Interaction-disorder competition in a spin system evaluated through the Loschmidt Echo
Pablo R. Zangara, Axel D. Dente, Aníbal Iucci, Patricia R. Levstein, Horacio M. Pastawski
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] N ov Interaction-disorder competition in a spin system evaluated through the LoschmidtEcho
Pablo R. Zangara, Axel D. Dente,
1, 2
An´ıbal Iucci, Patricia R. Levstein, and Horacio M. Pastawski Instituto de F´ısica Enrique Gaviola (IFEG), CONICET-UNC and Facultad de Matem´atica,Astronom´ıa y F´ısica, Universidad Nacional de C´ordoba, 5000 C´ordoba, Argentina INVAP S.E., 8403 San Carlos de Bariloche, Argentina Instituto de F´ısica La Plata (IFLP), CONICET-UNLP and Departamento de F´ısica,Universidad Nacional de La Plata, CC 67, 1900 La Plata, Argentina
The interplay between interactions and disorder in closed quantum many-body systems is rel-evant for thermalization phenomenon. In this article, we address this competition in an infinitetemperature spin system, by means of the Loschmidt echo (LE), which is based on a time reversalprocedure. This quantity has been formerly employed to connect quantum and classical chaos, andin the present many-body scenario we use it as a dynamical witness. We assess the LE time scalesas a function of disorder and interaction strengths. The strategy enables a qualitative phase dia-gram that shows the regions of ergodic and nonergodic behavior of the polarization under the echodynamics.
PACS numbers: 71.30.+h,03.65.Yz,72.15.Rn,05.70.Ln
The ergodic hypothesis of statistical mechanics impliesthe equivalence between time and ensemble averages. Itis expected that a conservative many-body system satis-fying such hypothesis would explore uniformly the wholeenergy shell. It is now a long time since Fermi, Pasta andUlam (FPU) questioned how the irregular dynamics in-duced by nonlinearities in a Hamiltonian may lead to en-ergy equipartition as a manifestation of ergodicity. Suchdynamics, referred to as thermalization , did not show upin their pioneering numerical simulations. Even thoughtheir striking results are now explained by the theory ofchaos , the solution of the quantum analogues are still inthe early stages .Thermalization in isolated, strongly interacting, quan-tum systems is defined relative to a certain set ofobservables . In particular, remarkable experiments havebeen performed to monitor momentum distributions ofcold atoms loaded in one-dimensional optical lattices ,where the integrability of the underlying dynamics isweakly broken. Accordingly, a fundamental question iswhether a nonergodic to ergodic transition threshold ex-ists as one may go parametrically from an integrable to anonintegrable quantum system. In the FPU problem onecan surely answer affirmatively, since the onset of dynam-ical chaos can play the role for such a transition . Forinteracting quantum systems in the presence of disorder,it is expected that a phase transition exists between de-localized states and a phase characterized by many-bodylocalization (MBL) . This would constitute the soughtthreshold between ergodic and nonergodic behavior .The MBL results from a quantum dynamical phasetransition between diffusive (ergodic) and localized (non-ergodic) dynamics . It occurs at nonzero (and even-tually infinite) temperature, and is manifested in dy-namical properties. The crucial idea is a competitionbetween interactions and Anderson disorder . In themany-body ergodic phase, the expectation values of ob-servables computed on a finite subsystem, using a sin- gle energy eigenstate of the whole interacting system,would coincide with those evaluated in the correspond-ing microcanonical thermal ensemble . In this condi-tion one may say that the system acts as its own heatbath. Quite on the contrary, in the nonergodic many-body phase, dynamics resembles a “glassy” behavior thatprecludes self-thermalization. In view of the difficulties ofaddressing a full many-body dynamics of specific correla-tion functions , much of the progress in assessing thetransition between the mentioned regimes relied mainlyon the evaluation of spectral properties .The dynamics of specific observables has proved usefulto monitor the onset of many-body chaos . In particu-lar, the Survival Probability (SP) of the eigenstates of anunperturbed Hamiltonian ˆ H