Coherent Forward Scattering Peak and Multifractality
Maxime Martinez, Gabriel Lemarié, Bertrand Georgeot, Christian Miniatura, Olivier Giraud
CCoherent Forward Scattering Peak and Multifractality
M. Martinez, G. Lemarié,
1, 2, 3
B. Georgeot, C. Miniatura,
2, 3, 4, 5, 6 and Olivier Giraud Laboratoire de Physique Théorique, IRSAMC, Université de Toulouse, CNRS, UPS, France MajuLab, CNRS-UCA-SU-NUS-NTU International Joint Research Unit,Singapore Centre for Quantum Technologies, National University of Singapore, Singapore Université Côte d’Azur, CNRS, INPHYNI, Nice, France Department of Physics, National University of Singapore, Singapore School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore Université Paris-Saclay, CNRS, LPTMS, 91405 Orsay, France (Dated: November 5, 2020)It has recently been shown that interference effects in disordered systems give rise to two non-trivial struc-tures: the coherent backscattering (CBS) peak, a well-known signature of interference effects in the presenceof disorder, and the coherent forward scattering (CFS) peak, which emerges when Anderson localization setsin. We study here the CFS effect in the presence of quantum multifractality, a fundamental property of sev-eral systems, such as the Anderson model at the metal-insulator transition. We find that the CFS peak shapeand its peak height dynamics are generically controlled by the multifractal dimensions D and D , and by thespectral form factor. We check our results using a 1D Floquet system whose states have multifractal propertiescontrolled by a single parameter. Our predictions are fully confirmed by numerical simulations and analyticperturbation expansions on this model. Our results provide an original and direct way to detect and characterizemultifractality in experimental systems. PACS numbers: 05.45.Df, 05.45.Mt, 71.30.+h, 05.40.-a
Introduction.
In the field of quantum transport, the coher-ent backscattering (CBS) effect is a well-known signature ofinterference effects surviving configuration average in time-reversal symmetric disordered systems [1–4]. It is visible asa peak in reciprocal space (i.e. momentum space for spatiallydisordered systems). Recently, it was discovered that, in thepresence of Anderson localization, CBS was further accompa-nied by the emergence of a coherent forward scattering (CFS)peak, leading to a twin-peak structure breaking ergodicity inthe long-time limit [5]. The CFS peak is in fact a smoking gunin reciprocal space of strong localization. Such a signature inreciprocal space is particularly interesting for cold atoms ex-periments. Indeed, working in reciprocal space allows a bettercontrol of the initial energy distribution, which greatly simpli-fies the analysis of data [6]. It is also the relevant space tostudy thermalization and classical wave condensation in thepresence of interactions [7, 8]. In the localized regime, theCFS peak has been described in numerous theoretical works[5, 9–15] and has recently been observed experimentally withcold atoms [16].It has also been shown that the CFS peak embodies the crit-ical properties of the metal-insulator Anderson transition [13],which takes place in dimension 3 and beyond. Importantly,the critical wavefunctions possess multifractal properties,i.e. scale-invariant fluctuations characterized by a continuousfamily of fractal dimensions D q . This fundamental propertyhas been extensively studied theoretically (see [17] andreferences therein), with however very few experimentalobservations [18]. In [13], it was argued that the criticalCFS peak also embodies these multifractal properties. In thisLetter, we show how the characteristic features of this peak,in particular its height dynamics and large-time shape generi-cally relate to the multifractal properties. More precisely, we show that the multifractal dimension D , characterizing theinverse participation ratio, controls the CFS peak contrast atlarge times, and that the multifractal information dimension D controls the peak contrast dynamics in the thermodynamiclimit. These general results are checked on a particularsystem, the Ruijsenaars-Schneider model [19], a dynamicalsystem where all states have multifractal properties controlledby a single parameter. Our results are very well corroboratedby numerical simulations and analytical perturbative expan-sions on this model. They should be useful for experimentalstudy of quantum multifractality, as they provide a directaccess, in reciprocal space, to signatures of a multifractalbehavior of eigenstates. We will present here our main resultsand refer to [20] for the more technical details. The Ruijsenaars-Schneider (RS) model.
