Spreading, Nonergodicity, and Selftrapping: a puzzle of interacting disordered lattice waves
aa r X i v : . [ n li n . C D ] D ec Spreading, Nonergodicity, and Selftrapping: apuzzle of interacting disordered lattice waves
Sergej Flach
Abstract
Localization of waves by disorder is a fundamental physical problem en-compassing a diverse spectrum of theoretical, experimental and numerical studiesin the context of metal-insulator transitions, the quantum Hall effect, light propa-gation in photonic crystals, and dynamics of ultra-cold atoms in optical arrays, toname just a few examples. Large intensity light can induce nonlinear response, ul-tracold atomic gases can be tuned into an interacting regime, which leads again tononlinear wave equations on a mean field level. The interplay between disorder andnonlinearity, their localizing and delocalizing effects is currently an intriguing andchallenging issue in the field of lattice waves. In particular it leads to the predictionand observation of two different regimes of destruction of Anderson localization- asymptotic weak chaos, and intermediate strong chaos, separated by a crossovercondition on densities. On the other side approximate full quantum interacting manybody treatments were recently used to predict and obtain a novel many body local-ization transition, and two distinct phases - a localization phase, and a delocalizationphase, both again separated by some typical density scale. We will discuss selftrap-ping, nonergodicity and nonGibbsean phases which are typical for such discretemodels with particle number conservation and their relation to the above crossoverand transition physics. We will also discuss potential connections to quantum manybody theories.
Sergej FlachCenter for Theoretical Physics of Complex Systems, Institute for Basic Science, Daejeon, Korea,e-mail: [email protected]
New Zealand Institute for Advanced Study, Centre for Theoretical Chemistry and Physics, MasseyUniversity, 0745 Auckland New Zealand. e-mail: [email protected]
In this contribution we will discuss the regimes of wave packet spreading in nonlin-ear disordered lattice systems, and its relation to quantum many body localization(MBL). We will consider cases when the corresponding linear (single particle) waveequations show Anderson localization, and the localization length is bounded fromabove by a finite value.There are several reasons to analyze such situations. Wave propagation in spa-tially disordered media has been of practical interest since the early times of studiesof conductivity in solids. In particular, it became of much practical interest for theconductance properties of electrons in semiconductor devices more than half a cen-tury ago. It was probably these issues which motivated P. W. Anderson to performhis groundbreaking lattice wave studies on what is now called Anderson localiza-tion [1]. With evolving modern technology, wave propagation became of importancealso in photonic and acoustic devices in structured materials [2, 3]. Finally, recentadvances in the control of ultracold atoms in optical potentials made it possible toobserve Anderson localization there as well [4].In many if not all cases wave-wave interactions can be of importance, or can evenbe controlled experimentally. Short range interactions hold for s-wave scattering ofatoms. When many quantum particles interact weakly, mean field approximationsoften lead to effective nonlinear wave equations. Electron-electron interactions insolids and mesoscopic devices are also interesting candidates with the twist of anew statistics of fermions. Nonlinear wave equations in disordered media are ofpractical importance also because of high intensity light beams propagating throughstructured optical devices induce a nonlinear response of the medium and subse-quent nonlinear contributions to the light wave equations. While electronic excita-tions often suffer from dephasing due to interactions with other degrees of freedom(e.g. phonons), the level of phase coherence can be controlled in a much better wayfor ultracold atomic gases and light.There is a fundamental mathematical interest in the understanding, how Ander-son localization is modified in the presence of quantum many body interactions,or classical nonlinear terms in the wave equations. All of the above motivates thechoice of corresponding linear (single particle) wave equations with finite upperbounds on the localization length. Then, the corresponding noninteracting quantumsystems, as well as linear classical waves, admit no transport. Analyzing transportproperties of nonlinear, respectively interacting quantum, disordered wave equa-tions allows to observe and characterize the influence of wave-wave interactions onAnderson localization in a straightforward way.The chapter is structured in the following way. We will introduce the classicalmodel, discuss its statistical properties, and in particular the nonGibbsean phase,and some of its generalizations. We will then come to wave packet spreading, aself-trapping theorem, and to results of destruction of Anderson localization. Fi-nally, we will discuss the relation of these results to many body localization for thecorresponding quantum many body problems. itle Suppressed Due to Excessive Length 3
