Dynamical many-body localization in an integrable model
Aydin Cem Keser, Sriram Ganeshan, Gil Refael, Victor Galitski
DDynamical many-body localization in an integrable model
Aydin Cem Keser, Sriram Ganeshan,
1, 2
Gil Refael, and Victor Galitski
1, 2, 4 Condensed Matter Theory Center, Department of Physics,University of Maryland, College Park, MD 20742, USA. Joint Quantum Institute, University of Maryland, College Park, MD 20742, USA. Institute of Quantum Information and Matter, Department of Physics,California Institute of Technology, Pasadena, CA 91125, USA. School of Physics, Monash University, Melbourne, Victoria 3800, Australia. (Dated: September 22, 2018)We investigate dynamical many-body localization and delocalization in an integrable system ofperiodically-kicked, interacting linear rotors. The linear-in-momentum Hamiltonian makes the Flo-quet evolution operator analytically tractable for arbitrary interactions. One of the hallmarks ofthis model is that depending on certain parameters, it manifests both localization and delocaliza-tion in momentum space. We present a set of “emergent” integrals of motion, which can serve asa fundamental diagnostic of dynamical localization in the interacting case. We also propose an ex-perimental scheme, involving voltage-biased Josephson junctions, to realize such many-body kickedmodels.
I. INTRODUCTION
Recently, there has been a lot of interest and progressin understanding Anderson-type localization propertiesof disordered, interacting many-body systems. Notably,the remarkable phenomenon of many-body localization(MBL) was discovered . In an isolated system, MBLmanifests itself in the localization of all eigenstates andleads to the breakdown of ergodicity and violation ofthe eigenvalue thermalization hypothesis , forcing torevisit the very foundations of quantum statistical me-chanics .In this work, we ask whether a driven interacting sys-tem can be dynamically many-body localized. We answerthis question in the affirmative and present an exactlysolvable model of a kicked chain of interacting linear ro-tors, which shows both dynamical MBL and delocalizedregimes.A quantum kicked-rotor is a canonical model of quan-tum chaos which exhibits dynamical localization inmomentum space. The time dependent Schr¨odingerequation for a general kicked rotor is given by (here andbelow, we set (cid:126) = 1 and the driving period T = 1), i ∂ t ψ ( θ, t ) = [2 πα ( − i ∂ θ ) l + K ( θ ) δ ( t − n )] ψ ( θ, t ) . (1)In the ground breaking paper , Fishman, Prange, andGrempel proved that the eigenvalue problem for theFloquet operator of a kicked rotor is equivalent tothat of a particle hopping in a (quasi)periodic potential((ir)rational α ) in one-dimension given by, (cid:88) r W m + r u r + tan[ ω − παm l ] u m = Eu m . (2)This mapping traced the origin of the dynamical localiza-tion in driven systems to Anderson localization in time-independent settings. While the quadratic rotor ( l = 2)is nonintegrable, the linear rotor model ( l = 1) in Eq. (1)was exactly solved in Refs. . The corresponding integrable lattice version in Eq. (2) with l = 1 is dubbedMaryland model (MM) (See section A 1 for techni-cal details of this mapping). For the linear rotor, bothclassical and quantum dynamics is integrable, and thedynamical localization (absence of chaos) is due to theexistence of a complete set of integrals of motion . How-ever, the incommensurate MM is an Anderson insulatorwith no classical interpretation even though the linearrotor manifests classical integrability. Thus, the dynam-ical localization for the quantum version of both linear( l = 1) and quadratic rotor ( l = 2) seems to stem fromthe Anderson mechanism .In this work, we generalize the linear kicked rotormodel by considering an interacting chain of driven rotorsin order to understand the anatomy of many-body local-ization in the dynamical space. One of the remarkablefeatures of this many-particle model is that it manifestsboth localization and delocalization in dynamical spacedepending on whether the components of the (cid:126)α are irra-tional or rational.Let us emphasize from the outset that the model weconsider is integrable for all parameters owing to thefirst order differential operator. This leads to integralsof motion (IOMs), which are local in the spatial (angu-lar) variables. The underlying integrability is a special,non-universal feature of our model (3), which allows us tosolve it exactly. However, the existence of the local-in- θ IOMs has no direct relation to dynamical MBL, whichoccurs in momentum space.As shown below, dynamical MBL is accompanied bythe appearance of additional integrals of motion boundedin momentum space – a central result of this work. Thenon-interacting version of these additional IOM’s for thelinear kicked rotor was pointed out by Berry in Ref.