A Appendix: Degeneracy as Intersection of Subspaces

Floquet Supersymmetry

Abstract

We introduce the notion of time-reflection symmetry in periodically driven (Floquet) quantum systems, and show that it enables a Floquet variant of quantum-mechanical supersymmetry. In particular, we find Floquet analogues of the Witten index that place lower bounds on the degeneracies of states with quasienergies and . We provide a simple class of disordered, interacting, and ergodic Floquet models with an exponentially large number of states at quasienergies , 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.

Quantum systems driven by time-periodic perturbations are ubiquitous in atomic, molecular, and optical physics Shirley (1965); Sambe (1973); Bukov et al. (2015a). In recent years, periodic driving has been exploited by theory Rahav et al. (2003); Eckardt et al. (2005); Oka and Aoki (2009); Lindner et al. (2011); Dalibard et al. (2011); D’Alessio and Polkovnikov (2013); Goldman and Dalibard (2014); Grushin et al. (2014); Iadecola et al. (2015) and experiment Lin et al. (2009); Aidelsburger et al. (2013); Miyake et al. (2013); Jotzu et al. (2014); Bordia et al. (2017) as a resource for quantum simulation; by varying certain control parameters periodically in time, intricate effective Hamiltonians can be realized for synthetic quantum systems that might be outlandish in the context of solid-state physics. However, the analogy between static and periodically-driven (Floquet) quantum matter only goes so far. Being time-dependent, Floquet systems do not conserve energy and generically heat up to infinite temperature. They lose any discernible phase structure D’Alessio and Rigol (2014); Lazarides et al. (2014a) unless some notion of integrability Lazarides et al. (2014b); Chandran and Sondhi (2016); Gritsev and Polkovnikov (2017), many-body localization (MBL) D’Alessio and Polkovnikov (2013); Huse et al. (2013); Chandran et al. (2014); Ponte et al. (2015); Khemani et al. (2016), or prethermalization Abanin et al. (2015); Bukov et al. (2015b); Mori et al. (2016); Kuwahara et al. (2016); Canovi et al. (2016); Abanin et al. (2017); Else et al. (2017); Vaj () is invoked.

With any of these three stabilizing mechanisms, Floquet systems have exhibited many new phases that lack equilibrium counterparts. These include a vast array of Floquet topological phases von Keyserlingk and Sondhi (2016); Else and Nayak (2016); Roy and Harper (2016); Potter et al. (2016); Potirniche et al. (2017) and the so-called “ spin glass” (SG) Khemani et al. (2016); von Keyserlingk et al. (2016) or “discrete time crystal” (DTC) Else et al. (2016); Yao et al. (2017) phase, which are the objectives of recent experiments Zhang et al. (2017); Choi et al. (2017). These Floquet phases share qualitative features that stem from their nonequilibrium nature; for example, in all such phases there are certain operators whose dynamics synchronizes robustly with the periodic drive in a nontrivial way. This is especially striking in the SG/DTC, where the local magnetization exhibits robust subharmonic response at half the driving frequency.

In this work, we introduce a distinct class of Floquet systems that also exhibits subharmonic response, but for fundamentally different reasons than the SG/DTC. We dub this phenomenon “Floquet supersymmetry” (FSUSY), as the underlying structure has many close parallels to quantum-mechanical supersymmetry (SUSY). For instance, while SUSY exchanges bosons and fermions, FSUSY exchanges forward and backward time evolution—its generator is a time-reflection symmetry. SUSY models are characterized by an invariant, the Witten index, which provides a lower bound on the ground-state degeneracy; similarly, FSUSY models are characterized by two invariants, which place lower bounds on the degeneracies of the “quasienergies” and .

After establishing this general framework, we present a simple class of interacting, disordered, and ergodic Floquet models exhibiting FSUSY. In these models, the degeneracies of the quasienergies are exponentially large—at least , where is the system size. We show that this exponentially large degeneracy is robust to any disorder and interactions preserving the underlying time-reflection symmetry. Such models show a distinct experimental signature of FSUSY. Local observables exhibit a subharmonic response; however, in stark contrast to the SG/DTC, the response is suppressed exponentially in system size. This finite-size scaling of the response serves as sharp evidence of FSUSY. It is remarkable that this subharmonic response occurs in an otherwise ergodic quantum system; FSUSY provides an example of a class of thermalizing Floquet systems which display nontrivial phenomena in a macroscopic subspace of the full Hilbert space.

We begin with some definitions. Consider a periodically driven system with the time-dependent Hamiltonian , with the driving period hereafter set to (along with ). Define the Floquet unitary , which evolves states by one period:

(1)

has eigenstates {} with corresponding eigenvalues {}; the quasienergies {} are defined modulo .

