there is an order-disorder thermodynamic phase transition at \(T=0\ .\) The model can be “solved” more formally and the same results obtained more directly following a nonlinear transformation to bond variables. Let \(s_1 \equiv S_1 (=\pm 1), s_2=s_1S_2 ,\ldots , s_n =s_{n-1}S_n\ ,\) … , each \(s_n=\pm 1\) independent of the others. Then \(H =-J \sum_{n-2}^N s_n\ .\) This is just an Hamiltonian of \(N-1\) noninteracting pseudo-spins in a pseudo external magnetic field \(J\ .\) Because \(s_1\) has two values but does not explicitly enter \(H\ ,\) each configuration \(\{ s_1; s_2, s_3 ,\ldots, s_N \}\) is two-fold degenerate. This accounts for \(k_B T \log 2\) in (1).

In \(d \geq 2\ ,\) Ising’s model exhibits a genuine phase transition; the second derivative of \(F\) is discontinuous at a finite temperature \(T\) identified as the “Curie point” \(T_c\ ,\) a quantity proportional to\(J\) that is also a function of \(d\ .\) Table I lists \(T_c\) to three decimal places in terms of \(z \equiv \sharp\) of nearest-neighbor sites on hypercubic lattices.

Table 1: Critical temperature of Ising ferromagnet on hypercubic lattices
Lattice type Coordination number z kTc/zJ
lc (d=1) 2 0
sq (d=2) 4 0.567
sc (d=3) 6 0.752
hypercubic (in d ≥ 4) 2d 1-0.596/d

More on Tc

The critical temperature of zero in \(d=1\) shown in Table I was obtained trivially. The nonvanishing values of \(T_c\) on any of the three standard lattices in \(d=2\) (the \(sq\ ,\) honeycomb and triangular) can also be found exactly using the duality relations of Kramers and Wannier (Kramers & Wannier, 1941). Duality is what relates \(K=J/k_B T\) on a spin lattice to \(K^{*}=J/k_B T^{*}\) on a dual lattice that is constructed on the bonds of said lattice. The \(sq\) lattice is self-dual with coordination number \(z=4\) and the triangular and honeycomb lattices are duals of each other with \(z=6\) and 3 respectively. The duality relations \(\tanh K^{*}=e^{-2K}\) and \(\tanh K=e^{-2K^{*}}\) were originally derived by comparing high temperature series expansions of the partition function with low \(T\) expansions, without needing to actually evaluate either sum. Armed with just this sort of information, Onsager showed that \(T_c(z)\) is given by \(\sinh 2K_c= \tan \frac{\pi}{z}\) for Ising ferromagnets on any one of the 3 principal lattices in 2D.

There is no such formula for the values of \(T_c\) listed in Table I for the Ising ferromagnet in \(d=3\) and \(d \geq 4\) dimensions, so there the values of \(T_c\) have to be obtained numerically.

Symmetry breaking

The exact eigenvalue of the transfer matrix of the Ising model on a \(sq\) lattice, hence the exact evaluation of the partition function and of the free energy in this model, including the particulars of the singular phase transition (both specific heat and magnetic susceptibility diverge at \(T_c\ ,\)) were first obtained by L. Onsager (Onsager, 1944) in the early 1940’s by the use of spinor algebra. A subsequent version based on a more familiar fermion field theory was constructed by T.D. Schultz, D.C. Mattis and E.H. Lieb (Schultz, Mattis & Lieb, 1964) and will be sketched below. But it seems that some properties of these exact solutions generalize to all model ferromagnets in arbitrary dimensions, viz.:

Above \(T_c\) in zero external magnetic field, the “up-down” symmetry is maintained perfectly. When cooling below \(T_c\) this symmetry is spontaneously broken by the onset of LRO. It is noteworthy that the application of an homogeneous external magnetic field (B), which also breaks the symmetry of the ferromagnet, also creates LRO in any dimension \(d\ .\) It follows that a finite real external magnetic field pushes the critical temperature all the way up to \(T_c \to \infty\ .\)

Stated otherwise: regardless of the ground state and of the nature of the low-temperature phase of a magnetic substance(and regardless of spatial dimension \(d = 1, 2, 3, \ldots\ ,\)) in the presence of a finite, real, external field \(B \ne 0\) there will be created LRO at all finite \(T \geq 0\ .\) The only interesting question is, how does this magnetic order behave as function of \(\left\vert B \right\vert\) in the limit \(\left\vert B \right\vert \to 0\ ?\) If it remains finite in that limit, we have spontaneous ferromagnetism; if in that limit it vanishes, the information to be sought is in the magnetic susceptibility, a quantity related to the short-range order.

Transfer matrix in d = 1 Ising model

In Gibbsian statistical mechanics, the partition function \(Z\) is related to the free energy \(F\) by \(Z=e^{-\beta F}=Tr \{ e^{-\beta H} \}\ ,\) hence knowledge of the one yields the other. The temperature is given by \(\beta = 1/k_B T\ .\) The trace (abbreviated\[Tr\]) is defined as the sum over all diagonal elements of the argument, treated as a matrix. Because the number of such terms is exponential in \(N\) this sum cannot be performed efficiently in the limit \(N \to \infty\ ,\) especially near \(T_c\ .\) (Otherwise one could obtain, for example, the energy as a function of temperature \(\langle E \rangle = {\partial (\beta F) \over \partial \beta}\) and other thermodynamic quantities, numerically.) The following calculation, carried out explicitly for the Ising \(lc\ ,\) shows how to get around this difficulty in 1D.

In the 1D model with \(B=0\ ,\) the quantity inside the \(Tr \{ \}\) operation can be written as \(e^{-\beta H}=e^{\beta JS_1S_2} e^{\beta JS_2S_3} \ldots e^{\beta JS_nS_{n+1}}, \ldots\ ,\) upon ordering interactions consecutively. We note that each factor has exactly the same form, \(V = \begin{pmatrix} e^{\beta J} & e^{-\beta J} \\ e^{-\beta J} & e^{\beta J} \end{pmatrix} = e^{\beta J} \mathbf{l} +e^{-\beta J} \delta_x\ .\) (Here \(\mathbf{l}\) is the unit \(2 X 2\) matrix.) It follows that,

\[\tag{3} Tr \{ e^{-\beta H} \} = Tr \{ V \cdot V \cdot \ldots V \} = Tr \{ V^N \} = \lambda_1^n + \lambda_2^n\]


where a dot "\(\cdot\)" indicates ordinary matrix multiplication. The \(\lambda\)'s are the two eigenvalues of the \(2X2\) “transfer matrix” V, viz., \(\lambda_1 = 2 \cos{h \beta} J\) and \(\lambda_2 = 2 \sin{h \beta} J\ .\) Then, \(Z = \lambda_1^n+ \lambda_2^n = \lambda_1^n (1+ \left ( \frac{\lambda_2}{\lambda_1} \right )^N )\ .\) Thus only the larger eigenvalue survives, given that the contribution of the smaller one to the partition function (and to F) is exponentially smaller when we proceed to the thermodynamic limit \(N \to \infty\ .\)

The largest eigenvalue also acquires a special significance if we identify the spinor \((p_j, 1-p_j)\) as the probability of the jth spin being “up” (\(p_j\)) and “down” \((1-p_j)\ .\) With \(p_j\) in the interval 0,1 each of these probabilities is positive and they add up to 1. The corresponding probability of the \(j+n_{th}\) spin being “up” or “down” is then \((p_j, 1-p_j) \cdot V_n \propto (p_{j+n}, 1-p_{j+n}) \ .\) We present this as a proportionality and not as an equation because the sum of the probabilities at \(j + n\) also need to be normalized. (In probability theory, by normalization is meant that each entry is positive and the sum of all entries is unity.) Therefore the correct equation is:

\[\tag{4} (p_j, 1-p_j) \cdot V_n = z(n) (p_{j+n}, 1-p_{j+n}) \]


with \(z(n)\) to be determined. In the present example it is found that the right-hand vector tends to (½,½ ) asymptotically at large \(n\ ,\) regardless of the initial \(p_j\ .\) If the initial state belonged to the largest eigenvalue of \(V\ ,\) which is \(\lambda_1 = 2 \cos{h \beta} J\ ,\) then \(p_j\) = ½ and the preceding result holds for all \(n\) and not just asymptotically. It follows that the function \(z(n)\) is \(z^n\) and that \(Z = z^N\ .\) The other eigenvalue of \(V\ ,\) \(2 \sin{h \beta} J\ ,\) belongs to a spinor (½ , –½ ) that cannot be interpreted in terms of probabilities.