[ 18].As argued below, these “emergent” IOMs and the dy-namical MBL are universal phenomena, which would sur-vive in non-integrable generalizations of the model (whichhowever is not analytically solvable).To capture the dynamical localization for this interact- a r X i v : . [ c ond - m a t . d i s - nn ] A ug ing model we monitor three indicators: energy growth atlong times, (momentum degrees of freedom at long times,and the existence of integrals of motion in the momentumspace. Below, we first present the detail of our model inSection II. The main analytical results are outlined inSection III, and their numerical analysis and key con-clusions are given in Section IV. An outline of technicalderivation of the results is given in Section V.The exper-imental proposal to realize our model (3) is explained inSection VI. In section VII we list our conclusions andSection VIII contains acknowledgements. Finally, Ap-pendix A 1 and Appendix A 2 are devoted to the sum-mary of the connection of the rotor problem to a latticemodel and the correspondence between our model and a d -dimensional lattice, respectively.Throughout the text, in all the summations (cid:80) i ... or (cid:80) ij ... we only consider i (cid:54) = j . So that, expressions like (cid:80) j / ( α i − α j ) are not divergent. The denominator van-ishes only when there is a resonance, such as α i → α j . II. THE MODEL
We consider a many body interacting generalization,ˆ H ( t ) = ˆ H + ˆ V ∞ (cid:88) n = −∞ δ ( t − n ) , with ˆ V = d (cid:88) i =1 K (ˆ θ i ) , ˆ H = 2 π d (cid:88) i =1 α i ˆ p i + 12 d − (cid:88) i (cid:54) = j J ij (ˆ θ i − ˆ θ j ) (3)of the linear rotor model.ˆ H is the static Hamiltonian describing d particles ona ring, each rotating 2 πα i radians per one period of thekick. ˆ θ i is the position operator for the i th particle onthe ring and ˆ p i is its angular momentum operator, whichhas integer eigenvalues in the (cid:126) = 1 units. These d parti-cles interact through a translationally invariant two-bodypotential J ij (ˆ θ i − ˆ θ j ). This form of the interaction en-sures conservation of momentum . The rotors are drivenby ˆ V ( t ) which contains periodic delta function impulses,where the strength of the impulse is given by a generalperiodic one-body potential K (ˆ θ i ). The local potentialsare periodic with 2 π and therefore the most general formcan be written as K ( θ j ) = (cid:80) m k m e i mθ j . Here, k m is the m th Fourier component of the potential that acts on the j th particle. The periodic form of the interaction poten-tial J can be written as J ij = (cid:80) m b ijm e i m ( θ i − θ j ) . Here, b ijm is the m th Fourier component of the interaction po-tential between the i th and j th particle. Reversing i and j and replacing m with − m in this Fourier expansion hasno effect on the components, therefore b ijm = b ji − m . Thisproperty will be handy while deriving formulas through-out the text.Note that in our model the localization is a conse-quence of incommensurate driving period and angularvelocities (cid:126)α . This situation is different from recent works using Floquet analysis to probe dynamical properties ofMBL states of disordered Hamiltonians (in these pa-pers, a Floquet perturbation is imposed on a state, many-body localized in coordinate space, while our goal is toinduce dynamical many-body localization in momentumspace by the Floquet perturbation). III. MAIN RESULTSA. Energy dynamics
Following the conjecture of D’Alessio andPolkovnikov , we test the dynamical localizationby computing the energy growth as a function of time atlong times. The average energy after N kicks (equivalentto time) can be written as, E ( N ) = (cid:104) ψ N | ˆ H | ψ N (cid:105) . Tocompute it, we write | ψ N (cid:105) = ˆ U NF | ψ (cid:105) , ˆ U F = e − i ˆ V e − i ˆ H , (4)where ˆ U F is the Floquet evolution operator, which cap-tures the state of the system after each kick. Owing tothe linear dependence of the Hamiltonian on the momen-tum, we can explicitly compute this expectation. E ( N ) = E (0) + d (cid:88) i =1 (cid:88) m πα i (cid:104) ˆΓ mi (cid:105) sin( mN πα i )sin( mπα i ) . (5)In the above expression, E (0) = (cid:104) ψ | H | ψ (cid:105) correspondsto the average many body energy over the initial state.ˆΓ mi = − i mk m e i m ( ˆ θ i + πα i [ N +1] ) depends on the chosenform of K (ˆ θ ) averaged over the initial state and due toits periodic nature is a bounded function of the numberof kicks N . The growth of energy for large N is thencompletely dependent on the nature of α i appearing inthe ratio sin( mNπα i )sin( mπα i ) . Note that this expression is com-pletely independent of interactions, which we prove inSection V A. This is a consequence of momentum con-serving interactions and this property is no longer validwhen the translational invariance of interactions is bro-ken. B. Momentum dynamics
