Daniel C. Mattis

Two Soluble Models of an Antiferromagnetic Chain

Two genuinely quantum mechanical models for an antiferromagnetic linear chain with nearest neighbor interactions are constructed and solved exactly, in the sense that the ground state, all the elementary excitations and the free energy are found. A general formalism for calculating the instantaneous correlation between any two spins is developed and applied to the investigation of short- and long-range order. Both models show nonvanishing long-range order in the ground state for a range of values of a certain parameter X which is analogous to an anisotropy parameter in the Heisenberg model. A detailed comparison with the Heisenberg model suggests that the latter has no long-range order in the isotropic case but finite long-range order for any finite amount of anisotropy. The unreliability of variational methods for determining long-range order is emphasized. It is also shown that for spin ½ systems having rather general isotropic Heisenberg interactions favoring an antiferromagnetic ordering, the ground state is nondegenerate and there is no energy gap above the ground state in the energy spectrum of the total system.

Ordering Energy Levels of Interacting Spin Systems

The total spin S is a good quantum number in problems of interacting spins. We have shown that for rather general antiferromagnetic or ferrimagnetic Hamiltonians, which need not exhibit translational invariance, the lowest energy eigenvalue for each value of S [denoted E(S)] is ordered in a natural way. In antiferromagnetism, E(S + 1) > E(S) for S > O. In ferrimagnetism, E(S + 1) > E(S) for S > S, and in addition the ground state belongs to S < S. S is defined as follows: Let the maximum spin of the A sublattice be S A and of the B sublattice S B; then S ≡ S A − S B. Antiferromagnetism is treated as the special case of S = O. We also briefly discuss the structure of the lowest eigen-functions in an external magnetic field.

Solvable spin systems with random interactions

Abstract It is shown that if the bonds connecting spins on a lattice are separable functions of random variables, the thermodynamic and magnetic parameters may be obtained using the known properties of a spin system with non-random bonds.

New wave‐operator identity applied to the study of persistent currents in 1D

We show that a large class of backward‐scattering matrix elements involving Δk ∼ ± 2kF vanish for fermions interacting with two‐body attractive forces in one dimension. (These same matrix elements are finite for noninteracting particles and infinite for particles interacting with two‐body repulsive forces.) Our results demonstrate the possibility of persistent currents in one dimension at T = 0, and are a strong indication of a metal‐to‐insulator transition at T = 0 for repulsive forces. They are obtained by use of a convenient representation of the wave operator in terms of density‐fluctuation operators.

Ground state energy of Hubbard model

Abstract We find rigorous upper and lower bounds to the ground state energy of the Hubbard-Gutzwiller model.

EXACT WAVE FUNCTIONS IN SUPERCONDUCTIVITY

The ground‐state wave function and some of the excited states of the BCS reduced Hamiltonian are found. In the limit of large volume, the boundary and continuity conditions on the exact wave function lead directly to the equations which Bardeen, Cooper, and Schrieffer found by a variational technique. It is also shown in what sense the BCS trial wave function may be considered asymptotically exact in this limit. Finite‐volume corrections are included in an appendix, and explicit calculations are carried out for a one‐step model of the kinetic energy which has possible applications to the problem of the finite nucleus.

Theory of Paramagnetic Impurities in Semiconductors

In this paper, a model of a paramagnetic impurity in a semiconductor (or of an F′ center in an alkali halide) is proposed. It is an exactly soluble form of the quantum‐mechanical 3‐body problem. Specifically, we deal with 2 interacting particles in any number of dimensions in an attractive external potential, and present the qualitative features of the resulting eigenvalues and eigenfunctions. We find algebraically the conditions for a magnetic moment to appear (e.g., for an F′ center to become unstable with respect to an F center) and discover that even a large 2‐body electronic repulsion U does not cause a moment to appear when the one‐electron bound state orbits about the impurity are sufficiently great. Conversely, in the case of small, tightly bound orbits, beyond a certain value of U, the impurity does in fact become magnetic in the ground state. Using the exact ground‐state solution, we show that a perturbation‐theoretic expansion in powers of U has a finite radius of convergence.

New aspects of polyconductivity

Abstract This paper is a review of some of our current work on diverse aspects of “switchable” materials. We discuss the instability of the metallic state against the insulating state in certain materials, and introduce the concept of an “energy gap of the second kind” to characterize such “polyconductors”. Interesting effects such as negative resistance, semiconductor technology with a variable energy gap, laser beam modulation, high temperature superconductivity, electronic switching, etc., are discussed in this context. Computer solutions of some of our nonlinear characteristics are shown.

Electron states near surfaces of solids

Abstract The mathematical theory of electron eigenstates near the surfaces of solids is developed in stages. The first stage is the study of the eigenstates of solids which terminate at a surface but are otherwise unperturbed; it is proved that near the surface the three-dimensional band edges are “softened” and that van Hoves singularities in the density of states are eliminated. A set of “vacuum states” lying primarily outside the solid is constructed out of plane waves orthogonalized to the eigenstates of the solid. This set of vacuum states is not orthonormal, and it must be orthonormalized by a procedure different from that of Schmidt (which is unsuitable). Effects of surface perturbations are studied. An exact method is elaborated for obtaining eigenfunctions in closed form, for a variety of perturbing potentials that extend arbitrary distances from the surface and include interband matrix elements. It consists of calculating the effects of one surface layer at a time and cumulating the results. The intrinsic instability of certain surfaces against the formation of bands of surface states is shown to be the consequence of the vanishing of a bulk quantity α zz , one of the components of the inverse-effective mass tensor, at certain points in the B.Z.

Exactly Soluble Model of Interacting Electrons

We diagonalize a many‐fermion Hamiltonian consisting of terms quadratic as well as quartic in the field operators. A dual spectrum of eigenstates is an interesting result. We also derive a formula for obtaining the free energy at finite temperature.