Thus, the evaluation of \(Z\) can be reduced to an ordinary eigenvalue problem subject to the following famous, if obvious, theorem:

Frobenius’ Theorem: “The largest eigenvalue of a matrix of arbitrary dimension, all elements of which are positive, belongs to an eigenvector that has only non-negative elements.”

For want of a better name we shall call this the “largest eigenvector”. Because all other eigenvectors must have one or more changes of sign in order to be orthogonal to the “largest eigenvector,” it is the only eigenvector that can be normalized according to the following rule: each of its entries must lie in the interval 0,1 and the sum of all its entries must = 1. Thus normalized, the “largest eigenvector” becomes the “reduced density matrix” and its entries are probabilities. Only this “largest eigenvector” is ever needed in the calculation of \(Z\) and of free energy F, whereas all eigenvectors are required in the evaluation of any nontrivial correlation function.

Application to Ising model on Sq lattice

In 2D statistical mechanics, the matrix “transferring” the \(n^{th}\) column of spins to \(n+1\) is a sort of “quantum” \(lc\ .\) The rows are labeled \(m=1,2, \dots ,M\ .\) All bonds \(-J' \sum_{m=1}^M S_{n,m} S_{n,m+1}\) that connect spins on rows \(m\) and \(m+1\) within a single vertical \(n^{th}\) column must be included. Combining these with horizontal transfers of the \(n_{th}\) column into the \(n+ 1^{st}\) we obtain a complete transfer operator \(V_n = \prod_m e^{\beta J' \sigma_{z,(n,m)} \sigma_{z,(n,m+1)}} (e^{\beta J} l_{(n,m)}+e^{-\beta J} \sigma_{x,(n,m)})\ .\) The horizontal contributions are given by the \(V\) defined just above Eq. (3). Because all references in this operator are to the \(n^{th}\) column we can omit the column index \(n\) for the sake of notational simplicity.

The second factor in the \(V\) shown above is exponentiated as follows\[(e^{\beta J} l_{m}+e^{-\beta J} \sigma_{x,m}) \equiv \sqrt{2 \sinh 2 \beta J} e^{K^* \sigma_{x,m}} \] after defining \(K= \beta J\ ,\) using a trivial Pauli operator identity, and defining \(\tanh K^* = exp -2K\) as before. After similarly defining \(K'= \beta J'\) for vertical bonds, we obtain the full 2D transfer matrix (all \(m\)) in the form:

\[\tag{5} W = C^M e^{K^* \sum_m \sigma_{x,m}} e^{K' \sum_m \sigma_{z,m} \sigma_{z,m+1}} \]

where \(C = \sqrt{2 \sinh 2K}\)   (5A)         

It “transfers” all the spins on the \(n^{th}\) column to \(n + 1\ .\) The exponent is congruent to a \(d=1\) Ising \(lc\) with nearest-neighbor bonds \(K'\) in a transverse magnetic field \(K^*\ ,\) an exactly solvable model. We therefore seek to solve the eigenvalue problem\[W \Psi = z \Psi\] for the largest possible value of \(z\ .\) This can be done following a sequence of simplifying transformations. The first of these is a global rotation about the y-axis by 90º, i.e., \(\sigma_{x,m} \Rightarrow \sigma_{z,m}\) and \(\sigma_{z,m} \Rightarrow - \sigma_{x,m}\ .\) After this (5)A) becomes:

\[W = C^M e^{K^* \sum_m \sigma_{z,m}} e^{K' \sum_m \sigma_{x,m} \sigma_{x,m+1}} \] (5B)

It is possible to express both the spin operators \(\sigma_{z,m} = 2 \sigma_m^+ \sigma_m^- -1 \) and \(\sigma_{x,m} = \sigma_m^+ + \sigma_m^- \) entirely in terms of the spinor raising/lowering operators, in such a way that the exponents are homogeneously quadratic in the \(\sigma^{\pm '} s\ .\) But because these operators are neither fermions (which anticommute) nor bosons (which commute), these exponents cannot be readily diagonalized. The eigenvalues of the exponentiated quadratic forms have no obvious significances.

In fact, the \(\sigma^{\pm '} s\) satisfy the following mixed commutation relations\[\tag{6} \sigma_j^+ \sigma_j^- + \sigma_j^- \sigma_j^+ = 1\ ,\]

whereas for \(j \ne l\ ,\) \(\sigma_j^+ \sigma_l^- + \sigma_j^- \sigma_l^+ = 0\ .\)    

To proceed we make use of a highly nonlinear “Jordan-Wigner” transformation. Such a mapping of fermions onto spins was originally invented in the 1920’s to prove that it was mathematically possible to construct a fermionic field theory out of an array of spins ½. Here we invert the construction, expressing each spin by a fermion operator \(c\) that carries an exponential wake made up of “earlier” fermion operators. The algebra of the fermions is postulated to be pure anticommutation\[\tag{7} c_j^{\dagger} c_k + c_k c_j^{\dagger} \equiv \{ c_j^{\dagger}, c_k \} = \delta_j^k, \{ c_j^{\dagger} c_k^{\dagger} \} = \{ c_j, c_k \} = 0\]


We note the following trivial identities\[c_j e^{\pm i \pi c_j^{\dagger} c_j} = -c_j, e^{\pm i \pi c_j^{\dagger} c_j} c_j = + c_j, e^{\pm i \pi c_j^{\dagger} c_j} = -1\] and \(e^{\pm 2 i \pi c_j^{\dagger} c_j} = 1\ .\)

We construct the Pauli spin operators in (5)B) out of such fermion field operators.

\(\tag{8} \sigma_m^+ = e^{i \pi \displaystyle \sum_{j<m} c_j^{\dagger} c_j}\)

and \(\sigma_m^- = c_m e^{-i \pi \displaystyle \sum_{j<m} c_j^{\dagger} c_j}\ .\)   

The reader will want to verify that this representation of the \(\sigma\) operators satisfies the mixed commutation relations in (6). Then, inserting (8) into (5)B) with the aid of the above identities yields the transfer operator as a product of two exponential forms, each quadratic in fermion operators. Thus, the eigenvalues of the quadratic forms in the following expressions are useful in the evaluation of \(Z\ .\)

\(\tag{9} W = C^M e^{K^* \sum_m (2 c_m^{\dagger} c_m -1)} e^{K' \sum_m (c_m^{\dagger} - c_m)(c_{m+1}^{\dagger} + c_{m+1})} \)

(9A)

We expand the local operators \(c_m\) in plane waves, \(c_m = \sqrt{\frac{1}{M}} \sum_{k=-\pi}^{\pi} e^{ikm} a(k)\ .\) (For didactic reasons we have imposed periodic boundary conditions on the \(lc\) of fermions, setting \(c_{m+N}=c_m\ ,\) but with little additional effort solutions can be found for more general or even for arbitrary boundary conditions, or for periodic boundary conditions on the original spin operators.) Because the Fourier expansion takes the form of a unitary transformation it preserves the algebra. Therefore the a’s satisfy the same set of anticommutation relations as the \(c\)’s in Eq. (7), viz.,

\(a^{\dagger}(k)a(q)+a(q) a^{\dagger}(k) \equiv \{ a^{\dagger}(k), a(q) \} = \delta_{k,q}\) with all other anticommutators = 0.

