Flow Equations for Disordered Floquet Systems
SSciPost Physics Submission
Flow Equations for Disordered Floquet Systems
S. J. Thomson , D. Magano , M. Schir´o Centre de Physique Th´eorique, CNRS, Institut Polytechnique de Paris, Route de Saclay,F-91128 Palaiseau, France Institut de Physique Th´eorique, Universit´e Paris-Saclay, CNRS, CEA, F-91191Gif-sur-Yvette, France JEIP, USR 3573 CNRS, Coll`ege de France, PSL Research University, 11 Place MarcelinBerthelot, 75321 Paris Cedex 05, France Instituto de Telecomunica¸c˜oes, Physics of Information and Quantum Technologies Group,Portugal Instituto Superior T´ecnico, Universidade de Lisboa, Portugal* [email protected] 8, 2020
Abstract
In this work, we present a new approach to disordered, periodically driven (Flo-quet) quantum many-body systems based on flow equations. Specifically, weintroduce a continuous unitary flow of Floquet operators in an extended Hilbertspace, whose fixed point is both diagonal and time-independent, allowing us todirectly obtain the Floquet modes. We first apply this method to a periodicallydriven Anderson insulator, for which it is exact, and then extend it to drivenmany-body localized systems within a truncated flow equation ansatz. In partic-ular we compute the emergent Floquet local integrals of motion that characterise aperiodically driven many-body localized phase. We demonstrate that the methodremains well-controlled in the weakly-interacting regime, and allows us to accesslarger system sizes than accessible by numerically exact methods, paving the wayfor studies of two-dimensional driven many-body systems.
Contents a r X i v : . [ c ond - m a t . d i s - nn ] S e p ciPost Physics Submission4 Flow Equations for Floquet Systems 8 A.1 Diagonalising the Hamiltonian 24A.2 Obtaining the eigenstates 24
B Details of the calculation 25
B.1 Defining and Computing the Generator 25B.2 Computing the flow of the running couplings 26
References 26
Understanding the nonequilibrium dynamics of quantum many-body systems is a key chal-lenge at the heart of modern research in condensed matter physics, motivated by a host ofrecent experimental developments in quantum simulation which have enabled unprecedentedlevels of control of strongly correlated quantum matter.Experimental advances in ultracold atomic gases, for example, have made it possible toengineer almost perfectly isolated quantum systems and have allowed transport properties andnonequilibrium dynamics to be probed with a high degree of control and resolution [1]. Onecurrent frontier is the use of time-periodic drive, such as a laser or time-varying magnetic field,to engineer effective Hamiltonians, leading to new states of matter far from equilbrium [2–4].This line of thought, known as Floquet engineering, has recently lead to a number of dra-matic experimental breakthroughs with cold atoms in dynamically modulated optical lattices,including the realization of non-trivial topological phases [5, 6], the control of magnetic corre-2 ciPost Physics Submission lations in strongly interacting fermionic gases [7], and the experimental realization of stronglydriven Fermi [8] and Bose Hubbard models [9, 10], which play important roles in the quantumsimulation of condensed matter systems.Another exciting contemporary development is the experimental realization of novel phasesof matter with no equilibrium counterpart, such as the Many-Body Localized (MBL) phase [11–14] seen in isolated disordered many-body quantum systems which fail to thermalize. Recentexperimental advances in quantum simulators have allowed highly precise control over disor-dered many-body systems and led to evidence of MBL behavior in a number of platforms,ranging from one and two dimensional arrays of ultracold atoms [15–19] to ion traps withprogrammable random disorder [20, 21] and dipolar systems made by nuclear spins [22, 23].The combination of disorder, interactions and periodic drive represent one of the currentfrontiers of the field. While ergodic quantum many-body systems are expected to reach ther-mal equilibrium of local observables at long times [24], which for driven systems lacking timetranslational invariance corresponds to infinite temperature [25, 26], MBL phases can avoidthermalisation even in presence of a periodic drive [27–29] and give rise to exotic nonequi-librium phases of matter such as quantum time crystals [30], which have been intensivelytheoretically explored in [31–40] and very recently experimentally observed [21, 41].From a theoretical perspective, the study of periodically driven - or
Floquet - quantum sys-tems has a long history, in the context of phenomena ranging from dynamical localization [42]and quantum dissipation [43] to quantum chaos [44]. Due to the time-periodic nature of theHamiltonian H ( t + T ) = H ( t ) and the linearity of Schrodinger equation, the problem can beanalyzed using the Floquet theorem, a basic result in the theory of linear ordinary differentialequations and the time-analogue of Bloch’s theorem in solid state physics. This formal anal-ogy, which has been a major force in the recent understanding of driven quantum many-bodyproblems [45], goes further with the introduction of an effective Floquet Hamiltonian , H F ,defined through the evolution operator over a period, e − iH F T ≡ U ( T,
0) where T is the driveperiod. This static Hamiltonian can be explicitly derived, and is therefore of practical rele-vance, only in certain regimes such as at high drive frequency [46–49] where energy absorptionis suppressed. In the more interesting regime of intermediate drive frequency one needs togo beyond the effective Floquet Hamiltonian [50, 51]. Numerical approaches based on exactdiagonalization of the full evolution operator allow access to the complete information of theFloquet eigenstates and eigenmodes, but are usually limited to very small system sizes. In thecontext of disordered systems, powerful methods such as the strong-disorder renormalizationgroup have been extended to the Floquet context [52–54].In this work, we set out and demonstrate the use of the flow equation approach to studyperiodically driven and disordered many-body quantum systems. Flow equations have beenused in recent years to study a wide variety of systems with time-independent Hamiltonians,including Kondo and impurity models [55, 56], quenches in the Hubbard model [57], andmore recently have gained a lot of attention in the context of quantum localization [58–65]. The method has also been successfully employed in the study of dissipative systems,either by decoupling the system from its environment (e.g. the spin-boson model studied inRefs. [66–68]) or more recently in a situation where the dynamics is generated by a MarkovianLindblad master equation [69] with a Lindbladian which can be diagonalised directly usingthis approach. In the context of Floquet systems previous works have used similar techniquesto systematically derive effective Hamiltonians [70, 71]. Here instead we focus on the Floquetevolution operator and devise a flow equation approach to diagonalize it, therefore obtainingthe quasienergies and Floquet eigenstates directly.3 ciPost Physics Submission This paper is organized as follows. In Section 2 we first summarise the use of flow equa-tions for static systems, discuss the generalisation of this approach to time-dependent systemsand demonstrate why periodic drive is particularly amenable to study with flow equations. InSection 3, we give a brief summary of Floquet theory and set out the mathematical formalismwe will go on to use in Section 4 where we discuss a number of different flow equation ap-proaches and show how our method differs from existing techniques. In Section 5 we presentan example of the Floquet flow equation technique applied to a system of non-interactingfermions in a disordered potential subject to periodic drive. We then go on to show how themethod may be extended to the case of fermions with weak interactions in Section 6, wherewe consider a driven many-body localized phase and show how our method gives insight asto the emergent local integrals of motion which characterise this phase, as well as an estimatefor the breakdown of the localization as a function of drive frequency. Finally, we concludewith an outlook towards the future and discuss possible applications of our method beyondthose which we present here.
Flow equation methods have a rich history as applied to time-independent models. Theywere originally introduced to condensed matter physicists by Wegner [72], independently inthe context of high-energy physics by Glazek and Wilson [73,74] under the name of ‘similaritytransforms’, and to mathematicians under the names ‘double bracket flow’ and ‘isospectralflow’ by Refs. [75–77]. For a detailed introduction to the method, we refer the reader toRefs. [56, 72], but here we will present a brief overview of the original formulation of themethod before discussing how it can be extended to time-dependent Hamiltonians.