We say that has time-reflection symmetry if there exists a unitary operator satisfying and

(2)

We hereafter set by redefining . Since maps the “forward” Floquet evolution operator to the “backward” Floquet evolution operator , it can be interpreted as reversing the direction of time. However, unlike the usual time-reversal operator, is unitary, hence the name “time-reflection symmetry.” Using Eq. (2), we can deduce the action of on the Floquet eigenbasis:

(3)

thus maps eigenstates of with quasienergy to eigenstates of with quasienergy . Hence,

(4)

In the eigenspaces, , so (2) implies that and share a common eigenbasis for the states. We will label the common eigenbasis for as {} where and is the degeneracy of the eigenspace. Because ,

(5)

These properties motivate the definition of two trace formulas which we will prove to be integers providing lower bounds for the degeneracies . Define as

(6)

where we have used (4). Moreover, (5) implies that both invariants are integers and ; these invariants thus provide a lower bound for the number of and quasienergy eigenstates, respectively.

Given time reflection symmetry, are topological invariants in the following sense. Consider any small perturbation to which preserves the time-reflection symmetry (2). We expect that the time reflection operator for which (2) holds will change continuously as is perturbed; we illustrate this later in a concrete model. Since the trace is a continuous function of and , small changes in the arguments must lead to small changes in . Because the latter are integers, they must remain invariant. Hence, any symmetry-respecting perturbation continuously connected to the identity operator will not change the trace invariants . We emphasize that the existence of these invariants and the subsequent properties depends essentially on the presence of time-reflection symmetry.

At this point, it is useful to draw contrasts and comparisons with ground states of static systems. At first glance, this symmetry-protected “pinning” of quasienergy eigenvalues to or may be reminiscent of the protection of certain zero-energy modes in symmetry protected topological phases Jackiw and Rebbi (1976); Su et al. (1979); Jackiw and Rossi (1981); Read and Green (2000); Teo and Kane (2010). In systems with topological defects or boundaries, zero modes may appear as bound states protected by index theorems that define topological invariants similar to Eq. (6Ati (); Weinberg (1981). However, in our Floquet setting there is no such bulk-boundary correspondence nor defects; the symmetry-protected and quasienergy modes are bulk entities.

In fact, the closest static analogues of these protected many-body degeneracies arise in SUSY Witten (1981), where the relevant topological invariant is the Witten index Witten (1982) , with the Hamiltonian, the fermion number, and the inverse temperature. The (integer) Witten index places a lower bound on the number of eigenstates at zero energy, and thereby on the ground-state degeneracy of . One remarkable phenomenon that can arise in certain SUSY models is “superfrustration,” where the Witten index scales exponentially with system size Fendley and Schoutens (2005); van Eerten (2005). This phenomenon typically arises in SUSY lattice models in two dimensions Huijse and Schoutens (2008); Huijse et al. (2012).

We now show that an exponentially large degeneracy arises in the context of FSUSY in one spatial dimension. For the sake of exposition, we begin with the simplest model below and add interactions later. Consider a spin- chain with sites and the two part drive

(7a)
(7b)
(7c)

Here, are Pauli operators on the site , and are random couplings. We impose that is even and choose periodic boundary conditions (identifying sites and ).

The model (7) has time-reflection symmetry, generated by the operator

(8)

To see that , one can rewrite , where . Using the fact that , one verifies that Eq. (2) holds with . Observe that depends explicitly on , just as the generator of SUSY depends explicitly on the Hamiltonian.

In fact, one can define another time-reflection operator

(9)

with the requisite properties, and . Again, this parallels SUSY, in which the Hamiltonian is constructed from the SUSY generators Witten (1981).

Having established time-reflection symmetries in this model, we calculate the trace invariants (6) and find

(10)

(for both ). Thus, there are an exponentially large number of states with quasienergy .

The above model can be rewritten in terms of free fermions via a Jordan-Wigner transformation. In fact, setting the random couplings places at a bicritical point separating four phases in the free-fermion phase diagram. In the spin language, these phases are a trivial paramagnet (PM), a Floquet paramagnet (PM, the precursor of the Floquet symmetry-protected topological phase in the disordered case), a trivial symmetry-broken phase (SB), and a Floquet symmetry-broken phase (SB, the precursor of the SG/DTC phase in the disordered case) Khemani et al. (2016).