The Gross-Pitaevskii equation is describing Bose-Einstein condensates (BEC) of in-teracting ultracold atoms in certain mean-field approximations. It is also known asthe nonlinear Schr¨odinger equation which is integrable in 1+1 dimensions, and hasmany further applications e.g. in nonlinear optics. Its discretized version - the dis-crete nonlinear Schr¨odinger equation (DNLS) or discrete Gross-Pitaevskii equation(DGP) - is typically nonintegrable, and is realized with a BEC loaded onto opticallattices [5]. Similar the discrete nonlinear Schr¨odinger equation (DNLS) is realizedfor various one- and two-dimensional networks of interacting optical waveguides[6].Additional disorder, either due to natural inhomogeneities, or intentionally im-planted, finally leads to the disordered discrete Gross-Pitaevsky (dDGP) or disor-dered discrete nonlinear Schr¨odinger equation (dDNLS) equation with Hamiltonian H = (cid:229) l e l | y l | + n | y l | − ( y l + y ∗ l + y ∗ l + y l ) (1)with complex variables y l , lattice site indices l and nonlinearity strength n ≥ e l with zero average ¯ e = lim N → ¥ N − ( (cid:229) l = Nl = e l ) = s e = lim N → ¥ N − ( (cid:229) l = Nl = e l ) are dis-tributed with some probability density distribution (PDF) P e . Here we will use thebox distribution P e ( x ) = W for | x | ≤ W and P e = y l = ¶ H D / ¶ ( i y ⋆ l ) : i ˙ y l = e l y l + n | y l | y l − y l + − y l − . (2)Eqs. (2) conserve the energy (1) and the norm A = (cid:229) l | y l | . Note that varying thenorm of an initial wave packet is strictly equivalent to varying n . Note also that thetransformation y l → ( − ) l y ∗ l , n → − n , e l → − e l leaves the equations of motioninvariant. Therefore the sign of the nonlinear coefficient n can be fixed without lossof generality to be positive. The existence of the second - in addition to the energy - conserved quantity A ≥ e l = y l into a Sergej Flach two-component one - a first component of high density localized spots and a secondcomponent of delocalized wave excitations with infinite temperature [7, 8, 9, 10,11]. The high density localized spots are conceptually very similar to selftrappingand discrete breathers [12].We will briefly compute the effect of nonvanishing disorder e l = A l ≥ ≤ f l ≤ p suchthat y l = p A l e i f l . (3)The Hamiltonian transforms into H = (cid:229) l e l A l − p A l A l + cos ( f l − f l + ) + n A l , (4)and the norm simply becomes A = (cid:229) l A l . (5)Assuming a large system of N sites the corresponding average densities become h = H / N , a = A / N . (6)From here on we will always implicitly consider the thermodynamic limit N → ¥ .We will as well express final results in terms of densities scaled with the nonlinearityparameter n [13]: y = n h , x = n a . (7)A number of questions can be posed. First, can any pair of realizable densities { x , y } be obtained by assuming a Gibbs distribution r G = Z e − b ( H + m A ) (8)where Z is the partition function, b the inverse temperature, and m the chemical po-tential ? And if not, what is the dynamics in the corresponding nonGibbsean phase?The lowest energy state E min is evidently given by f l = const : E min = (cid:229) l ( e l − ) A l + n A l , (9)In the absence of any potential e l = A l = A / n = a . With the notation of the inverse temperature b we conclude that the zerotemperature limit of the scaled energy density at a given value of the scaled normdensity is given by itle Suppressed Due to Excessive Length 5 y b → ¥ = − x + x / . (10)For a nonzero potential with finite variance the lowest energy density limit will belowered by a finite value. With the distribution P e ( x ) = W for | x | ≤ W we arrive atthe upper and lower bound s − ( + W ) x + x / ≤ y b → ¥ ≤ − x + x / . (11)At the same time the upper limit for the total energy of a finite system with N sitesis obtained by concentrating all the norm on one lattice site which yields E max = n A / y max = Nx /