In the case of a time-independent Hamiltonian, the flow equation formalism can be bestexplained by analogy with a Schrieffer-Wolff transform [78, 79]. A Schrieffer-Wolff transformis a unitary transform of the form:˜ H = e S H e − S = H + [ S, H ] + ..., (1)where, by choosing [
S, H ] = − V such that V contains the off-diagonal terms, the Hamilto-nian can be diagonalised to leading order. By expanding the exponential to higher ordersthe expansion can be made more accurate, at the cost of having to evaluate high-order com-mutators. Rather than making a single ‘large’ unitary transform, however, one can insteadimagine making an infinitesimal transform which can be made arbitrarily accurate: H ( l + d l ) = e η ( l )d l H ( l )e − η ( l )d l = H ( l ) + d l [ η ( l ) , H ( l )] . (2)where l is a fictional ‘flow time’ which runs from l = 0 to l = ∞ and η ( l ) is some (scale-dependent) anti-Hermitian generator for the transform, which we shall discuss in detail later.By making infinitely many of these unitary transforms, the diagonalization process can beexpressed as a single continuous unitary transform obeying the ‘equation of motion’: ∂H ( l ) = [ η ( l ) , H ( l )] . (3)4 ciPost Physics Submission By analogy with renormalization group, this equation is known as the ‘flow equation’ for theHamiltonian. By integrating the flow equation, which is typically done numerically, one canobtain a diagonal Hamiltonian . There are many possible choices for the generator η ( l ), asthe only requirements are that it is anti-Hermitian and diagonalises the Hamiltonian in thelimit l → ∞ . In this work we shall concentrate on the canonical choice (also known as the‘Wegner generator’ [72]) where we choose η ( l ) = [ H ( l ) , V ( l )] , (4)where H contains the diagonal components of the Hamiltonian and V = H − H contains theoff-diagonal terms. Other choices are possible [58, 64], however the canonical generator tendsto be a robust choice that can be stably numerically integrated. For reference, in Appendix Awe sketch the application of this method to a system of non-interacting fermions where it canbe applied straightforwardly and exactly.Flow equations have proven to be useful non-perturbative tools able to access unique pa-rameter regimes and system sizes inaccessible to most other state-of-the-art modern numericalmethods, such as two-dimensional many-body localized systems or systems with disorderedlong-range couplings [61, 65], and are the subject of active ongoing development due to theirflexibility and range of desirable properties. One key advantage of the flow equation methodover many other techniques is that flow equations treat all energy scales equivalently, and arenot restricted to perturbative situations or low-lying excitations. As compared with renormal-ization group calculations, it is important to note that flow equations retain all informationcontained in the Hamiltonian at all stages of the flow, as they are simply unitary transformsof the original problem: no decimation is required and no information is lost, as the transformcan always be reversed back into the original basis. A formal extension of the flow equation method to time-dependent Hamiltonian problems wasintroduced in Ref. [80] in the context of the time-dependent Kondo model. We briefly recall thebasic idea here, since it will serve as starting point for our Floquet flow. The aim is to performa time-dependent unitary transformation into a frame in which the Schr¨odinger equation i∂ t | ψ ( t ) (cid:105) = H ( t ) | ψ ( t ) (cid:105) is simplified. The dynamics in the rotated frame, | ˜ ψ ( t ) (cid:105) = U ( t ) | ψ ( t ) (cid:105) ,reads i∂ t | ˜ ψ ( t ) (cid:105) = ˜ H ( t ) | ˜ ψ ( t ) (cid:105) with a transformed Hamiltonian of the following form:˜ H ( t ) = U ( t ) [ H ( t ) − i∂ t ] U † ( t ) (5)In the spirit of time-independent flow equations, a series of infinitesimal time-dependent uni-tary transformations is introduced, parametrized by the scale l and with generator η ( l, t ).The time-dependent analog of Eq. 3 becomes [80]: ∂ l H ( l, t ) = [ η ( l, t ) , H ( l, t )] + i∂ t η ( l, t ) (6)This differs from the time-independent flow by the addition of the time derivative term, whichvanishes for a time-independent problem and reduces back to Eq. 3 . Similarly, the canonicalgenerator (Eq. 4) can be modified to: η ( l, t ) = [ H ( l, t ) , V ( l, t )] − i∂ t V ( l, t ) (7) In principle, one can also construct the unitary transform explicitly as a time-ordered exponential ˜ U = T l (cid:82) ∞ exp( η ( l ))d l , however this is typically difficult to evaluate and rarely of practical use. ciPost Physics Submission where H ( l, t ) represents the diagonal part of the Hamiltonian and V ( l, t ) = H ( l, t ) − H ( l, t )represents the off-diagonal part. Following Ref. [80], this form of generator can be shownto eliminate all (non-resonant) off-diagonal couplings in the l → ∞ limit, giving a diagonal,yet still time-dependent, Hamiltonian. However, for a general time-dependent protocol, thisflow can be extremely complicated to solve, requiring the solution of a set of coupled partialdifferential equations in both flow time l and real time t . In this work, we focus on situationswhere the time dependence of the microscopic Hamiltonian is periodic, and we find thatin this case the canonical generator (Eq. 7) can be implemented efficiently using Floquettheory (which we briefly introduce in the following section). This provides a starting point toformulate a flow equation approach directly in Floquet space which, differently from Eq. 7,eliminates both the off-diagonal terms as well as the time-dependence. We will discuss thedetails of this approach, which is the main result of this manuscript, in Section 4. Periodically driven systems, known as Floquet systems, are described by Hamiltonians whichare periodic in time, satisfying H ( t + T ) = H ( t ) where T is the period of the drive. FollowingFloquet’s theorem [81], which is analogous to Bloch’s theorem in solid state systems, period-ically driven systems admit a complete set of quasi-periodic solutions of the time-dependentSchr¨odinger equation, as ‘Floquet eigenstates’ | Ψ α ( t ) (cid:105) = e − iε α t/ (cid:126) | ψ α ( t ) (cid:105) (8)where | ψ α ( t + T ) (cid:105) = | ψ α ( t ) (cid:105) are states with the same periodicity as the drive, which satisfy( H ( t ) − i∂ t ) | ψ α ( t ) (cid:105) = ε α | ψ α ( t ) (cid:105) (9)while the phases ε α ∈ R are known as the Floquet quasi-energies . The quasi-energies donot depend on the microscopic time t , only the period of the drive T , and are associatedto corresponding Floquet eigenstates which form a complete orthonormal basis. Note thatbecause of the complex exponential in Eq. 8, the quasienergies are only uniquely definedup to a shift by an integer multiple of ω = 2 π/T . By analogy with crystal momentumin solid-state systems, one typically defines a ‘Brillouin zone’ such that all quasienergiesare uniquely determined in a given interval of width ω . Here, we use the convention that ε n ∈ [ − ω/ , ω/ ∀ n . In the following, we shall work in units where (cid:126) ≡
1. There areessentially two general approaches to obtain Floquet eigenstates and eigenmodes for quantummany-body systems, which we briefly sketch below.
In numerical studies based on exact solution of quantum dynamics the most efficient way toobtain Floquet eigenstates and eigenvectors is through the evolution operator over a driveperiod, U ( T,
0) = exp (cid:18) − i (cid:90) T dtH ( t ) (cid:19) (10)6 ciPost Physics Submission which satisfies several useful properties. For example, the evolution at integer multiples of thedrive satisfies U ( nT,
0) = ( U ( T, n , which implies that as long as one observes the systemonly stroboscopically the quantum evolution corresponds to applying n times the same evolu-tion operator U ( T, U ( T, ≡ exp ( − iT H F ).Furthermore, one can show that the Floquet modes are the eigenstates of the evolution oper-ator during a period T with eigenvalues e − iε α T/ (cid:126) which suggests the decomposition U ( t + T, t ) = (cid:88) α e − iε α T/ (cid:126) | ψ α ( t ) (cid:105) (cid:104) ψ α ( t ) | (11)The above results suggest several practical numerical approaches to solve the Floquet problem.One possibility involves computing the evolution operator from time t = 0 to time T , solvingthe exact dynamics, then diagonalizing it to find the Floquet modes at time t = 0 andthe quasi-energies. Then one can just propagate these states to get them in the full firstcycle. Otherwise one can solve for the evolution operator in the full first cycle U ( t,
0) andthen diagonalize it to obtain directly the Floquet modes over the first cycle. While theseapproaches are practically useful for exact numerics, here we choose a different path for ourFloquet flow equation approach using the expansion in an extended Hilbert space.
