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arXiv:1710.05927 · Physical Review Letters

Floquet Supersymmetry

Thomas Iadecola, Timothy H. Hsieh

Published May 24, 2018• 13 citations
arXivHTMLPDF

Abstract

We show that time-reflection symmetry in periodically driven (Floquet) quantum systems enables an inherently nonequilibrium phenomenon structurally similar to quantum-mechanical supersymmetry. In particular, we find Floquet analogs of the Witten index that place lower bounds on the degeneracies of states with quasienergies 0 and π. Moreover, we show that in some cases time-reflection symmetry can also interchange fermions and bosons, leading to fermion-boson pairs with opposite quasienergy. We provide a simple class of disordered, interacting, and ergodic Floquet models with an exponentially large number of states at quasienergies 0 and π, which are robust as long as the time-reflection symmetry is preserved. Floquet supersymmetry manifests itself in the evolution of certain local observables as a period-doubling effect with dramatic finite-size scaling, providing a clear signature for experiments.

DOI
10.1103/physrevlett.120.210603

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We show that time-reflection symmetry in periodically driven (Floquet) quantum systems enables an inherently nonequilibrium phenomenon structurally similar to quantum-mechanical supersymmetry. In particular, we find Floquet analogs of the Witten index that place lower bounds on the degeneracies of states with quasienergies 0 and π. Moreover, we show that in some cases time-reflection symmetry can also interchange fermions and bosons, leading to fermion-boson pairs with opposite quasienergy. We provide a simple class of disordered, interacting, and ergodic Floquet models with an exponentially large number of states at quasienergies 0 and π, which are robust as long as the time-reflection symmetry is preserved. Floquet supersymmetry manifests itself in the evolution of certain local observables as a period-doubling effect with dramatic finite-size scaling, providing a clear signature for experiments.
We propose a Floquet period-doubling time-crystal model based on a disordered interacting long-range spin chain where the periodic swapping of nearby spin couples is applied. This protocol can be applied to systems with any local spin magnitude sss and in principle also to systems with nonspin (fermionic or bosonic) local Hilbert space. We explicitly consider the cases s=1/2s=1/2s=1/2 and s=1s=1s=1, using analytical and numerical methods to show that the time-crystal behavior appears in a range of parameters. In particular, we study the persistence of period-doubling oscillations in time and the time-crystal properties of the Floquet spectrum (quasienergy $\ensuremath{\pi}$-spectral pairing and long-range correlations of the Floquet states), and introduce a quantity (the local imbalance) to assess what initial states give rise to a period-doubling dynamics. We also consider the quasienergy average level-spacing ratio, and find that the interval of parameters where the system dose not thermalize and persistent period doubling is possible corresponds to the one where the Floquet spectrum shows time-crystal properties.
The manifestations of topology are ubiquitous in condensed matter physics. One of the most striking examples is the integer quantum Hall effect, which served as a theoretical basis for developing many classes of new materials in the last years such as topological insulators. It has been realized that topology can manifest itself also in time-dependent quantum systems. The question that we try to address in this thesis is how much of these topological structures can be harnessed in the realistic dynamics of a periodically driven closed quantum system. We investigate two kinds of topological periodically driven systems Thouless pumping and Floquet Chern insulators. Our analysis shows that the nonequilibrium occupations of the eigenvalues of the Floquet Hamiltonian, the so called quasienergies, play a crucial role. In the case of Thouless pumping we analysed the pumped charge averaged over many cycles and discovered that the corrections to perfect quantization of the pumped charge are actually quadratic in the frequency of the driving for a suddenly switched-on driving. In the case of Floquet Chern insulators, we studied fermions on the honeycomb lattice subjected to a circularly polarised field. We found that while the quasienergy bulk states can be populated in a controlled way, edge states get unavoidable excitations due to exponentially small gaps preventing adiabaticity. These defects reflect in nonequilibrium currents flowing at the edges, that can be manipulated by tailoring the parameters of the periodic driving. Nevertheless, the nonequilibrium state undergoes a generalized form of topological transition, thanks to the controlled population of bulk states. We describe it with a generalized version of the Chern marker of Bianco and Resta
