Dynamical Detection of Level Repulsion in the One-Particle Aubry-André Model
DDynamical Detection of Level Repulsion in the One-Particle Aubry-Andr´e Model
E. Jonathan Torres-Herrera and Lea F. Santos Instituto de F´ısica, Benem´erita Universidad Aut´onoma de Puebla, Apt. Postal J-48, Puebla, 72570, Mexico Department of Physics, Yeshiva University, New York City, NY, 10016, USA
The analysis of level statistics provides a primary method to detect signatures of chaos in thequantum domain. However, for experiments with ion traps and cold atoms, the energy levels arenot as easily accessible as the dynamics. In this work, we discuss how properties of the spectrumthat are usually associated with chaos can be directly detected from the evolution of the numberoperator in the one-dimensional, noninteracting Aubry-Andr´e model. Both the quantity and themodel are studied in experiments with cold atoms. We consider a single-particle and system sizesexperimentally reachable. By varying the disorder strength within values below the critical point ofthe model, level statistics similar to those found in random matrix theory are obtained. Dynamically,these properties of the spectrum are manifested in the form of a dip below the equilibration point ofthe number operator. This feature emerges at times that are experimentally accessible. This workis a contribution to a special issue dedicated to Shmuel Fishman.
I. INTRODUCTION
There has been a surprising revival of interest in quantum chaos, especially from a dynamical perspective, with theexponential growth of out-of-time ordered correlators (OTOC) taken as a main indication of chaotic behavior [1–9].The more traditional approach to quantum chaos, however, focuses on the properties of the spectrum and uses levelstatistics as in random matrix theory (RMT) as its main signature [10–13]. There are several examples of cases wherea correspondence between the exponential growth of the OTOC and level repulsion as in RMT has been found [14–18],but exceptions also exist [19–21]. In the present work, we propose a way to directly detect the effects of level repulsionin the evolution of a quantum system. The quantity and model that we consider, namely, the number operator andthe Aubry-Andr´e model, are accessible to experiments with cold atoms [22].The Aubry-Andr´e model has quasiperiodic disorder [23–27], so is contrary to the Anderson model where the disorderis random, in one-dimension (1D), and for a single particle, it can present both localized and delocalized regimes. Allstates in the Aubry-Andr´e model become localized only above a critical disorder strength, while in the one-particle1D infinite Anderson model, all states are localized for any disorder strength [28–31]. Despite this difference, whenthe systems are finite and have small disorder strengths, they present similar level spacing distributions; namely, theyshow distributions as in RMT, the so-called Wigner–Dyson distributions [32, 33]. This is a finite-size effect, not asignature of chaos. Wigner–Dyson distributions in these non-chaotic 1D models emerge when the localization lengthis larger than the system size. However, these models can still be used as a way to demonstrate how the propertiesof the spectrum get manifested in the dynamics of realistic quantum systems. Here, we show how the level repulsionpresent in the finite one-particle 1D Aubry-Andr´e model affects its dynamics.In studies of many-body quantum systems, it has been shown that the survival probability, that is, the probabilityto find the system in its initial state later in time, decays below its saturation value in systems that present levelrepulsion [34–43]. This dip below saturation is commonly known as correlation hole [34–41]. In many-body quantumsystems, the time for its appearance grows exponentially with system size [44], which makes its experimental obser-vation very challenging even for relatively small systems. To circumvent this issue, one could employ systems withfew-degrees of freedom [33, 45]. However, two other problems remain: the correlation hole in systems with manyparticles emerges at extremely low values of the survival probability, and this quantity is non-local in real space, whileexperiments usually deal with local quantities (exceptions include [46]).To solve these problems, we consider the one-particle 1D Aubry-Andr´e model and study the evolution of thenumber operator. This is a local quantity routinely measured in experiments with cold atoms. In the presence oflevel repulsion, a correlation hole develops at times that grow