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arXiv:math-ph/0307051 · Markov Processes And Related Fields

The large-spin asymptotics of the ferromagnetic XXZ chain

Tom Michoel, Bruno Nachtergaele

Published Jul 24, 2003• 8 citations
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Abstract

We present new results and give a concise review of recent previous results on the asymptotics for large spin of the low-lying spectrum of the ferromagnetic XXZ Heisenberg chain with kink boundary conditions. Our main interest is to gain detailed information on the interface ground states of this model and the low-lying excitations above them. The new and most detailed results are obtained using a rigorous version of bosonization, which can be interpreted as a quantum central limit theorem.

DOI
10.48550/arxiv.math-ph/0307051

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We present new results and give a concise review of recent previous results on the asymptotics for large spin of the low-lying spectrum of the ferromagnetic XXZ Heisenberg chain with kink boundary conditions. Our main interest is to gain detailed information on the interface ground states of this model and the low-lying excitations above them. The new and most detailed results are obtained using a rigorous version of bosonization, which can be interpreted as a quantum central limit theorem.
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.
The exact expression for the density matrix of the kink ground state of the ferromagnetic XXZ chain is obtained. Utilizing this, we exactly calculate various correlation functions such as the longitudinal and transverse spin-spin correlation functions, and the ferromagnetic and antiferromagnetic string formation probabilities. The asymptotic behaviors of these correlation functions are also analyzed. As a consequence, we find that the spin-spin correlation functions decay exponentially for large distances, while the string formation probabilities exhibit Gaussian decay for large strings. We also evaluate the entanglement entropy, which shows interesting behaviors due to the lack of the translational invariance of the state.
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 review some recent advances in the computation of exact correlation functions of the XXZ-½ Heisenberg chain. We first give a general introduction to our method which is based on the algebraic Bethe ansatz and the resolution of the quantum inverse scattering problem, leading in particular to multiple integral representations for the correlation functions. Then we describe recently obtained compact formulas for the spin-spin correlation functions of the XXZ-½ Heisenberg chain. We outline how this leads to several explicit results including the known two point functions in the limit of free fermions, the so-called emptiness formation probability at anisotropy Δ=1/2 and its large distance asymptotic behaviour in the massless phase of the model.
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.
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.