Abstract
Thouless pumping provides one of the simplest manifestations of topology in quantum systems and has attracted a lot of recent interest, both theoretically and experimentally. Since the seminal works by David Thouless and Qian Niu in 1983 and 1984, it has been argued that the quantization of the pumped charge is robust against weak disorder, but a clear characterization of the localization properties of the relevant states, and the breakdown of quantized transport in the presence of interaction or out of the adiabatic approximation, has long been debated. Thouless pumping is also the first example of a topological phase emerging in a periodically driven system. Driven systems can exhibit exotic topological phases without any static analogue and have been the subject of many recent proposals both in fermionic and in bosonic systems. Recent experimental studies have been performed in diverse platforms ranging from cold atoms to photonics and condensed-matter systems. This Review serves as a basis to understand the robustness of the topology of slowly driven systems and also highlights the rich properties of topological pumps and their diverse range of applications. Examples include systems with synthetic dimensions or work towards understanding higher-order topological phases, which underline the relevance of topological pumping for the fast-growing field of topological quantum matter.
Key points
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A direct current is usually associated with a dissipative flow of electrons in response to an applied bias voltage. In quantum systems, however, coherent transport can be induced via adiabatic cyclic variation of at least two system parameters in the absence of any external bias.
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A Thouless pump is a ‘quantum’ pump, in which the amount of ‘charge’ pumped during one cycle is quantized according to the Chern number — a topological invariant.
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Quantized charge pumps are robust to perturbations, such as disorder or weak interactions, as long as they do not change the topology of the pump.
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One-dimensional topological pumps can be seen as dynamical versions of the 2D integer quantum Hall effect in (1+1)-D, in which time plays the role of one spatial dimension and can be understood as a synthetic dimension.
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The concept of synthetic dimensions opens the door towards realizations of exotic higher-dimensional systems, such as the 4D quantum Hall effect.
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Acknowledgements
The authors thank all of their collaborators on the topic of topological pumping and related work, with whom they have had many stimulating interactions and discussions. M.A. acknowledges funding from the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) via Research Unit FOR 2414 under project number 277974659 and under Germany’s Excellence Strategy — EXC-2111 — 390814868. R.C. acknowledges the project Quantox of QuantERA ERA-NET Cofund in Quantum Technologies (grant agreement number 731473).
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Glossary
- Anderson-localized
-
The absence of diffusion in a disordered medium, which enables electron localization in a lattice potential, provided that the degree of disorder is sufficiently large.
- Bare tunnelling
-
Probability of a particle jumping from one lattice site to another in the absence of interaction among the particles.
- Glide symmetry
-
In 2D, a glide reflection symmetry is a symmetry operation that consists in a reflection with respect to a line and then a translation along that line, combined into a single operation.
- Hardcore interaction
-
An infinite interaction between particles, so that particles cannot overlap in space; represented by a Dirac delta potential that diverges, when the positions of the particles coincide.
- Hardcore bosons
-
Bosons that cannot occupy the same quantum state, like fermions; they interact with a Dirac delta potential, but without antisymmetric exchange statistics.
- Kubo formula
-
A formula expressing the linear response of a system quantity to an external time-dependent perturbation, which can be used to calculate the susceptibility of particles systems in response to a field.
- Landau–Zener transition
-
The probability of finding the system in the upper energy branch, when changing the system parameters through an avoided crossing. In the adiabatic limit, there will be no excitation to the upper branch.
- Slater determinant
-
An expression that describes the fermionic many-body wavefunction, satisfying the Pauli principle and the antisymmetric condition under the exchange of two particles.
- Solitons and antisolitons
-
The soliton is an excitation (like a kink) that propagates at a constant velocity in a nonlinear medium. The antisoliton is a kink that propagates in the opposite direction.
- Wannier orbitals
-
Localized molecular orbitals of crystalline systems that belong to a complete set of orthogonal functions used in solid-state physics.
- Wannier-state formalism
-
A solid-state physics formalism that uses Wannier functions as a complete set of orthogonal functions.
- Winding number
-
An integer representing the number of times that the curve travels anticlockwise around a singularity. The winding number depends on the orientation of the curve.
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Citro, R., Aidelsburger, M. Thouless pumping and topology. Nat Rev Phys 5, 87–101 (2023). https://doi.org/10.1038/s42254-022-00545-0
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DOI: https://doi.org/10.1038/s42254-022-00545-0