Critical conditions for ferromagnetism in the localized-electron theory with nonorthogonality: a controlled microscopic determination

Van Vleck's 1936 analysis of nonorthogonality in the localized (Heitler–London) theory of ferromagnetism showed that inter-atomic overlap integrals enter the many-atom exchange couplings and render uncertain any attempt to deduce the exact critical conditions — the minimum number of interacting neighbors, the total-spin sector, and the lattice structure — necessary for ferromagnetic order. The surviving open core is the controlled microscopic determination of these critical conditions with nonorthogonality included: a treatment of the many-electron secular problem for arbitrary lattice and arbitrary total-spin sector in which the overlap-induced corrections to the effective spin Hamiltonian are bounded rather than heuristically estimated. The literature lineage shows that the orthogonal-effective-model side of the question (coordination thresholds and finite-temperature order) has been subsumed into rigorous statistical mechanics, where the controlling parameter is dimensionality rather than coordination number; but the microscopic question — whether the mapping from the nonorthogonal localized theory to an effective spin Hamiltonian is controlled at the overlaps relevant to real ferromagnets, and what critical conditions it implies — remains open, and is the local-moment instance of the still-open microscopic origin of ferromagnetism.

Background

In the Heitler–London or localized picture, a magnetic solid is modeled as electrons localized on atoms, with their spins coupled by exchange — an effective spin-spin interaction of purely quantum-mechanical origin, a consequence of the Coulomb repulsion combined with the Pauli exclusion principle. Heisenberg (1928) introduced this approach to ferromagnetism and computed the exchange coupling with the help of the Gaussian approximation, in which Gaussian-type functions replace the atomic orbitals so that the exchange integrals become tractable; within this method ferromagnetism requires at least eight interacting neighbors. Bloch (1930) reached a similar threshold from spin-wave theory, which describes a magnet's low-lying excitations as waves of spin deviation: for spin $S=1/2$ he required $z > 8$, where $z$ is the coordination number, the number of equivalent nearest neighbors of an atom. These calculations were questioned by Inglis (1934), who stressed that Heitler–London secular calculations for a very large number of electrons assume mutually orthogonal atomic wave functions, so that the nonorthogonality corrections may accumulate without limit as the system grows. Van Vleck's 1936 paper 'Nonorthogonality and Ferromagnetism' took up this objection: he proved that the accumulation is only apparent, but found that nonorthogonality still undermines the critical criteria. His model is 'a system of $n$ electrons in $n$ distinct states', one state per atom, treated as a problem of exchange degeneracy only: 'Every atom must then be in an s state, as otherwise there would be the spatial degeneracy to contend with', so each atom carries a single electron (per-atom spin $1/2$) and the varying spin quantity is the total spin $S'$ of the whole crystal. Coupling is retained only between adjacent atoms, and he represented the many-electron wave function as a linear combination of permutations of the localized orbitals, so the energy problem becomes the secular equation $\det K = 0$ — the finite matrix eigenvalue problem for the expansion coefficients — whose matrix $K$ (Eqs. (2)–(5)) is built from the exchange integrals $V^{(r)}$ and the nonorthogonality (overlap) integrals $\Delta^{(r)}$ associated with each permutation, with higher-order, multi-site exchange integrals entering through cycle expansions; permutations that do not factor into simple permutations with distinct atoms in each cycle are discarded, their $V$ and $\Delta$ integrals being set to zero (Eqs. (14)–(15), described in the paper as 'slight specializations of the model'). His central findings were: (i) the influence of nonorthogonality is sufficient to render uncertain any attempt to deduce exactly the critical conditions for ferromagnetism (in the abstract's words, 'minimum number of neighbors, etc.'); (ii) the correction factor $1+f(x)$ (Eqs. (27)–(28)), a multiplicative nonorthogonality correction, is inaccurate because it replaces the mean value of a quotient of two matrices by the quotient of the means — here $x$ is the variable defined in Eq. (26), related to but not equal to the relative magnetization $2S'/n$ (the ratio of the intensity of magnetization at temperature $T$ to the true saturation value), with $x=-1$ in the saturation case $S'=\tfrac12 n$, and the paper itself notes that the rescaled eigenvalue $y=W/(\tfrac12 nsJ)$ is a better argument for $f$; (iii) the factorization approximations for higher-order exchange integrals introduce errors of order $2z\delta$ (with $\delta$ the nearest-neighbor overlap integral) that are only heuristically argued to reduce to $2z\delta^2$ by a special choice of effective potential; and (iv) an alternative partition-function method, in which the thermodynamics is evaluated by organizing the permutation sum into cycles, gives a different and worse criterion than the Gaussian method ($z \ge 16$ under the orthogonality assumption, versus $z \ge 8$ for the Gaussian method), leaving open which approximation is more reliable. In modern language, the paper's central point is that the mapping from the microscopic localized theory to an effective orthogonal spin Hamiltonian (the Heisenberg model, acting only on spin degrees of freedom) is not controlled when the overlaps are non-negligible.