By translational invariance this procedure breaks the transfer matrix up into \(N/2\) noninteracting sectors, each labeled by \(k\) and containing a form bilinear in fermions\[W = C^M e^{K^* \sum_k (2 a^{\dagger}(k)a(q)-1)} e^{K' \sum_k e^{-ik} ( a^{\dagger}(-k)-a(k) )( a^{\dagger}(k)+a(-k) )} = \prod_{k>0} W(k)\] (9B)

Each factor on the rhs takes the form,

\(W(k) \propto e^{2K^* \big( a^{\dagger}(k)a(k) + a^{\dagger}(-k)a(-k) \big) } e^{K' \big( e^{-ik}(a^{\dagger}(-k)-a(k))(a^{\dagger}(k)+a(-k) ) + e^{ik} (a^{\dagger}(k)-a(-k))(a^{\dagger}(-k)+a(k)) \big) }\)

Because factors in different k-sectors commute, the individual 4X4 \(W(k)\)’s can be diagonalized individually. We need to find the largest solution \(\lambda_k\) of the equation \(W(k) \Psi = \lambda_k \Psi\) in each separate k-sector. Their product yields the partition function \(Z\ .\) Consequently the free energy \(F\ ,\) the logarithm of \(Z\ ,\) is explicitly a sum which turns into an integral in \(\lim M \to \infty\ ,\)

\(F = -kTN \sum_{k=0}^{\pi} \log \lambda_k = -kT \frac{NM}{2 \pi} \int_{0}^{\pi} dk \log \lambda_k \)

The thermodynamic properties are obtained from \(F\) by successive differentiations. These derivatives of \(F/T\) can be calculated in closed form in terms of elliptic functions (Mattis, 2006). Note that \(F\) is (correctly) extensive (proportional to the area \(NM\ .\)) It is easily shown that the identical integral would have been obtained in the large \(NM\) limit, had we transferred the rows instead of the columns. So, even though the procedure might have seemed asymmetric, actually all symmetries are preserved.

Without going into details of the evaluation, we find the free energy \(F\) has singular derivatives at \(T_c\) (the actual value of \(T_c\) is easily calculated and agrees with the earlier estimates.) Above \(T_c\) there is no LRO and the magnetization is zero. At or below \(T_c\) there develops LRO because of a two-fold degeneracy of the ground state of the transfer operator. Correlations can be calculated with the aid of Toeplitz matrix theory. Both the magnetization and the LRO increase with decreasing \(T\) until, at \(T=0\ ,\) all spins are precisely parallel, all “up” or all “down”.

The two-dimensional transfer matrix of the \(d = 3\) dimensional Ising model can also be written in the form of Eq. (5)B) provided the column label \(m\) is replaced by a planar label \((n,m)\ ,\) with bonds to \((n \pm 1,m)\) and \((n,m \pm 1)\ .\) That is, the transfer matrix for the 3D Ising model can be mapped onto a two-dimensional Ising model in a perpendicular field. Because only the largest eigenvalue is needed in the calculation of \(Z\) or \(F\ ,\) a variational approximation is useful, as is the renormalization group (RG).

Unfortunately the Jordan-Wigner transformation itself fails to be of help, because the exponential tails fail to cancel for half the bonds; therefore the quadratic form in spins on a 2D plane cannot be transformed into a quadratic form in either fermions or bosons. The transfer operator can, however, be reexpressed as a quartic form in fermions (Mattis, 2006; Chap. 3, §3.12). Thus the 3D Ising model falls into the realm of problems (\(\Phi^4\) field theories) that are generally well understood yet have not yet found an exact mathematical solution outside of approximate or RG procedures.

Other seemingly simple model ferromagnets that remain unsolved at the present time include the 2D Ising model in a real, finite, external magnetic field (whether homogeneous or staggered,) as well as most three-dimensional models of any kind.

More symmetry considerations

In the above, the spontaneous breaking of discrete up/down symmetry in the ground state at \(T = 0\) was of no particular consequence. But what if the symmetry had been continuous? Let us consider spins that can point into any direction according to either the O(2) (circular) or O(3) (spherical) symmetries. Then the excitation spectrum above the ground state becomes gapless. Consider the following magnetic polymer (\(lc\)) whose dynamics are described by the following Hamiltonian:

\[\tag{10} H = -J \sum_{n=1}^{N-1} \vec S_n \cdot \vec S_{n+1}\]


in which we assume the individual spins are themselves classical two- or three-dimensional vectors of unit length (all \(\vec S_n^2 = 1\)) and not operators. The dot product (\(\cdot\)) ensures the bond energies are scalar under rotations. This \(H\) is known as the “classical” Heisenberg Hamiltonian if the spin vectors are three-dimensional or as the “classical” “X-Y” model if the spins are constrained to lie in the x,y plane. We distinguish the classical spins here from any of the quantum versions discussed supra, in which the components of the individual spins fail to commute.

(The distinction between ferromagnetism in the extreme quantum limit of s=½ and the classical models of ferromagnetism may, in fact, be academic (note), as all interesting properties, correlations, etc. are qualitatively independent of the magnitudes of the spins; not so, the difference between models with Z(2), O(2) and O(3) symmetries. Such rotational symmetries, and not the quantum mechanics, seem to be a determining factor in the thermodynamics.)

The classical O(2) X-Y model is also known as the “plane rotator” model, given that bonds connecting nearest-neighbor sites \(i,j\) take the form \(-J \cos (\vartheta_i - \vartheta_j)\) of rigid coupled pendulums.

On a \(d = 2\) lattice, neither the O(2) nor the O(3) model can sustain LRO at any \(T > 0\ .\) The lack of spontaneous symmetry breaking at any finite \(T\) in both models is the result of a rigorous no LRO theorem first proved by Hohenberg and later generalized by Mermin and Wagner (Mermin & Wagner, 1966) to all systems having a continuous symmetry in \(d \le 2\) spatial dimensions. This theorem clearly does not apply to the Ising model because of its discrete symmetry; nor does it address the issues of the existence or nonexistence of a phase transition at finite \(T\ .\)

In fact, the X-Y model (but not the Heisenberg model!) does exhibit an unusual phase transition on a two-dimensional lattice, at a finite \(T_{K-T}\) approximately equal to \(0.9 J/k_B\) separating two disordered phases. This, the well-known “Kosterlitz-Thouless” phase transition, is a two-dimensional version of a liquid <–> vapor transition. There are many ways to examine its critical properties.

In one of them (Mattis, 1984), the transfer matrix of the classical model is mapped onto a one-dimensional anisotropic Heisenberg \(lc\) of spins ½ in which the J-matrix takes the form \(\begin{pmatrix} 0.9 & 0 & 0 \\ 0 & 0.9 & 0 \\ 0 & 0 & g \end{pmatrix}\ ,\) with the effective parameter \(g\) varying as \(1/k_T\ .\) The critical point is thus at \(g=0.9\ .\) Internal excitations (here physically interpreted as the spectrum of quantized clockwise or anticlockwise vortices) are gapped (bound) at low temperatures but have a continuous spectrum above \(T_{K-T}\ .\)

On the same two-dimensional lattice, the O(3) model remains in its high-temperature phase at all finite \(T\) without undergoing any phase transition whatever. The cause is, presumably, the high density of low-energy hedgehog-like excitations called skyrmions that are allowed in this model but not in the other.

Both models do support a gapless spectrum of spin waves at low \(T\ .\) In dimensions \(d \ge 3\ ,\) both O(2) and O(3) models exhibit rather ordinary order-disorder second-order phase transitions at a finite \(T_c\ .\) Now let us examine some details in \(d=1\) as the simplest example.

Here again the ground state energy is the same \(E_0\) but the excitation spectrum can now be vanishingly small, as a small twist \(\phi\) in the orientation \(\vartheta\) of each spin relative to its neighbor (say, \(\vartheta_{n+1} = \vartheta_n + \phi\) with \(\vartheta_1=0\)) only costs an energy \(1/2 J \phi^2\) per bond. The imposition of any boundary conditions – say, requiring that the first and last spins be parallel – causes \(\phi\) to become discretized, e.g. \(\phi = 2n \pi/N\ ,\) with \(n\) an integer \(\le N\ .\) The ratio \(N/n = \lambda\) is related to the wavelength of the excitation, in units of the lattice parameter \(a\ .\) We call this excitation a spin wave; its energy forms a quasi-continuum and vanishes as \(n^2/N\ .\) The lack of an energy gap in the large \(N\) limit is a feature of many field theories with continuous symmetries, as was first remarked in the 1950’s by Nambu, Goldstone, et al, by analogy with the spin waves that Bloch found 20 years earlier. Where the “Goldstone mode” relates to the spin wave spectrum, the Goldstone boson relates to the “magnon”, which is an elementary bosonic particle that results from further quantization of spin dynamics. It carries 1 unit of angular momentum \(\hbar\ .\)

But just as there is an exception in particle physics, e.g. for the Higgs boson, there is one in magnetism for integer quantum spins in a 1D antiferromagnetic Heisenberg chain, where the lowest excitations have to surmount a “mass” gap. We turn to this interesting anomaly next.