2. In the thermodynamic limit N → ¥ the upper limit for the energy density is diverging. We conclude that at any givennorm density x the energy density is bounded from below by the finite minimumvalue y b → ¥ , but is not bounded from above and can take arbitrary large values.The partition function Z = Z ¥ Z p (cid:213) m d f m dA m exp [ − b ( H + m A )] (12)depends on all amplitudes and phases. Integration over the phase variables f m re-duces the symmetrized partition function to Z = ( p ) N Z ¥ (cid:213) m dA m I ( b p A m A m + ) e − b ( H + m A ) (13)with the reduced Hamiltonian H = (cid:229) l e l A l + A l (14)depending only on the amplitudes, and with I being the Bessel function of 0th order.The strategy of finding a nonGibbsean phase is simply to find the line of infi-nite temperature b = b → bm → const . With thenotation m l = m + e l and after some simple algebra it follows for infinite temperatureln Z = N ln 2 p − N (cid:229) l = (cid:18) ln ( bm l ) + bb m l (cid:19) . (15)With the standard relations H = (( mb ¶¶m − ¶¶b ) ln Z and A = − b ¶¶m ln Z we obtain A = N bm , H = N b m + N (cid:229) l = e l bm l . (16) Sergej Flach
Note that we used m ∼ / b in the infinite temperature limit.The total norm is not affected by the presence of a potential. The energy is af-fected, however things are different for the densities. The second term in the energyexpression in (16) can be expanded as N (cid:229) l = e l b ( m + e l ) ≈ bm N (cid:229) l = e l − m N (cid:229) l = e l ! . (17)The first term on the rhs diverges as √ N for any finite variance of e l - too slow tocontribute to the final relation between the densities at the infinite temperature point(because we have to divide by N ). The second term on the rhs is proportional to N / m - the thermodynamic limit will remain a contribution in the density, howeverthe infinite temperature limit leads to 1 / m → y b = = x . (18)Since the energy density is not bounded from above, we conclude that for all densi-ties y > y b = the system will not be described by a Gibbs distribution.Let us discuss some consequences in the absence of disorder (the presence ofdisorder will not substantially alter them). First, at a given scaled norm density x ,a homogeneous state A l = const with constant phases f l = const will correspondto the lowest energy state which is in the Gibbsean regime. For staggered phases f l + = f l + p the homogeneous state A l = const yields a scaled energy density y st = x + x /
2. Therefore we find that for x ≥ A l = const ,regardless of their phase details, are launched in the Gibbsean regime. However for x < x there exist initial states which are inhomogeneousin the amplitudes such that the state will be located in the nonGibbsean regime.If an initial state is in the nonGibbsean regime, we can not conclude much aboutthe nature of its dynamics. It could remain to be strongly chaotic, and described bya negative temperature Gibbs distribution. It could be also nonergodic, less chaotic,or non-mixing.The reported dynamical regimes in the Gibbsean and nonGibbsean are remark-ably different [7, 8, 9, 10, 11, 12]. While the Gibbsean regime is characterized bya relatively quick decay into a thermal equilibrium on time scales which are pre-sumably inverse proportional to the largest Lyapunov coefficient, the nonGibbseanregime is very different. The dynamics is still chaotic, however the system relaxesinto a two-component state - condensed hot spots with concentrated norm in themand corresponding high energy, and cold low energy density regions between them.Some of the results seem to indicate that the system produces as much of a conden-sate fraction as is needed to keep the remaining noncondensed part in a Gibbseanregime with infinite temperature b =
0. There is no evident mixing and relaxation itle Suppressed Due to Excessive Length 7 in the condensate fraction. The condensed hot spots are similar to discrete breatherswhich are well known to exist in such models [12].Interestingly models without norm conservation also allow for discrete breathers[12]. With only one conservation law (energy) and one variational parameter (in-verse temperature) the equilibrium state of a Boltzmann distribution is always ca-pable of yielding the prescribed energy density. Still such systems can produce hotspots, or discrete breathers, in thermal equilibrium in certain control parameter do-mains [12]. The remarkable difference to the above cases is, that the condensed hotspots do have a finite life time, and mixing, ergodicity and thermal equilibrium areobtained after finite times.