under the evolution of thefull interacting Hamiltonian ˆ H + ˆΣ decays with a char-acteristic time scale τ . In such a case, a crossover froman exponential controlled by 1 /τ ∝ || ˆΣ || to a Gaussiancharacterized by 1 /τ ∝ || ˆΣ || is interpreted as evidenceof the onset of a chaotic structure in the eigenstates ofˆ H + ˆΣ . The first regime is described by the Fermigolden rule (FGR), and hence the breakdown of the FGRleads to dynamical chaos. Much in the spirit of this dy-namical approach, in this article we propose and ana-lyze the evolution of an experimentally accessible localobservable, as a way to assess the interaction-disordercompetition in a many-spin system. We resort to the Loschmidt echo (LE) , a measure of the revival thatoccurs when a time-reversal procedure is applied to ˆ H .The LE has been experimentally evaluated from the localpolarization of spin systems to quantify the role of per-turbations (i.e. ˆΣ) and the system’s own complexity ondynamical reversibility . Besides, in classically chaoticsystems, it is well-known that the LE of a semiclassicalexcitation undergoes a transition into a regime where itsdecay rate is given by the classical Lyapunov exponent .Here, we use the LE dynamics as a time-dependent au-tocorrelation function that quantifies the time scales andergodicity. Specifically, the time reversal procedure “fil-ters out” the irrelevant dynamics produced by an inte-grable ˆ H which would hide the sought information .Thus, the LE becomes a privileged dynamical witnessof the competition between the interactions and disorderand a potential experimental candidate. The spin model .- We consider a spin chain, whose dy-namics is given by the total Hamiltonian ˆ H = ˆ H + ˆΣ:ˆ H = N X i =1 J (cid:2) S xi S xi +1 + S yi S yi +1 (cid:3) (1)ˆΣ = N X i =1 ∆ S zi S zi +1 + N X i =1 h i S zi (2)where ∆ is the magnitude of the homogeneous Isinginteraction and h i are randomly distributed fields in aninterval [ − h, h ]. Periodic boundary conditions (ring)are imposed and, unless explicitly stated, N = 12.Notice that ˆ H can be mapped into two independentnoninteracting fermion systems by the Wigner-Jordantransformation , while ˆΣ includes both the two-bodyIsing interaction and the local fields (disorder). Sinceˆ H encloses single-particle physics, we consider it as theirrelevant part of the dynamics, a term that the LE man-ages to get rid of, allowing us to focus on interactions anddisorder. This fact justifies the idea of using the LE as afilter for the relevant physical processes. The spin autocorrelation function evaluated as a (local)Loschmidt echo .- We consider a high (infinite) tempera-ture state formally denoted by | Ψ eq i , which representsan ensemble average over all basis states with the samestatistical weights. We study the spin autocorrelationfunction at the particular site 1, M , (2 t R ) = (cid:10) Ψ eq (cid:12)(cid:12) ˆ S z (2 t R ) ˆ S z (0) (cid:12)(cid:12) Ψ eq (cid:11)(cid:10) Ψ eq (cid:12)(cid:12) ˆ S z (0) ˆ S z (0) (cid:12)(cid:12) Ψ eq (cid:11) (3)where the spin operator is written in the Heisenberg pic-ture as:ˆ S z (2 t R ) = ˆ U † + ( t R ) ˆ U †− ( t R ) ˆ S z ˆ U − ( t R ) ˆ U + ( t R ) . (4)The evolution operators are ˆ U + ( t R ) = exp[ − i ~ ( ˆ H +ˆΣ) t R ] and ˆ U − ( t R ) = exp[ − i ~ ( − ˆ H + ˆΣ) t R ]. There-fore, it is explicit that the echo procedure performedover ˆ H yields a global evolution operator ˆ U (2 t R ) =ˆ U − ( t R ) ˆ U + ( t R ). Using cyclic invariance of the trace, onecan replace the average over all basis states by an aver-age over all states that have spin 1 up-polarized . Addi-tionally, we replace the ensemble average by an averageover a few entangled states , which provides a quadraticspeedup of computational efforts. It yields equivalent re-sults provided that only local observables (e.g. Eq. 3) M , ( t ) t [ /J] Saturation
Further increase of interactions M , ( t ) t [ /J] Increasing interactions Further increase of disorder d)c) b)a) M , ( t ) t [ /J] Increasing disorder M , ( t ) t [ /J] FIG. 1. Color online. Loschmidt echo for different parameterregimes. The characteristic time scale is defined by the decayup to 2 /