Role of spin in d = 1 antiferromagnets

The antiferromagnetic Heisenberg Hamiltonian, i.e., the lc of Eq. (10) with \(J < 0\ ,\) in which spins are \(S = 1, 2, \ldots\) operators, as opposed to classical vectors considered earlier, has a magnon spectrum exhibiting a finite excitation gap even at the longest wavelengths. This is quite unlike the gapless spin wave spectrum \(\omega \propto k^2\) of the ferromagnet, \(J > 0\ ,\) and from the excitations of the S=½ antiferromagnet – the spectrum of which, \(\omega \propto \left | k \right |\ ,\) is calculated exactly by “Bethe’s ansatz” discussed at the end of this article. The spin wave excitations of \(S= 3/2, 5/2 \ldots lc\) antiferromagnets and of classical spin antiferromagnets also are all gapless. So what happens to integer spin operators of magnitude \(S(S+1) = 2, 6, 12, \ldots\ ,\) to make them that different?

We start with the proof that the excitation spectrum for all half-odd-integer spins \((1/2, 3/2, \ldots)\) is continuous and gapless, regardless whether the sign of \(J\) is positive or negative.

The proof in 1D is as follows: take the ground state wave function \(\Psi_0\) for a \(lc\) subject to periodic boundary conditions and operate on it by \(\Gamma = \prod_n \gamma_n \) in a way that distorts the \(n^{th}\) individual bond only by a small amount \(1/N\ ,\) thus each of the \(N\) bonds sees its energy rise in an amount \(O(1/N^2)\ .\) If \(\Gamma \Psi_0\) is orthogonal to \(\Psi_0\) we have constructed an excited state of total energy \(O(1/N)\) above the ground state. One may consider this as the variational calculation of the energy of a 1-magnon state. (The proof in \(d = 2\) and \(d = 3\) just extends the proof for the \(lc\) to finite-width strips or cylinders.) If the spins are higher but of the form half-odd-integer, an operator \(\Gamma\) having these properties is easily constructed. In the case of integer spins, however, the corresponding operator, when applied to the ground state wave function, fails to yield a state orthogonal to the ground state and the proof fails.

This energy gap in the spectrum of integer spin antiferromagnetic \(lc\)’s was first conjectured by D. Haldane and it is named after him; it turns out to exist only in 1D antiferromagnets and only if \(S\) is an integer\[S =1, 2 , 3, \ldots\ .\] The magnitude of Haldane’s gap goes to zero as \(S\) is increased. (This is expected if the model is to approach its gapless correspondence limit smoothly.) The gap also vanishes in any dimension \(d > 1\ .\) Thus we are dealing with a feature that is both interesting and fragile, which is optimum for spins \(S =1\) in \(d = 1\) and will be extensively revisited at the end of this article.

We can discern the dichotomy in a \(lc\) of as few as 3 spins \(S\) arrayed in a triangle. For 3 spins the Heisenberg Hamiltonian is diagonalizable\[H = \frac{-J}{2} \left[ T(T+1)-3S(S+1) \right]\ ,\] where \(T\) = the total combined spin. \(T\) has a maximum value 3/2 for spins S=½ and a maximum 3 for spins 1. In the ferromagnet (\(J > 0\)) the ground states for both these values of \(S\) belongs to their respective maxima and are similar in all aspects.

In the antiferromagnetic (\(J < 0\)) triangle of 3 spins, the ground states belong to a total spin minimum. The minima are \(T = 0\) for spins 1, and \(T\) = ½ for spins ½. After some back-of-the-envelope exact calculations, one determines that the ground state of the three spins ½ consist of two degenerate doublets, i.e., that it is 4-fold degenerate. Thus there is no energy gap separating the two lowest-lying states.

For the antiferromagnetic triangle made of spins \(S = 1\ ,\) however, the ground state belongs to a unique \(T = 0\) singlet state. All other eigenstates in this model lie at energies that are at least \(J\) higher so that here, the Haldane gap is \(J\ .\)

Exercise for the reader: contrast the eigenstates of \(S\)= ½ and \(S\)=1 Heisenberg antiferromagnetic chains of 4 spins when laid out on a single square plaquette, with or without diagonal linkages. (Either geometry can be done analytically in closed form.) The conclusions are quite similar. Discussion of the Haldane gap in the limit of large \(N\) is reprised near the end of this article. References to early and pertinent literature are given in Mattis, 2006.

The antiferromagnetic \(lc\) of spins \(s = 1\) exhibits an additional idiosyncracy at large \(N\ :\) the two ends, at \(n = 1\) and \(N\) respectively, act like free spins ½ in their response to external fields, as in paramagnetic resonance (EPR.) The characteristics of EPR allow one to determine the spin; the ends of a chain of spins 1 display spins ½ ! This surprising behavior was, in fact, predicted theoretically – and confirmed by experiment – almost simultaneously. Chains of \(S = 2\) spins would, presumably, have ends that exhibit the properties of spins 1, etc. The situation is similar to that in the defective antiferromagnets discussed below, given that dangling ends of the chain at \(n=1\) and \(N\) can be viewed as breaks in a longer chain, or as a symmetry-breaking disruption of translational invariance, caused by cutting the bond connecting the last spin at \(N\) to the first, in a chain with periodic boundary conditions.

No magnetism at finite T in 1D

In 1D one can also obtain the free energy of the classical O(3) Heisenberg model without separate calculations of energy and entropy, using a transfer matrix for the partition function \(Z = e^{-\beta F}\ .\) To within boundary terms its largest eigenvalue yields,

\[\tag{11} F=-(N-1)k_BT \log \Big( \frac{k_BT}{J} \sinh \frac{J}{k_BT} \Big) \]


an expression that translates to a (thermal averaged) angle between neighboring spins of \(< \left\vert \phi \right\vert > \approx \sqrt{\frac{k_BT}{J}}\) at low \(T\ .\) The correlations of two spins separated by a macroscopic distance \(na\) fall off \(\propto exp -n < \left\vert \phi \right\vert > \ .\) Thus, in this gapless one-dimensional model with continuous symmetry the ground state LRO disappears exponentially at any finite \(T > 0\ .\) Following the Mermin-Wagner theorem some such result was to be expected. It is also notable that many years earlier, L.D. Landau had already observed that no model with finite range interactions can sustain LRO in 1D at any finite \(T\ .\)

The nature of local moments

Given all these choices of models, what are the most physically plausible magnetic contents of a unit cell? Typically, spins and angular momenta that characterize an atom or ion disappear (are quenched) in solids. The earth elements are counter-examples, in that they have unfilled f-shells that can accommodate up to 7 electrons in an orbital that is somewhat smaller than typical inter-atomic distances and which are, therefore, not much disturbed by the crystal symmetry. A factor of 7 Bohr magnetons puts these spin magnitudes largely in the classical limit, such as was assumed in the classical Heisenberg model treated above.

