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arXiv:1612.00550 · Physical review. B./Physical review. B

Haldane phase on the sawtooth lattice: Edge states, entanglement spectrum, and the flat band

Benoît Grémaud, G. G. Batrouni

Published Apr 24, 2017• 19 citations
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Abstract

Using density matrix renormalization group numerical calculations, we study the phase diagram of the half filled Bose-Hubbard system in the sawtooth lattice with strong frustration in the kinetic energy term. We focus in particular on values of the hopping terms which produce a flat band and show that, in the presence of contact and near neighbor repulsion, three phases exist: Mott insulator (MI), charge density wave (CDW), and the topological Haldane insulating (HI) phase which displays edge states and particle imbalance between the two ends of the system. We find that, even though the entanglement spectrum in the Haldane phase is not doubly degenerate, it is in excellent agreement with the entanglement spectrum of the Affleck-Kennedy-Lieb-Tasaki (AKLT) state built in the Wannier basis associated with the flat band. This emphasizes that the absence of degeneracy in the entanglement spectrum is not necessarily a signature of a nontopological phase, but rather that the (hidden) protecting symmetry involves nonlocal states. Finally, we also show that the HI phase is stable against small departure from flatness of the band but is destroyed for larger ones.

DOI
10.1103/physrevb.95.165131

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Using density matrix renormalization group numerical calculations, we study the phase diagram of the half filled Bose-Hubbard system in the sawtooth lattice with strong frustration in the kinetic energy term. We focus in particular on values of the hopping terms which produce a flat band and show that, in the presence of contact and near neighbor repulsion, three phases exist: Mott insulator (MI), charge density wave (CDW), and the topological Haldane insulating (HI) phase which displays edge states and particle imbalance between the two ends of the system. We find that, even though the entanglement spectrum in the Haldane phase is not doubly degenerate, it is in excellent agreement with the entanglement spectrum of the Affleck-Kennedy-Lieb-Tasaki (AKLT) state built in the Wannier basis associated with the flat band. This emphasizes that the absence of degeneracy in the entanglement spectrum is not necessarily a signature of a nontopological phase, but rather that the (hidden) protecting symmetry involves nonlocal states. Finally, we also show that the HI phase is stable against small departure from flatness of the band but is destroyed for larger ones.
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 fourfold 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.
We investigate the entanglement properties of a nontrivial topological phase in the one-dimensional (1D) Bose-Hubbard model with additional nearest-neighbor repulsion. Employing the large-scale density-matrix renormalization group technique we show that a gapped insulating phase protected by lattice inversion symmetry, the so-called Haldane insulator, appears between the Mott and density wave phases in the intermediate-coupling regime. The phase boundaries were determined from the central charge via the von Neumann entropy. The Haldane insulator reveals a characteristic degeneracy in the entanglement spectra. Breaking the lattice inversion symmetry strongly affects the distinctive gapped dispersion of the dynamical charge response of the bosonic Haldane insulator.
Abstract The Haldane phase is the prototype of symmetry protected topological (SPT) phases of spin chain systems. It can be protected by several symmetries having in common the degeneracy of the entanglement spectrum (ES). Here we explore in depth this degeneracy for the spin-1 Affleck–Kennedy–Lieb–Tasaki and bilinear-biquadratic Hamiltonians and show the emergence of a bulk-edge correspondence that relates the low energy properties of the entanglement Hamiltonian of a periodic chain and that of the physical Hamiltonian of an open chain. We find that the ES can be described in terms of two spins-1/2 behaving as the effective spins at the end of the open chain. In the case of non-contiguous partitions, we find that the entanglement Hamiltonian is given by the spin-1/2 Heisenberg Hamiltonian, which suggests a relationship between SPT phases and conformal field theory. We finally investigate the string order parameter and the relation with the bulk-edge correspondence.
We investigate the two-leg Hubbard model with diagonal hopping to explore the interplay between geometrical frustration and strong electron-electron interactions. Using the Density Matrix Renormalization Group (DMRG) method, we demonstrate the emergence of a topological Haldane phase, which results explicitly from the complementary effects of diagonal hopping-induced frustration and strong on-site Coulomb repulsion. The topological phase transition from a trivial insulator to the nontrivial Haldane phase is characterized by significant changes in magnetic properties, edge correlations, and the appearance of a nonzero string order parameter. Furthermore, we confirm the topological nature of this phase through a detailed analysis of the spin gap and entanglement spectrum, demonstrating clear signatures of symmetry-protected topological order.
We systematically study the effects of higher-order interactions on the S=32,1S=\frac{3}{2},1S=23​,1 orthogonal dimer chains using exact diagonalization and density matrix renormalization group. Due to frustration and higher spin, there are rich quantum phases, including three Haldane phases, two gapless phases, and several magnetically ordered phases. To characterize these phases and their phase transitions, we study various physical quantities such as energy gap, energy-level crossing, fidelity susceptibility, spin correlation, entanglement spectrum, and central charge. According to our calculations, the biquadratic term can enhance the Haldane phase regions. In particular, we numerically identify that a Haldane phase in S=32S=\frac{3}{2}S=23​ case is adiabatically connected to the exact AKLT (Affleck, Kennedy, Lieb, and Tasaki) point when adding a bicubic term. Our study on the orthogonal dimer model, which is a 1D version of the Shastry-Sutherland model, provides insights into understanding the possible S=32,1S=\frac{3}{2},1S=23​,1 Haldane phases in quasi-1D and 2D frustrated magnetic materials.
We carry out an extensive study of the phase diagram of the extended Bose Hubbard model, with a mean filling of one boson per site, in one dimension by using the density matrix renormalization group and show that it can have Superfluid (SF), Mott-insulator (MI), density-wave (DW) and Haldane-insulator (HI) phases depending on the precise value of filling and how edge states are handled. We show that the critical exponents and central charge for the HI-DW, MI-HI and SF-MI transitions are consistent with those for models in the two-dimensional Ising, Gaussian, and Berezinskii-Kosterlitz-Thouless (BKT) universality classes, respectively; and we suggest that the SF-HI transition may be more exotic than a simple BKT transition.