Since the Floquet modes are periodic with period T it is possible to expand them in terms ofFourier harmonics at integer multiples of their fundamental frequency ω , and to associate eachof these harmonics with an element the space of T -periodic, square-integrable functions L T .This results into an enlarged Floquet Hilbert space (often referred to as a Sambe space [82])denoted F = H ⊗ L T which is given by the tensor product of the original Hilbert space of themicroscopic system H and L T .Explicitly, for a given time-periodic Floquet mode | ψ α ( t ) (cid:105) in H , we can write it in theextended Floquet Hilbert space F by: | ψ α ( t ) (cid:105) = (cid:88) n | ψ nα (cid:105) e inωt = (cid:88) n | ψ nα (cid:105) ⊗ σ n (12)where σ n | m (cid:105) = | n + m (cid:105) is a creation operator in ‘frequency space’ L T associated to the n th harmonic, and the sum over harmonics runs over both positive and negative values.The index n is often known as the ‘photon number’ by analogy with the quantum systemsdriven by coherent radiation [83], although here we shall refer to it as the ‘harmonic index’.Plugging this expansion in Eq. (9) allows us to cast the problem of finding Floquet modesinto a conventional eigenvalue problem in a higher dimensional space, a sort of synthetic extradimension [84, 85]. To see this we notice that the time derivative − i∂ t acting in H can berewritten in the extended Hilbert space F as: D = ⊗ ω ˆ n, (13)where ˆ n = (cid:80) n n | n (cid:105) (cid:104) n | is the number operator in frequency space. This can be verifiedby showing that − i∂ t exp( inωt ) = ωn exp( inωt ), and therefore − i∂ t → ω ˆ n in L T . Themain object of interest in this extended Hilbert space is known as the Floquet quasienergy7 ciPost Physics Submission operator [86], given by: K = H ( t ) − i∂ t = (cid:88) n H ( n ) ⊗ σ n + ⊗ ω ˆ n (14)whose eigenstates and eigenvalues give the Floquet modes. As noted in Ref. [86], althoughthe quasienergy operator K lives in a dramatically larger Hilbert space than the initial formu-lation of the problem, there is a considerable simplification involved in that we can now usestandard time-independent methods to diagonalize K . Moreover, we never need to constuct K explicitly, as we shall go on to show: this is important as the explicit form of K contains alot of redundant information due to the tensor product. In Section 4 we will discuss severaldifferent strategies to diagonalize K , before showing a concrete example in Section 5 for anon-interacting system, and demonstrate the extension to weak interactions in Section 6. Based on the discussion of previous sections we can now present the main aim of thismanuscript, which is to devise a flow equation approach to diagonalize the Floquet oper-ator K in the extended Floquet Hilbert space. This is achieved by applying a continuousunitary transform parameterized by a running scale l and with generator η ( l ) such that theflow of K towards a diagonal form is given byd K/ d l = [ η ( l ) , K ( l )] , (15)for a suitable choice of η . In the static case, the choice of unitary transform is not unique, andone has a lot of freedom to choose a generator with properties which suit the problem. Forexample, it is possible to choose a generator which preserves the sparsity of the Hamiltonian[58], although this turns out to be numerically difficult to integrate [64]. In the case ofdriven systems, however, there is an even richer variety of possible choices. Depending on therequirements, one may choose a generator which diagonalises K in H (i.e. a diagonal, but stilltime-dependent Hamiltonian), in L T (time-independent but not diagonal in real space) or inthe full Hilbert space F . Here we review several choices from the existing literature whichaccomplish the first two goals, and propose a new generator which cuts to the chase by actingdirectly in F to produce a diagonal, time-independent Floquet operator from which we candirectly obtain the quasi-energies and Floquet eigenstates.In each of the following cases, we shall split the Floquet quasienergy operator K into adiagonal piece K and an off-diagonal piece K off , and choose the generator to be η = [ K , K off ].We have the complete freedom to choose the form of these terms to be whatever we wish: weshall see that by making different choices, we can obtain qualitatively very different results.From here on, we suppress explicit dependence on the running scale l for clarity. The canonical generator (Eq. 7), corresponds to the time-dependent generator introduced inRef. [80], which we may expand in Floquet harmonics. As such diagonalises the Hamiltonianin space, but leaves the final form time-dependent, i.e. diagonal in H but not in L T . Theresult of this flow is shown schematically in Fig. 4b. Concretely, in order to compute the8 ciPost Physics Submission Figure 1: A schematic of different flow schemes for the Floquet quasienergy operator, heredenoted as K = (cid:80) i,j,n K ( n ) ij | i (cid:105) (cid:104) j | ⊗ σ n . Panel (a) shows the full Floquet quasienergy operator K - note that in all of the cases we consider, this is a sparse matrix, however the method canequally be applied to dense matrices. Panels (b), (c) and (d) show the form of the diagonalmatrix after the application of Type I, II and III generators respectively. Panel (b) showsthat Type I generators (Section 4.1) result in a Floquet matrix which is time-dependent butdiagonal in space, while (c) demonstrates that Type II generators (Section 4.2) result in ablock-diagonal form of K which is time-independent but not diagonal in real space. Panel (d)shows the result after applying a Type III (Section 4.3) generator which we focus on in therest of this manuscript, which is both diagonal in space and time-independent.unitary transform using this generator, we split the Floquet quasienergy operator K intodiagonal ( K ) and off-diagonal ( K off ) pieces: K = (cid:88) n (cid:88) i H ( n ) ii ⊗ σ n + ⊗ ω ˆ n (cid:124) (cid:123)(cid:122) (cid:125) K + (cid:88) n (cid:88) i (cid:54) = j H ( n ) ij ⊗ σ n (cid:124) (cid:123)(cid:122) (cid:125) K off (16)where the canonical generator Eq. 7 is then given by η = [ K , K off ]. This amounts to thechoice that the ‘off-diagonal’ elements to be removed by the unitary transform are the termswhich are off-diagonal in real space, regardless of their time-dependence.In order to obtain the quasi-energies, the time-dependence of the final result must beremoved by a further change of basis into a rotating frame such that the problem becomestime-independent. 9 ciPost Physics Submission It is also possible to use flow equations to obtain a time-independent effective Floquet Hamil-tonian which is not diagonal in real space. To do this, we may make a different decompositionof the Floquet operator into diagonal and off-diagonal parts, following Ref. [70]: K = H (0) ⊗ + ⊗ ω ˆ n (cid:124) (cid:123)(cid:122) (cid:125) K + (cid:88) n (cid:54) =0 H ( n ) ⊗ σ n (cid:124) (cid:123)(cid:122) (cid:125) K off (17)which, in contrast with the previous case, amounts to choosing the ‘off-diagonal’ terms whichwill be eliminated by the flow to be all terms with harmonic index n (cid:54) = 0 regardless of theirreal-space structure. With this choice, the generator becomes: η = [ K , K off ] = [ D , K off ] = (cid:88) n ωn (cid:88) ij H ( n ) ij ⊗ σ n (18)Contrary to the canonical generator, this choice leads to a Hamiltonian which is time-independent without being diagonal in space, i.e. diagonal in L T but not in H . In figureFig. 