Periodically driven quantum systems known as Floquet insulators can host topologically protected bound states known as “ π modes” that exhibit response at half the frequency of the drive. Such states can also appear in undriven lattice field theories when time is discretized as a result of fermion doubling, raising the question of whether these two phenomena could be connected. Recently we demonstrated such a connection at the level of an explicit mapping between the spectra of a continuous-time Floquet model and a discrete-time undriven lattice fermion model. However, this mapping relied on a symmetry of the single-particle spectrum that is not present for generic drive parameters. Inspired by the example of the temporal Wilson term in lattice field theory, in this paper we extend this mapping to the full drive parameter space by allowing the parameters of the discrete-time model to be frequency-dependent. The spectra of the resulting lattice fermion models exactly match the quasienergy spectrum of the Floquet model in the thermodynamic limit. Our results demonstrate that spectral features characteristic of beyond-equilibrium physics in Floquet systems can be replicated in static systems with appropriate time discretization. Published by the American Physical Society 2024
Although the statistical mechanics of periodically driven ("Floquet") systems in contact with a heat bath has some formal analogy with the traditional statistical mechanics of undriven systems, closer examination reveals radical differences. In Floquet systems all quasienergies epsilon_{j} can be placed in a finite frequency interval 0< or =epsilon_{j}<omega (with omega the driving frequency and variant Planck's over 2pi=1 ). Therefore, if we describe a Floquet system approximately by restricting its available state space to be spanned by a finite number N of basis states, then the number of near degeneracies ( |epsilon_{j}-epsilon_{k}|< or =delta for arbitrarily small fixed delta ) in this interval grows without limit as N is increased. As we noted in a previous paper, this leads to pathologies, including drastic changes in Floquet states, as N tends to infinity. In earlier work on Floquet systems in contact with a heat bath these difficulties were often put aside by fixing N while taking the coupling to the bath to be smaller than any quasienergy difference. This led to a simple explicit theory for the reduced density matrix, with some major differences from the usual time-independent statistical mechanics. We show that, for weak but finite coupling between system and heat bath, the accuracy of a calculation within the truncated space spanned by the N lowest energy eigenstates of the undriven system is limited, as N increases indefinitely, only by the usual Born-Markov approximation, which neglects bath memory effects. As we seek higher accuracy by increasing N , we inevitably encounter quasienergy differences smaller than the system-bath coupling. We therefore derive here the equations for the steady-state reduced density matrix without restriction on the size of quasienergy splittings. In general, this matrix is no longer diagonal in the Floquet states. We analyze, in particular, the behavior near a weakly avoided crossing, where near degeneracies of quasienergies appear. In spite of the Floquet state pathologies, the explicit form of our results for the density matrix gives a consistent prescription for the statistical mechanics of periodically driven systems in the limit as N approaches infinity.
We reformulate the Floquet theory for periodically driven quantum systems following a perfect analogy with the proof of the Bloch theorem. We observe that the current standard method for calculating the Floquet eigenstates using the quasienergy alone is incomplete and unstable and pinpoint an overlooked quantum number, the average energy. This new quantum number resolves many shortcomings of the Floquet method stemming from the quasienergy degeneracy issues, particularly in the continuum limit. Using the average-energy quantum number, we get properties similar to those of the static energy, including a unique lower-bounded ordering of the Floquet states, from which we define a ground state, and a variational method for calculating the Floquet states. This is a first step towards reformulating Floquet first-principles methods, which have long been thought to be incompatible due to the limitations of the quasienergy.
We derive Floquet theory from quantum geometry. We identify quasienergy folding as a consequence of a broken gauge group of the adiabatic gauge potential U ( 1 ) ↦ Z . Fixing instead the gauge freedom using the parallel-transport gauge uniquely decomposes Floquet dynamics into a purely geometric and a purely dynamical evolution. The dynamical average-energy operator provides an unambiguous sorting of the quasienergy spectrum, identifying a Floquet ground state and suggesting a way to define the filling of Floquet-Bloch bands. We exemplify the features of geometric Floquet theory using an exactly solvable X Y model and a nonintegrable kicked Ising chain. We elucidate the geometric origin of inherently nonequilibrium effects, like the π -quasienergy splitting in discrete time crystals or π -edge modes in anomalous Floquet topological insulators. The spectrum of the average-energy operator is a susceptible indicator for both heating and spatiotemporal symmetry-breaking transitions. Last, we demonstrate that the periodic lab-frame Hamiltonian generates transitionless counterdiabatic driving for Floquet eigenstates. This work directly bridges seemingly unrelated areas of nonequilibrium physics.