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arXiv:math-ph/0110017 · Advances in Theoretical and Mathematical Physics

The spectral gap for the ferromagnetic spin-J XXZ chain

Tohru Koma, Bruno Nachtergaele, Shannon L. Starr

Published Jan 1, 2001• 26 citations
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Abstract

We investigate the spectrum above the kink ground states of the spin J ferromagnetic XXZ chain with Ising anisotropy A. Our main theorem is that there is a non-vanishing gap above all ground states of this model for all values of J. Using a variety of methods, we obtain additional information about the magnitude of this gap, about its behavior for large A, about its overall behavior as a function of A and its dependence on the ground state, about the scaling of the gap and the structure of the low-lying spectrum for large J, and about the existence of isolated eigenvalues in the excitation spectrum.

DOI
10.4310/atmp.2001.v5.n6.a3

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We investigate the spectrum above the kink ground states of the spin J ferromagnetic XXZ chain with Ising anisotropy A. Our main theorem is that there is a non-vanishing gap above all ground states of this model for all values of J. Using a variety of methods, we obtain additional information about the magnitude of this gap, about its behavior for large A, about its overall behavior as a function of A and its dependence on the ground state, about the scaling of the gap and the structure of the low-lying spectrum for large J, and about the existence of isolated eigenvalues in the excitation spectrum.
We investigate the low-lying excited states of the spin J ferromagnetic XXZ chain with Ising anisotropy Δ and kink boundary conditions. Since the third component of the total magnetization, M , is conserved, it is meaningful to study the spectrum for each fixed value of M . We prove that for J ≥3/2 the lowest excited eigenvalues are separated by a gap from the rest of the spectrum, uniformly in the length of the chain. In the thermodynamic limit, this means that there are a positive number of excitations above the ground state and below the essential spectrum.
In this thesis we present three results about the ferromagnetic quantum XXZ model: 1) Existence of a spectral gap above all infinite-volume ground states in one dimension for any choice of spin S>1/2 (for S=1/2 this was already known); 2) Characterization of one-dimensional "droplet" states as ground states of the spin-1/2 model with up spin boundary fields; 3) An upper bound for the spectral gap above the spin-1/2 interface ground states in dimension two and higher. The upper bound vanishes as 1/R^2, where R is the linear size of the system. This thesis contains research performed in collboration with other authors: O. Bolina, P. Contucci, T. Koma, B. Nachtergaele.
We study the structure of the spectrum of the infinite XXZ quantum spin chain, an anisotropic version of the Heisenberg model. The XXZ chain Hamiltonian preserves the number of down spins (or particle number), allowing to represent it as a direct sum of N-particle interacting discrete Schrödinger-type operators restricted to the fermionic subspace. In the Ising phase of the model we use this representation to give a detailed determination of the band and gap structure of the spectrum at low energy. In particular, we show that at sufficiently strong anisotropy the so-called droplet bands are separated from higher spectral bands uniformly in the particle number. Our presentation of all necessary background is self-contained and can serve as an introduction to the mathematical theory of the Heisenberg and XXZ quantum spin chains.
For a class of Hamiltonians of XXZ spin chains in a uniform external magnetic field that are small quantum perturbations of an Ising Hamiltonian, it is shown that the spectral gap above the ground-state energy remains strictly positive when the perturbation is turned on, uniformly in the length of the chain. This result is proven for rather generic perturbations of both the ferromagnetic and the antiferromagnetic Ising Hamiltonian. In the antiferromagnetic case, the external magnetic field is required to be small. For a chain of an even number of sites, the two-fold degenerate ground-state energy of the unperturbed antiferromagnetic Hamiltonian may split into two energy levels separated by a very small gap. These results are proven by using a new, quite subtle refinement of a method developed in earlier work and used to iteratively block-diagonalize Hamiltonians of systems confined to ever larger subsets of a lattice by using strictly local unitary conjugations. The new method developed in this paper provides complete control of boundary effects on the low-energy spectrum of perturbed Ising chains uniformly in their length.
We study the structure of the spectrum of the infinite XXZ quantum spin chain, an anisotropic version of the Heisenberg model. The XXZ chain Hamiltonian preserves the number of down spins (or particle number), allowing to represent it as a direct sum of N-particle interacting discrete Schrödinger-type operators restricted to the fermionic subspace. In the Ising phase of the model we use this representation to give a detailed determination of the band and gap structure of the spectrum at low energy. In particular, we show that at sufficiently strong anisotropy the so-called droplet bands are separated from higher spectral bands uniformly in the particle number. Our presentation of all necessary background is self-contained and can serve as an introduction to the mathematical theory of the Heisenberg and XXZ quantum spin chains.