The subsequent literature developed three responses. First, Löwdin (1950) introduced symmetric orthogonalization, a canonical transformation that replaces the overlapping atomic basis by an orthogonal one at the cost of transformed, generally multi-center effective interactions; this route underlies tight-binding and Wannier-function methods (Wannier functions being localized lattice orbitals built from the single-electron band states) and replaces the nonorthogonal problem by an orthogonal one, without by itself supplying quantitative error bounds for the nonorthogonal treatment. Second, Herring (1962), in the definitive review of the Heitler–London method at large inter-atomic separation, gave a rigorous justification of the spin-Hamiltonian concept and a prescription for exact a priori calculation of the limiting exchange coupling, supplementing the Lieb–Mattis theorem (1962) — for spin-independent interactions the ground state has the minimal total spin — and the exact solution of the one-dimensional spin-$1/2$ chains (Lieb, Schultz and Mattis 1961) with a surface-integral sign analysis showing that antiferromagnetic couplings should be the rule and ferromagnetic ones the exception; but his results hold only in the small-overlap regime of well-separated atoms. Third, the rigorous statistical mechanics of the resulting effective orthogonal models established that, at the model level, finite-temperature ordering is governed by dimensionality rather than coordination number: the fully aligned state is a ground state for any lattice with ferromagnetic coupling (each bond is minimized by parallel spins), continuous-symmetry order is rigorously absent in one and two dimensions at $T>0$ (Mermin–Wagner 1966), and low-temperature order is established in $d\ge 3$ — for classical counterparts by the infrared-bound and reflection-positivity methods of rigorous statistical mechanics (Fröhlich–Simon–Spencer 1976; Fröhlich–Israel–Lieb–Simon 1978), and for the quantum Heisenberg model by Dyson–Lieb–Simon (1976, 1978). Separately, van Vleck's own 1945 survey recorded that the $z\ge 8$-type criteria are artifacts of the approximation schemes — ferromagnetic alloys being known on the simple cubic lattice with $z=6$ — and that higher approximations depend on structural details beyond the neighbor count. The underlying question, the microscopic origin of ferromagnetism — whether spin-independent Coulomb interactions in an electron system can produce strong ferromagnetic order — remains open in the modern rigorous literature, as documented in Tasaki's reviews of the Hubbard model (the minimal model of interacting electrons on a lattice, with hopping between sites and on-site repulsion $U$): rigorous ferromagnetism is known only in special limits (Nagaoka's theorem for a single hole at $U=\infty$; flat-band models, whose lowest single-electron band is dispersionless; and multi-band models with a rigorous example of metallic ferromagnetism, as in Tanaka–Tasaki 2007), and 'neither of these two approaches [the localized Heisenberg picture and the itinerant Bloch/Hartree-Fock picture] has yet produced a truly convincing explanation about the origin of ferromagnetism' (Tasaki 1998).

