We consider the highly anisotropic ferromagnetic spin 1/2 Heisenberg chain with periodic boundary conditions. In each sector of constant total z component of the spin, we develop convergent expansions for the lowest band of eigenvalues and eigenfunctions. These eigenstates describe droplet states in which the spins essentially form a single linear droplet which can move. Our results also give a convergent expansion for the dispersion relation, i.e., the energy of the droplet as a function of its momentum. The methods used are from math-ph/0208026 and cond-mat/0104199, and this short paper should serve as a pedagogic introduction to those papers.
We investigate the effect of a magnetic field supported at a single lattice site on the low-energy spectrum of the ferromagnetic Heisenberg XXZ chain. Such fields, caused by impurities, can modify the low-energy spectrum significantly by pinning certain excitations, such as kink and droplet states. We distinguish between different boundary conditions (or sectors), the direction and also the strength of the magnetic field. E.g., with a magnetic field in the z direction applied at the origin and ++ boundary conditions, there is a critical field strength
Bc (which depends on the anisotropy of the Hamiltonian and the spin value) with the following properties: for
B<Bc there is a unique ground state with a gap, at the critical value
Bc there are infinitely many (droplet) ground states with gapless excitations, and for
B>Bc there is again a unique ground state but now belonging to the continuous spectrum. In contrast, any magnetic field with a nonvanishing component in the
xy plane yields a unique ground state, which, depending on the boundary conditions, is either an (anti)kink, or an (anti)droplet state. For such fields, i.e., not aligned with the z axis, excitations always have a gap and we obtain a rigorous lower bound for that gap.
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.
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.
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 entanglement spectrum (ES) of a finite XXZ spin- chain in the limit Δ → −1 + for both open and periodic boundary conditions. At Δ =− 1 (ferromagnetic point) the model is equivalent to the Heisenberg ferromagnet and its degenerate ground state manifold is the S U (2) multiplet with maximal total spin. Any state in this so-called 'symmetric sector' is an equal weight superposition of all possible spin configurations. In the gapless phase at Δ >− 1 this property is progressively lost as one moves away from the Δ =− 1 point. We investigate how the ES obtained from the states in this manifold reflects this change, using exact diagonalization and Bethe ansatz calculations. We find that in the limit Δ → −1 + most of the ES levels show divergent behavior. Moreover, while at Δ =− 1 the ES contains no information about the boundaries, for Δ >− 1 it depends dramatically on the choice of boundary conditions. For both open and periodic boundary conditions the ES exhibits an elegant multiplicity structure for which we conjecture a combinatorial formula. We also study the entanglement eigenfunctions, i.e. the eigenfunctions of the reduced density matrix. We find that the eigenfunctions corresponding to the non-diverging levels mimic the behavior of the state wavefunction, whereas the others show intriguing polynomial structures. Finally we analyze the distribution of the ES levels as the system is detuned away from Δ =− 1.