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 exact SO(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. Table of Contents 1. Introduction 478 2. The One-Dimensional Model (1.1) 2.1. The Ground State 481 2.2. The Ground State Two-Point Correlation Function 485 2.3. The Energy Gap 487 2.4. The Infinite Chain 492 3. The Spin 3/2 Model on the Hexagonal Lattice 3.1. The Ground State 496 3.2. Properties of the Ground State 499 3.3. Proof of Exponential Decay 500 4. VBS States on an Arbitrary Lattice 4.1. The Ground States 504 4.2. The VBS State on the Cayley Tree 506 4.3. SU{n) VBS State and Random Loop Representation 510 4.4. SU(n) Quantum "Spin" System 514

Derivation of the canonical (or Boltzmann) distribution based only on quantum dynamics is discussed. Consider a closed system which consists of mutually interacting subsystem and heat bath, and assume that the whole system is initially in a pure state (which can be far from equilibrium) with small energy fluctuation. Under the "hypothesis of equal weights for eigenstates", we derive the canonical distribution in the sense that, at sufficiently large and typical time, the (instantaneous) quantum mechanical expectation value of an arbitrary operator of the subsystem is almost equal to the desired canonical expectation value. We present a class of examples in which the above derivation can be rigorously established without any unproven hypotheses.It is often said that the principles of equilibrium statistical physics have not yet been justified. It is not clear, however, what statement should be regarded as the ultimate justification. Recalling the astonishingly universal applicability of equilibrium statistical physics, it seems likely that there are many independent routes for justification which can be equally convincing and important [1,2]. In the present paper, we concentrate on one of the specific scenarios for obtaining canonical distributions from quantum dynamics [3].Outline of the work: Let us outline our problem and the main result. We consider an isolated quantum mechanical system which consists of a subsystem and a heat bath. The subsystem is described by a Hamiltonian H S which have arbitrary nondegenerate eigenvalues ε 1 , . . . , ε n . For convenience we let ε j+1 > ε j and ε 1 = 0. The heat bath is described by a Hamiltonian H B with the density of states ρ(B). The inverse temperature of the heat bath at energy B is given by the standard formula β(B) = d log ρ(B)/dB. We assume (as usual) β(B) is positive and decreasing in B. The density of states ρ(B) is arbitrary except for a fine structure that we will impose on the spectrum of H B .The coupling between the subsystem and the heat bath is given by a special Hamiltonian H ′ which almost conserves the unperturbed energy and whose magnitude is H ′ ∼ λ. We assume ∆ε ≫ λ ≫ ∆B, where ∆ε is the minimum spacing of the energy levels of H S , and ∆B is the maximum spacing of that of H B . These conditions guarantee a weak coupling between the subsystem and the bath, as well as macroscopic nature of the bath. The Hamiltonian of the whole system is H = H S ⊗ 1 B + 1 S ⊗ H B + H ′ , where 1 S and 1 B are the identity operators for the subsystem and the heat bath, respectively.

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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 exact SO(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. Table of Contents 1. Introduction 478 2. The One-Dimensional Model (1.1) 2.1. The Ground State 481 2.2. The Ground State Two-Point Correlation Function 485 2.3. The Energy Gap 487 2.4. The Infinite Chain 492 3. The Spin 3/2 Model on the Hexagonal Lattice 3.1. The Ground State 496 3.2. Properties of the Ground State 499 3.3. Proof of Exponential Decay 500 4. VBS States on an Arbitrary Lattice 4.1. The Ground States 504 4.2. The VBS State on the Cayley Tree 506 4.3. SU{n) VBS State and Random Loop Representation 510 4.4. SU(n) Quantum "Spin" System 514

No abstract