Actually, the Hamiltonian in Heisenberg’s 1928 model of magnetism, or what is commonly understood today to be the Heisenberg Hamiltonian, is similar to Eq. (4) in form but uses operators for the components of individual vector spins. As we know, in quantum theory all angular momenta satisfy an algebra \(\vec S \times \vec S=i\hbar\vec S\) with 3 operator components \(\vec S = (S_x,S_y,S_z)\ .\) In the extreme quantum limit, for a single electron, the \(S\)’s have an irreducible representation in \(2 \times 2\) Pauli spin matrices that anticommute with one another but commute with operators at all other sites. For higher spins the irreducible representations are \(2s+1\) dimensional, where \(\vec S^2 = \hbar^2 s(s+1)\ ;\) the various components are operators that satisfy \(\left[ S_x,S_y \right] =2i\hbar S_z \ ,\) which is equivalent to the generic \(\vec S \times \vec S=i\hbar\vec S\ .\)

Despite the introduction of operators into the problem, the ground state of the s = ½ quantum ferromagnet (\(J > 0\)) remains simple\[\Psi_0 = \prod_j \begin{bmatrix} 0 \\ 1 \end{bmatrix}_j\ .\] The product form is maintained even for higher spins, \(\Psi_0 = \prod_j \psi_j^{(0)}\ .\) Low-lying excitations – spin waves – of the Heisenberg ferromagnet, were derived by F. Bloch from this explicit translational invariance. (The amplitude of an excitation has to be constant, therefore only the phase can advance. This is, by definition, a plane wave.) However, once there are many excitations present, as happens at finite \(T\ ,\) translational invariance is lost and the ensuing nonlinear problem becomes quite more difficult.

Nevertheless, in 1D, and only for spins ½, a complete set of solutions encompassing all compound excitations in the Heisenberg model was conjectured by H. Bethe (Bethe, 1931). The “free fermions” of E. Lieb, T. Schultz and D. Mattis (Lieb, Schultz & Mattis, 1961) are merely a simplified form of Bethe’s ansatz that yields all the states of the spin ½ “X-Y” model (but again, only in 1D.) Before recapitulating the solutions of what are, after all, very special models, it is fruitful to take a step back and reexamine, where does the sign and magnitude of the nearest-neighbor interactions \(J\) connecting individual spins come from? We proceed from the point of view of the more fundamental many-electron physics.

Nature of magnetic interactions in cells and in metals

Itinerant electrons occupying unfilled or partly filled energy bands are principally responsible for chemical binding, electronic conductivity and any significant magnetic correlations in metallic materials. The “band theory” of electrons is fundamental to answering questions about interactions among magnetic species (even when it is appropriate to use quasi-classical localized spins in the theory, as in the aforementioned rare earths,) especially when the interactions among them are mediated by itinerant electrons.

There are at present two acceptable ways to treat itinerant electrons: either by way of Schrödinger’s wave equation or by using the “tight-binding approximation.” The first approach is the simpler one for present purposes, so it is used here. (The second is more descriptive when, as in the rare earth metals, some electrons occupy degenerate, localized, orbitals in each unit cell while other electrons are itinerant, thereby requiring labels for each.)

In 1D, the nonrelativistic Hamiltonian of N purely itinerant electrons is, quite simply,

\[\tag{12} H = - \frac{\hbar}{2m} \sum_{j=1}^N \frac{\partial^2}{\partial x_j^2} + V(x_1,\ldots,V_N)\]


The differential equation \(H \psi (x1;\xi_1;\ldots; x_N;\xi_N) = E \psi (x1;\xi_1;\ldots; x_N;\xi_N)\ ,\) is to be solved for its full complement of energies \(E\) and eigenfunctions \(\psi\ .\) The potential energy is a function \(V\) that is almost an arbitrary function of the coordinate variables \(x_j\) as long as it is symmetric under their permutation in the interval \(0 <x_j<L\) (because the particles are indistinguishable the potential energy cannot distinguish among them.)

Although the set of \(\xi\) labeling the electrons’ (discrete, ±½) spin coordinates is absent from the Hamiltonian in Eq. (12), it is present in the eigenfunctions. Implementation of the Pauli principle requires both, because a particle is identified by specifying both its spatial and spin coordinates.

Pauli Principle. The interchange of 2 fermion particles, that is, of both the space and spin coordinates of any 2 particles in \(\psi\ ,\) causes the latter to change its sign.

All \(\psi\) are constrained to be totally antisymmetric under the group of permutation of particles, i.e. to change sign under an odd number of interchanges of both the spatial and spin coordinates of the particles. This requirement reflects a fundamental tenet of quantum theory that is most often stated (imprecisely) as, “no two fermions with the same spin can occupy the same state.”

The two-body forces in \(V\) could conceivably be made sufficiently strong that not more than one electron can occupy a given cell – but they need not be. The only approximations in writing an Hamiltonian in the form of (12) come from using the non-relativistic form of the kinetic energy and, in the potential energy, neglecting any further terms that depend on the particles’ momenta (including relativistic effects as spin-orbit coupling, generally thought to be negligible in many materials.) In this approximation, \(H\) commutes with the total spin operator and therefore the eigenstates of \(H\) can be labeled by the magnitude \(S\) of the total spin.

With these caveats, E.H. Lieb and the present author proved the following so-called “Lieb-Mattis theorem” in 1962. Let us denote it as “LM I”. It governs the eigenstate spectrum of the Hamiltonian in Eq. (12) and can be stated, in part, as follows,

If \(E_0(S)\) is the lowest energy eigenvalue belonging to total spin \(S\) of \(N\) electrons in 1D, then \(\tag{13} E_0(S) \le E_0(S + 1)\ .\)


The inequality (<) applies generally but the equality (=) applies only if the two-body potential is sufficiently singular. LM I can be tweaked to higher dimensions in the possible, albeit implausible, cases of potentials that are separable in the Cartesian coordinates x, y, … of the particles.

(Note: there is no contradiction between LM I and the so-called “exchange” correction to the Coulomb interaction that favors the largest possible spin in the ground state of partly occupied spherical shells of angular momentum l ≥ 1. This ferromagnetic exchange mechanism forms the theoretical basis for Hund’s rules; it is quantitatively verified in the spectra of d-and f-shell transition series atoms and ions, among others. Transition-series shells contain 2l + 1 degenerate spatial states for each particle; the exchange corrections lift the degeneracies for 2 or more particles in the shell, favoring the maximum total spin. There is no such degeneracy to be lifted in a one-dimensional model.)

To restate LM I more succinctly: in a 1D system of electrons it is impossible to lower the total energy by increasing the total magnetization. Thus Eq. (13) precludes ferromagnetism in 1D – not just at finite temperature, because of the lack of LRO – but also in the ground state at T = 0. This result applies to lc’s of lengths \(N=2,\ldots,\infty\ .\) It follows that to describe magnetic properties of some material, using a simpler but less fundamental Ising or Heisenberg Hamiltonian formalism governing a smaller set of dynamical variables, any physically correct nearest-neighbor interaction parameters \(J\) must have a sign appropriate to antiferromagnetism – and not to ferromagnetism.

The principal virtue of LM I is that it reduces the number of plausible theories of ferromagnetism to just those few models that, paradoxically, reconcile long-ranged ferromagnetic order with a short-range tendency to antiparallelism. In this article we discuss three such mechanisms.

Antiferromagnetism and ferrimagnetism

In the 1930’s Néel had already postulated the existence of antiferromagnetism in some magnetic salts. Note that this fits in well with the aforementioned theorem. Néel’s insight explained why such substances did not exhibit a macroscopic magnetic moment despite indirect evidence of internal spin dynamics.

Late in the 1960’s W. Heisenberg, together with H. Wagner and K. Yamasaki (Heisenberg, Wagner & Yamasaki, 1969) reprised the study of his eponymous antiferromagnet in 3D. They note that when viewed as a field theory, Heisenberg’s antiferromagnet has a spectrum similar to QED. In both instances the ground state has zero angular momentum and elementary excitations (magnons or photons) are transversal to the direction of propagation and (at long wavelengths) have dispersion linear in the momenta \(\left | k \right |\ .\)

Earlier in the 1960’s a second theorem by Lieb and Mattis addressed antiferromagnetism, ferro- and ferri-magnetism, all within this Heisenberg model, in all dimensions d. Denote this theorem “LM II”. (Lieb generalized it subsequently, to the Hubbard model of interacting electrons, in the special case of a half-filled band.)