4c we show a pictorial representation of this type of flow - it leaves K block-diagonal,i.e. diagonal in frequency (time-independent) but not diagonal in space. This is useful if onewishes to obtain the effective Floquet Hamiltonian H F for stroboscopic evolution, or studythe form of the effective Hamiltonian in the context of Floquet engineering. This is similarin spirit to the method of Ref. [71] designed to construct effective Floquet Hamiltonians, al-though the precise choice of generator is different. If one is interested in the quasi-energiesand Floquet eigenstates then one must then employ a second transformation to diagonalize H F . Motivated by the two previous choices, we here propose a modified generator combiningaspects of both which we call the ‘Wegner-Floquet’ (WF) generator. Essentially, it is givenby Wegner’s canonical choice (Eq. 4) applied to the Floquet quasienergy operator (i.e. actingin F ) rather than to the bare Hamiltonian (i.e. acting only in H ). The end result is thatthe WF generator will completely diagonalize the problem in the full extended Hilbert space F = H ⊗ L T . The crucial difference is that in this choice of generator, we choose the ‘off-diagonal’ terms to include both the off-diagonal terms in H as well as the n (cid:54) = 0 harmonicsof all terms (‘off-diagonal’ in L T ), fulfilling our goal of finding a generator which can directlyobtain the quasi-energies. A schematic representation is shown in Fig. 4d, where the finalform of K is diagonal in both space and time.We write the Floquet quasienergy operator as: K = (cid:88) i H (0) ii ⊗ + ⊗ ω ˆ n (cid:124) (cid:123)(cid:122) (cid:125) K + (cid:88) i (cid:88) n (cid:54) =0 H ( n ) ii ⊗ σ n + (cid:88) i (cid:54) = j (cid:88) n H ( n ) ij ⊗ σ n (cid:124) (cid:123)(cid:122) (cid:125) K off (19)where we explicitly separate the zero-frequency diagonal components ( K , which becomesthe Floquet quasi-energies in the limit l → ∞ ) from the rest of the Floquet matrix, defining K off as all other components (comprising finite-frequency terms which are diagonal in real10 ciPost Physics Submission space, and all off-diagonal terms at all frequencies). As before, we choose the generator to be η = [ K , K off ], and the end result of our diagonalization process will be a matrix of the form˜ K = ˜ H ⊗ + ⊗ ω ˆ n (20)where ˜ H = (cid:80) i ˜ h (0) i n i are the Floquet quasi-energies.In the following, we shall focus on this Wegner-Floquet generator, as it is new to theliterature and allows us to directly obtain the Floquet quasienergies with a single unitarytransform. One feature that the Wegner-Floquet generator inherits from the static Wegnergenerator is that it works on the principle of energy-scale separation, and therefore is iden-tically zero when applied to a translationally invariant system in real-space. Here, we willconsider the case of disordered systems in real space where the on-site potential is stronglyinhomogeneous, although we note that the method still works for clean systems, either withsome small ‘seed’ randomness to break the homogeneity or when applied in momentum space. We shall first consider a prototype system of non-interacting fermions. It is worth empha-sising up front that while for non-interacting systems the flow equation method may seemlike an elaborate way to solve a problem which can be more efficiently treated by exact di-agonalization, the real advantage of this method lies in its ability to non-perturbatively solve interacting quantum systems on system sizes far larger than accessible to numerically exactmethods. We shall return to interacting systems in Section 6 after first demonstrating themethod and establishing our notation.
We now turn to a concrete example, namely a one-dimensional chain of length L of non-interacting fermions in a disordered potential subject to periodic drive, i.e. a driven Andersoninsulator [87–89]. In the absence of drive, the eigenstates of the Anderson insulator areexponentially localized in space, however we find that the model can exhibit a transition as afunction of driving frequency to a delocalized phase. The Hamiltonian is given by: H ( t ) = F ( t ) L (cid:88) i =1 h i c † i c i + G ( t ) L − (cid:88) i =1 J (cid:16) c † i c i +1 + c † i +1 c i (cid:17) (21)where the on-site energies are drawn from a box distribution h i ∈ [0 , W ], J is the nearest-neighbour hopping amplitude and the functions F ( t ) and G ( t ) are some T -periodic functionswhich we leave as arbitrary for the moment.We can expand these coefficients in terms of Fourier harmonics, which allows us to writedown an ansatz in the full Floquet Hilbert space F for the form of the running (i.e. scale-dependent) Floquet operator K ( l ). It takes the form: K ( l ) = (cid:88) n (cid:88) i h ( n ) i c † i c i + (cid:88) ij J ( n ) ij c † i c j ⊗ σ n + ⊗ ω ˆ n (22)11 ciPost Physics Submission with time-dependent coefficients determined by the relations h i ( t, l ) = h i F ( t ) = (cid:80) n h ( n ) i ( l ) e inωt and J ij ( t, l ) = J ij G ( t ) = (cid:80) n J ( n ) ij ( l ) e inωt . We choose the time-independent coefficients h ( n ) i and J ( n ) ij to be real, and they satisfy the constraints h ( n ) i = h ( − n ) i and J ( n ) ij = J ( − n ) ji respectively.For non-interacting systems, this form of K ( l ) turns out to be exact. Note that despite theinitial Hamiltonian only containing nearest-neighbour hopping terms, in the running Floquetoperator we must allow for arbitrarily long-range hopping terms J ( n ) ij ∀ i (cid:54) = j - one significantdownside of Wegner-type generators is that they do not preserve the sparsity of matricesto which they are applied. In the present case, this results in the generation of long-rangehopping terms at early stages of the flow, which decay to zero in the l → ∞ limit but mustnonetheless be kept track of, as well as the excitation of higher harmonics not present in theoriginal microscopic model which likewise will decay at the end of the procedure. Following the Wegner-Floquet procedure outlined in Eq. 19, we can split the Floquet quasienergyoperator explicitly into diagonal and off-diagonal pieces as: K = (cid:34)(cid:88) i h (0) i c † i c i (cid:35) ⊗ + ⊗ ω ˆ n (23) K off = (cid:88) n (cid:54) =0 (cid:88) i h ( n ) i c † i c i ⊗ σ n + (cid:88) n (cid:88) ij J ( n ) ij c † i c j ⊗ σ n (24)where the superscripts are harmonic indices. From this, we can compute the generator η =[ K , K off ] which we can use to diagonalize the matrix K : η = (cid:88) n (cid:88) ij J ( n ) ij ( h (0) i − h (0) j ) c † i c j ⊗ σ n + (cid:88) n (cid:54) =0 ωn (cid:88) i h ( n ) i c † i c i + (cid:88) ij J ( n ) ij c † i c j ⊗ σ n (25)The flow of the Floquet operator towards diagonal form is given by d K/ d l = [ η, K ]. Fromthis, following some algebra, we can obtain flow equations for the running couplings:d h ( n ) i d l = − n ω h ( n ) i + 2 (cid:88) m (cid:88) j J ( n − m ) ij J ( m ) ji ( h (0) i − h (0) j ) + (cid:88) m ω ( n − m ) (cid:88) j (cid:16) J ( n − m ) ij J ( m ) ji − J ( m ) ij J ( n − m ) ji (cid:17) (26)d J ( n ) ij d l = − nωJ ( n ) ij ( h (0) i − h (0) j + nω ) + (cid:88) m J ( n − m ) ij ( h (0) i − h (0) j )( h ( m ) j − h ( m ) i )+ (cid:88) m (cid:88) k J ( n − m ) ik J ( m ) kj ( h (0) i + h (0) j − h (0) k )+ (cid:88) m ω ( n − m ) (cid:16) J ( m ) ij ( h ( n − m ) i − h ( n − m ) j ) + J ( n − m ) ij ( h ( m ) j − h ( m ) i ) (cid:17) + (cid:88) m ω ( n − m ) (cid:88) k (cid:16) J ( n − m ) ik J ( m ) kj − J ( m ) ik J ( n − m ) kj (cid:17) (27)12 ciPost Physics Submission which reduce back to the static flow equations for non-interacting fermions in the limit of ω →
