Spectral and Entanglement Properties of the Bosonic Haldane Insulator

Abstract

We discuss the existence of a nontrivial topological phase in one-dimensional interacting systems described by the extended Bose-Hubbard model with a mean filling of one boson per site. Performing large-scale density-matrix renormalization group calculations we show that the presence of nearest-neighbor repulsion enriches the ground-state phase diagram of the paradigmatic Bose-Hubbard model by stabilizing a novel gapped insulating state, the so-called Haldane insulator, which, embedded into superfluid, Mott insulator and density wave phases, is protected by the lattice inversion symmetry. The quantum phase transitions between the different insulating phases were determined from the central charge via the von Neumann entropy. The Haldane phase reveals a characteristic four-fold degeneracy of the entanglement spectrum. We finally demonstrate that the intensity maximum of the dynamical charge structure factor, accessible by Bragg spectroscopy, features the gapped dispersion known from the spin-1 Heisenberg chain.PACS numbers: 05.30. Jp, 75.10.Pq, 64.70.Tg, A quarter-century after Haldane's conjecture of an appearance of finite gap in the integer-spin chain [1], the so-called Haldane phase protected by the lattice inversion symmetry attracts renewed attention from a topological point of view. Such a topological protected state, characterized by symmetries and a finite bulk gap, is termed now as a symmetry-protected topological (SPT) ordered phase [2,3]. In higher dimensions, the so-called Kane-Mele topological band insulator of noninteracting fermions [4,5] exhibits a SPT state protected by U (1) and time-reversal symmetries. Since particles in real materials normally interact, it is not sufficient to study SPT order for non-interacting systems. To analyze SPT states in interacting systems two main approaches have been proposed. The first is based on the definition of appropriate topological invariants within a Green function scheme [6]. It has been successfully applied to the one-dimensional (1D) Peierls-Hubbard model [7,8]. The second uses the entanglement spectrum as a fingerprint of topological order [9]. Here the lowest entanglement level reflects the degree of degeneracy corresponding to symmetries and the edge states of the system. This has been worked out for various spin chains [3,10,11].Interestingly a hidden SPT phase was also found in interacting boson systems with long-range repulsion [12]. This phase resembles the Haldane gapped phase of the quantum spin-1 Heisenberg chain. Indeed, assuming that the site occupation of an 1D extended Bose-Hubbard model (EBHM) with nearest-neighbor interaction is restricted to n j = 0, 1 or 2, the system can be described by an effective spin-1 model with S z j = n j − ρ for a mean boson filling factor ρ = 1. The Haldane insulator (HI) then appears between the conventional Mott insulator (MI) and the density wave (DW) phases at intermediate couplings [12,13]. Field theory predicts the MI-HI transition to be in the Luttinger liquid universality class with central charge c ...

Phase diagrams of the extended Bose-Hubbard model in one dimension by Monte-Carlo simulation with the help of a stochastic-series expansion

“…For the case of n c = 2, our results are in good agreement with those in Ref. [14,21]. However, the SF exists between the ρ = 1 MI and HI as in Fig.1 and Fig.3, whereas the SF does not exist there in the phase diagram obtained in Refs.…”

“…However, the SF exists between the ρ = 1 MI and HI as in Fig.1 and Fig.3, whereas the SF does not exist there in the phase diagram obtained in Refs. [14,21]. The phase diagram of V = 4.0 with n c = 3 in Fig.3 is in good agreement with that obtained by DMRG in Refs.…”

Quantum phase transitions in the dimerized extended Bose-Hubbard model

“…As a result, an SPT Haldane insulator appears between the MI and DW phases for intermediate couplings [11,20], which resembles the gapped Haldane phase of the quantum spin-1 Heisenberg chain [8]. We note that the HI phase continues to exist if one includes higher boson numbers n b > 2 [21,22]. In the DMRG calculations, a finite maximum number of bosons per site n b must be used.…”

Density waves in strongly correlated quantum chains

“…For the pure EHM both ∆ c2 and ∆ n vanish at the continuous CDW-BOW QPT. For a nonzero dimerization δ, the neutral gap closes whereas ∆ c2 remains finite, indicating that the CDW-SPTI QPT belongs to the Ising universal- At criticality, the central charge c can easily be extracted from the entanglement entropy [177,178]. For periodic boundary conditions, conformal field theory predicts the von Neumann entropy to be S L ( ) = c 3 ln L π sin π L + s 1 where s 1 is a nonuniversal constant [179].…”