LM II states in part that if in any dimension \(d\) a space lattice is bipartite, (i.e., if it can be subdivided into 2 sublattices, say \(A\) and \(B\ ,\) such that spins \(s\) on the \(A\) sublattice interact antiferromagnetically but only with spins \(s\) on the \(B\) sublattice, and vice-versa,) the maximum spin in the ground state is exactly \(S_{total}= s \left | N_A-N_B \right |\ .\)

\(S_{total}\) is only identically zero if \(N_A = N_B\) but otherwise, the ground state spin can be finite. Thus there can be ferromagnetic LRO in the ground state even though the magnetization only attains the smallest value possible and decreases further with increasing temperature \(T\ .\) In \(d \ge 3\) lattices this LRO can persist over the entire range of \(T < T_c\) but in \(d \le 2\) lattices, we note once again that, because of continuous symmetry, \(T_c =0\ .\)

Ferrimagnets

In bipartite lattices where \(N_A = N_B\) but the magnitude of the individual spins \(s_A\) on sites of the \(A\) sublattice ≠ that of the \(s_B\ ,\) LM II shows that even though the A-B couplings are antiferromagnetic the ground state can have a magnetic moment corresponding to maximum spin \(S_{total}= N \left | s_A-s_B \right |\ .\) More generally, the actual proof of LM II indicates that the ground state spin on bipartite lattices can attain \(S_{total}= \left | N_As_A-N_Bs_B \right |\ .\) Ferrimagnetism is a good example found in nature.

This phenomenon is named after the ferrites; in magnetic iron ore, magnetite (\(Fe^3O^4\)), a spin 1 \(Fe^{2+}\) is neighbor to a spin ½ \(Fe^{3+}\ .\) Despite being antiparallel, their spins do not compensate. In other oxides or salts forming bipartite lattices, \(N_A = 2N_B\) (or other multiples) are possible. In any event, the macroscopic magnetism in a ferrimagnet is spread over the entire lattice just as in a ferromagnet, and like this last, it loses LRO at any finite \(T > 0\) in d=1 or 2 dimensions in the absence of an ordering external magnetic field; however, the situation is different in d ≥ 3.

In d ≥ 3 the ferrimagnet can, just like the hypothetical ferromagnet it resembles, exhibit spontaneously broken symmetry and LRO that gradually decreases with increasing \(T\) and vanishes only at a finite Curie temperature \(T_c\ .\) (Again, for \(T \ge T_c\) all LRO is extinguished.) Its elementary excitations, the magnons, exhibit ferromagnetic-type dispersion \(\omega \propto \left | \vec k \right |^2\) at long wavelengths. This dispersion reverts to an antiferromagnetic-type dispersion \(\omega \propto \left | \vec k \right |\) for short wavelength excitations.

Although it is an instructive model, one cannot consider ferrimagnetism as providing a general picture of ferromagnetism, given that the model applies principally to magnetic salts with localized spins and provides no explanation for LRO in metals such as iron.

The Kondo lattice

A different mechanism, derived from the model single-spin impurity “Kondo effect,” has been invoked by many authors to explain “heavy fermion” ferromagnetism in rare earth solids. This so-called “Kondo lattice” provides a more general framework in which to situate ferromagnetism. Imagine itinerant electrons (if they are confined to a narrow band they are “heavy,” otherwise not) that interact locally with the localized spin of an f-shell (magnitude s = ½ or greater.)

Hund’s rule teaches that within the \(f\)-shell the interactions align the spins of the electrons into a total spin S; therefore the low-lying Hilbert space of each atom or ion is limited to the corresponding \(2S+1\) states of \(S_z\) ranging from \(- S\) to \(+S\ .\)

The interaction between the itinerant and \(f\)-shell electrons can be expressed in Heisenberg form with, typically, \(J < 0\ ,\) given that itinerant and localized electrons belong to different shells and that intershell intraatomic interactions typically – but not necessarily – favor antiparallelism. We also consider \(J > 0\) to investigate exceptional cases. The Hamiltonian that includes both the motion of the band electrons and their interactions with localized spins is inherently quantum mechanical. It is,

\[\tag{14} H = H_0 - J \sum_i \vec S_i \cdot \vec \sigma_i\]


The operators governing localized spin angular momenta are the components of \(\vec S_j = (S_{x,j},S_{y,j},S_{z,j})\) whose irreducible representations are \(2s+1\) dimensional matrices. The spin of the itinerant electron can be expressed in the Wannier creation/destruction operators localized at site\( R_j\ .\) It is \(\vec \sigma_j = \frac{1}{2} (\psi_{j,\uparrow}^{\dagger},\psi_{j,\downarrow}^{\dagger}) \cdot \vec \sigma \cdot \begin{pmatrix} \psi_{j,\uparrow} \\ \psi_{j,\downarrow} \end{pmatrix}\ .\)

The Hamiltonian of the itinerant particles is given by \(H_0 = -t \sum_{(i,j)} \sum_{\xi=-1/2}^{+1/2} (\psi_{j,\xi}^{\dagger} \psi_{i,\xi}+H.c.)\ ,\) otherwise known as the “tight-binding” energy-band Hamiltonian. (Sites \(i\) and \(j\) are nearest neighbors.) Expanding the Wannier operators in a series of Bloch operators on a cubic lattice\[\psi_{j,\xi}= \sqrt{1/N} \sum_k e^{-ik \cdot R_j} c_{k,\xi}\ ,\] the energy-band Hamiltonian becomes:

\[\tag{15} H_0 = -2t \sum_{k,\xi} (\cos k_1 + \ldots + \cos k_D) c_{k,\xi}^{\dagger} c_{k,\xi}\]


where \(k\) spans the “first Brillouin Zone,” i.e, \(-\pi/a < k_x < \pi/a\ ,\) … , \(-\pi/a < k_D < \pi/a\ .\) The band structure on these simple lattices is just the sum of \(\cos k_j\) terms. The eigenstates of (14) are complicated functions of the number of itinerant electrons \(N\ ,\) the dimensions \(d\ ,\) and the ratio \(J/t\ .\) Interactions break the translational invariance. So while typical eigenstates for either sign of J are generally metallic (owing to the itinerancy) and magnetic, an insulating phase is also possible. The ground state at \(T=0\) can sustain magnetic LRO in all dimensions d ≥1.

If \(N=N\) precisely (i.e., in the case of a half-filled band,) the ground state, which normally is metallic, is instead, an insulator with an energy gap O(J). For J < 0 each cell has spin | s–½ |, which vanishes if the \(f\)-shell contains only a single electron but is otherwise non-zero. If J > 0 each cell has spin | s+½ | in the ground state. Nevertheless at half-filling the effective coupling between nearest neighbor spins due to motional energy \(H_0\) is itself always antiferromagnetic, hence the half-filled band is an antiferromagnetic insulator.

Now consider the special case s = ½, J < 0 , and suppose the number of electrons is N ≠ N. Define the excess number: n=N – N. This quantity can be as positive as +N or as negative as –N.

In 1D, it has been proved that the ground state spin is ½\(\left | n \right |\) for all values of \(n\ .\) This suggests that the ground state is a ferrimagnet of some sort, and that the A and B sublattices consist of the itinerant and the localized electrons respectively. This may not be the case in higher dimensions, and the model appears to lean more to ferrimagnetism in d=1 than in higher dimensions (note).

Paradoxically, for J > 0 (but not too large compared to t,) and |n| arbitrary (but small compared with N,) the individual atoms acquire maximum spin but the overall spontaneous magnetization vanishes.

Starting some 3 decades ago, a voluminous literature in the Kondo lattice model has been motivated by interest in the heavy fermions, high-temperature superconductivity and other magnetic phenomena. The topic deserves further investigation. The phase diagram as a function of \(s\ ,\) \(J/t\) and \(n\) in d dimensions is obviously very complex, with ferromagnetic, spiral magnetic and superconducting phases all within the realm of possibilities.

Nagaoka mechanism of ferromagnetism

The simplest model displaying ferromagnetism is named after Nagaoka. Its description follows\[N\] electrons in a nondegenerate band (\(H_0\) of Eq. (14) is a good example) are subject to a large, repulsive on-site potential such that no two electrons, even though they have opposite spins, can occupy the same site. The hopping matrix element that connect 2 neighboring sites is \(t\ .\) Thus the Hamiltonian, known as the Hubbard Hamiltonian if \(U^*\) is finite, is:

\[\tag{16} H_{nagao} =H_0 + U^* \sum_j c_{\uparrow}^{\dagger} (R_j) c_{\uparrow} (R_j) c_{\downarrow}^{\dagger} (R_j) c_{\downarrow} (R_j)\]


The interaction \(U^*\) is a local repulsive potential. When taken in \(\lim U^* \to +\infty\) the second term in (16) acts as a projection operator to prevent double occupancy of any of the \(N\) given sites, each of which can still be empty or occupied by a single electron.

