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arXiv:2204.14144 · Physical review. B./Physical review. B

Splitting of topological charge pumping in an interacting two-component fermionic Rice-Mele Hubbard model

Eric Bertok, Fabian Heidrich‐Meisner, Armando A. Aligia

Published Jul 27, 2022• 37 citations
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Abstract

A Thouless pump transports an integer amount of charge when pumping adiabatically around a singularity. We study the splitting of such a critical point into two separate critical points by adding a Hubbard interaction. Furthermore, we consider extensions to a spinful Rice-Mele model, namely, a staggered magnetic field or an Ising-type spin coupling, further reducing the spin symmetry. The resulting models additionally allow for the transport of a single charge in a two-component system of spinful fermions, whereas in the absence of interactions, zero or two charges are pumped. In the SU(2)-symmetric case, the ionic Hubbard model is visited once along pump cycles that enclose a single singularity. Adding a staggered magnetic field additionally transports an integer amount of spin while the Ising term realizes a pure charge pump. We employ real-time simulations in finite and infinite systems to calculate the adiabatic charge and spin transport, complemented by the analysis of gaps and the many-body polarization to confirm the adiabatic nature of the pump. The resulting charge pumps are expected to be measurable in finite-pumping speed experiments in ultracold atomic gases for which the $\mathrm{SU}(2)$ invariant version is the most promising path. We discuss the implications of our results for a related quantum-gas experiment by Walter et al. [arXiv:2204.06561].