The RS model[19] is a variant of the kicked rotor [21, 22], a paradig-matic model of quantum chaos which exhibits Anderson lo-calization in momentum space. It is a 1D Floquet sys-tem whose corresponding Hamiltonian reads H = p / − π a [ x ( mod 2 π )] ∑ ∞ n = − ∞ δ ( t − n ) , featuring a periodically-kicked sawtooth potential with strength 2 π a . The notice-able difference with the kicked rotor comes from the spa-tial discontinuities of the sawtooth potential, inducing long-range hopping between momentum basis states, like in thepower-law random banded matrix (PRBM) model [17, 23–25], and in turn multifractality [26–29]. The dynamics ofsuch periodically-kicked systems is captured by introducingthe Floquet operator U over one period, whose eigenvec-tors | ϕ α (cid:105) are associated with quasi-energies ω α ∈ [ − π , π [ ,so that U t | ϕ α (cid:105) = e i ω α t | ϕ α (cid:105) . For the RS model, U can bewritten explicitly in the momentum basis | p (cid:105) (with integer p , a r X i v : . [ c ond - m a t . d i s - nn ] N ov P o s i t i o n x x π π S c a l e d t i m e τ Λ ( x , x , t ) . . . . . Figure 1. Contrast of the CFS peak, Eq. (2), as a function of thescaled time τ = π t / N for the RS model with N =
128 and a = . τ =
25. The blacksolid line gives numerical results and the blue solid line correspondsto Eq. (9) rescaled (see text). ≤ p ≤ N −
1) as (cid:104) p | U | p (cid:48) (cid:105) = U pp (cid:48) = e i φ p N − e π ia − e π i ( p (cid:48) − p + a ) / N , (1)where the dynamical phases φ p can be taken as randomly dis-tributed over [ , π [ . In the following, we will denote disorderaverage by bracketed terms (cid:104) ( · · · ) (cid:105) . This random matrix en-semble breaks time-reversal symmetry, so that the usual CBSeffect is destroyed [16, 30]. Its spectrum displays interme-diate statistics [31]. It has been intensively studied in manybranches of theoretical physics and mathematics [19, 31–39].The eigenstates of RS are multifractal in momentum space[27, 28]. This can be characterized by the anomalous scalingof the disorder-averaged moments of the wavefunctions atlarge N , (cid:104) ∑ p | ϕ α ( p ) | q (cid:105) ∼ N − D q ( q − ) , where ϕ α ( p ) ≡ (cid:104) p | ϕ α (cid:105) and 0 ≤ D q ≤ a controls the nature of the eigenstates[27], in particular when a goes from 0 to 1 the system goesfrom the regime of strong multifractality ( D q (cid:28)
1) to theregime of weak multifractality ( D q ∼ The CFS contrast.
The RS model above is only an ex-ample of a more general class of systems which can be de-scribed by an evolution operator U with multifractal eigen-states. For such systems, the CFS interference phenomenontakes place in reciprocal space. The expressions we derive be-low apply to this general situation; However, for definiteness,we use the notation corresponding to the RS model, whereeigenvectors are multifractal in momentum space and the CFSpeak appears in position space. Position space is spanned bystates | x (cid:105) with x = π n / N , 0 ≤ n ≤ N −
1. These states di-rectly relate to the momentum states by Fourier transform, (cid:104) x | p (cid:105) = exp ( ipx ) / √ N . We consider the time evolution of thesystem starting from some position state | x (cid:105) . The disorder-averaged position distribution after t iterations of the map U is (cid:104)| (cid:104) x | U t | x (cid:105) | (cid:105) and after an initial transient regime, it fea-tures a peak around the initial value x = x , the CFS peak. Tosingle out this interference effect resisting disorder average,we introduce the contrast Λ ( x , x , t ) as the relative differencebetween the quantum distribution of probability (cid:104)| (cid:104) x | U t | x (cid:105) | and the long-time limit classical one 1 / N Λ ( x , x , t ) = (cid:104)| (cid:104) x | U t | x (cid:105) | (cid:105) − / N / N . (2)The time behavior of the contrast is illustrated in Fig. 1 in thecase of the RS model. A peak emerges at short times around x = x , its height oscillates (left projection in Fig. 1) and even-tually stabilizes (right projection).Expanding over eigenstates of U , the contrast (2) simplywrites Λ ( x , x , t ) = N ∑ αβ (cid:68) e i ω αβ t ϕ ∗ α ( x ) ϕ α ( x ) ϕ ∗ β ( x ) ϕ β ( x ) (cid:69) − , (3)where ϕ α ( x ) ≡ (cid:104) x | ϕ α (cid:105) and ω αβ = [ ω α − ω β ]( mod 2 π ) ∈ [ − π , π [ . Note that in Eq. (3), t can be considered a contin-uous variable: In the following we shall therefore resort to theusual Fourier transform rather than the discrete one.At long times, only the diagonal part α = β in Eq. (3) sur-vives, giving, for fixed system size N , the stationary limit Λ ∞ ( x , x ) = N ∑ α (cid:10) | ϕ α ( x ) | | ϕ α ( x ) | (cid:11) − . (4)The time dependence of Λ ( x , x , t ) is fully encapsulated inthe off-diagonal terms α (cid:54) = β . The function F ( x , x , t ) = Λ ( x , x , t ) − Λ ∞ ( x , x ) that governs the time dynamics of thecontrast is given by the inverse Fourier transform ofˆ F ( x , x , ω ) = π N ∑ α (cid:54) = β (cid:10) δ ( ω − ω αβ ) ϕ ∗ α ( x ) ϕ α ( x ) ϕ ∗ β ( x ) ϕ β ( x ) (cid:11) . (5)In what follows, we will first analyze the stationary (i.e. t → ∞ ) contrast Λ ∞ ( x , x ) for finite N , discuss its peak valueat x = x and its shape around x . Then, we will analyze thetime dynamics of the peak at x = x , given by F ( x , x , t ) , inthe thermodynamic limit N → ∞ . As we shall see, the limitsof large times t and large system sizes N do not commute andlead to different regimes with distinct properties. These sta-tionary distribution and time dynamics are illustrated in Fig. 1for the RS model. For both regimes, the connection with mul-tifractal properties will be achieved by expressing them as afunction of the 4-point correlator C αβ ( p , p (cid:48) , p , p (cid:48) ) = (cid:10) ϕ α ( p ) ϕ ∗ α ( p (cid:48) ) ϕ β ( p ) ϕ ∗ β ( p (cid:48) ) (cid:11) (6)between two eigenstates in the momentum representation,using the spatial Fourier transform. Following the rationalebehind random matrix theory (RMT), we will make theassumption that the phases and norms of each wavefunctionsin the correlator C αβ are independent random variables, sothat only terms where phase factors cancel do survive thedisorder average. Stationary contrast and D . The stationary contrast Λ ∞ ( x , x ) defined in Eq. (4) can be expanded in the mo-mentum basis as N ∑ α ∑ p p (cid:48) p p (cid:48) C αα e i [( p − p (cid:48) ) x +( p − p (cid:48) ) x ] − p = p (cid:48) , p = p (cid:48) and p = p (cid:48) , p = p (cid:48) . Taking care of double counting ( p = p (cid:48) = p = p (cid:48) ) and making use of normalization ( ∑ p | ϕ α ( p ) | = Λ ∞ ( x , x ) = N ∑ α ∑ p (cid:54) = p (cid:10) | ϕ α ( p ) | | ϕ α ( p ) | (cid:11) e i ( p − p )( x − x ) . (7)The contrast at the tip of the peak , Λ ∞ ≡ Λ ∞ ( x , x ) , can beevaluated by rewriting Eq. (7) for x = x as a sum over p , p and subtracting its diagonal part. Using the multifractal scal-ing of the inverse participation ratio 1 / ∑ p (cid:104)| ϕ α ( p ) | (cid:105) ∼ N D ,this gives Λ ∞ = − γ N − D , (8)where γ is some numerical factor. For the RS model, thisgeneral prediction is very well verified (see Fig. 2) with γ oforder unity and that depends only weakly on x and a [20].The contrast around the peak can be obtained from Eq. (7)by using the multifractal scaling of the correlation function, (cid:104)| ϕ α ( p ) | | ϕ α ( p ) | (cid:105) ∼ | p − p | D − / N D + [40, 41], whichyields Λ ∞ ( x , x ) ∝ N N − ∑ p = cos [ p ( x − x )] (cid:16) − pN (cid:17) (cid:16) pN (cid:17) D − . (9)Again, this prediction is fully general and only depends onthe model via the multifractal dimension D . As shown inFig. 2, Eq. (9) applied to the RS model is also in very goodagreement with numerical results and reproduce quite well thespatial profile of the contrast in the CFS region around x = x .Remarkably, the behavior Eq. (8) can even be checked ana-lytically in the RS model. Indeed, using a perturbation expan-sion at finite N in the regime of strong multifractality a (cid:28) a [20] the expressions D = a and Λ ∞ ( x , x ) = D N − ∑ p = π ( − pN ) N sin p π N sin [ p ( x + π N )] sin [ p ( x + π N )] , (10)which for x = x leads to Λ ∞ ∼ D log N . Since 1 − γ N − D ∼ D log N for a (cid:28)
1, Eq. (8) is verified analytically at first orderin a for the RS model. Note that the dip at x = − x in Fig. 2c,which is an idiosyncrasy of our model, is well-described byEq. (10) (see [20] for details).Our analysis shows that, in the stationary limit t → ∞ takenat finite system size N , the spatial profile of the CFS peakfor a system with multifractal eigenstates is controlled by themultifractal dimension D . More precisely, Eq. (8) shows that N − Λ ∞ ( a ) a = . a = . a = . . . D ipr2 . . D fi t ( b ) ( c ) P o s i t i o n x x π π D Λ ∞ ( x , x ) x − δ x x + δ Λ ∞ ( x , x ) Λ ∞ ( d ) x − δ x x + δ Figure 2. Contrast of the CFS peak at infinite time. (a) Scaling of Λ ∞ defined in Eq. (4) with system size N , for different values of a . (b) D extracted from a fit of Λ ∞ as a function of N vs D extracted fromthe scaling of the inverse participation ratio with N . (c) CFS peakcontrast for different values of D for fixed size N = x = x , black dotted line is Eq. (10). (d)Zoomed-in CFS spatial profiles around x for a = .