The existence of the second conserved quantity A has also a nontrivial consequencefor the decay of localized initial states or simply wave packets [14]. Consider acompact localized initial state with finite norm A and energy H . Such a state hasnonzero amplitudes inside a finite volume only, and strictly zero amplitudes out-side. Note that the theorem can be easily generalized to non-compact localized ini-tial states with properly, e.g. exponentially, decaying tails. The theorem addressesthe question whether such a state can spread into an infinite volume and dissolvecompletely into some homogeneous final states. To measure the inhomogeneity ofstates we use the participation number (PN) P = A (cid:229) l A l . (19)This measure is bounded from below P ≥ P ≤ N where N isthe number of available sites. The lower bound is achieved by concentrating all theavailable norm onto one single site A l = A d l , l . The upper bound is achieved by acompletely homogeneous state A l = A / N . Note that a typical value of the PN in athermalized state is about N /
2, due to inavoidable fluctuations in the amplitudes ondifferent sites.For an infinite system N → ¥ , the PN is unbounded from above. A localizedinitial state has a finite PN. If this state evolves and stays localized, its PN stayslocalized as well. If it manages to spread into the infinitely large reservoir of thesystem, and if the densities become on all sites of order A / N , then the PN will beof order N . If the PN stays finite, then a part of the excitation is said to stay local-ized - either in the area of the initial excitation spot, or in other, possible migrating,locations. Therefore, the participation number turns to be a useful measure of in-homogeneity of a state, including localized distributions on zero or also nonzerodelocalized backgrounds. It is the more useful as its inverse, up to a constant, isprecisely the anharmonic energy share of the full Hamiltonian (1). Sergej Flach
The selftrapping theorem [14] uses the existence of the second integral of mo-tion - the norm. We split the total energy H = h y | L | y i + H NL into the sum of itsquadratic term of order 2 and its nonlinear terms of order strictly higher than 2.Then, L is a linear operator which is bounded from above and below. In our specificexample, we have h y | L | y i ≥ w m h y | y i = w m A where 2 + W ≥ w m ≥ − − W and w m is an eigenvalue of L .If the wavepacket amplitudes spread to zero at infinite time, lim t → ¥ ( sup l | y l | ) =
0. Then lim t → ¥ ( (cid:229) l | y l | ) < lim t → ¥ ( sup l | y l | )( (cid:229) l | y l | ) = A = (cid:229) l | y l | istime invariant. Consequently , for t → ¥ we have H NL = H ≤ w m (cid:229) l | y l | = w m A . Since H and A are both time invariant, this inequality should be fulfilled atall times. However when the initial amplitude √ A of the wavepacket is large enough,it cannot be initially fulfilled because the nonlinear energy diverges as A while thetotal norm diverges as A only. Thus such an initial wavepacket cannot spread tozero amplitudes at infinite time. This proof is valid for any strength of disorder W including the ordered case W =
0, and any lattice. Note that the opposite is not true- i.e., if the wavepacket does not fulfill the criterion for selftrapping according to theselftrapping theorem, we can only conclude that it may not selftrap in the course ofspreading. We will still coin this regime non-selftrapping.Let us consider two examples. First, take a single site intial state y l = √ A d l , .The energy is H = e A + n A , and the norm A = A . A zero amplitude final statehas an upper energy bound of ( W / + ) A . For an amplitude A > A c with n A c = W + − e the initial state can not spread into a final one with infinite PN. The PNis bounded from above by P max with P − max = ( e − W − ) / ( n A ) + L with average scaled energy density y and norm density x . Theselftrapping theorem tells that selftrapping will persist for y > y c with y c = (cid:18) W + (cid:19) x . (20)Let us discuss the connection between selftrapping and Gibbs-nonGibbs regimesfor the second example of a spreading wave packet excited initially on many sites.At any time during its spreading (including the initial time) we can trap it with fixedboundaries at its edges, and address its thermodynamic properties. We also note thatif a wave packet spreads, its width L will increase with time, and the densities y and x will correspondingly drop keeping a linear dependence y = y x x . Then, for y ≤ y > itle Suppressed Due to