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Physical review. B, Condensed matter

Exact ground state of a class of quantum spin systems

Ido Kanter

Published Apr 1, 1989• 17 citations
Publisher

Abstract

We present some exact results for the ground-state wave function and the energy of disordered quantum-mechanical spin systems in any dimensions. The system contains all nearest- and some of the next-nearest-neighbor interactions. The solution is also available for some range of anisotropy. In all the cases the ground-state wave function is insensitive to the disorder.

DOI
10.1103/physrevb.39.7270

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We present some exact results for the ground-state wave function and the energy of disordered quantum-mechanical spin systems in any dimensions. The system contains all nearest- and some of the next-nearest-neighbor interactions. The solution is also available for some range of anisotropy. In all the cases the ground-state wave function is insensitive to the disorder.
We discuss the ground state of the Heisenberg spin-1/2 chain in one dimension, considering nearest neighbor interactions. In the thermodynamic limit the energy is calculated analytically in several ways. We start by applying the Bethe ansatz to calculate the exact ground state energy of the XXX Heisenberg model. Next we study small perturbations of the spin anisotropy, the parameter for the interaction in the z-direction for the XXZ model, by applying time-independent perturbation theory. In addition we expand an exact integral expression of the energy of the XXZ model around the spin anisotropy in the thermodynamic limit. The results are compared to numerical solutions, which are obtained by means of exact diagonalization up to twelve lattice sites. We find that for ten or more lattice sites the numerics agree well with the expectation from the analytical calculations. Finally the concept of integrability is discussed in the context of the Heisenberg model.
The critical values of the ratio of interaction energies at which the ordering of spins in the ground state is indeterminate (and probably Tc=0{T}_{c}=0Tc​=0) are calculated exactly for a variety of Ising lattices with antiferromagnetic next-nearest-neighbor coupling, and for anisotropic nearest-neighbor coupling. Using these results and the exact two-dimensional solutions, the general dependence of the critical point on interaction ratio is sketched. The complex case of the antiferromagnetic fcc is reconsidered.
We investigate the existence and the properties of fully separable (fully factorized) ground states in quantum spin systems. Exploiting techniques of quantum information and entanglement theory, we extend a recently introduced method and construct a general, self-contained theory of ground state factorization in frustration-free quantum spin models defined on lattices in any spatial dimension and for interactions of arbitrary range. We show that, quite generally, nonexactly solvable translationally invariant models in presence of an external uniform magnetic field can admit exact, fully factorized ground state solutions. Unentangled ground states occur at finite values of the Hamiltonian parameters satisfying well-defined balancing conditions between the applied field and the interaction strengths. These conditions are analytically determined together with the type of magnetic orderings compatible with factorization and the corresponding values of the fundamental observables such as energy and magnetization. The method is applied to a series of examples of increasing complexity, including translationally invariant models with short, long, and infinite ranges of interaction, as well as systems with spatial anisotropies, in lower and higher dimensions. We also illustrate how the general method, besides yielding a large series of exact results for complex models in any dimension, recovers, as particular cases, the results previously achieved on simple models in low dimensions exploiting direct methods based on factorized mean-field ansatz.