just sublinearly with the system size. In addition, forsystems that are not too large, the minimum point of the hole occurs at values that are not very small, and therefore,do not require extraordinary precision for detection. All these factors should make the experimental observation ofthe correlation hole viable in this model.Before proceeding with the presentation of our results, we note that this work is a contribution to a special issuededicated to Shmuel Fishman. As such, we find it pertinent to mention that Griniasty and Fishman studied ageneralization of the Aubry-Andr´e model in [47]. We expect our results to be valid in this broader picture also. a r X i v : . [ c ond - m a t . d i s - nn ] J a n II. FINITE ONE-PARTICLE ONE-DIMENSION AUBRY-ANDR´E MODEL
We study the one-particle 1D Aubry-Andr´e model with open boundaries described by the following Hamiltonian, H = L (cid:88) j =1 h cos[( √ − πj + φ ] c † j c j − J L − (cid:88) j =1 ( c † j c j +1 + c † j +1 c j ) . (1)Above, c † j ( c j ) is the creation (annihilation) operator on site j . The first term defines the quasiperiodic onsite energieswith disorder strength h ; φ is a phase offset chosen randomly from a uniform distribution [0 , π ]; the second term isresponsible for hopping the particle along the chain (we choose J = 1), and L is the number of sites.The basis vectors | ϕ j (cid:105) that we use to write the Hamiltonian matrix correspond to states that have the particleplaced on a single site j , such as | . . . (cid:105) . The eigenvalues of the matrix are denoted by E α and the correspondingeigenstates are | ψ α (cid:105) = (cid:80) j C ( j ) α | ϕ j (cid:105) , where C ( j ) α = (cid:104) ϕ j | ψ α (cid:105) = (cid:16) C ( j ) α (cid:17) ∗ = (cid:104) ψ α | ϕ j (cid:105) . A. Level Statistics
To study the degree of short-range correlations between the eigenvalues, we consider the level spacing distribution P ( s ), which requires unfolding the spectrum [11, 48], and the ratio ˜ r α between neighboring levels [49, 50], which doesnot require unfolding the spectrum. To detect long-range correlations, we look at the level number variance [11, 48],which also requires unfolding the spectrum.The unfolding procedure consists of locally rescaling the energies. The number of levels with energy less thanor equal to a certain value E is given by the staircase function N ( E ) = (cid:80) n Θ( E − E n ), where Θ is the unit stepfunction. N ( E ) has a smooth part N sm ( E ), which is the cumulative mean level density, and a fluctuating part N fl ( E ).By unfolding the spectrum, one maps the energies { E , E , . . . } onto { (cid:15) , (cid:15) , . . . } , where (cid:15) n = N sm ( E n ), so that themean level density of the new energy sequence becomes one. Statistics that measure long-range correlations are moresensitive to the unfolding procedure than short-range correlations [51]. In this paper, we discard 20% of the energiesfrom the edges of the spectrum, and obtain N sm ( E ) by fitting the staircase function with a polynomial of degree 7.
1. Short-Range Correlations
In the spectra of full random matrices, neighboring levels repel each other and P ( s ) follows the Wigner–Dysondistribution. The exact form of the distribution depends on the symmetries of the Hamiltonian. P WD ( s ) = a β s β exp( − b β s ) (2)has β = 1 for the Gaussian orthogonal ensemble (GOE), where the full random matrices are real and symmetric; β = 2 for the Gaussian unitary ensemble (GUE), where the full random matrices are Hermitian; and β = 4 for theGaussian symplectic ensemble (GSE), where the full random matrices are written in terms of quaternions. The valuesof the constants for a β and b β are found, for example, in [48]. The degree of correlation between the eigenvaluesincreases from GOE to GUE to GSE.In contrast with the spectra of RMT, one may find systems with uncorrelated eigenvalues, where the level spacingdistribution is Poissonian and systems with eigenvalues that are more correlated than in random matrices and nearlyequidistant, as in the ”picket-fence”-kind of spectra [52, 53] and the Shnirelman’s peak [54].The ratio ˜ r α between neighboring levels is defined as [49, 50]˜ r α = min (cid:18) r α , r α (cid:19) , where r α = s α s α − , (3)and s α = E α +1 − E α is the spacing between neighboring levels. The average value (cid:104) ˜ r (cid:105) over all eigenvalues varies asfollows: (cid:104) ˜ r (cid:105) ≈ .
39 for the Poissonian distribution, (cid:104) ˜ r (cid:105) ≈ .
54 for the GOE, (cid:104) ˜ r (cid:105) ≈ .
60 for the GUE, (cid:104) ˜ r (cid:105) ≈ .