Another indicator for dynamical localization is thespread in the momentum degrees of freedom. The i thmomentum after N kicks, p i ( N ) = (cid:104) ψ N | ˆ p i | ψ N (cid:105) is givenby the following expression, p i ( N ) = (cid:104) ˆ p i (cid:105) + (cid:88) m (cid:104) ˆΓ mi (cid:105) sin( mN πα i )sin( mπα i )+ (cid:88) mj (cid:104) ˆΓ intmij (cid:105) sin( mN π ∆ α ij ) mπ ∆ α ij . (6)We have defined ∆ α ij = α i − α j . In the above expression,the first term is the i th momentum in the initial eigen-state (cid:104) p i (cid:105) = (cid:104) ψ | ˆ p i | ψ (cid:105) . The second term correspondsto the kicking potential as defined in Eq. (5). The lastterm depends explicitly on the form of interaction viaˆΓ intmij = − i mb ijm e i m (ˆ θ i − ˆ θ j + πN ∆ α ij ) and is a bounded func-tion of N . The growth of momenta at long times cor-responding to the last term is completely determined bythe ratio sin( mNπ ∆ α ij ) mπ ∆ α ij . C. Integrals of motion
Recent works have shown that the existence of inte-grals of motion (IOM) can be used as a diagnostic toquantify both non-interacting Anderson localization and many body localization . In the context of dy-namical localization for the model at hand, we work inthe momentum basis and search for existence of IOMs inthis basis. We begin by constructing IOMs by identifyingoperators ˆ C i that satisfies [ ˆ C i , ˆ U F ] = 0 and [ ˆ C i , ˆ C j ] = 0.The existence of these IOMs encode information aboutdynamical localization. Operators that commute withˆ U F satisfy the property (cid:104) ˆ C i (cid:105) N +1 = (cid:104) ˆ C i (cid:105) N and are givenbyˆ C i = ˆ p i + 12 (cid:88) m mk m e i m (ˆ θ i + πα i ) sin( mπα i ) + 12 (cid:88) mj b ijm e i m (ˆ θ i − ˆ θ j ) π ( α i − α j ) . (7)This expression is a generalization of the constant of mo-tion given by Berry . The derivation of this expressionis given in Sec. V B. Since ˆ C i is an IOM, ˆ p i is bounded intime as long as the series in the last two terms converges.The delocalization of ˆ p i with time will occur due to thediverging denominators in the ˆ θ dependent terms.We reiterate that the model always contains d inte-grals of motion, ˆ B i = ˆ θ i /α i (mod 2 π ) ([ ˆ B i , ˆ U F ] = 0)which results in integrability for both localized and de-localized cases. For example, if α i = r i /s i is rational,the existence of ˆ B i results in integrability even thoughˆ C ...d do not exist. Since ˆ B i is momentum independent,its existence cannot bind p i ( N ). For the fully localizedcase, we have additional d IOM’s ˆ C ...d given in Eq. (7)that explicitly depend on ˆ p i and thus constrain it. Thus,ˆ C i ’s can be thought of as “emergent” IOM’s constrainingthe momentum growth, resulting in dynamical MBL and“emergent” superintegrability. IV. DYNAMICAL MANY-BODYLOCALIZATION AND THE STRUCTURE OF (cid:126)α
Equipped with three analytical expression for the en-ergy growth, momentum growth and IOMs [ eqs. (5)to (7)], we now diagnose the dynamical localization forthree distinct cases with varying structure of (cid:126)α . We note that there are cases where mα i can get arbitrar-ily close to integers (Liouville numbers) and total energyand momentum are no longer bounded. In the single ro-tor case, such a situation yields interesting consequenceslike marginal resonance and a mobility edge in the mo-mentum lattice, . In this study we exclude this pos-sibility and refer to generic irrationals only. A. Case I: α . . . α d are distinct generic irrationals For irrational values of α i , the total energy in Eq. (5)is always a bounded function of N since mα i / ∈ integer.In Fig. 1(AI), we fix initial states to be momentum eigen-states ψ ( (cid:126)θ ) which is e i (cid:126)p · (cid:126)θ up to a normalization factor.For momentum eigenstates, (cid:104) ˆΓ mi (cid:105) = 0 as the expec-tation value involves the integral (cid:82) d θe i mθ over a circle.Thus, we plot the root mean square (RMS) of system’senergy σ ( E ) = (cid:113) (cid:104) ˆ H (cid:105) − (cid:104) ˆ H (cid:105) as a function of N . Fig. 1(AI) shows boundedness in the spread of the energy asa function N . The momentum growth shown in Eq. (6)has contributions from both interactions and the kick-ing potential. In Fig. 1 (AII) we