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Haldane Phase in Spin-1 Bosons with a Flat Band

Hong Wei Yang, Hayate Nakano, Hosho Katsura

Published Mar 3, 2020• 0 citations
OpenAlex

Abstract

We demonstrate that the Haldane phase can be realized with ultracold spin-1 atoms loaded on a lattice which has a flat band at the bottom of the band structure. The many-body ground state of the model Hamiltonian is exactly solvable in a special case, which turns out to serve as a representative of the Haldane phase. In this system, the Haldane phase is protected by $\mathbb{Z}_2 \times \mathbb{Z}_2$ spin rotation symmetry or time-reversal symmetry. Depending on several parameters, the Hamiltonian can either be in a gapped Haldane or a gapless critical phase. In a certain limit, the phase diagram is determined analytically by perturbation theory, while for general cases, the phase diagram is studied numerically by the variational uniform matrix product state (VUMPS) algorithm.

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We demonstrate that the Haldane phase can be realized with ultracold spin-1 atoms loaded on a lattice which has a flat band at the bottom of the band structure. The many-body ground state of the model Hamiltonian is exactly solvable in a special case, which turns out to serve as a representative of the Haldane phase. In this system, the Haldane phase is protected by Z2×Z2\mathbb{Z}_2 \times \mathbb{Z}_2Z2​×Z2​ spin rotation symmetry or time-reversal symmetry. Depending on several parameters, the Hamiltonian can either be in a gapped Haldane or a gapless critical phase. In a certain limit, the phase diagram is determined analytically by perturbation theory, while for general cases, the phase diagram is studied numerically by the variational uniform matrix product state (VUMPS) algorithm.
We theoretically demonstrate that interacting symmetry-protected topological (SPT) phases can be realized with ultracold spinful bosonic atoms loaded on the lattices which have a flat band at the bottom of the band structure.Ground states of such systems are not conventional Mott insulators in the sense that the ground states possess not only spin fluctuations but also non-negligible charge fluctuations.The SPT phases in such systems are determined by both spin and charge fluctuations at zero temperature.We find that the many-body ground states of such systems can be exactly obtained in some special cases, and these exact ground states turn out to serve as representative states of the SPT phases.As a concrete example, we demonstrate that spin-1 bosons on a sawtooth chain can be in an SPT phase protected by Z 2 × Z 2 spin rotation symmetry or time-reversal symmetry, and this SPT phase is a result of spin fluctuations.We also show that spin-3 bosons on a kagome lattice can be in an SPT phase protected by D 2 point group symmetry, but this SPT phase is, however, a result of charge fluctuations.
Recent proposals of topological flat band models have provided a new route to realize the fractional quantum Hall effect without Landau levels. We study hard-core bosons with short-range interactions in two representative topological flat band models, one of which is the well-known Haldane model (but with different parameters). We demonstrate that fractional quantum Hall states emerge with signatures of an even number of quasidegenerate ground states on a torus and a robust spectrum gap separating these states from the higher energy spectrum. We also establish quantum phase diagrams for the filling factor 1/2 and illustrate quantum phase transitions to other competing symmetry-breaking phases.
We investigate disordered phases realized in one-dimensional quantum spin systems.We revisit an effective field theory proposed by Schulz [Phys.Rev. B 34, 6372 (1986)], which encapsulates several essential properties in low energy, from the point of view of symmetry.Although this effective theory was originally derived by perturbation theory, we find that it is fully consistent with the symmetry of the Hamiltonian and thus uniquely identified in the non-perturbative sense.This theory explains many possible phase transitions known to occur among valence-bond-solid phases, which are typical disordered phases in one-dimensional spin systems.Furthermore, this theory indicates that the phase transitions only occur under one of four symmetries: time reversal, bond-centered inversion, dihedral group of spin rotations, and site-centered inversion combined with a spin rotation.Surprisingly, the first three symmetries are consistent with those protecting the Haldane phase in spin-1 chains, which have already been found by a completely different approach using the matrix-product state (MPS).On the other hand, the last one is not known so far.To understand it, we explicitly construct a microscopic model that has two distinct disordered phases protected by the combined inversion symmetry.We examine this model and confirm the existence of those disordered phases by perturbation theory and numerical simulations.For completeness, we further give a more rigorous proof of their existence, based on the non-translation-invariant MPS.We also propose an extension of the Lieb-Schultz-Mattis theorem for the absence of a unique gapped ground state in half-odd-integer spin systems with site-centered inversion symmetry.
