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arXiv:1908.08062 · Physical Review Letters

Half-Integer Quantized Topological Response in Quasiperiodically Driven Quantum Systems

Philip J. D. Crowley, Ivar Martin, Anushya Chandran

Published Aug 31, 2020• 41 citations
arXivHTMLPDF

Abstract

A spin strongly driven by two harmonic incommensurate drives can pump energy from one drive to the other at a quantized average rate, in close analogy with the quantum Hall effect. The pumping rate is a nonzero integer in the topological regime, while the trivial regime does not pump. The dynamical transition between the regimes is sharp in the zero-frequency limit and is characterized by a Dirac point in a synthetic band structure. We show that the pumping rate is half-integer quantized at the transition and present universal Kibble-Zurek scaling functions for energy transfer processes. Our results adapt ideas from quantum phase transitions, quantum information, and topological band theory to nonequilibrium dynamics, and identify qubit experiments to observe the universal linear and nonlinear response of a Dirac point in synthetic dimensions.

DOI
10.1103/physrevlett.125.100601

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cond-mat.quant-gasCross-listedarXiv:2603.22173•Mar 24, 2026
A spin strongly driven by two harmonic incommensurate drives can pump energy from one drive to the other at a quantized average rate, in close analogy with the quantum Hall effect. The pumping rate is a nonzero integer in the topological regime, while the trivial regime does not pump. The dynamical transition between the regimes is sharp in the zero-frequency limit and is characterized by a Dirac point in a synthetic band structure. We show that the pumping rate is half-integer quantized at the transition and present universal Kibble-Zurek scaling functions for energy transfer processes. Our results adapt ideas from quantum phase transitions, quantum information, and topological band theory to nonequilibrium dynamics, and identify qubit experiments to observe the universal linear and nonlinear response of a Dirac point in synthetic dimensions.
Quasiperiodically driven fermionic systems can support topological phases not realized in equilibrium. The fermions are localized in the bulk, but support quantized energy currents at the edge. These phases were discovered through an abstract classification, and few microscopic models exist. We develop a coupled layer construction for tight-binding models of these phases in $d\ensuremath{\in}{1,2}$ spatial dimensions, with any number of incommensurate drive frequencies DDD. The models exhibit quantized responses associated with synthetic two- and four-dimensional quantum Hall effects in the steady state. A numerical study of the phase diagram for (d+D)=(1+2)(d+D)=(1+2)(d+D)=(1+2) shows (i) robust topological and trivial phases separated by a sharp phase transition, (ii) charge diffusion and a half-integer energy current between the drives at the transition, and (iii) a long-lived topological energy current which remains present when weak interactions are introduced.
The paradigm of Floquet engineering of topological states of matter can be generalized into the time-quasiperiodic scenario, where a lower-dimensional time-dependent system maps onto a higher-dimensional one by combining the physical dimensions with additional synthetic dimensions generated by multiple incommensurate driving frequencies. Differently from most previous works in which gapped topological phases were considered, we propose an experimentally realizable, one-dimensional chain driven by two frequencies, which maps onto a gapless Weyl semimetal in a synthetic dimension. Based on analytical reasoning and numerical simulations, we find that the nonadiabatic quantum dynamics of this system exhibit energy pumping behaviors characterized by universal functions. We also numerically find that such behaviors are robust against a considerable amount of spatial disorder.
Few-level quantum systems driven by nf{n}_{\mathrm{f}}nf​ incommensurate fundamental frequencies exhibit temporal analogs of noninteracting phenomena in nf{n}_{\mathrm{f}}nf​ spatial dimensions, a consequence of the generalization of Floquet theory in frequency space. We organize the fundamental solutions of the frequency lattice model for nf=2{n}_{\mathrm{f}}=2nf​=2 into a quasienergy band structure and show that every band is classified by an integer Chern number. In the trivial class, all bands have zero Chern number and the quasiperiodic dynamics is qualitatively similar to Floquet dynamics. The topological class with nonzero Chern bands has dramatic dynamical signatures, including the pumping of energy from one drive to the other, chaotic sensitivity to initial conditions, and aperiodic time dynamics of expectation values. The topological class is however unstable to generic perturbations due to exact level crossings in the quasienergy spectrum. Nevertheless, using the case study of a spin in a quasiperiodically varying magnetic field, we show that topological class can be realized at low frequencies as a prethermal phase, and at finite frequencies using counterdiabatic tools.
When a system is driven across a continuous phase transition, the density of topological defects demonstrates a power-law scaling behavior versus the quenching rate, as predicted by Kibble-Zurek mechanism. In this study, we generalized this idea and address the scaling of quantum state transport in a one-dimensional topological system subject to a linear drive through its topological quantum phase transition point. We illustrate the power-law dependencies of the quantum state's transport distance, width, and peak magnitude on the driving velocity. Crucially, the power-law exponents are distinct for the edge state and bulk state. Our results offer a novel perspective on quantum state transfer and enriches the field of Kibble-Zurek behaviors and nonadiabatic quantum dynamics.
When a system is driven across a continuous phase transition, the density of topological defects demonstrates a power-law scaling behavior versus the quenching rate, as predicted by the Kibble-Zurek mechanism. In this study, we generalize this idea and address the scaling of quantum state transport in a one-dimensional topological system subject to a linear drive through its topological quantum phase transition point. We illustrate the power-law dependences of the quantum state's transport distance, width, and peak magnitude on the driving velocity. Crucially, the power-law exponents are distinct for the edge state and bulk state. Our results offer a different perspective on quantum state transfer and enrich the field of Kibble-Zurek behaviors and nonadiabatic quantum dynamics.
Quantum anomalies arise when symmetries of a classical theory cannot be preserved upon quantization, leading to unconventional topological responses. A prominent example is the parity anomaly of a single two-dimensional Dirac fermion, which enforces a half-quantized Hall response. Anomaly inflow mechanism allows this effect to be observed at the surfaces of three-dimensional topological insulators, however, its realization in a genuinely two-dimensional system has remained elusive. Here we report the observation of a parity-anomalous Hall response at the critical point of a quantum Hall topological phase transition in a synthetic two-dimensional system of ultracold dysprosium atoms. By coupling a continuous spatial dimension to a finite synthetic dimension encoded in atomic spin states, we engineer tunable Chern bands with C = 0 and 1. At the transition, the bulk gap closes at a single Dirac point, where we observe a robust half-quantized Hall drift despite strong non-adiabatic excitations. We show that this response originates from the global structure of the band topology, is protected by an emergent parity symmetry at criticality, and disappears when parity is explicitly broken. Our work establishes synthetic quantum systems as a powerful platform to probe quantum anomalies and their interplay with topology and non-equilibrium dynamics.