0, or in the case where all coefficients with harmonic index n (cid:54) = 0 are zero (i.e. thetime-independent case). Details of this calculation are shown in Appendix B.Note that for discontinuous drive protocols, the expansion in terms of Fourier harmonicsinvolves in principle infinitely many components, which is clearly impractical, so we mustretain only a finite number of harmonics. In fact, even for continuous drive protocols, theequations above are not closed for a finite number of harmonics, meaning that we mustmake a suitable truncation and choose to keep only N h harmonics (with the harmonic index n ∈ [ − N h , N h ]). The choice of a suitable value for N h will depend on the problem in question.Careful inspection of the above equations reveals that for large harmonic index n the equationsreduce to d h ( n ) i / d l ≈ − n ω h ( n ) i and d J ( n ) ij / d l ≈ − n ω J ( n ) ij and so these coefficients will decayexponentially in flow time for large values of n . This means that terms with large harmonicindices typically do not play a significant role except at very low frequencies, and only thelowest-order harmonics in the Fourier expansion need be retained in order for the procedureto be accurate. Bearing in mind that not all couplings are independent (e.g. J ( n ) ij = J ( − n ) ji , intotal this results in L ( L + 1) / N h × L ) coupled ordinary differential equations (where L is the number of real-space sites), which can be straightforwardly solved numerically.An interesting feature of the above flow equations is that in the limit of very high frequency ω → ∞ , the harmonics with n > h ( n ) i / d l ≈ − n ω h ( n ) i and d J ( n ) ij / d l ≈− n ω J ( n ) ij and decay exponentially to zero, with essentially zero feedback on the n = 0harmonics. In other words, it is explicit from the above equations that at high frequencies theproblem reduces to a static problem with the couplings given by their time-averaged ( n = 0)values, as one might reasonably expect. Also, if we take the case of J ij = 0 ∀ i, j thenthe flow equations return the time-averaged h i , i.e. just the zeroth harmonic h (0) i : all theharmonics h ( n ) for n > h (0) term and decay exponentially to zero. For the following results, we use a discontinuous step-like drive, as this represents the mostchallenging case for the harmonic expansion we employ. The Hamiltonian we consider is: H ( t ) = (cid:40)(cid:80) Li =1 h i c † i c i , if T / ≤ t < T / (cid:80) L − i =1 J (cid:16) c † i c i +1 + c † i +1 c i (cid:17) , otherwise (28)where the function F ( t ) is now chosen to be a step function to match the above Hamiltonian,with G ( t ) = 1 − F ( t ) in Eq. 21. We will compute the Fourier harmonics of this drive (whichhas been chosen to have purely real Fourier components), and keep N h of the harmonicsfor the flow equation procedure, such that the driving protocol is described by a function F ( t, N h ). Our purpose in choosing a step drive is partly because this is a common choice inthe simulation of driven systems (see, e.g., Refs. [32, 45]), and partly because it representsa worst-case scenario in terms of computational complexity for the method, as formally werequire infinite many Fourier components to exactly realise the discontinuous form of thedrive. The form of F ( t, N h ) is shown in Fig. 2, which demonstrates how well the step drive isapproximated as a function of the number of harmonics N h .In Fig. 3 we show the quasienergies and the flow of the couplings at two different drivefrequencies for system size L = 12, disorder strength h i ∈ [0 , W ] with W = 5 and hopping13 ciPost Physics Submission . . . . . . t/T . . . F ( t , N h ) N h = 2 N h = 4 N h = 6 N h = 8 N h = 10 N h = 12 N h = 14 N h = 16 Figure 2: A comparison of the drive for varying N h showing how the harmonic expansionapproximates the discontinuous form of the drive as N h is increased. The discontinuous stepdrive ( N h → ∞ ) approximated by F ( t, N h ) is indicated by the black dashed line. J = 0 .
5. We compare the quasienergies obtained with the flow equation (FE) approach withthose computed using exact diagonalization (ED) using the QuSpin package [90, 91]. In theED results, we use the exact Hamiltonian given by Eq. 28, i.e. we implement the step driveexactly with no approximation. For illustrative purposes we keep the system size small, toprevent the plots from becoming cluttered with too many couplings, and we show the flowup to l max = 150, though typically we use a larger l max ∼ to ensure convergence. Toreduce the numerical load, we set all off-diagonal couplings to zero when they decay below10 − . In principle, to reduce the computational cost further, one could dynamically reducethe number of equations when the off-diagonal couplings have decayed, however we did notfind it necessary to implement this feature here.In Fig. 4a we demonstrate the accuracy versus number of retained harmonics N h for alarger range of frequencies and system sizes by plotting the relative error, defined as: δε = 1 L (cid:88) α (cid:12)(cid:12)(cid:12)(cid:12) ε EDα − ε F Eα ε EDα (cid:12)(cid:12)(cid:12)(cid:12) (29)where ε ED/F Eα are the eigenvalues obtained with exact diagonalization (ED) and flow equa-tions (FE) respectively. This is then averaged over N s = 128 disorder realisations.As a further, far more demanding check of our method, we also compute the level spacingstatistics as a function of frequency, a metric which has been extensively used in studies ofmany-body localization to distinguish localized and delocalized phases [92]. Specifically, wecompute the following quantities: δ α = | ε α − ε α +1 | (30) r α = min( δ α , δ α +1 ) / max( δ α , δ α +1 ) (31)and we compute r = (cid:104) r n (cid:105) , where the (cid:104) ... (cid:105) represents the average within a sample and theoverline the average over disorder realisations. In a localized phase, the energy level spacingswill be distributed according to a Poisson distribution P ( δ ) = exp( − δ ), corresponding to r ≈ .
39. In a delocalized phase, however, the energy levels will exhibit level repulsion:while for undriven systems in a delocalized phase the spacings will follow the Wigner-Dyson14 ciPost Physics Submission
Figure 3: Two representative samples of the quasienergies and flow of the couplings for twodifferent disorder realisations at different drive frequencies (left and right columns respec-tively), with L = 12, W = 5, J = 0 . N h = 5 harmonics. The left column shows theresults for a drive frequency ω = 2 π which is much larger than the disorder bandwidth W ,while the right column shows results for ω = π/ ε n obtained by the FE method (redcrosses) compared with the results from ED (orange dots). The higher frequency result ismore accurate; more harmonics are required at lower frequencies. Panels (c) and (d) showthe flow of the zero-frequency terms - the solid lines are the h (0) i coefficients, while the dashedlines are the J (0) ij . Panels (e) and (f) show the flow of the finite-frequency harmonics, whichdecay to zero extremely quickly at high drive frequencies (note the logarithmic scale) butmuch more slowly for small ω .distribution P ( δ ) = ( π/ δ exp( πδ /
2) (leading to r ≈ . driven systems where theFloquet eigenvalues lie in the folded range [ − ω/ , ω/ r ≈ .
53 in the delocalized phase. Theresults for a system size L = 12 with N h = 10 harmonics are shown in Fig. 4b, and demonstrategood agreement between exact diagonalization and flow equation results.15 ciPost Physics Submission N h . . . " ! = 0 . ⇡! = 0 . ⇡! = 0 . ⇡! = 0 . ⇡! = 1 . ⇡! = 1 . ⇡ N h " ( a )( b ) ! . . r Figure 4: (a) Relative error in the quasienergies (as compared with exact diagonalization)with N h for L = 12, disorder strength W = 5 and hopping J = 0 . N h = 10 harmonics. The grey dashed lines represent r ≈ .
39 and r ≈ .
53 corresponding toPoisson and Circular Ensemble [25, 26] level statistics respectively.