Using the exact-diagonalization (ED) and mean-field (MF) approaches, we investigate the ground-state phase diagram of the interacting Haldane model on the honeycomb lattice, incorporating spin-dependent sublattice potentials ${\mathrm{\ensuremath{\Delta}}}_{\ensuremath{\sigma},\ensuremath{\alpha}}$. Here $\ensuremath{\alpha}=\text{A},\text{B}$ and $\ensuremath{\sigma}=\ensuremath{\uparrow},\ensuremath{\downarrow}$ denote the sublattice and spin components, respectively. Setting ${\mathrm{\ensuremath{\Delta}}}_{\ensuremath{\sigma},\text{A}}=+\mathrm{\ensuremath{\Delta}}$ $(\ensuremath{-}\mathrm{\ensuremath{\Delta}})$ and ${\mathrm{\ensuremath{\Delta}}}_{\ensuremath{\sigma},\text{B}}=\ensuremath{-}\mathrm{\ensuremath{\Delta}}$ $(+\mathrm{\ensuremath{\Delta}})$ for $\ensuremath{\sigma}=\ensuremath{\uparrow}$ $(\ensuremath{\downarrow})$ results in the system favoring a spin ordered state. Conversely, introducing the nearest-neighbor Coulomb interaction can induce charge ordering in the system. Due to the competition between these factors, we observe that in both ED and MF approaches, an exotic state with Chern number C=1C=1C=1 survives amidst two locally ordered phases and a topologically ordered phase with C=2C=2C=2. In the ED method, various properties, such as the fidelity metric, the excitation gaps and the structure factors, are employed to identify critical points. In the MF method, using a sufficiently large lattice size, we define the local order parameters and band gaps to characterize the phase transitions. The interacting Haldane model and the spin-dependent lattice potential may be experimentally realized in an ultracold atom gas, providing a potential means to detect this intriguing state.
Abstract Topology in quantum many-body systems has profoundly changed our understanding of quantum phases of matter. The model that has played an instrumental role in elucidating these effects is the antiferromagnetic spin-1 Haldane chain 1,2 . Its ground state is a disordered state, with symmetry-protected fourfold-degenerate edge states due to fractional spin excitations. In the bulk, it is characterized by vanishing two-point spin correlations, gapped excitations and a characteristic non-local order parameter 3,4 . More recently it has been understood that the Haldane chain forms a specific example of a more general classification scheme of symmetry-protected topological phases of matter, which is based on ideas connected to quantum information and entanglement 5–7 . Here, we realize a finite-temperature version of such a topological Haldane phase with Fermi–Hubbard ladders in an ultracold-atom quantum simulator. We directly reveal both edge and bulk properties of the system through the use of single-site and particle-resolved measurements, as well as non-local correlation functions. Continuously changing the Hubbard interaction strength of the system enables us to investigate the robustness of the phase to charge (density) fluctuations far from the regime of the Heisenberg model, using a novel correlator.
We reexamine the one-dimensional spin-1 XXZXXZXXZ model with on-site uniaxial single-ion anisotropy as to the appearance and characterization of the symmetry-protected topological Haldane phase. By means of large-scale density-matrix renormalization group (DMRG) calculations the central charge can be determined numerically via the von Neumann entropy, from which the ground-state phase diagram of the model can be derived with high precision. The nontrivial gapped Haldane phase shows up in between the trivial gapped even Haldane and N\'eel phases, appearing at large single-ion and spin-exchange interaction anisotropies, respectively. We furthermore carve out a characteristic degeneracy of the lowest entanglement level in the topological Haldane phase, which is determined using a conventional finite-system DMRG technique with both periodic and open boundary conditions. Defining the spin and neutral gaps in analogy to the single-particle and neutral gaps in the intimately connected extended Bose-Hubbard model, we show that the excitation gaps in the spin model qualitatively behave just as for the bosonic system. We finally compute the dynamical spin structure factor in the three different gapped phases and find significant differences in the intensity maximum which might be used to distinguish these phases experimentally.