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arXiv:2003.14261 · New Journal of Physics

Topological phases arising from attractive interaction and pair hopping in the extended Hubbard model

Roman Rausch, Matthias Peschke

Published Jun 12, 2020• 6 citations
arXivHTMLPDF

Abstract

Abstract The extended Hubbard model with an attractive density–density interaction, positive pair hopping, or both, is shown to host topological phases, with a doubly degenerate entanglement spectrum and interacting edge spins. This constitutes a novel instance of topological order which emerges from interactions. When the interaction terms combine in a charge-SU(2) symmetric fashion, a novel partially polarized pseudospin phase appears, in which the topological features of the spin degrees of freedom coexist with long-range η-wave superconductivity. Thus, our system provides an example of an interplay between spontaneous symmetry breaking and symmetry-protected topological order that leads to novel and unexpected properties.

DOI
10.1088/1367-2630/ab9c65

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•Jan 1, 2002•17 citations
Abstract The extended Hubbard model with an attractive density–density interaction, positive pair hopping, or both, is shown to host topological phases, with a doubly degenerate entanglement spectrum and interacting edge spins. This constitutes a novel instance of topological order which emerges from interactions. When the interaction terms combine in a charge-SU(2) symmetric fashion, a novel partially polarized pseudospin phase appears, in which the topological features of the spin degrees of freedom coexist with long-range η-wave superconductivity. Thus, our system provides an example of an interplay between spontaneous symmetry breaking and symmetry-protected topological order that leads to novel and unexpected properties.
We study a one-dimensional model of interacting bosons coupled to a dynamical Z2{\mathbb{Z}}_{2}Z2​ field, the Z2{\mathbb{Z}}_{2}Z2​ Bose-Hubbard model, and analyze the interplay between spontaneous symmetry breaking and topological symmetry protection. In a previous work, we showed how this model exhibits a spontaneous breaking of the translational symmetry through a bosonic Peierls transition. Here we find how, at half filling, the resulting phase also displays topological features that coexist with the presence of long-range order and yields a topological bond order wave. Using both analytical and numerical methods, we describe the properties of this phase, showing that it cannot be adiabatically connected to a topological phase of noninteracting bosons, and thus constitutes an instance of an interaction-induced symmetry-breaking topological insulator.
The Bose-Hubbard model (BHM) has been widely explored to develop a profound understanding of the strongly correlated behavior of interacting bosons. Quantum simulators not only allow the exploration of the BHM but also extend it to models with interesting phenomena such as gapped phases with multiple orders and topological phases. In this work, an extended Bose-Hubbard model involving a dimerized one-dimensional model of long-range interacting hard-core bosons is studied. Bond-order density wave phases (BODW) are characterized in terms of their symmetry breaking and topological properties. At certain fillings, interactions combined with dimerized hoppings give rise to an emergent symmetry-breaking leading to BODW phases, which differs from the case of non-interacting models that require an explicit breaking of the symmetry. Specifically, the BODW phase at filling $\ensuremath{\rho}=1/3$ possesses no analog in the noninteracting model in terms of its symmetry-breaking properties and the unit cell structure. Upon changing the dimerization pattern, the system realizes topologically trivial BODW phases. At filling $\ensuremath{\rho}=1/4$, on-site density modulations are shown to stabilize the topological BODW phase. Our work provides the bridge between interacting and noninteracting BODW phases and highlights the significance of long-range interactions in a dimerized lattice by showing unique BODW phases that do not exist in the noninteracting model.
Topological insulators have attracted numerous attentions recent years, where the Su-Schrieffer-Heeger (SSH) model is one of the most studied models. While the interacting version of it has been explored recently, the interplay between interactions and long-range hoppings merit further investigations. In this work, we uncover a rich phase diagram of the interacting SSH model with extended-range hoppings, in which it consists of several topological phases, two novel superconducting-like (SC-like) phases and five distinct charge-density-wave (CDW) phases. We substantiate that the SC-like and two CDW phases are direct consequences of imbalanced interactions and extended-range hoppings. We derive the order parameters (OPs) for each of the phases and verify them in large-system simulations, finding consistency with the entanglement entropy and the fidelity in capturing the phase transitions. In contrast to the non-interacting case where the favored hoppings are unidirectional in the topological phases, the derived OPs suggest non-unidirectional hoppings are possible under the influence of interactions.