In addition to the quasienergies, in order to have a complete solution to the problem onemust compute the Floquet eigenstates. For a non-interacting system, in the l = ∞ basiswhere the Floquet Hamiltonian ˜ H F is diagonal, the eigenstates are the single-particle statesof ˜ H F , given by | ψ i (cid:105) = c † i ( l = ∞ ) | (cid:105) for i ∈ [1 , L ]. For static systems (see Appendix A.2), inorder to obtain the eigenstates we simply need to invert the unitary transform to transform c † i ( l = ∞ ) backwards from l = ∞ to l = 0. An additional complication in the case of Floquetsystems is that if we want to compute the Floquet eigenstates, then we do not want toobtain the eigenstates in the initial microscopic basis, where they will be time-dependent.Strictly speaking, obtaining the Floquet eigenstates is a two-step process: the first step isthe inversion of the Wegner-Floquet unitary transform to obtain time-dependent eigenstatesin the initial basis, followed by a Type II unitary transform (see Sec. 4.2) to get rid of thetime-dependence. We can nonetheless approximate the result of this two-step procedure witha single quasi-unitary transform by reversing only the flow of the zero-frequency terms in the16 ciPost Physics Submission Hamiltonian. We make the following ansatz for the flow of the creation operator: c † i ( l ) = (cid:88) j β (0) i,j ( l ) c † j (32)with initial condition β (0) i,j ( l = ∞ ) = δ i,j and we obtain the flow equation for this operator byconsidering only the zero frequency components of the generator:d c † i ( l )d l = (cid:88) j β (0) i,j ( l )[ η (0) , c † j ] = (cid:88) jk J (0) jk ( h j − h k ) β (0) i,k c † j (33)giving a flow equation for the coefficients β i,j β i,j = (cid:88) k J (0) jk ( h j − h k ) β (0) i,k (34)which we can use to approximately construct the Floquet eigenstates. This procedure worksbest at high frequency, however for all but the lowest frequencies the results are of reasonableaccuracy, with an overlap of close to unity for frequencies ω > π/ n = 0 terms deviates stronglyfrom unitarity, hence the larger error.)In Fig. 5a, we show the overlap between the eigenstates generated by the flow equationprocedure and the output of an exact diagonalization calculation. At high drive frequencies,this approximate method is able to reproduce the eigenstates with extremely high fidelity(up to an unimportant arbitrary phase factor), however at frequencies much lower than thedisorder bandwidth the approximation fails and the full unitary transform must be used.We emphasise that while this can be done exactly as a two-step process using two differentgenerators, as explained above, we do not explore this option here, as in this work we aremore interested in exploring what may be obtained from a single unitary transform using theWegner-Floquet generator. Any initial state can be decomposed in terms of Floquet eigenstates, and with the completeknowledge of the eigenstates and eigenvalues, we can perform this decomposition and time-evolve any initial state we choose. Specifically, an arbitrary state | φ ( t ) (cid:105) at a time t = 0 canbe expressed in terms of the Floquet eigenstates | ψ α ( t ) (cid:105) like so: | φ ( t = 0) (cid:105) = (cid:88) α | ψ α ( t = 0) (cid:105) (cid:104) ψ α ( t = 0) | φ ( t = 0) (cid:105) = (cid:88) α c α | ψ α ( t = 0) (cid:105) (35)with c α = (cid:104) ψ α ( t = 0) | φ ( t = 0) (cid:105) . The time evolution of this state for some time t > | φ ( t ) (cid:105) = (cid:88) α c α e − iε α t | ψ α (0) (cid:105) (36)where t is an integer multiple of the drive period T . This is known as stroboscopic evolution.As an example, the expectation value of the density on site i is given by: (cid:104) n i ( t ) (cid:105) = (cid:104) φ ( t ) | n i | φ ( t ) (cid:105) = (cid:88) αβ c ∗ α c β exp[ i ( ε α − ε β ) t ] (cid:104) ψ α (0) | n i | ψ β (0) (cid:105) (37)17 ciPost Physics Submission ! . . . h E D | F E i N h = 0 N h = 2 N h = 4 N h = 6 N h = 8 t/T . . h n L / ( t ) i ( b )( a ) Figure 5: (a) Overlap between the Floquet eigenstates obtained with exact diagonalizationand approximate states obtained with the flow equation method, again with L = 12, W = 5and J = 0 . N s = 128 disorder realisations. Error bars representing thevariance over the disorder realisations are smaller than the plot markers. As described in thetext, the approximation used here is better at higher frequencies. Note that there is very littledependence on harmonic number N h , with the lines for N h ≥ L = 12 with N h = 8 harmonics at a frequency ω = π . The flow equation result is shown in blue (solid line), with the result from exactdiagonalization in orange (dashed) : the agreement is excellent.The results of the stroboscopic evolution from an initial charge density wave state | ... (cid:105) are shown in Fig. 5b. The agreement between exact diagonalization and flow equation resultsis almost perfect. As the system we consider here is a free fermion Anderson insulator whichdoes not exhibit dephasing, the late time states do not synchronise with the drive [45], howeverwe nonetheless show the results of this calculation as a proof of principle that this methodallows one to recover the full dynamical behavior in addition to the quasienergies. Now that we have covered driven non-interacting systems and demonstrated how to completelysolve them using the flow equation approach, let us discuss how to add interactions into the18 ciPost Physics Submission picture. We now start from a time-dependent, interacting fermionic Hamiltonian of the form: H = (cid:88) i h i ( t ) c † i c i + (cid:88) ij J ij ( t ) c † i c j + 12 (cid:88) ij ∆ ij ( t ) c † i c † j c j c i (38)where the coefficients h i ( t ), J ij ( t ) and ∆ ij ( t ) all have some arbitrary (periodic) time-dependence,and here we allow for arbitrary long-range couplings from the start. The static form of thisHamiltonian has been studied using flow equation methods in the context of many-body lo-calization (MBL) in Ref. [61] in the case of short-range couplings and in Ref. [65] in thecase of long-range couplings. In the static case, the diagonal Hamiltonian can be written asa series of mutually commuting n -body interaction terms, ˜ H = (cid:80) i h i n i + (cid:80) ij ˜∆ ij n i n j + (cid:80) ijk Γ ijk n i n j n k + ... , which requires us to keep track of a prohibitively large number of termsduring the diagonalization process for any system of realistic size. Fortunately, in certaincases, one can perform a truncation of this series and keep only the lowest-order terms. Fol-lowing the ansatz described in Refs. [61, 65], we will here assume a scale-dependent Floquetoperator of the form: K ( l ) = (cid:88) n (cid:88) i h ( n ) i c † i c i + (cid:88) ij J ( n ) ij c † i c j + 12 (cid:88) ij ∆ ( n ) ij c † i c † j c j c i ⊗ σ n + ⊗ ω ˆ n (39)and discard all newly-generated terms outside of this variational manifold. For the staticsystem, this ansatz is valid either in the strongly disordered, many-body localized regime [61]or in the weakly-interacting regime (where the generation of higher-order terms is suppressedby powers of the interaction strength, which we choose here to be small) [65]. Here, we willassume weak interactions and work in the regime where the static Hamiltonian would bein the many-body localized phase. By analogy with the non-interacting system in Section 5,we may expect to see a delocalization transition as a function of the drive frequency; we willexamine the evidence for this later. The final fixed-point diagonal Floquet operator will beof the form ˜ K = ˜ H F ⊗ + ⊗ ω ˆ n , with ˜ H F given by:˜ H F = (cid:88) i ˜ h i n i + 12 (cid:88) ij ˜∆ ij n i n j (40)from which we can construct the many-body quasienergies simply by applying this Hamilto-nian to each of the 2 L many-body states.In previous works, we have provided a direct estimate of the error in the ansatz given byEq. 39, simply given by taking quantities which are conserved by unitary transforms (known inthis context as ‘invariants of the flow’) and computing them at the start and the end of the flowto give a measure of the deviation from unitarity and hence the error in the approximation.In the present case, the situation is more complicated: what is conserved here are traces ofinteger powers of the Floquet quasienergy operator I n = Tr[ K n ]. This matrix contains a largenumber of duplicated (and therefore redundant) terms due to the tensor product structurewith the basis of T -periodic functions, and so direct construction of the flow invariant isnot practical. It remains an open question for future work whether conserved quantities ofthe flow of K can be practically computed for interacting systems. Here, we focus on theweakly-interacting case where by construction the higher-order terms are heavily suppressed. By comparison with previous work [61, 62, 65], here we use the convention where the normal-orderingcorrections [93] are computed with respect to the vacuum and evaluate to zero. We will address the effect ofstronger interactions and non-zero normal-ordering corrections in more detail in a future work. ciPost Physics Submission The Floquet quasienergy operator may again be separated into diagonal and off-diagonal partsas K = K + K off , now with: K = (cid:34)(cid:88) i h (0) i c † i c i (cid:35) ⊗ + ⊗ ω ˆ n + 12 (cid:88) ij ∆ (0) ij c † i c † j c j c i ⊗ (41) K off = (cid:88) n (cid:54) =0 (cid:88) i h ( n ) i c † i c i ⊗ σ n + (cid:88) n (cid:88) ij J ( n ) ij c † i c j ⊗ σ n + 12 (cid:88) n (cid:54) =0 (cid:88) ij ∆ ( n ) ij c † i c † j c j c i ⊗ σ n (42)The calculation proceeds similarly to the non-interacting case, but now the generator acquiresadditional terms due to the interactions. These new contributions lead to the flow equationfor the interaction term:d∆ ( n ) ij d l = − ω n ∆ ( n ) ij + 2 (cid:88) m (cid:88) k (cid:54) = i,j (cid:104) J ( n − m ) ik J ( m ) ik (∆ (0) ij − ∆ (0) kj ) + J ( n − m ) jk J mjk (∆ (0) ij − ∆ (0) ik ) (cid:105) (43)where we refer the reader to Refs. [61, 65] for details on how to compute the flow of theinteracting terms. For the interacting system within the approximation of Eq. 39, there arenow L + ( N h × L (3 L − /
2) coupled differential equations to solve. Note that the totalnumber of equations scales polynomially in the system size (and linearly in N h ), in contrastwith exact diagonalization where the size of the Hilbert space scales exponentially with thesystem size. In regimes where this approximation can applied, flow equations can thereforeaccess far larger system sizes than numerically exact methods are capable of reaching. We firstdemonstrate that the approximation of Eq. 39 is accurate before providing a proof-of-conceptdemonstration of a quantity that cannot be computed with any other method. We again compare the eigenvalues obtained using the FE method with the results of anexact diagonalization calculation. Here we use the same driving protocol as in Section 5, andchoose the interaction term to be a time-independent, nearest-neighbour coupling satisfying∆ ij ( t ) = δ i,i ± ∆ . Fig. 6 shows the results for L = 5, N h = 3, ω = 2 π , d ∈ [0 , W ] with W = 5and ∆ = 0 .
