1.1.1 Motivation
Periodic driving is an indispensible part of the toolkit used to engineer nontrivial topo- logical phases in a variety of experimental settings. Most famously, it has been used in cold-atom systems to generate the artificial gauge fields necessary to realize the Harper- Hofstadter model [15, 16], which describes particles in a periodic potential subject to a magnetic field [17, 18], and the Haldane model [3], a prototypical instance of a Chern in- sulator [19]. Theoretical treatments of such systems rely heavily on Floquet theory, which provides systematic ways to derive time-independent effective Hamiltonians for quantum systems driven periodically in time (for a review, see Ref. [20]). By design, these effective Hamiltonians bear a close resemblance to those of the original (static) models [21–24].
This resemblance is misleading, however, because it ignores the fact that the periodically- driven system is not actually at equilibrium. For example, even the relatively simple project of realizing band insulators of noninteracting particles (let alone topologically nontrivial ones) in periodically-driven systems is itself highly nontrivial: the definition of such a state of matter relies on the equilibrium assumptions that energy is a good quantum number and
that a chemical potential can be defined, neither of which is true for a quantum system with a generic time-dependent Hamiltonian. More generally, there is the fundamental compli- cation that the quasienergies (essentially the eigenvalues of the time-independent Floquet Hamiltonian) are only defined modulo the driving frequency ω; in other words, there is no distinction between a Floquet eigenstate with quasienergy and one with quasienergy + n ω for any integer n. Thus, for an isolated Floquet system, there is no meaningful notion of a “ground state,” as there is at equilibrium.
Several approaches to these problems may be taken. First, in an isolated quantum system, one can try to prepare the system in a single Floquet eigenstate, for example by starting from the ground-state of an undriven system and adiabatically ramping the driving amplitude up from zero [25–27]. One problem with this approach, which also appears in static systems, is that in order to ramp between initial and final states that exhibit qualitatively different behavior, one usually has to cross a quantum critical point at which the gap closes and the adiabatic criterion fails. This necessarily involves the creation of excitations, the density of which can be estimated using the scaling theory of Kibble [28] and Zurek [29]. The prominence of this problem is magnified in Floquet systems, as one must consider the additional possibility of complications arising from quasienergy degeneracies modulo ω [30]. Despite these problems, this method of Floquet state preparation has been used successfully to prepare noninteracting topological states in the laboratory [17–19], and successful theories of this approach are being developed [31, 32].
The second approach to this problem follows the modus operandi of equilibrium quan- tum statistical mechanics. In this approach, dubbed “periodic thermodynamics” by Walter Kohn [33], the periodically-driven system is coupled to a thermal reservoir (or heat bath), and the balance between energy absorption due to the drive and dissipation due to the reservoir allows the system to reach a steady state at long times. This approach is espe- cially relevant to the study of periodically-driven solid-state systems, as electrons in real solids generally couple nontrivially to their environment (e.g., to vibrations in the ionic lat- tice background). The inclusion of a reservoir remedies the issue that Floquet systems have
no ground state a priori, with the steady-state density matrix and associated distribution functions defining the physical state of the system. However, owing to the breakdown of detailed balance in such systems, the resulting steady state is generically nonthermal (i.e., is not a Boltzmann or other equilibrium distribution) and depends on the details of the system-bath coupling [34].
The bulk of Part I (namely, Chapters 2–6) of this dissertation concerns itself with periodic thermodynamics and its implications. The questions of interest include:
• Can periodically-driven open quantum systems reach steady states that “look” ther- mal (i.e. that are well-described by an equilibrium statistical ensemble with a fixed temperature and/or chemical potential)?
• Are there examples of interesting (topological) quantum phases that realize such quasithermal steady states, and, if so, what kinds of steady-state behaviors can they exhibit?
• Are there constraints on the driving protocol, the bath, and the system-bath coupling which, if satisfied, ensure that the system will reach a quasithermal steady state? • Are there simple examples of truly nonthermal steady states whose distribution func-
tions can be calculated analytically? If so, what interesting nonequilibrium steady- state properties do they exhibit?
The work detailed in this dissertation manages to answer (at least partially) all of these questions, as we outline in Sec. 1.1.2.