We investigate the quantum phases of a half-filled generalized interacting Su-Schrieffer-Heeger model with intracell, nearest-neighbor, and next-nearest-neighbor intercell hoppings, together with an on-site inter-sublattice interaction. In the noninteracting limit, the model hosts one topologically trivial phase and two symmetry-protected topological (SPT) phases, distinguished under periodic boundary conditions by different winding numbers and under open boundary conditions by two-fold and four-fold entanglement-spectrum degeneracies, respectively. When interactions are introduced, these free-fermion SPT phases evolve into distinct interacting topological phases that retain characteristic signatures such as entanglement-spectrum degeneracy structures, boundary modes, and nonzero string order parameters. For strong repulsive interactions, a symmetry-breaking phase with unequal but spatially uniform sublattice densities appears between the trivial and topological regimes. For strong attractive interactions, period-2 and period-4 charge-density-wave phases emerge from particle clustering. At intermediate attractive interactions, the competition between interaction-induced localization and hopping-induced delocalization gives rise to a Luttinger liquid phase, a paired Luttinger liquid phase, and a gapless symmetry-protected topological (gSPT) phase. The gSPT phase is characterized by a gapless charge mode together with symmetry-protected current-carrying edge states. We further characterize the gapless phases and the associated quantum phase transitions through central charges and critical exponents.
The phase diagrams and superconducting properties of the extended Hubbard model with pair hopping interaction, i.e., the Penson-Kolb-Hubbard model are studied. The analysis of the model is performed for d-dimensional hypercubic lattices, including d=1d=1d=1 and $d=\ensuremath{\infty},$ by means of the (broken symmetry) Hartree-Fock approximations and, for $d=\ensuremath{\infty},$ by the slave-boson mean-field method. For d=1,d=1,d=1, at half filling the phase diagram is shown to consist of nine different phases including two superconducting states with center-of-mass momentum q=0q=0q=0 and q=Qq=Qq=Q $(\ensuremath{\eta}$ pairing), site and bond-located antiferromagnetic and charge-density wave states as well as three mixed phases with coexisting site and bond orderings. The stability range of the bond-type orderings shrinks with increasing lattice dimensionality d and for $d=\ensuremath{\infty}$ the corresponding diagram consists of four phases only, involving exclusively site-located orderings. Comparing the pair hopping model with the attractive Hubbard model we found in both cases gradual evolution from the BCS-like limit to the tightly bound pairs regime and a monotonic increase of the gap in the excitation spectrum with increasing coupling. However, the dynamics of electron pairs in both models is qualitatively different, which results in different dependences of condensation energies and critical temperatures on interaction parameters as well as in different electrodynamic properties, especially in a strong coupling regime.
A one-dimensional model of interacting electrons with on-site U, nearest- neighbor V, and pair-hopping interaction W is studied at half-filling using the continuum limit field theory approach. The ground state phase diagram is obtained for a wide range of coupling constants. In addition to the insulating spin-density wave (SDW) and charge-density wave (CDW) phases for large U and V, respectively, we identify a bond-charge-density-wave (BCDW) phase W < 0, \U - 2V\ < \2W\ and a bond-spin-density-wave (BSDW) for W > 0, \U - 2V\ < W. The possibility of bond-located ordering results from the site-off-diagonal nature of the pair-hopping term and is a special feature of the half-filled band case. The BCDW phase corresponding to an enhanced Peierls instability in the system. The BdSDW is an unconventional insulating magnetic phase, characterized by a gapless spin excitation spectrum and a staggered magnetization located on bonds between sites. The general ground state phase diagram including insulating, metallic, and superconducting phases is discussed. A transition to the eta(pi) -superconducting phase at \U - 2V\ much less than 2t less than or equal to W is briefly discussed.