田﨑 晴明

We study a class of Hubbard models in which the corresponding single-electron ground states have bulk degeneracy. We prove that the ground states of the models exhibit ferromagnetism when the electron filling factor is not more than and sufficiently close to rho-0 = Absolute value of V /2 Absolute value of LAMBDA (where Absolute value of V is the dimension of degeneracy and Absolute value of LAMBDA is the number of sites), and exhibit paramagnetism when the filling factor is sufficiently small. This is the first time that a three-dimensional itinerant-electron system is proved to exhibit ferromagnetism in a finite range of the electron filling factor.

We study the S = 1/2 spin chain with the Hamiltonian H = SIGMA(j=1)L-1 {sigma-2j(x)sigma-2j+1x+sigma-2j(y)sigma-2j+1y+lambda-sigma-2j(z)sigma-2j+1} + beta-SIGMA(j=1)L(1-sigma-2j-1.sigma-2j), where sigma(i) = (sigma(i)x, sigma(i)y, sigma(i)z) are the Pauli matrices. We find that a nonlocal unitary transformation reveals the hidden Z2 X Z2 symmetry of the system. It has been argued that a similar hidden Z2 X Z2 Symmetry of the S = 1 chain is fully broken when and only when the system exhibits the Haldane pp. We prove that the present system exhibits both an excitation gap and a full breaking of the hidden Z2 X Z2 symmetry in a range of the parameter space including the line beta = 0, lambda > -1. We argue that the range with such properties indeed extends to the limit beta --> infinity in which the present model reduces to the S = 1 spin chain. This observation provides support of Hida's conclusion that the Haldane gap in the S = 1 chain is continuously connected to the gap in the decoupled S = 1/2 system with beta = 0.

We study the phase diagram of S = 1 antiferromagnetic chains with particular emphasis on the Haldane phase. The hidden symmetry breaking measured by the string order parameter of den Nijs and Rommelse can be transformed into an explicit breaking of a Z2 x Z2 symmetry by a nonlocal unitary transformation of the chain. For a particular class of Hamiltonians which includes the usual Heisenberg Hamiltonian, we prove that the usual Neel order parameter is always less than or equal to the string order parameter. We give a general treatment of rigorous perturbation theory for the ground state of quantum spin systems which are small perturbations of diagonal Hamiltonians. We then extend this rigorous perturbation theory to a class of "diagonally dominant" Hamiltonians. Using this theory we prove the existence of the Haldane phase in an open subset of the parameter space of a particular class of Hamiltonians by showing that the string order parameter does not vanish and the hidden Z2 x Z2 symmetry is completely broken. While this open subset does not include the usual Heisenberg Hamiltonian, it does include models other than VBS models.

We study the phase diagram of S = 1 antiferromagnetic chains with particular emphasis on the Haldane phase. The hidden symmetry breaking measured by the string order parameter of den Nijs and Rommelse can be transformed into an explicit breaking of a Z2 x Z2 symmetry by a nonlocal unitary transformation of the chain. For a particular class of Hamiltonians which includes the usual Heisenberg Hamiltonian, we prove that the usual Neel order parameter is always less than or equal to the string order parameter. We give a general treatment of rigorous perturbation theory for the ground state of quantum spin systems which are small perturbations of diagonal Hamiltonians. We then extend this rigorous perturbation theory to a class of "diagonally dominant" Hamiltonians. Using this theory we prove the existence of the Haldane phase in an open subset of the parameter space of a particular class of Hamiltonians by showing that the string order parameter does not vanish and the hidden Z2 x Z2 symmetry is completely broken. While this open subset does not include the usual Heisenberg Hamiltonian, it does include models other than VBS models.

We show that the Haldane phase of the S = 1 antiferromagnetic chain is closely related to the breaking of a hidden Z2 X Z2 symmetry. When the chain is in the Haldane phase, this Z2 X Z2 Symmetry is fully broken, but when the chain is in a massive phase other than the Haldane phase, e.g., the Ising phase or the dimerized phase, this symmetry is broken only partially or not at all. The hidden Z2 X Z2 symmetry is revealed by introducing a nonlocal unitary transformation of the chain. This unitary transformation also leads to a simple variational calculation which qualitatively reproduces the phase diagram of the S = 1 chain.

We study the S = 1 antiferromagnetic Heisenberg chain with uniaxial anisotropy, which exhibits the Haldane gap phenomenon. We argue that, when a magnetic field in a direction transverse to the anisotropy axis is applied, a sharp phase transition takes place at a critical value of the field. The transition is characterized by vanishing of the hidden antiferromagnetic order which can be measured by the string order parameter of den Nijs and Rommelse. The existence of such a transition distinguishes a Haldane gap system from other disordered quantum spin systems.

The author studies the S=1 antiferromagnetic Heisenberg chain with uniaxial anisotropy, which exhibits the Haldane gap phenomenon. He argues that, when a magnetic field in a direction transverse to the anisotropy axis is applied, a sharp phase transition takes place at a critical value of the field. The transition is characterized by vanishing of the hidden antiferromagnetic order which can be measured by the string order parameter of den Nijs and Rommelse. The existence of such a transition distinguishes a Haldane gap system from other disordered quantum spin systems.

A quasi-one-dimensional S=1 antiferromagnet with XXZ-type intrachain couplings and Ising-type interchain couplings is studied. It has been suggested that when the interchain couplings are sufficiently small such a system has no long-range order, even at T=0, because of strong quantum fluctuation caused by the Haldane gap. Treating the problem in a restricted Hilbert space which describes the low-energy behavior of S=1 antiferromagnets rather accurately, we prove two theorems which establish the existence of a unique disordered ground state with a gap. © 1990 The American Physical Society.

We consider (spin-s generalizations of) Anderson's short-range resonating-valence-bond (RVB) state and derive rigorous results that suggest that the properties of the RVB state depend crucially on the basic lattice structures. In an arbitrary bipartite lattice with coordination number z, we prove that the spin-s RVB state has exponentially decaying correlation function when z1+ 2s+1. We also prove that the spin-s RVB state on the Cayley tree with coordination number z exhibits long-range Néel order when z&gt 2s+2 but does not when z2s+2. © 1989 The American Physical Society.