There is a third approach to studying Floquet systems that is more natural for describ- ing quantum systems that are isolated from their environment (e.g. cold atomic gases). In this approach, one leaves aside the problem of preparing a Floquet eigenstate, and focuses instead on evolving an easy-to-prepare initial state (a product state, say) for a time t and then measuring local observables in the time-evolved state. Here, the eigenstates of the Floquet Hamiltonian simply play the role of a convenient basis in which to perform the
quantum evolution. Indeed, for a generic Floquet system, the initial product state is highly out-of-equilibrium with respect to the eigenstates of the Floquet Hamiltonian—that is to say, it generically has nonvanishing overlap with a large fraction of these eigenstates in the thermodynamic limit. Hence, such experiments can be quite useful in that they reveal information contained in highly excited states of the Floquet Hamiltonian; recent work has shown that these many-body excited states can contain quantum-coherent informa- tion protected by many-body localization [35, 36] or integrability [37, 38]. Moreover, such protocols are quite feasible in quantum simulators, and have been implemented sucessfully in a number of recent experiments [39–41]. In Chapter 7, we investigate the feasibility of using such finite-time dynamics as a tool to diagnose the presence of topological edge states associated with SPT phases. This work also demonstrates how to use periodic driving to systematically engineer the interactions needed to stabilize such topological phases, and therefore serves as an additional demonstration of the power of using periodic driving to design topologically nontrivial quantum states.
1.1.2 Outline
Chapter 2 introduces a particularly simple example of a periodically-driven condensed matter system and elucidates some of its properties. Starting from a model of electrons hopping on a honeycomb lattice that is shaken by a particular phonon mode, we show that the low energy theory consists of two species of Dirac fermions coupled by a rotating mass term. This model is crucial to Chapter 3, where it is studied in more detail, Chapters 4 and 5, where it is generalized, and to Chapter 6, where it serves as an important point of contrast. It also provides an example of how topologically protected behavior, in this case the quantized pumping of fractional charge, can be induced by periodic driving.
In Chapter 3, the model introduced in Chapter 2 is studied in greater detail, with a focus on its relationship to Floquet theory. A symmetry analysis unveils a deep reason for the exact solvability of the model, and in particular for the fact that the nonequilibrium steady state of the periodically driven system (in the presence of coupling to a heat bath) is
described by an equilibrium statistical ensemble. We return to the study of nonequilibrium steady states of periodically-driven open quantum systems in Chapters 5 and 6.
In Chapter 4, we present generalizations of the model introduced in Chapter 2 that arise from the coherent excitation of different zone-boundary phonon modes of the honeycomb lattice. The resulting low-energy theories involve momentum-dependent scattering between the two Dirac fermion species, in contrast to the case of Chapter 2, where this scattering was momentum-independent. The generalized class of models studied here is exactly solvable for the same reason as the model of Chapter 2. This allows for a rigorous determination of the phase diagram and associated transport properties of these models. For certain parameter regimes, we find that the system becomes a Chern insulator, with topologically protected gapless edge states having quantized conductance. This Chern insulating phase is distinct from the one realized by the celebrated Haldane model [3], which serves to underscore the rich variety of topological bands that can be realized with periodic driving. Chapter 5 takes a deeper look at the statistical mechanics of periodically-driven quan- tum systems coupled to thermal reservoirs. The class of models studied here is again a generalization of the class of models studied in Chapters 2–4. However, in a departure from the studies of previous chapters, the treatment here focuses on a class of examples where the periodically-driven system fails to thermalize, in the sense that the nonequilibrium steady state resulting from coupling the system to an external bath is not described by a Boltzmann distribution. This failure to thermalize is brought about by coupling the system to a particle reservoir (e.g. to fermionic leads or a metallic substrate) rather than to a bath of acoustic phonons. Despite the nonthermal nature of the resulting steady state, this class of problems is still sufficiently simple that the associated distribution functions can be cal- culated analytically. In this sense, the models studied in this chapter interpolate between the special case studied in Chapters 2–4, where the asymptotic steady state is thermal, and the general case, where the asymptotic steady state is not calculable analytically.
Chapter 6 abandons the exactly solvable models of the previous chapters and focuses on generic periodically-driven noninteracting quantum systems coupled to zero-temperature
reservoirs. The question under investigation is when and whether such systems reach a steady-state distribution characterized by fully filled Floquet bands, or, equivalently, by the absence of particle-hole excitations. The answer depends in a detailed way on the interplay of band-structure effects in the periodically-driven system and the density of states of the reservoir. The results of this chapter highlight the fact that the notion of “designer quantum matter” has limitations—using periodic driving to modify the behavior of a static quantum system generically yields unavoidable nonequilibrium effects that must be carefully taken into account.
Chapter 7 leaves aside the problem of open Floquet systems, and instead focuses on the possibility of realizing exotic many-body states in isolated quantum systems. In particular, this work shows how to realize nonequilibrium analogues of symmetry-protected topological (SPT) phases in periodically-driven interacting spin systems. These states of matter are enabled by periodic driving, which allows for the design of multispin interactions that would otherwise be heavily suppressed at equilibrium. Another novel element is the use of finite-time quantum dynamics to diagnose the topological character of the underlying stroboscopic evolution. This avenue of exploration offers many exciting prospects for the future, which will be discussed briefly in Chapter 10.