Entanglement spectrum of one-dimensional extended Bose-Hubbard models

Abstract

The entanglement spectrum provides crucial information about correlated quantum systems. We show that the study of the block-like nature of the reduced density matrix in number sectors and the partition dependence of the spectrum in finite systems leads to interesting unexpected insights, which we illustrate for the case of a 1D extended Hubbard model. We show that block symmetry provides an intuitive understanding of the spectral double degeneracy of the Haldane-insulator, which is remarkably maintained at low on-site interaction, where triple or higher site occupation is significant and particle-hole symmetry is broken. Moreover, surprisingly, the partition dependence of the spectral degeneracy in the Haldane-isulator, and of a partial degeneracy in the Mott-insulator, are directly linked to the, in principle unrelated, density-density correlations, and presents an intriguing periodic behavior in superfluid and supersolid phases.Ultra-cold gases in optical lattices [1,2] allow for the controlled study of Hubbard-like models [3], fundamental for the description of strongly correlated systems. Although until recently only on-site interactions have been relevant, inter-site interactions may play a crucial role in new experiments on gases with dipolar interactions [4]. Polar lattice gases may allow for the simulation of extended Hubbard models, which play an important role in e.g. the analysis of high-T c superconductivity [5], being the simplest models that capture the interplay between strong correlations and charge-ordering effects.Most remarkably, it has been recently shown that 1D polar lattice bosons, in addition to conventional locallyordered phases, may allow for the so-called Haldaneinsulator (HI) phase [6-8] (a Haldane-liquid phase has been recently proposed for polar fermions [9]). The HI lacks local order, but presents non-local string-like order [10], resembling to a large extent the Haldane phase in integer spin chains [11]. The Haldane phase is one of the paradigms of topological quantum phases, which due to the lack of local order parameter fall beyond the Landau paradigm of condensed-matter physics. Topological phases have attracted a growing interest in recent years, most relevantly their entanglement properties. In particular, topological phases present an anomalous entanglement entropy, characterized by a topology-dependent universal additive constant [12,13].The entanglement entropy is defined as the von Neumann entropy, S = −T rρ A log ρ A , associated to the reduced density matrix, ρ A , obtained after partitioning the system into two parts, A and B, and tracing out B. However, ρ A contains much more information than its mere entanglement entropy. In particular, it has been shown that the set of eigenvalues of ρ A , the so-called entanglement spectrum (ES), provides key insights concerning quantum correlations in many-body systems [14]. The intriguing properties of the ES in different physical systems have attracted recently a major interest [15][16][17][18]. Interestingly, it has been shown...

Entanglement spectrum of the Heisenberg XXZ chain near the ferromagnetic point

“…In particular there has been a growing interest in studying the properties of the so called entanglement spectrum (ES) [2,3,4,5,6,7,8,9,10]. The original motivation for the study of the ES was that the ES lies at the heart of the density matrix renormalization group (DMRG) algorithm [2].…”

Entanglement spectra of Heisenberg ladders of higher spin

“…These comprise quantum Hall monolayers at fractional filling [3,4,5,6,7,8,9,10,11,12,13,14,15,16], quantum Hall bilayers at filling factor ν = 1 [17], spin systems of one [18,19,20,21,22,23,24,25,26] and two [27,28,29,30] spatial dimensions, and topological insulators [31,32]. Other topics recently covered include rotating Bose-Einstein condensates [33], coupled Tomonaga-Luttinger liquids [34], and systems of Bose-Hubbard [35] and complex paired superfluids [36].…”

Using entanglement to discern phases in the disordered one-dimensional Bose-Hubbard model

“…In recent years the use of entanglement properties as a means of deciphering phase has become commonplace [29][30][31][32][33][34]. Entanglement is a measurement of a wave function's non-locality and as such it is an ideal means of analysing various phases.…”

“…In each DMRG run, bipartitioning the chain into system and environment blocks is done routinely to compute singular-value decompositions [35]. These singular values, s a , can themselves be used to obtain information regarding the phase [31][32][33] without the need for multiple DMRG runs, thus saving substantial numerical costs. The most common such measure is the entanglement entropy or Von Neumann entropy defined as…”