3, shown by the horizontal dotted line. ( a ) h = 0,0 < ∆ . . J . Smooth decay, until saturation is reached(horizontal dashed line). ( b ) h = 0, 1 . J < ∆ < . J . LElong tails destroys saturation. The SP given by Eq. 7 de-termines the limit time scale (shaded region). ( c ) ∆ = 0,0 < h . . J . The localization length remains bigger thanthe system size. ( d ) ∆ = 0, 1 . J < h < . J . The local-ization length turns to be smaller than the system’s size, andthe polarization keeps around site 1. are evaluated. Thus, we consider: | Ψ neq i = |↑ i ⊗ N − X r =1 √ N − e i ϕ r | β r i , (5)where ϕ r is a random phase and {| β r i} are state vectorsin the computational Ising basis of the N − M , is now written in the Schr¨odinger pictureas: M , (2 t ) = 2 h Ψ neq | ˆ U † + ( t ) ˆ U †− ( t ) ˆ S z ˆ U − ( t ) ˆ U + ( t ) | Ψ neq i (6)The explicit time dependence of M , (2 t ) is evaluatedby means of a fourth order Trotter-Suzuki decompositionwithout any Hilbert space truncation, implemented ongeneral purpose graphical processing units . Results .- We start the evaluation of Eq. 6 with h = 0(no disorder), increasing the interaction strength ∆ fromzero, see Fig. 1 ( a ) and ( b ). The LE short-time scalingis 1 − M , ( t ) ∝ t , which can be analytically verifiedby expanding the evolution operators up to fourth order.Beyond the short-time window, it has a smooth decayproduced by the nonreversed terms ˆΣ . As we do nothave an explicit functional form for this regime, we definean effective characteristic time τ for the plotted curvesas the decay up to 2 /
3. The rates 1 /τ are plotted in Fig.2 as function of ∆, for different disorder magnitudes h .Beyond the decay regime, as shown in Fig. 1-( a ),the LE saturates at 1 /N , which means that the initialpolarization (local excitation defined by Eq. 5) is uni-formly spread over the whole spin set. This is indeedthe standard picture of decoherence process leading toan irreversible spread. But, if ∆ is further increased, theLE-decay slows down showing long tails, see Fig. 1-( b ).These destroy the saturation at least in the time-windowanalyzed in the present work. Such regime may be as-sociated with a glassy polarization dynamics, i.e. theprevalence of the freezing effect of the Ising terms overthe spreading induced by the XY ones.If we consider ∆ = 0 and let the disorder h increase,the picture is indeed the standard Anderson localizationproblem. In such a case, the localization length mustbe compared to the finite size of the system. Hence, forvery weak disorder, the LE degrades smoothly, as the lo-calization length is longer than the system’s size. Whenthe disorder is strong enough, the localization length issmaller than the system’s size and thus the initial lo-cal excitation remains around the site 1. In fact, thecrossover between these two physical situations can bequantified equating the system’s size with the localiza-tion length λ ≃ J /h , given in a FGR estimation .This yields h = √ J , in fairly good agreement with thebehavior shown in Figs. 1 ( c ) and ( d ).Notice that in any case, neither Ising interactions norAnderson disorder, can produce a LE-decay faster than awell defined time scale (see Figs. 1 ( b ) and ( d )). This isspecifically determined by the SP of the local excitationunder the evolution given by ˆ H , P , (2 t ) = 2 h Ψ neq | exp (cid:16) i ˆ H t/ ~ (cid:17) ˆ S z exp (cid:16) − i ˆ H t/ ~ (cid:17) | Ψ neq i , (7)where we emphasize that there is no dependence on ˆΣ.Such an ˆ H -controlled decay resembles the perturba-tion independent decay experimentally observed in spinsystems and the Lyapunov regime of classically chaoticsystems .In Fig. 2 we show the role of the time-scale determinedby Eq. 7, acting as the limit for LE decay rates. Theseresults are quite general, similar behavior is obtained fornext nearest neighbors interactions (both in ˆ H and ˆΣ),provided that ˆΣ has only Ising terms or Anderson disor-der.In order to analyze the ergodicity of the polarizationdynamics observed in our finite system, we evaluate themean LE, ¯ M , :¯ M , ( T ) = 1 T Z T M , ( t ) dt. The standard analysis of the Anderson localizationproblem should imply, in the present case, to computelim T →∞ ¯ M , ( T ). Instead, we analyze ¯ M , ( T ) at T =12 ~ /J which, as shown in Fig. 1, is long enough to al-low for a uniform spreading of the polarization, providedthat ∆ is strong. Also, a rigorous upper bound for the h=0 h=0.35 J h=1.0 J h=1.75 J / J / ] Interaction [J]
