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arXiv:1301.3735 · Physical Review B

Topological pumping in the one-dimensional Bose-Hubbard model

Davide Rossini, Marco Gibertini, Vittorio Giovannetti, Rosario Fazio

Published Feb 25, 2013• 22 citations
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Abstract

By means of time-dependent density-matrix renormalization-group calculations, we study topological quantum pumping in a strongly interacting system. The system under consideration is described by the Hamiltonian of a one-dimensional extended Bose-Hubbard model in the presence of a correlated hopping which breaks lattice inversion symmetry. This model has been predicted to support topological pumping [E. Berg, M. Levin, and E. Altman, Phys. Rev. Lett. 106, 110405 (2011)]. The pumped charge is quantized and of a topological nature. We provide a detailed analysis of the finite-size scaling behavior of the pumped charge and its deviations from the quantized value. Furthermore, we also analyze the nonadiabatic corrections due to the finite frequency of the modulation. We consider two configurations: a closed ring where the time dependence of the parameter induces a circulating current and a finite open-ended chain where particles are dragged from one edge to the opposite edge, due to the pumping mechanism induced by the bulk.

DOI
10.1103/physrevb.87.085131

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By means of time-dependent density-matrix renormalization-group calculations, we study topological quantum pumping in a strongly interacting system. The system under consideration is described by the Hamiltonian of a one-dimensional extended Bose-Hubbard model in the presence of a correlated hopping which breaks lattice inversion symmetry. This model has been predicted to support topological pumping [E. Berg, M. Levin, and E. Altman, Phys. Rev. Lett. 106, 110405 (2011)]. The pumped charge is quantized and of a topological nature. We provide a detailed analysis of the finite-size scaling behavior of the pumped charge and its deviations from the quantized value. Furthermore, we also analyze the nonadiabatic corrections due to the finite frequency of the modulation. We consider two configurations: a closed ring where the time dependence of the parameter induces a circulating current and a finite open-ended chain where particles are dragged from one edge to the opposite edge, due to the pumping mechanism induced by the bulk.
Interacting lattice bosons at integer filling can support two distinct insulating phases, which are separated by a critical point: the Mott insulator and the Haldane insulator [E. G. Dalla Torre, E. Berg, and E. Altman, Phys. Rev. Lett. 97, 260401 (2006).]. The critical point can be gapped out by breaking lattice inversion symmetry. Here, we show that encircling this critical point adiabatically pumps one boson across the system. When multiple chains are coupled, the two insulating phases are no longer sharply distinct, but the pumping property survives. This leads to strict constraints on the topology of the phase diagram of systems of quasi-one-dimensional interacting bosons.
A topological charge pump is a one-dimensional, dynamical analogue of a quantum Hall system that is particularly well suited for studying topology, both theoretically and experimentally, since no synthetic gauge fields are required and efficient numerical algorithms exist to treat one-dimensional interacting systems in the form of matrix-product state methods. Instead of a robustly quantized Hall conductivity, a topological charge pump pumps a quantized amount of charge in each cycle of a parameter modulation that is given by the same topological invariant as in the quantum Hall effect. The goal of this thesis is to understand the stability of quantized transport in various contexts that are relevant in ultracold quantum gases, such as disordered systems, quantum systems where energy can be exchanged with a bath and in the presence of genuine many-body interactions. The results are presented as part of four scientific papers. In the first paper, we characterize the breakdown of quantized pumping as a function of disorder strength by employing various instantaneous and time-dependent measures. We argue that in disorder averages, the full energy gap distributions should be considered. In the second paper, we study the breakdown of quantized pumping due to electron-phonon coupling and find a resonant breakdown when the pump frequency equals the phonon frequency. The third paper introduces the possibility of a splitting of critical points in a Rice-Mele Hubbard model, which allows for interaction-induced pumping. The final paper presents experimental data of an ultracold-atom setup studying this interaction-induced pumping, as well as numerical calculations detailing the breakdown due to the crossing of a spin-gapless line.
We show that a topological pump in a one-dimensional insulator can induce a strictly quantized transport in an auxiliary chain of noninteracting fermions weakly coupled to the first. The transported charge is determined by an integer topological invariant of the fictitious Hamiltonian of the insulator, given by the covariance matrix of single-particle correlations. If the original system consists of noninteracting fermions, this number is identical to the Thouless, Kohmoto, Nightingale, and den Nijs (TKNN) invariant of the original system and thus the coupling induces a transfer of topology to the auxiliary chain. When extended to particles with interactions, for which the TKNN number does not exist, the transported charge in the auxiliary chain defines a topological invariant for the interacting system. In certain cases this invariant agrees with the many-body generalization of the TKNN number introduced by Niu, Thouless, and Wu. We illustrate the topology transfer to the auxiliary system for the Rice-Mele model of noninteracting fermions at half filling and the extended superlattice Bose-Hubbard model at quarter filling. In the latter case the induced charge pump is fractional.
We study the effect of different open boundary conditions on the insulating ground states of the one-dimensional extended Bose-Hubbard model at and near unit filling. To this end, we employ the density-matrix renormalization-group method with system sizes up to 250 sites. To characterize the system, various order parameters and entanglement entropies are calculated. When opposite edge potentials are added to the two ends of the chain, the inversion symmetry is explicitly broken, and the regular bulk phases appear. On the other hand, simple open boundary conditions often exhibit nondegenerate ground states with a domain wall in the middle of the chain, which induces a sign flip of an order parameter. Such a domain wall can lead to an algebraic behavior of the off diagonals of the single-particle density matrix. We show that this algebraic behavior adds only a finite contribution to the entanglement entropy, which does not diverge as the system size increases. Therefore, it is not an indication of a superfluid phase. We confirm this picture by analytical calculations based on an effective Hamiltonian for a domain wall.
Interaction-induced topological systems have attracted a growing interest for their exotic properties going beyond the single-particle picture of topological insulators. In particular, the interplay between strong correlations and finite doping can give rise to nonhomogeneous solutions that break the translational symmetry. In this work, we report the appearance of a topological stripe state in an interaction-induced Chern insulator around half-filling. In contrast to similar stripe phases in nontopological systems, here we observe the appearance of chiral edge states on top of the domain wall. Furthermore, we characterize their topological nature by analyzing the quantized transferred charge of the domains in a pumping scheme. Finally, we focus on aspects relevant to observing such phases in state-of-the-art quantum simulators of ultracold atoms in optical lattices. In particular, we propose an adiabatic state preparation protocol and a detection scheme of the topology of the system in real space.
An extended Bose-Hubbard model (EBHM) with three- and four-body constraints can be feasible in cold atoms in an optical lattice. A rich phase structure including various symmetry-protected topological (SPT) phases is obtained numerically with suitable parameter settings and particle filling. The SPT phase is characterized by the Berry phase as a local topological order parameter and the structure of the entanglement spectrum (ES). Based on the presence of various topological phases, separated by gapless phase boundaries, the EBHM exhibits various bosonic topological pumps, which are constructed by connecting the different SPT phases without gap closing. The bulk topological pumps exhibit the plateau transitions characterized by many-body Chern numbers. For the system with boundary, the center of mass (CoM) under grand canonical ensemble elucidates the contributions of multiple edge states and reveals the topology of the system. We demonstrate that the interacting bosonic pumps obey the bulk-edge correspondence.