plot RMS deviation ofmomenta σ ( p ) . . . σ ( p d ) (we used d = 10 for the specificsimulation) and show that the spread in the momenta isbounded as a function of N . For this case the integralsof motion in Eq. (7) exist and convergent. Thus, all thediagnostics for this case point towards a true many bodydynamical localization. B. Case II: α = α and α . . . α d are distinct genericirrationals For this case, the energy remains a bounded func-tion of N since mα i are not integers. Thus, the RMSdeviation σ ( E ) is bounded as shown in Fig. (1 BI).However, the second term in the momentum expres-sion (Eq. (6)) develops a resonance (for the momenta p and p ) since α → α . Due to this resonanceterm, sin( mN π ∆ α ) / ( mπ ∆ α ) ∼ N as α → α .This resonant growth is reflected in the RMS deviationof σ ( p ) and σ ( p ) growing linearly with N , while the σ ( p ) . . . σ ( p ) remain bounded as shown in Fig. 1 (BII).This is a striking scenario where the resonant momentaare not localized even if the total energy is bounded forlarge N . However, the delocalization of p and p in timeis reflected in the break down of IOMs ˆ C and ˆ C due todiverging denominators in the resonant limit α → α .For this case, the bounded total energy fails to diagnosedelocalization as shown in Fig. 1 (BI, BII). We note thatthis scenario has no analogue in the non-interacting limit.Notice that interactions we considered possess transla-tional invariance, i.e. in the form J ( θ i − θ j ) given inEq. (3) and therefore interactions conserve momentum.The dichotomy between the energy growth and momen-tum growth is a result of conservation of momentum and (AI) (cid:126)α ∈ irrational, α i (cid:54) = α j (BI) (cid:126)α ∈ irrational, α = α (CI) α = 1 / α ,.., ∈ irrational N Σ (cid:72) E (cid:76) N Σ (cid:72) E (cid:76) N Σ (cid:72) E (cid:76) (AII) (cid:126)α ∈ irrational, α i (cid:54) = α j (BII) (cid:126)α ∈ irrational, α = α (CII) α = 1 / α ,.., ∈ irrational N Σ (cid:72) p (cid:76) ,.., Σ (cid:72) p (cid:76) N Σ (cid:72) p (cid:76) ,.., Σ (cid:72) p (cid:76) N Σ (cid:72) p (cid:76) ,.., Σ (cid:72) p (cid:76) FIG. 1. (Top panel) Plots showing evolution of the root mean square deviation of energy, σ ( E ) = (cid:112) (cid:104) H (cid:105) − (cid:104) H (cid:105) , as a functionof time, labeled by the number of kicks, N . (Bottom Panel) Plots showing evolution of the root mean square deviation ofindividual momenta in an ensemble of 10 particles σ ( p i ) = (cid:112) (cid:104) p i (cid:105) − (cid:104) p i (cid:105) as a function of time, labeled by the number of kicks, N . The initial states of the rotors are assumed to be definite momentum states. The total number of particles in this interactingensemble is d = 10 and circular boundary conditions apply. We consider two body interaction term to be J ij = cos( θ i − θ j ).The periodic kicking potential is given by K ( θ ) = (cid:80) ∞ m =1 k m cos( mθ ) for all particles. The m -th Fourier coefficient is fixed by k m = ˜ z/m , where ˜ z is a random complex number with real and imaginary parts uniformly distributed in the interval [0,1].We make sure that K ( θ ) is real. We choose (cid:126)α corresponding to three scenarios: Figs. (AI, AII): If ϕ denotes the golden ratio, α j = ( j/d ) ϕ − (1 / ϕ are all irrationals for j = 1 . . .
10. Figs. (BI, BII): We consider α j distinct irrationals as in Case A, butset α = α , which results in the resonant growth of σ ( p ) and σ ( p ), while σ ( E ) is bounded. Figs. (CI, CII): We consider α j as in Case A and set α = 1 /
2, which results in the resonant growth of σ ( E ) and σ ( p i ). the linear dependence of energy on momentum. If we al-low interactions that break translational invariance, mo-mentum is no longer conserved and resonance due to in-teractions triggers unbounded growth (delocalization) ofboth momenta and the total energy. C. Case III: α = 1 / and α . . . α d are distinctgeneric irrationals For rational α = 1 /
2, the system develops a differentkind of resonance compared to Case II. We consider akicking potential with ( k (cid:54) = 0). The resonance condition mα ∈ integer can be satisfied for m = 2 and the ratio sin( mNπα )sin( mπα ) grows as N . It results in energy growth anddelocalization of momenta p as shown in the RMS de-viations in Fig. 1 (CI, CII). This is a converse situationto Case II where the energy delocalizes with the delocal-ization of p even though the momenta p , p , . . . p arebounded. This situation is again captured by the IOMs,where ˆ C does not exist, while ˆ C . . . ˆ C are well definedand convergent as seen in Eq. (7).We can generalize the above representative cases. For each rational α i , its integral of motion ˆ C i breaks downand the corresponding momentum and energy divergewith time. For each pair α i = α j both ˆ C i and ˆ C j breakdown and the corresponding momenta diverge in oppositedirections, therefore ˆ C i + ˆ C j is still an IOM and the totalmomentum and energy of the pair are bounded.Other than the behaviour of energy in Case II in Sec-tion IV B, the rest of our analysis apply equally to theinteractions that break translational symmetry. V. DERIVATION OF MAIN RESULTS