FIG. 2. Color online. LE decay rates as a function of theinteraction strength ∆, for different disorder magnitudes h .Horizontal dashed line represents the ( ˆ H ) SP time scale, i.e.Eq. 7. Data for ∆ = 0 was obtained from single-particlephysics and 500 disorder realizations, while data for ∆ > integration time T must be the system’s Heisenberg time T H , at which finite-size recurrences show up. In Fig.3 we show a level plot of ¯ M , as a function of the in-teraction ∆ and disorder strength h . This results in aqualitative phase-diagram which evidences the competi-tion between such physical magnitudes. When both ∆and h are weak, ¯ M , remains near 1, since the systemis almost reversible. Thus, the parametric region at thebottom left corner may be associated with decoherence ,i.e. the system is weakly perturbed by uncontrolled de-grees of freedom. If either ∆ or h are further increased,the system enters in a diffusive regime where the initiallocal polarization rapidly spreads irreversibly all acrossthe spin system. Consistently, this bluish region is as-sociated with an ergodic behavior for the polarization,since it is equally distributed along the spin system. Forlow disorder ( h . . J ), increasing ∆ leads to an Isingpredominance, which freezes the polarization dynamics.We interpret such behavior as a glassy dynamics, withlong relaxation times. This localization keeps ¯ M , high.Analogously, increasing h for a fixed value of ∆ evidencesa smooth crossover to a localized phase, where the po-larization does not diffuse considerably.Notice that localization by disorder is weakened when1 . J . ∆ . . J , since the ergodic region seems tounfold for larger h . In fact, such values of interactionstrength correspond to a faster arrival to the 1 /N satu-ration, as shown in Figs. 1 ( a ) and ( b ) (when h = 0).Additionally, data around the ∆ axis show that the disor-der tends to abruptly destroy the quenching produced byIsing interactions. This seems suggestive of a parameterregion where the interaction-disorder competition leadsto a sharp transition between glassy and ergodic phases. b I n t e r ac ti on [ J ] Disorder h [J] a FIG. 3. Color online. Phase diagram for ¯ M , ( T ) at T =12 ~ /J . Data point ( a ) is given for the MBL transition inRef. , ∆ = 1 . J , h c = (2 . ± . J . Data point ( b ) is givenfor the MBL transition in Ref. , ∆ = 1 . J , h c = (3 . ± . J .These points are slightly shifted in the plot from ∆ = 1 . J inorder to avoid their overlap. However, a reliable finite size scaling of this regime wouldrequire excessively long times to capture how a vitreousdynamics is affected by disorder.Previous numerical results of the SP and an analysisof the many-body eigenstates of the same spin modelhave identified critical values for the MBL transition.Quite remarkably, they lie precisely at the crossover be-tween the ergodic and the localized phases of the LE; seedata-points ( a ) and ( b ) in Fig. 3. In our simulations,increasing N (e.g. 10, 12 and 14) enables a larger in-tegration time T , since T . T H ∝ N . In fact, when∆ ∼ . J , it can be verified that both sides of the tran- sition are well behaved since ¯ M , ∼ /N in the ergodicregime, while ¯ M , ∼ /λ for h strong enough (regardlessof N ), and d ¯ M , /dh increases with N . However, this fi-nite size scaling of ¯ M , ( T ) within our accessible rangeis not enough to yield precise critical values for the MBLtransition.In summary, we were able to draw a qualitative phase-diagram for the polarization under the LE dynamics,identifying ergodic, localized and glassy regimes. It dis-plays a nontrivial geography with a deep penetrationof the ergodic phase into the glassy domain separatingit from the localized region. Besides, while in finite1-d systems Anderson localization is indeed a smoothcrossover, it seems to develop into a ergodic-localizedtransition for nonzero interactions. Additionally, our re-sults suggest that the glassy-ergodic transition is a bet-ter candidate for a sharp phase transition. In spite ofthe fact that the local nature of the observable consti-tutes a limitation to perform a reliable finite size scaling,our strategy seems promising to analyze different under-lying topologies and different ways to breakdown inte-grability. 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