Problem Statement

Determine, within the localized (Heitler–London) microscopic theory of a ferromagnet, the critical conditions for the appearance of ferromagnetic order when nonorthogonality (nonzero inter-atomic overlap) is controllably included. The microscopic model is that of the 1936 paper: $n$ atoms on a lattice, one electron per atom in an $s$ state (so the per-atom spin is $1/2$ and the spin variable of the crystal is its total spin $S'$), nonorthogonal localized orbitals, coupling retained between adjacent atoms, nearest-neighbor overlap integral $\delta$ and coordination number $z$; the many-electron problem is the secular equation $\det K = 0$ over permutations of the localized orbitals, with exchange and overlap integrals $V^{(r)}$, $\Delta^{(r)}$ as matrix elements. Concretely: (1) determine, for an arbitrary lattice of this class and arbitrary total-spin sector $S'$, whether the overlap-induced corrections to the effective spin Hamiltonian admit a rigorous bound in a treatment of the many-atom secular problem, and, if they do, construct that treatment: the quantity being bounded and the dependence of the bound on $n$, $z$, $\delta$ and $S'$ must be stated explicitly, and the bound must be uniform in the number of atoms $n$ at fixed $z$, $\delta$ and relative magnetization $2S'/n$ (this is the content of the Inglis 'catastrophe' that the 1936 paper resolved only by heuristic estimates). A determination that the corrections cannot be so bounded at the overlaps relevant to real ferromagnets likewise resolves this part of the question. The approximations whose errors are to be controlled, rather than assumed away, include the idealizations of the 1936 scheme — in its own formalism, the correction factor $1+f(x)$ of Eqs. (27)–(28), obtained by replacing the mean value of a quotient of two matrices by the quotient of the means, with $x$ the variable of Eq. (26) ($x=-1$ at saturation, $S'=\tfrac12 n$; distinct from the relative magnetization $2S'/n$); the term of order $k/n$ omitted from the cycle recurrence (30), where $k$ is the number of constituent simple permutations (each involving different atoms) into which a retained permutation is factored, so that $k$ labels the permutations of degree $2k$; and the factorization error for higher-order exchange integrals, currently argued only heuristically to be of order $2z\delta^2$ rather than $2z\delta$. A treatment that does not use the permutation/cycle formalism at all is admissible provided it delivers bounds on the overlap-induced corrections to the effective spin Hamiltonian in the sense just stated. (2) When the corrections admit such a bound, derive from the controlled treatment the critical conditions for ferromagnetic order that it implies to within that bound — the minimum number of interacting neighbors (if such a minimum exists), the role of the spin (in this model the total spin $S'$ of the crystal, equivalently the relative magnetization $2S'/n$, since the per-atom spin is fixed at $1/2$), and the lattice structure — replacing the mutually inconsistent criteria $z \ge 8$ (Heisenberg's Gaussian method) and $z \ge 16$ (the partition-function method under the orthogonality assumption). Throughout, 'ferromagnetic order' means spontaneous magnetization at finite temperature in the thermodynamic limit — the existence, at zero applied field, of a nonzero solution of the magnetization equation, equivalently a nonzero Curie temperature — which is the sense in which the two 1936 criteria were derived (both follow from requiring $dQ/dx \ge 1$ at $x=-\tfrac12$, the paramagnetic limit $S'/n\to 0$, for the transcendental magnetization equation (33) of the paper). The question is open for the physically relevant regime where the overlap is not negligible: for the 1936 paper this is $\delta$ of order $0.1$ with $z=8$ or $12$, 'a value not unreasonable in ferromagnetic applications'. Bounds that hold only in the limit of vanishing overlap (equivalently, arbitrarily large lattice spacing) do not answer it — that small-overlap regime is settled by Herring's rigorous spin-Hamiltonian theory — and the orthogonal effective model alone, where any coordination number orders at $T=0$ and dimensionality governs finite-temperature order, does not answer whether nonorthogonality changes the mapping and hence the critical conditions.

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