If N =N–1 there is one “hole.” If the arbitrary disposition of spins around the hole is different for each position of the hole, then the hole can diffuse but its bandwidth \(4 \left | t \right | d\) is reduced to \(4 \left | t_{eff} \right | d\) with \(t_{eff}\) smaller than \(t\ ,\) its value depending on the details of the spin distribution. There is, however, one exception: If all spins are parallel – i.e. maximally ferromagnetic – then, because of translational invariance, \(t_{eff} \to t\ .\) So the energy of a single hole is lowest, at \( -2 \left | t \right | d\ ,\) in the ferromagnetic state.

Given that the ferromagnetic states(s) are the lowest in energy in the presence of 1 (or a few) holes, this must change when the density of holes exceeds a critical value. To lower their kinetic energy the electrons must cease to have all parallel spins (Gulacsi & Vollhardt, 2005), and an ordinary Fermi liquid of electrons, half of which have spins up and the other half down, is recovered, with any residual interactions then presumably given by the Ruderman-Kittel-Yosida mechanism. The critical number of holes beyond which the ferromagnetism is lost is estimated at a few percent of N in 3D (there are calculations for various lattices but no exact formula) and zero in d < 3.

Frustration

Frustration is a phenomenon whereby antiferromagnetic bonds cannot all be satisfied, thereby causing nontrivial symmetry breaking. The simple hypercubic lattices are bipartite and unfrustrated. But if there were antiferromagnetic interactions among members of the \(A\) sublattice and/or among members of the \(B\) sublattice, in addition to the antiferromagnetic bonds connecting spins on one sublattice with the other, it would become difficult or impossible for the spins on either of the sublattices to remain parallel to one other in the ground state. This is what happens in geometrical frustration. For this reason the abovementioned LM II simply does not apply to geometrically frustrated lattices.

Conceptually the simplest example of geometrical frustration is the triangular lattice. As the simplest and smallest example of geometrical frustration, consider a single triangle of 3 spins that are coupled antiferromagnetically. Whether the spins are in the quantum or classical limits, all three bonds cannot be simultaneously satisfied. This is to be distinguished from frustration caused by the random signs of the interactions in spin glasses.

One achieves a “sort of” geometrical frustration in the 1D Heisenberg antiferromagnet by augmenting the nearest-neighbor antiferromagnetic bonds \(J\) by second-neighbor antiferromagnetic bonds \(J'\ ;\) at some ratio of\( J'/J=O(1)\) the ground state solution ceases to exhibit LRO and breaks translational invariance spontaneously, turning itself into products of localized singlet (spin zero) clusters. Frustration raises the energies of the ground state and low-lying states and, typically, their degeneracies also.

In 2D there exist macroscopic models in which the free energy can be solved exactly at all \(T\) provided the interactions are restricted to nearest-neighbors. Consider the triangular Ising antiferromagnet (\(J < 0\ ,\)) a prime example of geometric frustration. While the ferromagnet on a triangular lattice has a finite Curie temperature, the antiferromagnet on the same lattice has none (as first shown by G. Wannier.) Its entropy remains macroscopic even at \(T = 0\ ,\) in flagrante delicto of the Third Law of thermodynamics.

The Ising model on a unfrustrated \(sq\) lattice, in which random \(\pm J\) interactions are frozen-in with equal probability, can also be solved explicitly and exactly (Forgacs, 1980; Mattis & Swendsen, 2008). Although the geometry is not frustrated, many of the plaquettes (all those with an odd number of antiferromagnetic bonds) are. This stochastic system – the prototype of what is often called a spin glass – also has a macroscopic entropy at \(T = 0\ ,\) in violation of the Third Law. The Curie temperature is lowered by this disorder and the maximally frustrated Ising spin glass has \(T_c = 0\ .\) Thus, distinct sources of frustration may have similar outcomes.

Defective antiferromagnets

In an Heisenberg bipartite antiferromagnet of spins ½, if one just plucks out a single site from the \(B\) sublattice at random the resulting ground state belongs to total spin \(S\)=½. If two sites on the same sublattice are plucked out, the total spin is 1 but if the plucked sites are nearby but on distinct sublattices, the resulting spin is again \(S=0\) (see LM I and II.)

Quantum Monte-Carlo calculations in 2D have shown a surprising but logical result: each missing spin is locally compensated (“screened”) by spin deviations carved out of a finite but large neighborhood of the defect, with an amplitude that drops off with distance. Being so spread out, its differential thermodynamical properties (e.g., the free energy of the defective lattice less the free energy of the same lattice without defect) re-appear as those of a classical vector spin \(S\ .\) Thus, even though the underlying eigenstates are all quantum mechanical the differential magnetic susceptibility near the defects is that of a classical spin \(S\ :\) \(\chi_{imp} = \frac{S^2}{T}+\ldots\) (and not \(\frac{S(S+1)}{T}\) +…) (Nagosa, 1989), where “…” indicate logarithmic corrections higher-order in \(T\ .\)

Decomposition of 1D quantum antiferromagnets into fermions

With ferromagnetic interactions, the ground state is typically a product function and the excited states are, in general, spin waves. On the other hand, the ground state of antiferromagnets is always complex and the excited states may or may not follow a simple pattern. In some artificial cases with nearest- and next-nearest neighbor interactions, models of quantum spins can be solved, i.e. can have some or all their eigenstates retrieved. But it is only in d = 1 dimension with nearest-neighbor interactions that there sufficient simplification to obtain all the eigenstates, thermodynamics, and any other observable property. Consider a reasonably anisotropic Heisenberg model Hamiltonian on the \(lc\ ,\)

\[\tag{17} H = \sum_n \left ( \frac{1}{2} (S_n^+ S_{n+1}^- +H.c.) + gS_{z,n}S_{z,n+1} \right )\]


It is an X-Y model if g = 0, the Heisenberg antiferromagnet if g =1 and the Heisenberg ferromagnet for g=–1. For spins ½ the Bethe ansatz yields the ground state and spectrum of excited states at all g. The idea is that if we start from the “vacuum” (all spins “down”,) a 2 spin-wave excitation takes the form, \(\sum_i \sum_j f_{i,j} S_i^+ S_j^+ | all \downarrow >\) where\[f_{i,j} = \exp (i(ki+k'j+\frac{1}{2} + \psi_{k,k'}) + \exp (i(k'i+kj+\frac{1}{2} - \psi_{k,k'})\] (un-normalized) for i > j, a sum of the only two product states having the same motional energy and the same momentum \((k+k')\ .\) If i < j, changes sign. Instead of scattering, which is what occurs in d≥2 dimensions, in 1D solvable models there is only refraction.

In the spin ½ X-Y model, the interaction g=0 and the boundary condition that enforces \((S_n^{\pm})^2 \equiv 0\) requires the phase factor to be \(\psi = \pm \pi\ ,\) independent of \(k,k'\ .\) Then \(f\) can be identified as a determinantal function, one that is easily generalized to accommodate any number of spin-wave excitations.