However, the properties we have found are not artifacts of free fermions; they are robust to any interaction which preserves time-reflection symmetry. To illustrate this, we add interactions to the transverse field part of the drive:

(11)

where parameterizes the strength of interaction and we demand that anticommutes with . Given this condition, the modified system continues to be time-reflection-symmetric, with the modified time-reflection operator . As a result, the trace invariants (and exponentially large degeneracies) remain the same even in the presence of these interactions. In the Supplementary Material, we provide an alternative way to understand the degeneracy as arising from the intersection of two large subspaces; this derivation also explains why the model’s properties are robust to interactions of the above type.

For the purpose of numerics, we specify to the choice

(12)

and we draw the random couplings and uniformly from the interval .

Figure 1: (Color online) Statistics of quasienergy levels outside of the degenerate subspaces in the model (7), as measured by the parameter defined in Ref. Pal and Huse (2010). All data points are averaged over disorder realizations and quasienergy (see main text). Gray dashed lines at and indicate the expected values for the Poisson and Wigner-Dyson distributions, respectively.
Figure 2: (Color online) A representative time-series of the magnetization starting from an initial state with all spins polarized in the direction, for a single disorder realization in the interacting version of the model (7) with and [see Eqs. (11) and (12)]. Period- oscillations around the expected infinite-temperature value are clearly visible at late times. Inset: power spectrum for the same and , averaged over disorder realizations. The dominant coherent structure in the power spectrum is the peak at , which results from the period- oscillations.

One might wonder whether the symmetry constraint (2), which is evidently strong enough to protect exponentially large degeneracies, is also strong enough to constrain the many-body spectrum outside of the degenerate subspaces. Given the presence of strong disorder, are there signatures of many-body localization in this system? To answer these questions, we performed exact diagonalization at system sizes and (for , , and disorder realizations, respectively) and computed the disorder-averaged level statistics of the states outside the subspaces. We computed the parameter Pal and Huse (2010); given an ordered list of quasienergies, is defined in terms of the quasienergy gaps as the average of the quantity over quasienergy () and disorder realizations. Even at these very small system sizes, we see level statistics consistent with the Wigner-Dyson distribution for (see Fig. 1). Thus, apart from the protected degeneracies, the model (7) appears to be a generic ergodic system.

Figure 3: (Color online) Magnitude of the peak in at as a function of system size, with an exponential fit (gray, dashed line) and the estimate (Floquet Supersymmetry) (orange) plotted for reference. The model used to generate the data is that used in Fig. 2. Data are averaged over , , , and disorder realizations for , , , and , respectively.

Nonetheless, we now show that the protected degeneracies give rise to a distinct subharmonic response which serves as a direct signature of Floquet supersymmetry. In particular, the time evolution of the expectation values of certain operators exhibit period- oscillations. This follows directly from the existence of states, which are protected by FSUSY. Assume there is at least one protected pair of states with quasienergy , and denote by the space spanned by the two states. Then the Floquet operator restricted to can be represented by , where is the projection onto . Hence, the operator will flip sign every period. Note that in this general discussion may be non-local operators; however, in the model (7), there is a local operator, namely the on-site , which flips sign under the Floquet evolution restricted to the degenerate subspaces (see Supplementary Material).

Therefore, in the time evolution of , the latter piece will decay to zero because the complement of is generically ergodic, while the former piece contributes the period- oscillations. However, the ratio of the size of to that of the entire Hilbert space decreases exponentially with system size . Hence, if one evolves from a random initial state, then the amplitude of such period- oscillations will decrease exponentially with , a phenomenon that distinguishes FSUSY from the SG/DTC phase. In fact, such dependence on system size also occurs in signatures of SUSY in Majorana models with translation symmetry Hsieh et al. (2016).

We have checked these properties in the above model by computing the time evolution of the total magnetization starting from an initial state with all spins polarized in the direction. A representative time series for is shown in Fig. 2. Plots of the expectation values of single-site operators look similar. A useful figure of merit for quantifying this subharmonic response is the power spectrum , obtained by taking the modulus-squared of the Fourier transform of , which displays a peak at if exhibits period- oscillations. We indeed find such behavior in the power spectrum; averaging over disorder realizations, we find a single peak at , and all other structure washes out (see Fig. 2 inset).

For a typical initial state, which has overlap with all eigenstates of , we can estimate (up to a multiplicative prefactor) that

(13)

where is the overlap of eigenstates with the initial state . This exponential upper bound on the finite-size scaling of results from the fact that the degenerate quasienergy eigenstates constitute a fraction of order of all eigenstates of . We see finite-size scaling of the disorder-average of in exact diagonalization that is consistent with this estimate (see Fig. 3). Our simulations were carried out at , so that the energy levels outside the degenerate subspaces are approximately Wigner-Dyson-distributed. It is interesting that even in this chaotic regime, there are still coherent period- oscillations. Although this effect disappears in the thermodynamic limit due to the exponential suppression described above, it should be accessible in quantum simulation experiments, which are performed at a variable finite size.