Excessive Length 9 not thermalize quickly enough in its core, and therefore never enters a nonGibbseanregime despite the fact that it would do so at equilibrium. For n = y l = B l exp ( − i l t ) Eq. (2) is reduced to the linear eigenvalue problem l B l = e l B l − B l − − B l + . (21)All eigenstates are exponentially localized, as first shown by P.W. Anderson [1].The normal modes (NM) are characterized by the normalized eigenvectors B ¯ n , l ( (cid:229) l B n , l = ) . The eigenvalues l ¯ n are the frequencies of the NMs. The width ofthe eigenfrequency spectrum l ¯ n of (21) is D = W + l ¯ n ∈ (cid:2) − − W , + W (cid:3) .While the usual ordering principle of NMs is with their increasing eigenvalues, herewe adopt a spatial ordering with increasing value of the center-of-norm coordinate X ¯ n = (cid:229) l lB n , l .The asymptotic spatial decay of an eigenvector is given by B ¯ n , l ∼ e −| l | / x ( l ¯ n ) where x ( l ¯ n ) is the localization length and x ( l ¯ n ) ≈ ( − l n ) / W for weak disor-der W ≤ V ¯ n ∼ x ¯ nu (for details see[16]). The average spacing d of eigenvalues of neighboring NMs within the range ofa localization volume is of the order of d ≈ D / V , which becomes d ≈ D W /
300 forweak disorder. The two scales d ≤ D are expected to determine the packet evolutiondetails in the presence of nonlinearity.Due to the localized character of the NMs, any localized wave packet with size L which is launched into the system for n = In the presence of nonlinearity the equations of motion of (2) in normal mode spaceread i ˙ f ¯ n = l ¯ n f ¯ n + n (cid:229) ¯ n , ¯ n , ¯ n I ¯ n , ¯ n , ¯ n , ¯ n f ∗ ¯ n f ¯ n f ¯ n (22)with the overlap integral I ¯ n , ¯ n , ¯ n , ¯ n = (cid:229) l B ¯ n , l B ¯ n , l B ¯ n , l B ¯ n , l . (23)The variables f ¯ n determine the complex time-dependent amplitudes of the NMs.The frequency shift of a single site oscillator induced by the nonlinearity is d l = n | y l | ≈ x . As it follows from (22), nonlinearity induces an interaction betweenNMs. Since all NMs are exponentially localized in space, each normal mode iseffectively coupled to a finite number of neighboring NMs, i.e. the interaction rangeis finite. However the strength of the coupling is proportional to the norm density n = | f | . Let us assume that a wave packet spreads. In the course of spreading its normdensity will become smaller. Therefore the effective coupling strength between NMsdecreases as well. At the same time the number of excited NMs grows. One possibleoutcome would be: (I) that after some time the coupling will be weak enough to beneglected. If neglected, the nonlinear terms are removed, the problem is reducedto an integrable linear wave equation, and we obtain again Anderson localization.That implies that the trajectory happens to be on a quasiperiodic torus - on whichit must have been in fact from the beginning. It also implies that the actions of thelinear wave equations are not strongly varying in the nonlinear case, and we areobserving a kind of anderson localization in action subspace. Another possibilityis: (II) that spreading continues for all times. That would imply that the trajectorydoes not evolve on a quasiperiodic torus, but instead evolves in some chaotic partof phase space. This second possibility (II) can be subdivided further, e.g. assumingthat the wave packet will exit, or enter, a Kolmogorov-Arnold-Moser (KAM) regimeof mixed phase space, or stay all the time outside such a perturbative KAM regime.In particular if the wave packet dynamics will enter a KAM regime for large times,one might speculate that the trajectory will get trapped between denser and densertorus structures in phase space after some spreading, leading again to localizationas an asymptotic outcome, or at least to some very strong slowing down of thespreading process.Published numerical data [17, 18, 19, 20, 21, 22] (we refer to [16, 23] for moreoriginal references and details of the theory) show that finite size initial wave pack-ets i) stay localized if x ≪ d and display regular-like (i.e. quasiperiodic as in theKAM regime ) dynamics, which is numerically hardly distinguishable from veryslow chaotic dynamics with subsequent spreading on unaccessible time scales); ii)spread subdiffusively if x ∼ d with the second moment of the wave packet m ∼ t a and chaotic dynamics inside the wave packet core; segregate into a two componentfield with a selftrapped component, and a subdiffusively spreading part for x > D .Spreading wave packets reduce their densities x , y in the course