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arXiv:1309.1471 · Physical Review A

Fine structures in the spectrum of the open-boundary Heisenberg chain at large anisotropies

Auditya Sharma, Masudul Haque

Published Apr 14, 2014• 11 citations
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Abstract

At large anisotropies, the spectrum of the Heisenberg XXZ spin chain separates into ``bands'' with energies largely determined by the number of domain walls. The band structure is richer with open-boundary conditions: there are more bands and the bands develop intricate fine structures. We characterize and explain these structures and substructures in the open-boundary chain. The fine structures are explained using degenerate perturbation theory. We also present some dynamical consequences of these subband structures, through explicit time evolution of the wave function from initial states motivated by the fine-structure analysis.

DOI
10.1103/physreva.89.043608

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At large anisotropies, the spectrum of the Heisenberg XXZ spin chain separates into ``bands'' with energies largely determined by the number of domain walls. The band structure is richer with open-boundary conditions: there are more bands and the bands develop intricate fine structures. We characterize and explain these structures and substructures in the open-boundary chain. The fine structures are explained using degenerate perturbation theory. We also present some dynamical consequences of these subband structures, through explicit time evolution of the wave function from initial states motivated by the fine-structure analysis.
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.
Abstract We focus in this work on a numerical DMRG study of the one dimensional X X Z Heisenberg spin chain. We discuss the dependence of the energy spectrum, level crossings, magnetization, static structure factor, etc as a function of the magnetic field for finite size chains with periodic boundary condition and open boundary condition(OBC). We show that finite size effects lead to the emergence of a series of magnetization steps related to crossing of energy levels with unit change in the total z component of the spin. We also show interesting relations between the emergence of the magnetization steps and the appearance of domain walls in the local magnetization profiles with a fixed number of nodes. We connect the domain walls in the OBC case with the solitons of the sine-Gordon model with OBC.
Abstract: We consider the ground states of the ferromagnetic XXZ chain with spin up boundary conditions in sectors with a fixed number of down spins. This forces the existence of a droplet of down spins in the system. We find the exact energy and the states that describe these droplets in the limit of an infinite number of down spins. We also prove that the droplet states, which were known to be separated by an energy gap from the ground state, are also separated by an energy gap from the remainder of the spectrum, i.e., they form an isolated band. We also prove the analogous results for finite chains with periodic boundary conditions and for the infinite chain.
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.