049 (left) and a = .
322 (right). The spatial range is given by δ = . π / the peak height value Λ ∞ = N with an exponent D . Time dynamics of the CFS peak height and D . We now aimat describing the temporal evolution of the CFS peak height Λ ( x , x , t ) = Λ ∞ + F ( x , x , t ) . Starting from Eq. (5) and as-suming eigenvector and eigenvalue decorrelation under disor-der average when x = x , we getˆ F ( x , x , ω ) = π N ˆ R ( ω ) ∑ α (cid:54) = β (cid:10) δ ( ω − ω αβ ) (cid:11) , (11)where the correlator ˆ R ( ω ) = N (cid:10) | ϕ α ( x ) | | ϕ β ( x ) | (cid:11) ω αβ = ω only involves eigenfunctions whose quasi-energies are exactlyseparated by ω and does not depend on the labels α and β be-cause of disorder averaging. This implies that we can writethe contrast as a convolution product Λ ( x , x , t ) = Λ ∞ + [( K N − ) ⊗ R ]( t ) , (12)where R ( t ) is the inverse Fourier transform of ˆ R ( ω ) and K N ( t ) = (cid:104) N | tr U t | (cid:105) = + N (cid:104) ∑ α (cid:54) = β e i ω αβ t (cid:105) is the spectral formfactor.To compute R ( t ) in Eq. (12), we follow the same steps asin the previous section : We expand the correlator ˆ R ( ω ) = ∑ p p (cid:48) p p (cid:48) C αβ ( p , p (cid:48) , p , p (cid:48) ) e i [( p − p (cid:48) + p − p (cid:48) ) x ] in momentumspace, where C αβ in Eq. (6) is computed for eigenfunctionswith ω αβ = ω , and only keep terms surviving disorder av-erage. We find ˆ R ( ω ) = − ∑ p (cid:10) | ϕ α ( p ) | | ϕ β ( p ) | (cid:11) ω αβ = ω .Writing ∑ p (cid:10) | ϕ α ( p ) | | ϕ β ( p ) | (cid:11) ω αβ = ω = ∑ p (cid:10) | ϕ α ( p ) | (cid:11) ˆ C ( ω ) ,three regimes can be identified for multifractal wavefunctions[17, 40]:ˆ C ( ω ) = C ω < ω ( ω / ω ) D − ω ≤ ω ≤ ω N D − ( ω / ω ) − ω ≤ ω , (13)where ω is proportional to the mean level spacing 2 π / N , ω ∝ N ω , and C is some numerical factor. We checked nu-merically that such an ω -dependence is well-verified in theRS model with ω = π a / N and the caveat that only the lasttwo regimes are visible [20] since there are no eigenstates sep-arated by ω < ω for this model [31].Before we go any further, note that the form factor K N ( t ) that plays a key role in Eq. (12) is given for large N by K N ( t ) = δ ( τ ) + K reg ( τ ) [42] where the only N dependence is in thescaled time τ = π t / N . For the RS model, K reg ( τ ) can beobtained analytically [31] and reads K reg ( τ ) = ( − a ) ( a τ ) a ( − cos a τ ) + ( a sin a τ + ( − a ) a τ ) . (14)The sinusoidal terms in the denominator of Eq. (14) comefrom the existence of a nonzero minimal level spacing in theRS model and are actually responsible for the oscillations ofthe contrast.We now discuss the general behavior of Λ ( x , x , t ) , first forfinite (but large) N , then in the thermodynamic limit N , t → ∞ at fixed τ , and finally for N → ∞ at fixed t .Firstly, for finite (but large) N , the dynamical behavior iscontrolled at large times t (cid:29) ω − (or equivalently τ (cid:29) τ ,with τ = / a for the RS model), by the limiting valueˆ C ( ω ) = C . In this regime, we get Λ ( x , x , τ ) ≈ K reg ( τ ) + ∆ N ( τ ) , (15)where ∆ N ( τ ) = ( Λ ∞ − )( C K reg ( τ ) + − C ) scales like N − D . In Fig. 3, we show that for different N , all curves Λ ( x , x , τ ) − ∆ N ( τ ) indeed collapse onto K reg ( τ ) when plot-ted as a function of τ , as soon as τ (cid:29) τ . For shorter times τ < τ , larger frequencies start to play a non-trivial role in thedynamics, so that the approximation ˆ C ( ω ) ≈ C is no longervalid.Secondly, when N , t → ∞ at fixed τ (the thermodynamiclimit ), ˆ R ( ω ) →