of time, andmay reach regime i), without much change in their spreading dynamics. Thereforewe can conclude that regime i) is at best a KAM regime, with a finite probabilityto launch a wave packet on a KAM torus, and a complementary one to observespreading [24]. Anderson localization is restored in that probabilistic way, as theprobability to stay on a KAM torus will reach value one in the limit of vanishingnonlinearity. Spreading wave packets, when launched in a domain of positive energydensities y , are either from the beginning in the nonGibbsean regime, or have toreach it at some later point in time. Assuming that the wave packet has enough itle Suppressed Due to Excessive Length 11 time to develop nonGibbsean structures, one should observe selftrapping. Publishednumerical studies did not focus on this issue, in particular for parameter valueswhich do not satisfy the seltrapping theorem. The reported numerical selftrappingdynamics is in full accord with the results of the selftrapping theorem.The most interesting result concerns the spreading dynamics and the exponent a . The assumption of strong chaos - i.e. the dephasing of normal modes on timesscales much shorter than the spreading time scales - leads to a = / a = , x > d : strong chaos . (24)In the asymptotic regime of small densities, instead the regime of weak chaos isobserved: a = , x < d : weak chaos . (25)Perturbation theories show that in that regime not all normal modes are resonantand chaotic, but only a fraction of them [18, 19, 25, 23]. Correcting the theory ofstrong chaos with the probability of resonances P r , the asymptotic value a = / D ∼ ( P r ( x ) x h I i ) , P r = − e − Cx , C ∼ x h I i d . (26)The corresponding nonlinear diffusion equation [26] for the norm density distri-bution (replacing the lattice by a continuum for simplicity, see also [27]) uses thediffusion coefficient D ( x ) : ¶ t x = ¶ ¯ n ( D ¶ ¯ n x ) . (27)In the regime of strong chaos it follows D ∼ x , and in the regime of weak chaos - D ∼ x . It is straightforward to show that the exponent a satisfies the relations (24)and (25) respectively [25, 28, 29, 23]. We can also conclude that a large system atequilibrium will show a conductivity which is proportional to D . Let us discuss the expected dynamical regimes of an infinitely large lattice excitedto some finite densities. In the nonGibbsean regime y > x the dynamics is known tobe nonergodic up extremely long time scales. At the same time the Gibbsean phaseis characterized by two different regimes - strong chaos at large densities, and weakchaos at small densities. Strong chaos implies that all normal modes are resonant andlead to chaos and mixing. Therefore we could assume that strong chaos is ergodic.For small enough density x we enter the weak chaos regime, where not all normalmodes are resonant and lead to chaos and mixing at the same time. Therefore it could be possible that this regime is nonergodic, despite the fact that it characterized by afinite conductivity.Now we recall the main statements from many body localization. This theorydeals with a quantum system of many interacting particles in the presence of dis-order. It was initially developed for fermions [30]. The system is considered in aspatially continuous system. At a given particle density, the conductivity is pre-dicted to be zero up to a nonzero critical energy density. Above this transition point,the system exhibits many body states which conduct in a nonergodic (multifractal)fashion. At even larger energy densities the system finally exhibits ergodic metallicstates. Later this theory was also developed for bosons, which is the quantum coun-terpart of our classical model [31]. In that case, again the system is characterizedby a finite energy density (at fixed particle density) below which all states are manybody localized. Above that critical density states are again extended but nonergodicand fractal.The cricital many body localization energy density scales to zero in the classicallimit. This is consistent with the fact that a wave packet spreads to infinity (assumingthat it indeed does so), as the wave packet is characterized by finite densities at anyfinite time, and approaches zero density in the limit of infinite time. It is thereforetempting to associate the regime of weak chaos with the nonergodic but metallicregime of the quantum theory. The test of this possibility is therefore one of thechallenging future tasks to be explored. References
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