An easier way to obtain all the solutions is to take advantage of the Jordan-Wigner transformation introduced earlier, and a-priori express the spins in terms of fermions \(c_i\) with “tails” on a finite \(lc\) that starts at n=1 and ends at \(N\ .\) We start by transforming the Hamiltonian into its fermion representation,

\[\tag{18} H = \sum_n \left ( \frac{1}{2} (c_n^{\dagger} c_{n+1} +H.c.) + g (c_n^{\dagger} c_n +1/2)(c_{n+1}^{\dagger} c_{n+1} +1/2) \right )\]


(the exponential tails cancel in the case of nearest-neighbor interactions.) This expresses the similarity between a one-dimensional gas of spinless fermions with the anisotropic Heisenberg model in 1D. One-fermion states can be expanded in \(\sin (kn)\) functions\[c_n = \sqrt{\frac{2}{N}} \sum_k a(k) \sin (kn)\] where the \(k = \frac{\pi}{N+1}m\ ,\) with \(m\) an integer in the range \(1, N\ ,\) to satisfy \(c_{N+1} = 0\ .\) Then the X-Y part of the Hamiltonian reduces to diagonal form with \(a^{\dagger}(k)a(k)=n(k)\) the only remaining operators. (The thermal average occupancy number is the Fermi function\( <n(k)>=\frac{1}{e^{\epsilon/kT + 1}}\) with \(\epsilon(k) = \cos k\ .\))

If g≠0 the calculations are more elaborate. Phase shifts take the nearest-neighbor interactions into account (repulsion for g > 0 or attraction for g < 0) and must be summed over all pairs of particles and their N! permutations. Thus,

\[\tag{19} f_{i,j,m,n,\ldots} = \sum_{permutations P} (-1)^P \exp i(k_1 i+k_2 j+\ldots+\frac{1}{2} \sum_{r<t} \psi(k_r,k_t))\]


The phase shifts are self-consistent functions of \(g, k\) and \(k'\ .\) After some lengthy algebra one finds that for \(g\) in the range \(-1 \le g \le +1\ ,\) the quasi-particle spectrum of excitations is gapless, mapping onto that of the X-Y model. For \(|g| > 1\ ,\) the spectrum is gapped. The thermodynamics is essentially that of an Ising ferromagnet if \(g\) << –1 and of an antiferromagnet if \(g\) >> +1.

Decades after these solutions were understood for spins ½ and the various features that depend on the nature of the excitations spectrum (magnetic susceptibility, specific heat and other thermodynamic properties that can be obtained from the correlation functions) had been calculated explicitly, these results were (incorrectly) generalized by several authors to higher spins on the \(lc\ ,\) notably to integer spins 1, etc. What invalidates all such generalizations are the subsidiary conditions discovered by C.N. Yang and R. Baxter, the Yang-Baxter (YB) relations, satisfied for three or more spins ½ on a \(lc\) but not for any of the higher spins.

Briefly, the \(YB\) relations deal with the effects of repeated scatterings. If scattering of \(k_1\) and \(k_2\) (abbreviated 1 & 2) followed by 1 & 3 leads to a phase shift that differs from scattering of 1 & 2 followed by 2 & 3, Bethe’s ansatz is invalid and scattering theory must be used.

More simply, we can just point out that the transformation of spins to fermions fails for quantum spins \(s\) > ½, hence Eq. (18) is no longer equivalent to the original Eq. (17) and the fermionic solutions to Eq. (18) become irrelevant. Haldane found that the categories of integer spins and half-odd-integer spins are distinct inasmuch as they could be mapped onto distinct field theories (Haldane, 1983). Even in the parameter range \(|g| \le 1\) that maps onto \(XY\ ,\) the \(lc\) of integer spins has a gapped spectrum while half-odd-integer spins have a gapless spectrum. All dynamical and thermodynamical properties then differ to that extent.

Conclusion

Physics and mathematics have always had a close relation, but none closer than the calculation of magnetism using concepts in analysis, number theory, algebra and group theory. We have tried to show this in the present article by concentrating on just a few of the topics that have come up in an evolving theory of magnetism. Hopefully, an even broader understanding of magnetic phenomena will follow new mathematics and mathematical concepts.

References

While not numbered, references/footnotes are listed in order of their appearance in the text.

  • A. Levinovitz and N. Ringertz, Eds., The Nobel Prize, the first 100 Years, Imperial College Press, London, 2001
  • Daniel C. Mattis, The Theory of Magnetism Made Simple: an introduction to physical concepts and to some useful mathematical methods, World Scientific Publ. Co., Singapore, 2006; the development of various aspects of magnetism and of its associated theories is the subject of chapters 1 and 2 and an extensive bibliography leads to the original documentation.
  • much of this material is discussed in greater depth in the various chapts. 3 – 9 of ref. 2.
  • P.A.M. Dirac, Proc. Roy. Soc. (London) A123, 60 (1931), also see pp. 76,78, ref. 2. The magnetic monopole has never been observed, but the existence of just one such monopole would necessitate quantization of all electric charges in the universe, a known fact of nature – and one that is otherwise unexplained.
  • ref. 2, p. 31 recounts the 1920’s history of this concept that culminated in the constant \(\mu_B =\frac{e \hbar}{2m_ec} = 0.927 x 10^{-20}\ ,\) in which \(m_e\) is the mass of the electron
  • ref. 2, chapt. 8.
  • H. Kramers and G. Wannier, Phys. Rev. 60, 252, 263 (1941)
  • L. Onsager, Phys. Rev. 65, 117 (1944)
  • T. Schultz, D. Mattis and E. Lieb, Rev. Mod. Phys. 36, 856 (1964)
  • see ref. 2, chapt. 3, §3.12
  • As we shall see later, this statement does not apply to the special case of one-dimensional antiferromagnets, in which the magnitude of the individual spins plays an important role.
  • N. Mermin and H. Wagner, Phys. Rev. Lett. 17, 1133 and 1307 (1966)
  • D. Mattis, Phys. Lett. 104, 357 (1984)
  • H. Bethe, Zeit. f. Physik 71, 205 (1931), reprinted in English translation in D. Mattis, The Many-Body Problem, an encyclopedia of exactly solved models in one dimension, World Scientific Publ., Singapore, 2009 (3rd Printing with revisions and corrections.)
  • E. Lieb, T. Schultz and D. Mattis, Ann. Phys. (NY) 16, 407 (1961), also reprinted in its entirety in The Many-Body Problem cited above. Among recent applications, note J. Jing and H. Ma, Level Crossing and Quantum Phase Transition of XY Ring, Mod. Phys. Lett. B22, 535 (2008)
  • W. Heisenberg, H. Wagner and K. Yamazaki, Nuov. Cim. LIX A, (1 Feb. 1969)
  • We list some early papers: P. W. Anderson, Heavy-electron superconductors, spin fluctuations and triplet pairing, Phys. Rev. B30, 1549 (1984), G. Baskaran and P.W. Anderson, Gauge theory of high-temperature superconductors and strongly correlated Fermi systems, Phys. Rev. B37, 580 (1988), P. Fazekas and E. Muller-Hartmann, Magnetic and nonmagnetic ground states of the Kondo lattice, Zeit. f. Phys. B85, 285 (1991), M. Sigrist, H. Tsunetsuga and K. Ueda, Rigorous results for the one-electron Kondo lattice model, Phys. Rev. Lett. 67, 2211 (1991), J.A. White, Numerical exact diagonalization of the one-dimensional symmetric Kondo lattice, Phys. Rev. B46, 13905 (1992), P.Paul and D. Mattis, Exctinction of spin interactions in the 2D Kondo lattice, Int. J.Mod. Phys. B24, 3199 (1995), H. Tsunetsuga, M. Sigrist and K. Ueda, The ground state phase diagram of the one dimensional Kondo lattice model, Rev. Mod. Phys. 69, 809-864 (1997), S. Capponi and F.F. Assaad, Spin and charge dynamics of the ferromagnetic and antiferromagnetic two-dimensional half-filled Kondo model, Phys. Rev. B63, 155114 (2001)
  • Z. Gulacsi and D. Vollhardt find this in a similar model (the periodic Anderson model) that can be solved exactly; see arXiv:cond-mat/0504174v1 (7 April, 2005)
  • G. Forgacs, Phys. Rev. B22, 4473 (1980). See also D. Mattis and R. Swendsen, Statistical Mechanics Made Simple, 2nd Edition, World Scientific Publ. Co, Singapore, 2008, §8.12, for the explicit solution of the transfer matrix in this example and for a discussion of unfrustrated (separable model) spin glasses.
  • N. Nagosa, Y. Hatsugai and M. Imada, J. Phys. Soc. Jpn. 58, 978 (1989), with further refs. given in ref 2., chapt. 5.
  • F.D.M. Haldane, Phys. Lett. A93, 454 (1983) and Phys. Rev. Lett. 50, 1153 (1983), see also: D. Controzzi and E. Hawkins, Int. J. Mod. Phys. B9, 4449 (2005)

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