01. Again, we keep the system size small for illustrative purposes, however wewill show results for larger systems in Section 6.4. We find that in the weakly interactingregime and at high frequencies, the agreement is excellent.The accuracy becomes significantly worse at low frequencies, moreso than in the non-interacting system. In that case, there was a transition to a delocalized state as a function offrequency. It is reasonable to assume that the interacting system will likewise go through aphase transition to a delocalized phase as the driving frequency is lowered, and at this pointthe ansatz (Eq. 39) used for the many-body Hamiltonian will break down. In the next section,we shall examine this in more detail from the perspective of local integrals of motion.
The final fixed-point diagonal Hamiltonian for the weakly-interacting problem, as shown inEq. 40, now contains only mutually commuting terms and is given by ˜ H F = (cid:80) i ˜ h i n i +20 ciPost Physics Submission Figure 6: A comparison of the results for weakly-interacting systems at two different fre-quencies, ω = 2 π and N h = 3 (left column) and ω = π with N h = 5 (right column), with L = 5, W = 5, J = 0 . = 0 .
01. As in Fig. 3, panels (a) and (b)show the quasienergies obtained with exact diagonalization and flow equation methods, whilepanels (c) and (d) show the flow of the zero-frequency terms (with additional dot-dashed linesshowing the ∆ (0) ij terms) and panels (e) and (f) show the flow of the terms with n >
0. Evenwith the increased N h , the agreement between FE and ED quasienergies starts to deviate atlower frequencies. We conjecture that this is due to the proximity to a phase transition wherethe ansatz of Eq. 39 breaks down; in Section 6.4 we explore this in further detail. (cid:80) ij ˜∆ ij n i n j . In previous works on static MBL [60, 61, 65, 94, 95], the real-space decay ofthe ˜∆ ij terms has been associated with the dephasing length of the local integrals of motion(or l -bits) that characterise MBL matter. These terms are expected to decay exponentiallyas a function of distance for localized systems with short-range couplings [61, 94, 95] andmay exhibit more exotic behavior for systems with long-range couplings [65]. In presence ofperiodic drive one would expect these l -bits to remain well defined at least for sufficientlyhigh driving frequency [27]. Here we explicitly construct the Floquet l -bits and study theirinteraction as a function of frequency.The results are shown in Fig. 7 for two system sizes, L = 36 and L = 48, retaining N h = 3harmonics and averaged over disorder realisations. We find that at high drive frequencies,these ‘Floquet integrals of motion’ (or Floquet l -bits) decay exponentially with distance as21 ciPost Physics Submission
10 20 30 | i − j | − − l og [ ˜ ∆ i j ] ω = 0 . πω = 0 . πω = 0 . πω = 0 . πω = 1 . πω = 1 . πω = 1 . π Figure 7: Decay of the Floquet l -bit couplings with distance for several different drivingfrequencies ω , and system sizes L = 36 (solid lines with square markers) and L = 48 (dashedlines with circular markers) , N h = 3. W = 5, J = 0 . = 0 .
01 both averagedover N s = 256 disorder realisations. Independently of system size, there appears to be alocalisalisation/delocalization transition as a function of the driving frequency ω , visible bythe change of the decay of the Floquet l -bits from an approximately frequency-independentexponential decay to a much slower exponential decay below ω c ≈ . π . Note that at thelowest frequencies shown, the results must be considered qualitative only.in the conventional static many-body localized system, however at low drive frequencies theyappear to flatten out, consistent with the system becoming less localized and potentially un-dergoing a transition to a delocalized phase. We caution that our results in the low frequencyregime are qualitative only: in the absence of normal-ordering corrections [56,61,65,93] Eq. 43can exhibit an unphysical exponential growth in flow time l in the delocalized regime for small | i − j | , visible in Fig. 7 at the smallest driving frequencies and at short distances. This isunrelated to our choice of N h and as such retaining more harmonics would not improve theaccuracy in this region. Our aim in this work was to present the simplest possible form of theflow for the driven interacting system as a proof-of-concept: we will explore the consequencesof including these normal-ordering corrections in a future work dedicated to interacting sys-tems. In conclusion, we have demonstrated a new method for the diagonalization of Floquet quasien-egy/evolution operators based around continuous unitary transforms, or ‘flow equations’. Thismethod is a generalisation of the Wegner flow which has been used in a variety of contexts inthe years since its development [56,72]. Here we make use of the Floquet theorem to rewrite aperiodically driven system as a time-independent system in a larger composite Hilbert spaceamenable to treatment with flow equation techniques. We have briefly reviewed a variety ofdifferent choices of generator and shown where our choice fits in, and we have given severalexamples of both non-interacting and weakly interacting quantum systems where this methodis capable of reproducing exact numerical results as well as going beyond the state-of-the-artaccessible to other methods, which we have demonstrated by computing for the first time the‘Floquet l -bits’ of a driven many-body localized system on length scales inaccessible to all22 ciPost Physics Submission other methods.The strengths of this method go beyond what we have shown here, however, and we closethis article with a look to the future. Flow equations have previously been used to studymany-body localization in two-dimensional systems [61] as well as in coupled chains [62],and a generalisation of the results shown in Section 6 to two-dimensional driven systems isstraightforward. Equally, flow equation methods are not limited to the models consideredhere: one particularly intriguing application is the study of time crystal behavior in driventransverse field Ising models in one and two dimensions, something which we will addressseparately in a forthcoming work. Another clear avenue for further improvement is the treat-ment of more strongly interacting quantum systems, through the incorporation of advancednon-perturbative techniques [56, 61, 65, 93] and an improved ansatz for the running Hamilto-nian in the ‘Floquet-MBL’ phase, perhaps incorporating higher-order terms in the operatorexpansion beyond the truncation considered here. As our method lends itself naturally tocontinuous drive protocols, as well as to discontinuous drives when expanded in terms of har-monics, it may also be interesting to use Floquet flow equations to examine continuous drivesin more detail: this is a particular strength of the method, as having to retain a large numberof harmonics in order to capture a discontinuous drive imposes a severe limit on the systemsizes achievable. By considering smooth drive protocols and retaining only a small number ofharmonics, even larger system sizes are available to this method and will be explored in otherworks in the near future. Acknowledgements
All numerical computations were performed on the Coll`ege de France IPH computer cluster.We acknowledge use of the QuSpin exact diagonalization library for benchmarking our flowequation code [90, 91].
