Elliott H. Lieb

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.

Proof of the strong subadditivity of quantum-mechanical entropy

We prove several theorems about quantum‐mechanical entropy, in particular, that it is strongly subadditive.

Valence bond ground states in isotropic quantum antiferromagnets

Haldane predicted that the isotropic quantum Heisenberg spin chain is in a “massive” phase if the spin is integral. The first rigorous example of an isotropic model in such a phase is presented. The Hamiltonian has an exactSO(3) symmetry and is translationally invariant, but we prove the model has a unique ground state, a gap in the spectrum of the Hamiltonian immediately above the ground state and exponential decay of the correlation functions in the ground state. Models in two and higher dimension which are expected to have the same properties are also presented. For these models we construct an exact ground state, and for some of them we prove that the two-point function decays exponentially in this ground state. In all these models exact ground states are constructed by using valence bonds.

Theory of monomer-dimer systems

We investigate the general monomer-dimer partition function,P(x), which is a polynomial in the monomer activity,x, with coefficients depending on the dimer activities. Our main result is thatP(x) has its zeros on the imaginary axis when the dimer activities are nonnegative. Therefore, no monomer-dimer system can have a phase transition as a function of monomer density except, possibly, when the monomer density is minimal (i.e.x=0). Elaborating on this theme we prove the existence and analyticity of correlation functions (away fromx=0) in the thermodynamic limit. Among other things we obtain bounds on the compressibility and derive a new variable in which to make an expansion of the free energy that converges down to the minimal monomer density. We also relate the monomer-dimer problem to the Heisenberg and Ising models of a magnet and derive Christoffell-Darboux formulas for the monomer-dimer and Ising model partition functions. This casts the Ising model in a new light and provides an alternative proof of the Lee-Yang circle theorem. We also derive joint complex analyticity domains in the monomer and dimer activities. Our considerations are independent of geometry and hence are valid for any dimensionality.

The Thomas-Fermi theory of atoms, molecules and solids

Abstract We place the Thomas-Fermi model of the quantum theory of atoms, molecules, and solids on a firm mathematical footing. Our results include: (1) A proof of existence and uniqueness of solutions of the nonlinear Thomas-Fermi equations as well as the fact that these solutions minimize the Thomas-Fermi energy functional, (2) a proof that in a suitable large nuclear charge limit, the quantum mechanical energy is asymptotic to the Thomas-Fermi energy, and (3) control of the thermodynamic limit of the Thomas-Fermi theory on a lattice.

Exact Solution of a Many-Fermion System and Its Associated Boson Field

Luttinger’s exactly soluble model of a one-dimensional many-fermion system is discussed. We show that he did not solve his model properly because of the paradoxical fact that the density operator commutators [p(p), p(−p′)], which always vanish for any finite number of particles, no longer vanish in the field-theoretic limit of a filled Dirac sea. In fact the operators p(p) define a boson field which is ipso facto associated with the Fermi-Dirac field. We then use this observation to solve the model, and obtain the exact (and now nontrivial) spectrum, free energy, and dielectric constant. This we also extend to more realistic interactions in an Appendix. We calculate the Fermi surface parameter , and find: (i.e., there exists a sharp Fermi surface) only in the case of a sufficiently weak interaction.

Harmonic maps with defects

Two problems concerning maps ϕ with point singularities from a domain Ω C ℝ3 toS2 are solved. The first is to determine the minimum energy of ϕ when the location and topological degree of the singularities are prescribed. In the second problem Ω is the unit ball and ϕ=g is given on ∂Ω; we show that the only cases in whichg(x/|x|) minimizes the energy isg=const org(x)=±Rx withR a rotation. Extensions of these problems are also solved, e.g. points are replaced by “holes,” ℝ3,S2 is replaced by ℝN,SN−1 or by ℝN, ℝPN−1, the latter being appropriate for the theory of liquid crystals.

The Finite Group Velocity of Quantum Spin Systems

AbstractIt is shown that if Φ is a finite range interaction of a quantum spin system,τtΦ the associated group of time translations, τx the group of space translations, andA, B local observables, then

On the superradiant phase transition for molecules in a quantized radiation field: the dicke maser model

\mathop {\lim }\limits_{\begin{array}{*{20}c} {|t| \to \infty } \\ {|x| > \upsilon |t|} \\ \end{array} } ||[\tau _t^\Phi \tau _x (A),B]||e^{\mu (\upsilon )t} = 0