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arXiv:2308.09316 · Physical review. B./Physical review. B

Nonadiabatic effect in topological and interacting charge pumping

Fan Yang, Xingyu Li, Hui Zhai

Published Feb 2, 2024• 5 citations
arXivHTMLPDF

Abstract

Topological charge pumping occurs in the adiabatic limit, and the nonadiabatic effect due to finite driving velocity reduces the pumping efficiency and leads to deviation from quantized charge pumping. In this paper, we discuss the relation between this deviation from quantized charge pumping and the entanglement generation after a pumping circle. In the simplest setting, we show that the purity $\mathcal{P}$ of the half-system reduced density matrix is equal to $\mathcal{R}$, defined as ${(1\ensuremath{-}\ensuremath{\kappa})}^{2}+{\ensuremath{\kappa}}^{2}$, where $\ensuremath{\kappa}$ denotes the pumping efficiency. In generic situations, we argue that $\mathcal{P}<\mathcal{R}$ and the pumping efficiency can provide an upper bound for purity and, therefore, a lower bound for generated entanglement. To support this conjecture, we propose a solvable pumping scheme in the Rice-Mele-Hubbard model, which can be represented as a brick-wall-type quantum circuit model. With this pumping scheme, numerical calculation of charge pumping only needs to include at most six sites, and therefore the interaction and the finite-temperature effects can be both included reliably in the exact diagonalization calculation. The numerical results using the solvable pumping circle identify two regimes where the pumping efficiency is sensitive to driving velocity and support the conjecture $\mathcal{P}<\mathcal{R}$ when both interaction and finite-temperature effects are present.

DOI
10.1103/physrevb.109.064301

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Topological charge pumping occurs in the adiabatic limit, and the nonadiabatic effect due to finite driving velocity reduces the pumping efficiency and leads to deviation from quantized charge pumping. In this paper, we discuss the relation between this deviation from quantized charge pumping and the entanglement generation after a pumping circle. In the simplest setting, we show that the purity P\mathcal{P}P of the half-system reduced density matrix is equal to R\mathcal{R}R, defined as (1\ensuremath−\ensuremathκ)2+\ensuremathκ2{(1\ensuremath{-}\ensuremath{\kappa})}^{2}+{\ensuremath{\kappa}}^{2}(1\ensuremath−\ensuremathκ)2+\ensuremathκ2, where \ensuremathκ\ensuremath{\kappa}\ensuremathκ denotes the pumping efficiency. In generic situations, we argue that P<R\mathcal{P}<\mathcal{R}P<R and the pumping efficiency can provide an upper bound for purity and, therefore, a lower bound for generated entanglement. To support this conjecture, we propose a solvable pumping scheme in the Rice-Mele-Hubbard model, which can be represented as a brick-wall-type quantum circuit model. With this pumping scheme, numerical calculation of charge pumping only needs to include at most six sites, and therefore the interaction and the finite-temperature effects can be both included reliably in the exact diagonalization calculation. The numerical results using the solvable pumping circle identify two regimes where the pumping efficiency is sensitive to driving velocity and support the conjecture P<R\mathcal{P}<\mathcal{R}P<R when both interaction and finite-temperature effects are present.
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.
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.
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.
Experiments with bosonic atoms in optical superlattices allow for the interesting possibility to study the adiabatic quantized pumping of bosonic atoms in the presence of interactions. We investigate this exotic phenomenon for bound bosonic pairs in the paradigmatic Su-Schrieffer-Heeger model where the ground state exhibits topological properties due to dimerized hoppings. We show that at unit filling there exist crossovers and phase transitions to bond-order phases of paired bosons known as the pair-bond-order phase for attractive interactions. The pair-bond-order phase is found to exhibit effective topological properties such as the presence of polarized paired edge states. This is further analyzed by studying the emergence and breakdown of the Thouless charge pumping of bound bosonic pairs by a parametric extension to the famous Rice-Mele model. Finally, we discuss how the pumping of paired bosons or different regimes of breakdown of charge pumping can be probed by state-of-the-art experiments, for example, with repulsively bound bosons.
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.
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.