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 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.
At the moment, the most efficient method to compute the state of a periodically driven quantum system is using Floquet theory and the Floquet eigenbasis. The wide application of this basis set method is limited by a lack of unique ordering of the Floquet eigenfunctions, an ambiguity in their definition at resonance, and an instability against infinitesimal perturbation at resonance. We address these problems by redefining the eigenbasis using a revised definition of the average energy as a quantum number. As a result of this redefinition, we also obtain a Floquet-Ritz variational principle, and justify the truncation of the Hilbert space.
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.
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 XY model and a non-integrable 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.
Clean, interacting, periodically driven systems are believed to thermalize to featureless, infinite-temperature states that preclude the formation of stable ordered states. Here, we show stable quantum order, distinct from prethermal order, and its phase transitions in the ground state of the average-energy operator. Decomposing the time evolution into geometric and average-energy dynamics, we apply Geometric Floquet theory to identify the Floquet ground state. We derive a general variational principle to compute the Floquet ground state in large interacting periodically driven systems, and use it to develop a matrix product state algorithm. In driven nonintegrable chains of more than 100 interacting spins, our numerical simulations exhibit a gapped, area-law entangled Floquet ground state in extended parameter regimes; the ground-state manifold can support both dressed equilibrium order (e.g., Floquet antiferromagnets) and inherently nonequilibrium order (e.g., ground-state time crystals). This quantum order is protected by a finite average-energy gap, which closes at phase transition points upon changing model parameters. We decompose Floquet phases of matter into their equilibrium and nonequilibrium elements: whereas the Floquet ground state at any fixed time is a representative of an equilibrium phase of matter, the nonequilibrium content is entirely produced by the drive holonomy and is geometric. Finally, we identify quasilocal counterdiabatic driving as a mechanism for the stability against heating of gapped Floquet ground states in nonintegrable quantum systems, and argue why Floquet ground states evade the Eigenstate Thermalization Hypothesis. Our work enables the study of interacting nonequilibrium matter deep into the many-body regime, and lays the foundations for the engineering of strongly correlated ordered quantum states.
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.
Periodically driven quantum systems exhibit many fascinating phenomena absent in equilibrium systems, but their simulation is more challenging than that of static systems. Consequently, quantum simulation of these systems offers greater opportunity for achieving quantum advantage. To build the foundation for simulating time-periodic Hamiltonians, we utilize the Floquet-Hilbert space formalism to transform the Hamiltonian into a time-independent form and provide its Pauli decomposition. Within this formalism, we develop a hybrid quantum-classical algorithm, Floquet-ADAPT-VQE, that produces three key quantities that govern periodically driven quantum dynamics: the quasienergy, the associated Floquet eigenstates, and expectation values of observables with respect to these eigenstates. The latter is achieved by applying a fixed-depth quantum circuit to an initial Floquet eigenstate—an efficient construction made possible by the Floquet-Hilbert space formalism. We also identify a suitable auxiliary initial state whose squared Floquet energy is independent of the number of auxiliary qubits as well as the driving frequency, which leads to better convergence with fewer ADAPT iterations. We demonstrate our algorithm by performing numerical simulations on a periodically driven XYZ model with a magnetic field. We also explore potential applications of our algorithm for studying various non-equilibrium phenomena in periodically driven systems.