There are several interesting avenues to pursue regarding both Floquet supersymmetry and the particular class of models presented. FSUSY provides an alternative mechanism for achieving subharmonic response; whereas the robustness in the discrete time crystal relies on the rigidity of eigenstates (long-range correlations in space), the robustness in FSUSY relies on the rigidity of the eigenvalues pinned to , a consequence of the underlying time-reflection symmetry. Moreover, FSUSY provides a mechanism whereby a protected subspace can exhibit nontrivial phenomena (e.g., period- oscillations) despite being embedded in a thermal system. Thus, even though non-integrable systems without many-body localization may heat to infinite temperature, it may be possible that a subspace (whose dimension can grow exponentially with system size) can behave nontrivially, as FSUSY illustrates.

The most pressing question concerning the model (7) is that of the nature of the degenerate states—aside from their fixed quasienergy, do they have any special properties that are not shared by the rest of the eigenstates of ? The Supplementary-Material derivation of the degenerate states as the intersection of two large subspaces suggests that the degenerate states may be highly entangled, but it would be useful to quantify the amount of entanglement. It would also be interesting to consider whether the protected macroscopic degeneracy could be useful for quantum information processing. Having access to an exponentially large number of exactly degenerate eigenstates could aid in the coherent storage and manipulation of quantum information.

We especially thank C.-M. Jian for useful discussions, as well as A. Chandran and V. Khemani. T.I. gratefully acknowledges the hospitality of the KITP during the course of this work. T.I. was supported by the National Science Foundation Graduate Research Fellowship Program under Grant No. DGE-1247312, the DOE under Grant No. DE-FG02-06ER46316, the Laboratory for Physical Sciences, Microsoft, and a JQI Postdoctoral Fellowship. T.H.H. is supported by the Gordon and Betty Moore Foundation’s EPiQS Initiative through Grant No. GBMF4304.

Supplementary Material for “Floquet Supersymmetry”

Appendix A Appendix: Degeneracy as Intersection of Subspaces

Here we provide an alternative proof for the degeneracy and show that it arises from the intersection of two large subspaces of the Hilbert space. For convenience, we assume is a multiple of ; it is straightforward to extend the proof below to other even . First observe that the operator has eigenvalues . This is because in any product state in the basis, produces a phase for every domain wall and otherwise. Since, the total number of domain walls and non-domain walls must add to , a multiple of , the total phase produced is .

Define to be the span of all product states in the basis on which the two operators and have identical action:

(A1)

Both and have eigenvalues , so to determine the dimension of , we need to count how many eigenvalues of are 1. This is achieved by evaluating

(A2)

where we have used the fact that any operator string involving a Pauli operator is traceless. Since the total number of eigenvalues is , we deduce that the number of eigenvalues 1, and hence the dimension of , is .

Consider the projection into the subspace . By appending this projection to the time evolution , we can trade for the simpler :

(A3)

where

(A4)

For the last equality, we have used the fact that all interactions in the argument of

(A5)

are assumed to anticommute with . Hence, is simply a rotated operator and thus has two -dimensional eigenspaces with eigenvalues .

Returning to (A3), we see that any state in will be an eigenstate of with eigenvalue . Thus finding a lower bound on the dimension of will provide the desired lower bound on the degeneracies. Finding this lower bound on the intersection is a pigeonhole-principle-type argument, which we make precise below.

We decompose

(A6)

where is the orthogonal complement of in . Now, define , the projector into . For any state , ; if this were not true, then it would follow that , which is a contradiction. This means that for any ,

(A7)

since otherwise the state would be annihilated by . Hence is injective as a function from into . It follows that . Since , we conclude that . A similar argument can be applied to lower-bound . Thus, we have proved that the eigenspaces at quasienergy and are each at least -fold degenerate.

This proof provides some intuition on the nature of the degenerate states: since they arise from the intersection of two extensive subspaces, one expects them to be generically highly entangled.

Appendix B Appendix: Dynamics of Restricted Floquet Evolution

Denote the union of the degenerate subspaces at quasienergies and by , and denote the projector onto by . From the above appendix, the Floquet evolution restricted to is given by . Recall that, up to a sign, is rotated by a unitary . It then follows that flips sign once per period of the projected Floquet evolution. In the simplest model (7), commutes with , so the on-site operator flips sign every period of the projected Floquet evolution.

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