1, so that Eq. (12) reduces to Λ ( x , x , τ ) ≈ K reg ( τ ) at any τ (and in particular for τ < τ ). This behavioris clearly seen in Fig. 3, where Λ ( x , x , τ ) approaches K reg ( τ ) at any fixed τ when N increases (note that ∆ N ( τ ) → N → ∞ at fixed but nonzero t , the previous dis-cussion still holds but with τ = π t / N → + , then the CFScontrast is given in this limit by the level compressibility χ = K reg ( τ → + ) (equivalently this limit τ → + may bereached with N , t → ∞ and t / N → + ). It was conjectured τ
25 50 75 100Scaled time τ . . . . . Λ ( x , x , τ ) − ∆ N ( τ ) τ χ = 1 − D N = 128 N = 512 N = 2048 N = 8192 K reg ( τ ) Figure 3. Contrast of the CFS peak at x = x as a function of thescaled time τ (see text) for a = .
25 ( τ = N and smoothed over a timewindow ∆ τ = .
3. The value of the constant C = .
46 in Eq. (13)is extracted from the numerically-computed ˆ C ( ω ) (data not shown).The black dashed line is the theoretical form factor K reg ( τ ) , Eq. (14). [43] and checked both numerically and analytically that for d -dimensional multifractal systems the level compressibility isrelated to the information dimension D via the simple iden-tity χ = − D / d . For 1D systems, this identity, together withthe above considerations, implies that the CFS contrast at any fixed nonzero time t goes to χ for N → ∞ and gives thus directaccess to D : lim N → ∞ Λ ( x , x , t ) = χ = − D . (16)This result was already put forward and numerically testedin [13] for spatially disordered systems. In the RS case, theconjecture χ = − D is verified very well numerically (andanalytically in the perturbative regime) [31]. Eq. (16) is con-firmed numerically in Fig. 3: When N increases, the contrastat small times gets closer and closer to the expected value χ . Conclusion.
In this Letter, we have shown that the CFSpeak, a distinctive signature of Anderson localization in re-ciprocal space, is also a marker of quantum multifractality, afundamental property of several systems, such as the Ander-son model at the metal-insulator transition. More specifically,we have shown that the contrast of the CFS peak is controlledby the multifractal dimension D in the stationary limit t → ∞ at finite system size N , and, assuming a well-verified conjec-ture, by D in the limit N → ∞ at finite t . These results werechecked numerically and analytically with the RS model, a 1DFloquet system analogous to power-law random banded ma-trices models, whose states have multifractal properties con-trolled by a single parameter. Our results offer an originalobservable to study multifractality and pave the way to exper-iments in reciprocal space, e.g. with cold atoms, to detect andcharacterize multifractality in quantum systems. ACKNOWLEDGMENTS.
C.M. wishes to thank S. Cristofari for stimulating dis-cussions and Laboratoire Collisions Agrégats Réactivité andLaboratoire de Physique Théorique (IRSAMC, Toulouse) fortheir kind hospitality. This work was supported by Pro-gramme Investissements d’Avenir under the program ANR-11-IDEX-0002-02, reference ANR-10-LABX-0037-NEXT,and research funding Grants No. ANR-17-CE30-0024, ANR-18-CE30-0017 and ANR-19-CE30-0013. We thank Calculen Midi-Pyrénées (CALMIP) for computational resources andassistance. [1] E. Akkermans, and G. Montambaux,
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