Having established the physical understanding of local-ization for this model, we now sketch the brief derivationleading to the final results in eqs. (5) to (7). In the fol-lowing we consider the expectation of a generic operatoras a function of time, X ( N ) = (cid:104) ψ N | ˆ X | ψ N (cid:105) . We writethe explicit evolution of the many body wave functionbetween two successive kicks, | ψ N (cid:105) = e − i ˆ V e − i ˆ H | ψ N − (cid:105) .Notice that ˆ H = 2 π(cid:126)α · ˆ (cid:126)p + (cid:80) d − i (cid:54) = j J ij (ˆ θ i − ˆ θ j ) con-tains the many-body interaction term which may seemdaunting, however, the linear momentum term allowsa factorization in the Floquet operator. The Baker-Campbell-Hausdorff formula ˆ Z = ln( e ˆ X e ˆ Y ) becomestractable when [ ˆ X, ˆ Y ] = s ˆ Y . In this case, the resultsimply reads ˆ Z = ˆ X + s ˆ Y / (1 − e − s ). Now fix m andlet ˆ X = i π ( α ˆ p + α ˆ p ) and ˆ Y = i ˜ b m exp( i m [ˆ θ − ˆ θ ])where ∆ α = α − α . The result of the commutatorreads [ ˆ X, ˆ Y ] = i π ∆ α m ˆ Y , precisely the tractable casediscussed above. Replacing s with i π ∆ α m in the for-mula ˆ Z = ˆ X + s ˆ Y / (1 − e − s ) and inverting both sides ofthe equality, we obtain the following factorization e − i π(cid:126)α · (cid:126)p − i ˜ b m exp( i m [ θ − θ ]) = e − i b m exp( i m [ θ − θ ]) e − i π(cid:126)α · (cid:126)p . (8)For this to hold, the Fourier coefficients must satisfy˜ b ijm = sin( πm ∆ α ij ) πm ∆ α ij b ijm e − i mπ ∆ α ij . (9)This argument can be generalized to more particles andFourier coefficients. Due to linearity summations overparticle indices i, j and Fourier indices m are introduced.All in all, we can factorize the evolution operator for ourmodel in Eq. (3) as, | ψ N (cid:105) = e − i ˆ V − i (cid:80) ij ˜ J ij (ˆ θ i − ˆ θ j ) e − i π(cid:126)α · ˆ (cid:126)p | ψ N − (cid:105) , (10)where we have defined the modified interaction term as,˜ J ij = (cid:88) m ˜ b ijm e i m ( θ i − θ j ) (11a)= (cid:88) m sin( πm ∆ α ij ) πm (∆ α ij ) b ijm e i m ( θ i − θ j − π ∆ α ij ) . (11b)The advantage of this factorization is that the operator e − i π(cid:126)α · ˆ (cid:126)p is a translation operator in the position basis.We can rewrite Eq. (10) in the position basis by actingwith (cid:104) θ | from the left. We define (cid:104) θ | ψ N (cid:105) = ψ N ( θ ) andexpress (cid:104) θ | e − i π(cid:126)α · (cid:126)p | ψ N − (cid:105) = ψ N − ( (cid:126)θ − π(cid:126)α ). For a singlekick we then have, ψ N ( (cid:126)θ ) = e − i V ( (cid:126)θ ) − i (cid:80) ij ˜ J ij ( θ i − θ j ) ψ N − ( (cid:126)θ − π(cid:126)α ) . (12)The above equation can be recursively iterated to yield, ψ N ( (cid:126)θ ) = e − i N − (cid:80) n =0 [ V ( (cid:126)θ − πn(cid:126)α )+ (cid:80) ij ˜ J ij ( (cid:126)θ − πn(cid:126)α ) ] × ψ ( (cid:126)θ − πN (cid:126)α ) . (13)Here ˜ J ij ( (cid:126)θ − πn(cid:126)α ) is a short hand notation for ˜ J ij ( θ i − θ j − πn ∆ α ij ). A. Derivation of the evolution of energy andmomentum averages
Now consider a generic operator ˆ X ≡ X (ˆ p ... ˆ p d ; ˆ θ ... ˆ θ d ). The expectation value of this operator after N kicks is X ( N ) = (cid:90) d(cid:126)θ ψ ∗ N ( (cid:126)θ ) X (cid:18) ∂dθ ... ∂dθ d ; θ ...θ d (cid:19) ψ N ( (cid:126)θ ) . (14)If we substitute ˆ X = ˆ p k , we get p l ( N ) = (cid:104) ˆ p l (cid:105) − N (cid:88) n =1 (cid:42) ∂ l V ( (cid:126)θ + 2 πn(cid:126)α ) + ∂ l (cid:88) ij ˜ J ij ( (cid:126)θ + 2 πn(cid:126)α ) (cid:43) . (15)Here, ∂ l ≡ ∂/∂θ l and (cid:104) ... (cid:105) ≡ (cid:82) d(cid:126)θ... | ψ | . The contribu-tion from the kicking potential is: (cid:42) − ∂ l (cid:88) n V ( (cid:126)θ + 2 πn(cid:126)α ) (cid:43) = (cid:88) m (cid:104) ˆΓ ml (cid:105) sin( mN πα l )sin( mπα l ) . (16)The contribution from the interaction potential is (cid:42) − ∂ l (cid:88) n (cid:88) ij ˜ J ij (ˆ θ i − ˆ θ j + 2 πn [ α i − α j ]) (cid:43) (17)= (cid:88) jm (cid:104) ˆΓ intmlj (cid:105) sin( mN π ∆ α li ) πm ∆ α li . Putting together above expressions, we obtain the ex-pression for the momentum dynamics presented inSec. (III B). p i ( N ) = (cid:104) ˆ p i (cid:105) + (cid:88) m (cid:104) ˆΓ mi (cid:105) sin( mN