Author contributions
The central formalism in Section 4 was developed by DM underthe supervision of SJT and MS. The results in Sections 5 and 6 were obtained by SJT. Theproject was conceived by and directed by MS. All authors contributed towards the writing ofthe final manuscript.
Funding information
SJT acknowledges financial support from the DIM SIRTEQ grantDynDisQ. DM acknowledges support from Funda¸c˜ao para a Ciˆencia e a Tecnologia (Portugal)through project UIDB/EEA/50008/2020 and partial support from LabEx ENS-ICFP (ANR-10-IDEX-0001-02). MS acknowledges financial support from the ANR grant “NonEQuMat”(ANR-19-CE47-0001).
A Example: Flow Equations for Static Hamiltonians
In this Appendix, we sketch the application of the flow equation technique to a static, non-interacting Hamiltonian to illustrate the exact application of the method for readers unfamiliarwith the technique. Additionally, we show for the first time how to obtain the eigenstatesof the non-interacting system using flow equation method, something we did not include in23 ciPost Physics Submission previous work on the topic [64].
A.1 Diagonalising the Hamiltonian
We start from a Hamiltonian describing a one-dimensional chain of non-interacting fermionsof length L : H ( l ) = (cid:88) i h i ( l ) n i + (cid:88) i (cid:54) = j J ij ( l ) c † i c j (44)where the coefficients h i ( l = 0) and J ij ( l = 0) are arbitrary. We split this into the diago-nal component H ( l ) = (cid:80) i h i ( l ) n i and off-diagonal component V ( l ) = H ( l ) − H ( l ). Thecanonical generator for this problem is given by: η ( l ) = [ H ( l ) , V ( l )] = (cid:88) ij J ij ( l )( h i ( l ) − h j ( l )) c † i c j (45)We can compute the flow of the Hamiltonian from d H/ d l = [ η ( l ) , H ( l )] and read off the flowequations for the running coupling constants:d J ij d l = − J ij ( h i − h j ) − (cid:88) k J ik J kj (2 h k − h i − h j ) (46)d h i d l = 2 (cid:88) j J ij ( h i − h j ) (47)where we have suppressed the dependence on the flow time l for clarity. A detailed introductionto the flow equation method as applied to a system of non-interacting fermions may be foundin Ref. [64]. A.2 Obtaining the eigenstates
The procedure for obtaining the eigenstates is simpler for static systems than for Floquetsystems. We apply the same logic, namely the fact that the eigenstates are the single-particlestates of the diagonal Hamiltonian, and make an ansatz for the form of the running creationoperator: c † i ( l ) = (cid:88) j α i,j c † k (48)again with α i,j ( l = ∞ ) = δ i,j . From this, we can obtain the flow equation:d c † i d l = [ η ( l ) , c † i ( l )] = (cid:88) jk J jk ( h j − h k ) α i,k c † j (49)which allows us to compute the single-particle eigenstates in the microscopic l = 0 basis. Thisprocedure can also be generalised to interacting systems with additional approximations.24 ciPost Physics Submission B Details of the calculation
A few useful identities that we will need are as follows: (cid:104) c † α c β , c † γ c δ (cid:105) = c † α { c β , c † γ } c δ − c † γ { c δ , c † α } c β = δ βγ c † α c δ − δ δα c † γ c β (50) σ n σ m = σ n + m (51)[ σ n , σ m ] = 0 (52)[ˆ n, σ n ] = nσ n (53)[ A ⊗ B, C ⊗ D ] = ( AC ) ⊗ ( BD ) − ( CA ) ⊗ ( DB ) (54) B.1 Defining and Computing the Generator
Following Eq. 19, we can write the Floquet operator explicitly as K = K + K off , with:: K = (cid:34)(cid:88) i h (0) i c † i c i (cid:35) ⊗ + ⊗ ω ˆ n (55) K off = (cid:88) n (cid:54) =0 (cid:88) i h ( n ) i c † i c i + (cid:88) n (cid:88) ij J ( n ) ij c † i c j ⊗ σ n (56)where the superscripts are harmonic indices. From this, the generator is defined as η =[ K , K off ]. The calculation of the generator can be separated into two non-zero parts: i ) (cid:88) k h (0) k c † k c k ⊗ , (cid:88) n (cid:88) ij J ( n ) ij c † i c j ⊗ = (cid:88) ij J ( n ) ij ( h (0) i − h (0) j ) c † i c j ⊗ σ n (57) ii ) [ ⊗ ω ˆ n, K off ] = (cid:88) n (cid:54) =0 ωn (cid:88) i h ( n ) i c † i c i + (cid:88) ij J ( n ) ij c † i c j ⊗ σ n (58)which, when put back together, reduce to η = [ K , K off ] = η + η with: η = (cid:88) n (cid:88) ij J ( n ) ij ( h (0) i − h (0) j ) c † i c j ⊗ σ n (59) η = (cid:88) n (cid:54) =0 ωn (cid:88) i h ( n ) i c † i c i + (cid:88) ij J ( n ) ij c † i c j ⊗ σ n (60)25 ciPost Physics Submission B.2 Computing the flow of the running couplings
The calculation of the flow of the Floquet operator d K/ d l = [ η, K ] can be separated into fourpieces: i ) [ η , ⊗ ω ˆ n ] = (cid:88) n (cid:88) ij J ( n ) ij ( h (0) i − h (0) j ) c † i c j ⊗ σ n , ⊗ ω ˆ n = − (cid:88) n nω (cid:88) ij J ( n ) ij ( h (0) i − h (0) j ) c † i c j ⊗ σ n (61) ii ) η , (cid:88) m (cid:88) ij H ( m ) ij ⊗ σ m = (cid:88) n (cid:88) ij J ( n ) ij ( h (0) i − h (0) j ) c † i c j ⊗ σ n , (cid:88) m (cid:88) k h ( m ) k c † k c k + (cid:88) kq J ( m ) kq c † k c q ⊗ σ m = (cid:88) n,m (cid:88) ij J ( n ) ij ( h (0) i − h (0) j )( h ( m ) j − h ( m ) i ) c † i c j ⊗ σ n + m + (cid:88) n,m (cid:88) ijk J ( n ) ik J ( m ) kj ( h (0) i + h (0) j − h (0) k ) c † i c j ⊗ σ n + m (62) iii ) [ η , ⊗ ω ˆ n ] = (cid:88) n (cid:54) =0 ωn (cid:88) i h ( n ) i c † i c i + (cid:88) ij J ( n ) ij c † i c j ⊗ σ n , ⊗ ω ˆ n = − (cid:88) n (cid:54) =0 n ω (cid:88) i h ( n ) i c † i c i + (cid:88) ij J ( n ) ij c † i c j ⊗ σ n (63) iv ) η , (cid:88) m (cid:88) ij H ( m ) ij ⊗ σ m = (cid:88) n ωn (cid:88) i h ( n ) i c † i c i + (cid:88) ij J ( n ) ij c † i c j ⊗ σ n , (cid:88) m (cid:88) k h ( m ) k c † k c k + (cid:88) kq J ( m ) kq c † k c q ⊗ σ m = (cid:88) n,m nω (cid:88) ij (cid:34) J ( m ) ij ( h ( n ) i − h ( n ) j ) + J ( n ) ij ( h ( m ) j − h ( m ) i ) + (cid:88) k ( J ( n ) ik J ( m ) kj − J ( m ) ik J ( n ) kj ) (cid:35) c † i c j ⊗ σ n + m (64)Putting these terms back together, we obtain the flow equations shown in the main text References [1] T. Langen, R. Geiger and J. Schmiedmayer,
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