DOI
10.1103/physrevb.106.045141

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A Thouless pump transports an integer amount of charge when pumping adiabatically around a singularity. We study the splitting of such a critical point into two separate critical points by adding a Hubbard interaction. Furthermore, we consider extensions to a spinful Rice-Mele model, namely, a staggered magnetic field or an Ising-type spin coupling, further reducing the spin symmetry. The resulting models additionally allow for the transport of a single charge in a two-component system of spinful fermions, whereas in the absence of interactions, zero or two charges are pumped. In the SU(2)-symmetric case, the ionic Hubbard model is visited once along pump cycles that enclose a single singularity. Adding a staggered magnetic field additionally transports an integer amount of spin while the Ising term realizes a pure charge pump. We employ real-time simulations in finite and infinite systems to calculate the adiabatic charge and spin transport, complemented by the analysis of gaps and the many-body polarization to confirm the adiabatic nature of the pump. The resulting charge pumps are expected to be measurable in finite-pumping speed experiments in ultracold atomic gases for which the SU(2)\mathrm{SU}(2)SU(2) invariant version is the most promising path. We discuss the implications of our results for a related quantum-gas experiment by Walter et al. [arXiv:2204.06561].
We investigate topological charge pumping in a system of interacting bosons in the tight-binding limit, described by the Rice-Mele model. An appropriate topological invariant for the many-body case is the change of polarization per pump cycle, which we compute for various interaction strengths from infinite-size matrix-product-state simulations. We verify that the charge pumping remains quantized as long as the pump cycle avoids the superfluid phase. In the limit of hardcore bosons, the quantized pumped charge can be understood from single-particle properties such as the integrated Berry curvature constructed from Bloch states, while this picture breaks down at finite interaction strengths. These two properties---robust quantized charge transport in an interacting system of bosons and the breakdown of a single-particle invariant---could both be measured with ultracold quantum gases extending a previous experiment [Lohse et al., Nat. Phys. 12, 350 (2016)]. Furthermore, we investigate the entanglement spectrum of the Rice-Mele model and argue that the quantized charge pumping is encoded in a winding of the spectral flow in the entanglement spectrum over a pump cycle.
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.
Interaction induced splitting of Dirac monopoles in the topological Thouless pumping of strongly interacting Bosons and SU(NNN) Fermions
Motivated by the observation of the breakdown of quantization for the Thouless pump in the presence of strong interaction by ETH [Walter et. al. Nat. Phys. 19, 1471 (2023), Viebahn et. al. arXiv:2308.03756], we study the interplay of strong interaction and topology in the (1+1)-dimensional interacting Rice-Mele model. We point out that the quantization of the interacting Thouless pump is dictated by the Chern number, i.e., the Dirac monopoles enclosed by the generalized Brillouin zone of the many-body wave function. By analyzing the change of location monopoles due to interaction, we predict the Thouless charge pump for strongly interacting Bose and SU(NNN) Fermi gases in optical lattices and explain the ETH experiment.
Recent experiments with ultracold quantum gases have successfully realized integer-quantized topological charge pumping in optical lattices. Motivated by this progress, we study the effects of static disorder on topological Thouless charge pumping. We focus on the half-filled Rice-Mele model of free spinless fermions and consider random diagonal disorder. We use both static and time-dependent simulations to characterize the charge pump. The set of measures that we compute in the instantaneous basis include the polarization, the entanglement spectrum, and the space-integrated local Chern marker. As a first main result, we conclude that the space-integrated local Chern marker is best suited for a quantitative determination of topological transitions in a disordered system. In the time-dependent simulations we use the time-integrated current to obtain the pumped charge in slowly periodically driven systems. As a second main result, we observe and characterize a disorder-driven breakdown of the quantized charge pump. There is an excellent agreement between the static and the time-dependent ways of computing the pumped charge. The topological transition occurs well in the regime where all states are localized on the given system sizes. This observation is consistent with previous studies and the topological transition is therefore not tied to a delocalization-localization transition of Hamiltonian eigenstates. For individual disorder realizations, the breakdown of the quantized pumping occurs for parameters where the spectral bulk gap inherited from the band gap of the clean system closes, leading to a globally gapless spectrum. As a third main result and with respect to the analysis of finite-size systems, we show that the disorder average of the bulk gap severely overestimates the stability of quantized pumping. A much better estimate is the typical value of the distribution of energy gaps, also called mode of the distribution. We discuss our results in the context of recent quantum-gas experiments that realized charge pumps.
Counterdiabatic (CD) driving suppresses the excitations of fast protocols by adding an auxiliary Hamiltonian, but in many-body and topological systems the exact CD term is generally non-local and multi-body. We study this obstruction for Thouless pumping in the Rice-Mele model, first without and then with interactions. For noninteracting fermions, the exact CD term of the pump is non-local, but the variational framework of Sels and Polkovnikov yields accurate nearest-neighbor approximations. For the spinful Rice-Mele chain with onsite and nearest-neighbor repulsion, in the regime where the competition of the two interactions stabilizes quantized pumping, this approach fails: the infinite-temperature variational action only penalizes the CD amplitudes, and the resulting two-parameter nearest-neighbor term does not prevent the pump from breaking down when the cycle time is reduced from 50/t50/t50/t to 2/t2/t2/t. We trace the failure to the structure of the exact many-body gauge potential, whose amplitude for each hop depends on the interaction energy that the hop costs. Resolving the nearest-neighbor currents by this energy, and fixing the coefficients with a ground-state-weighted action, gives a local ansatz with 14 coefficients that keeps the interacting pump quantized for cycle times T=2/tT=2/tT=2/t and T=1/tT=1/tT=1/t, with an accuracy per site that does not degrade between rings of 8 and 12 sites.
We show that local Hubbard correlations can split the open-boundary spectral flow of an otherwise integer Thouless pump. Using density-matrix renormalization group calculations for a period-three Hubbard chain with a sliding onsite-repulsion modulation, we identify at filling ρ=2/3ρ=2/3ρ=2/3 a correlated insulating pump with many-body Chern number C=+2C=+2C=+2, corresponding to two units of charge transported per cycle. A spin-degenerate Hartree/Aubry--André--Harper reference model realizes the same integer response through simultaneous edge spectral flow in two independent spin channels. In the full Hubbard model, however, this integer bulk pump is implemented differently at an open boundary: the edge flow separates into two boundary events, and the intervening low-energy state is a nearly charge-neutral triplet edge excitation. Thus the quantized bulk charge transport remains integer, but local correlations reorganize its boundary realization into separated charged and neutral spinful sectors.