πα i )sin( mπα i ) + (cid:88) mji (cid:104) ˆΓ intmij (cid:105) sin( mN π ∆ α ij ) mπ ∆ α ij . (18)Now we derive the expression for the energy growth.Substituting ˆ X = ˆ H in Eq. (14), we have E ( N ) = E (0) + (cid:88) mi πα i (cid:104) ˆΓ mi (cid:105) sin( mN πα l )sin( mπα l )+ 12 (cid:88) ij (cid:104) J ij (ˆ θ i − ˆ θ j + 2 πN ∆ α ij ) − J ij (ˆ θ i − ˆ θ j ) (cid:105) + (cid:88) ijm πα i (cid:104) ˆΓ intmij (cid:105) sin( mN π ∆ α ij ) mπ ∆ α ij . (19)In the above equation, a cancellation occurs between theinteraction terms in the last two lines of Eq. (19) owingto the following relation,12 (cid:88) ij (cid:104) J ij (ˆ θ i − ˆ θ j + 2 πN ∆ α ij ) − J ij ( θ i − θ j ) (cid:105) = − (cid:88) ijm πα i (cid:104) ˆΓ intmij (cid:105) sin( mN π ∆ α ij ) mπ ∆ α ij . (20)This completes the derivation of the energy growthshown in Sec. (19) E ( N ) = E (0) + d (cid:88) i =1 (cid:88) m πα i (cid:104) ˆΓ mi (cid:105) sin( mN πα i )sin( mπα i ) . (21)The dropping out of interaction terms from the totalenergy can be interpreted in the following way. As seenfrom Eq. (18), the contribution of interaction to the mo-mentum average picks up a negative sign when i and j areinterchanged. This means interaction transfers momen-tum from one particle to the other at each kick, in otherwords momentum is conserved for each couple of rotors.Since the energy is linear in momenta, when the mo-menta of rotors are summed, the contribution of interac-tions vanishes. We emphasize that when the interactionsbreak translational invariance, they no longer conservemomentum and in that case energy growth depends oninteractions too. B. Construction of integrals of motion
In this section, we outline the derivation involved inthe construction of integrals of motion. By inspectingEq. (15) and using the identity sin( mπα ) = [exp( i mπα ) − exp( − i mπα )] / (2 i ), we can write p l ( N + 1) − p l ( N ) =12 (cid:88) m mk m sin( mπα l ) (cid:18)(cid:68) e i m (ˆ θ l + πα l ) (cid:69) N − (cid:68) e i m (ˆ θ l + πα l ) (cid:69) N +1 (cid:19) + 12 (cid:88) mj b ljm π ∆ α lj (cid:18)(cid:68) e i m (ˆ θ l − ˆ θ j ) (cid:69) N − (cid:68) e i m (ˆ θ l − ˆ θ j ) (cid:69) N +1 (cid:19) , (22)noting that, the expression is valid whenever the denom-inators sin( mπα l ) and ∆ α lj = α l − α j are non-zero, inother words, whenever the resonances are avoided. Theabove expression can be organized in a way that it man-ifests the IOMs, (cid:42) ˆ p i + 12 (cid:88) m mk m e i m (ˆ θ i + πα i ) sin( mπα i ) + 12 (cid:88) mj b ijm e i m (ˆ θ i − ˆ θ j ) π ( α i − α j ) (cid:43) N +1 = (cid:42) ˆ p i + 12 (cid:88) m mk m e i m (ˆ θ i + πα i ) sin( mπα i ) + 12 (cid:88) mj b ijm e i m (ˆ θ i − ˆ θ j ) π ( α i − α j ) (cid:43) N (23)If we use the definition in Eq.(7) we get (cid:104) ˆ C l (cid:105) N +1 = (cid:104) ˆ C l (cid:105) N , thereby proving that ˆ C l is an integral of motion. C. Floquet Hamiltonian and quasienergyeigenstates
A special case of our model is when there are no reso-nances, namely, all IOMs associated with the momentum localization ˆ C ..d are intact. In this case, of particular im-portance is the following combination of the integrals ofmotion ˆ H F = (cid:88) i πα i ˆ C i . (24)Where ˆ H F is known as the Floquet Hamiltonian and isdefined as, e − i H F = e − i ˆ V e − i ˆ H = ˆ U F . (25)The quasienergy wavefunctions ψ ω are simultaneouseigenstates of ˆ H F and ˆ C i ’s. The quasienergy- ω state cen-tered around momenta (cid:104) ˆ (cid:126)p (cid:105) = (cid:126)M is ψ ω ( (cid:126)θ ) = (2 π ) − N/ exp (cid:26) i (cid:126)M · (cid:126)θ − (cid:88) jm k m e i m ( θ j + πα j ) mπα j ) − (cid:88) ijm b ijm πm ( α i − α j ) e i m ( θ i − θ j ) (cid:27) . (26)This satisfies ˆ C i ψ ω = M i ψ ω . By writing ˆ H F ψ ω = ωψ ω and using Eq. (24), we see that the eigenvalue equationis satisfied when, ω = 2 π(cid:126)α · (cid:126)M (mod 2 π ).We can also compute the momentum-momentum cor-relator, (cid:104) ˆ p i ˆ p j (cid:105) − (cid:104) ˆ p i (cid:105)(cid:104) ˆ p j (cid:105) for i (cid:54) = j over quasienergy eigen-states. The correlator over this state follows as (cid:104) ˆ p i ˆ p j (cid:105) − (cid:104) ˆ p i (cid:105)(cid:104) ˆ p j (cid:105) = − (cid:88) m | b ijm | π ( α i − α j ) . (27)In the case of a resonance α i → α j , this correlator clearlydiverges, while in the localized case it remains finite. VI. EXPERIMENTAL PROPOSAL
A natural venue for realizing the interacting kickedrotor model is superconducting grains. The Hamilto-nian, Eq. (3) could be implemented using a chain ofvoltage-biased superconducting grains coupled to eachother using Josephson junctions. Consider a chain ofgrains gated by a ground plane which is resistively con-nected to ground (see Fig. 2). We then supply a gatevoltage V i to each grain, and connect them to each otherby Josephson junctions J ij .Because the voltage on the i -th grain is locked to be V i , its phase winds with an angular velocity ˙ φ i = 2 eV i / (cid:126) .Therefore, the resistance and gate capacitor can be ig-nored when writing the effective stationary part of theHamiltonian: H = (cid:88) i q i V i − (cid:88) ij J ij cos( φ i − φ j ) . (28)In addition, the “kick” term is obtained by connect-ing each grain to a common macroscopic superconductor V i-1 V i V i+1 CR CR CR
Macroscopic Superconductor K i-1 K i K i+1 J i-1 J i J i+1 Stroboscopic Josephson Junction Coupling
FIG. 2. Schematic of chain of voltage-biased ( V i ) supercon-ducting grains coupled to each other ( J i ) using Josephsonjunctions. Each grain is connected ( K i ) to a grounded com-mon macroscopic superconductor stroboscopically through astrong Josephson coupling. (which is itself grounded), for a short time and througha strong Josephson coupling. V ( t ) = − (cid:88) i, n K ( t − nT ) cos φ i , (29)with K ( t ) = K when | t | < τ . To make the kick termas close to a delta function as possible, we must have2 eV i τ (cid:28) (cid:126) π , and Kτ ∼ (cid:126) , and τ (cid:28) T . A diagram ofthe circuit for a nearest-neighbor interaction is shown inFig. 2. Identifying, φ i = θ i , 2 πα i = 2 eV i / (cid:126) and p i = (cid:126) q i / e we see that we indeed obtain the Hamiltonian ofEq. (3). VII. CONCLUSIONS
In this work, we introduced the concept of dynam-ical many-body localization and presented an exactly-solvable model of driven linear rotors, which exhibits thisphenomenon. Although the model possesses a full set ofintegrals of motion, it is shown that dynamical MBL isaccompanied by the emergence of additional integrals ofmotion, local in momentum space. We believe that thisobservation has important general implications for under-standing dynamics of interacting many-body systems.We have shown that these integrals of motion breakdown due to two types of resonances indicating delo-calization in momentum space. One type of resonanceoriginates from commensuration of the external drivingperiod with the parameters of the system and the otherfrom the static interactions. An interesting feature ofthis model is that the total energy in the system, thatis linear in momenta, fails to be a good indicator of dy-namical localization, since when momentum conservinginteractions delocalize momentum, momenta of interact-ing pair grow in opposite directions.Moreover, we have shown, (see Appendix) by utilizingthe lattice mapping introduced by Fishman and Grempel and Prange, that our model maps into a disordered lat-tice with as many dimensions as there are rotors. Basedon this observation, we argue that what is observed is anAnderson type localization, particularly of the type seenin correlated disorder systems.We also have proposed an experimental setup com-posed of Josephson junctions and superconducting grainsto realize the model Hamiltonian.Finally, we emphasize that the results presented herecan apply to more generic non-integrable systems. Forexample, a recent work considers interacting kickedDirac particles with individual Hamiltonians, H =2 πασ x p + M σ z , and provides a simple argument thatthis non- integrable system also exhibits MBL. First, thismodel also exhibits localization when α ’s are generic dis-tinct irrationals. Second, although the number of inter-acting particles that can be considered numerically is lim-ited, note that at large momenta the Dirac model crossesover to the linear model considered here. This suggeststhat MBL should be robust to a class of non-integrablegeneralizations, for any number of interacting rotors. VIII. ACKNOWLEDGEMENTS
This work was supported by US-ARO, Australian Re-search Council, and Simons Foundation (A.C.K. andV.G.). SG gratefully acknowledges support by NSF-JQI-PFC and LPS-MPO-CMTC. GR is grateful for supportfrom the IQIM and the Moore Foundation, and the NSFunder grant DMR-1410435. The authors are grateful toSankar Das Sarma and Efim Rozenbaum for useful dis-cussions.
Appendix A: Mapping between Quantum KickedRotor and the tight-binding model
In the first part of this section, we derive the known lat-tice mapping of the one dimensional kicked rotor . Oncewe establish this derivation, we show that the many-bodylinear kicked rotor (3) also admits a lattice mapping ofa particle on a d-dimensional lattice. We emphasize thatexistence of such a mapping in the many-body case islimited to the case of linear p model ( l = 1).
1. Lattice model of single quantum kicked rotor
In the introduction we considered the time dependentSchr¨odinger equation for a kicked rotor. The kinetic partwas considered to be ˆ p l . For the quadratic kicked rotorcase l = 2 and the linear kicked rotor that we consider inthis work is l = 1. Also (cid:126) = 1 and kicking period T = 1are used in this equation. i ∂ t ψ ( θ, t ) = [2 πα ( − i ∂ θ ) l + K ( θ ) (cid:88) n δ ( t − n )] ψ ( θ, t ) . (A1)The above equation can be solved for ψ ( θ, t ) = e − iωt u ( θ, t ), where the function u has the unit period-icity of the driving. Let u ± defines the state just beforeand after the kick and are connected by the evolutionoperator in the following way, u + = e − i K ( θ ) u − , u − = e i ( ω − πα ˆ p l ) u + (A2)Define the following: ¯ u = u + + u − , exp( − i ˆ K ( θ )) =(1 − i ˆ W ( θ )) − (1 + i ˆ W ( θ )), exp( − i [2 πα ˆ p l − ω ]) = (1 − i ˆ T ( θ )) − (1 + i ˆ T ( θ )). Based on the above definitions, u ± = (1 ∓ i ˆ T ( θ ))¯ u and[ ˆ T ( θ ) + ˆ W ( θ )]¯ u = 0 (A3)is obtained.Fourier transforming the above expression, we get thefollowing tight binding model, (cid:88) n (cid:54) = m W m − n u n + T m u m = Eu m . (A4)Here, the energies and hoppings are: W m − n = − Eδ m,n − π (cid:90) π e − i ( m − n ) θ (cid:26) tan (cid:18) K ( θ )2 (cid:19)(cid:27) d θ, (A5a) T m = tan (cid:18)
12 [ ω − παm l ] (cid:19) . (A5b)This completes the derivation for the lattice mappingfor the single rotor case. In the following section, wegeneralize this derivation to demonstrate the existenceof a lattice mapping for the interacting rotor model ofEq. (3). d -dimensional lattice model In this section we show that there exists a d − dimen-sional lattice model corresponding to the d particle inter-acting version of the kicked rotor model in Eq. (3). Sucha mapping has been previously identified for the case of a d rotors driven by an interaction potential . Notice thatin our case, the interactions are encoded in the statingHamiltonian ˆ H . However, we show that for the linearmomentum dependence of the kinetic term, static inter-actions can expressed as the driven interactions and therest of the lattice mapping simple follows from Ref. [17].In order to establish this mapping, we use the factoriza-tion of the Floquet operator discussed in Sec. (V), see eqs. (8) to (13). The time dependent Hamiltonian inEq. (3) produces the same Floquet operator as,ˆ H ( t ) = 2 π d (cid:88) i =1 α i ˆ p i + ˆ V F ∞ (cid:88) n = −∞ δ ( t − n ) , (A6)where we have defined,ˆ V F = d (cid:88) i =1 K (ˆ θ i ) + 12 (cid:88) i (cid:54) = j ˜ J ij (ˆ θ i − ˆ θ j ) (A7)= d (cid:88) j =1 (cid:88) m k m e im ˆ θ j + 12 (cid:88) i (cid:54) = j (cid:88) m ˜ b ijm e im (ˆ θ i − ˆ θ j ) . (A8)The factorization enables to treat on site and interactionpotentials on equal footing. The ˜ b ij is defined in Eq. (9).Moreover, we write:ˆ V F = (cid:88) (cid:126)m V F(cid:126)m e i(cid:126)m · (cid:126)θ . (A9)The equivalence between Eq. (A7) and Eq. (A9) is con-ceptually very simple, but somewhat harder to put intowords. It is probably best to give an example. Take thetwo rotor case i.e. d = 2. Now (cid:126)m are two dimensionalvectors with integer components. Let us fix m ∈ Z . If (cid:126)m = ( m,
0) then V F(cid:126)m = k m . If (cid:126)m = (0 , m ) then V F(cid:126)m = k m .If (cid:126)m = ( m, − m ) then V F(cid:126)m = b m /
2. For all other vectors, V F(cid:126)m = 0. The summation in Eq. (A9) is over all possiblevectors (cid:126)m . Generalizing this notation to d dimensions,it is possible for both the differences of angles and an-gles themselves to appear as (cid:126)m · (cid:126)θ . Treating α ’s and p ’sas d -dimensional vectors as well, the Hamiltonian can besuccinctly as:ˆ H ( t ) = 2 π(cid:126)α ˆ (cid:126)p + (cid:88) (cid:126)m V (cid:126)m e i(cid:126)m · ˆ (cid:126)θ ∞ (cid:88) n = −∞ δ ( t − n ) . (A10)Following Ref. [17], the above driven Hamiltonian canbe mapped on to a d -dimensional lattice model, whichis closely related to the lattice mapping outlined in theprevious section for the single rotor case. H (cid:126)m,(cid:126)n u (cid:126)n = T (cid:126)m u (cid:126)m + (cid:88) (cid:126)n W (cid:126)m,(cid:126)n u (cid:126)n = Eu (cid:126)m . (A11)Here, (cid:126)m and (cid:126)n are vectors that contain integers that cor-respond to the quantized eigenvalues of the angular mo-mentum operator. The hopping and onsite terms aredefined as, W (cid:126)m − (cid:126)n = − Eδ (cid:126)m,(cid:126)n +1(2 π ) d (cid:90) π e − i ( (cid:126)m − (cid:126)n ) · (cid:126)θ (cid:40) − tan (cid:32) V F ( (cid:126)θ )2 (cid:33)(cid:41) d (cid:126)θ, (A12a) T (cid:126)m = tan (cid:18)
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