68 forthe GSE, and (cid:104) ˜ r (cid:105) ≈ h is large. As the disorderstrength decreases towards zero, where the eigenvalues become nearly equidistant, P ( s ) passes through all formsmentioned above, from Poisson to GOE-like, from GOE-like to GUE-like, from GUE-like to GSE-like, and finallyfrom GSE-like to the picket-fence case, with all the intermediate distributions between each specific case. This isshown in Figure 1a,b.In Figure 1a, we show the values of β obtained with the expression [55], P β ( s ) = A (cid:16) πs (cid:17) β exp (cid:20) − β (cid:16) πs (cid:17) − (cid:18) Bs − β πs (cid:19)(cid:21) , (4)where A and B come from the normalization conditions (cid:90) ∞ P β ( s ) ds = (cid:90) ∞ sP β ( s ) ds = 1 . (5)The values of β are shown as a function of the ratio ξ = 1 / ( h L ). This scaling factor collapses the curves for differentsystem sizes on a single curve. In Figure 1b, we depict (cid:104) ˜ r (cid:105) as a function of ξ . While both β and (cid:104) ˜ r (cid:105) capture thecrossovers from the Poissonian distribution up to the picket-fence spectrum as ξ increases, it is evident that there isnot an exact one-to-one correspondence between the two, but a more systematic comparison of the two quantitiestogether with a careful unfolding procedure is worth doing. In this case, various different models should be taken intoaccount, including true chaotic models.It is important to emphasize that the different level spacing distributions obtained with the model are not linkedwith the symmetries of the Hamiltonian. The Hamiltonian matrix used here is real and symmetric for any value of h ≥
0. The different forms of the distributions are rather a consequence of the changes in the level of correlations asone goes from uncorrelated eigenvalues for large disorder to nearly equidistant levels for the clean chain.There are other theoretical studies where level statistics as in RMT were generated [56–58]. Those approaches aredifferent from the one taken in the present work, where we do not build the matrix elements with the purpose ofgenerating specific level statistics; instead, they emerge due to finite size effects.
2. Long-Range Correlations
The analysis of long-range correlations can be done with the level number variance; that is, the variance Σ ( (cid:96) ) ofthe unfolded levels in the interval (cid:96) . For uncorrelated eigenvalues, Σ ( (cid:96) ) grows linearly with (cid:96) . In the case of fullrandom matrices, we have for the GOE [11],Σ ( (cid:96) ) = 2 π (cid:18) ln(2 π(cid:96) ) + γ e + 1 − π (cid:19) , (6)for the GUE, Σ ( (cid:96) ) = 1 π (ln(2 π(cid:96) ) + γ e + 1) , (7)and for the GSE, Σ ( (cid:96) ) = 12 π (cid:18) ln(4 π(cid:96) ) + γ e + 1 + π (cid:19) , (8)where γ e = 0 . . . . is Euler’s constant. For equidistant levels, as in the case of the harmonic oscillator, Σ ( (cid:96) ) = 0.The plot of Σ ( (cid:96) ) in Figure 1c makes it clear that the level of rigidity of the spectrum of the finite one-particle 1DAubry-Andr´e model is not equivalent to that for full random matrices. There is agreement for very small (cid:96) , but then,for an interval of values of (cid:96) , the correlations are stronger in the Aubry-Andr´e model, until this behavior switches atlarge values of (cid:96) (compare the light and dark curves). As for the picket-fence spectrum for the clean chain (bottomlight curve), we attribute the oscillations and the latter growth with (cid:96) to imperfections in the unfolding procedureand in the calculation of the level number variance, and to the fact that the eigenvalues are not exactly equidistant. III. EVOLUTION OF THE NUMBER OPERATOR
Let us prepare the system in a state | Ψ(0) (cid:105) = | ϕ j (cid:105) , where the particle is either on the first site of the chain, j = 1,or on the middle one, j = L/
2. We then evolve it under H (1), | Ψ( t ) (cid:105) = e − iHt | Ψ(0) (cid:105) . The quantity used in theanalysis of the dynamics is the number operator, n ,L/ ( t ) = (cid:104) Ψ( t ) | c † ,L/ c ,L/ | Ψ( t ) (cid:105) . (9) -6 -4 -2 ξ < r ~> -6 -4 -2 ξ -2 -1 β l Σ (a) (b)(c) FIG. 1. Level repulsion parameter β ( a ) and average ratio of spacings between consecutive levels (cid:104) ˜ r (cid:105) as a function of ξ = 1 / ( h L ),and level number variance ( c ). ( a , b ) Four system sizes are considered, L = 100 , , , and 4000. The four curves overlap,except for the smallest one in panel ( a ). The horizontal dot-dashed lines indicate the values for the Gaussian orthogonal ensemble(GOE), Gaussian unitary ensemble (GUE), and Gaussian symplectic ensemble (GSE) from top to bottom. ( c ) Numerical resultsfor L = 4000 (light color) and analytical curves for GOE, GUE, and GSE (dark color). The numerical curves from top tobottom have values of ξ that, according to β , lead to the Poissonian distribution, GOE shape, GUE shape, GSE shape, andthe picket-fence spectrum for h = 0. In all panels: averages over 10 random realizations. The results for n ( t ) and n L/ ( t ) are shown in Figure 2 on the top [(a), (c), (e)] and bottom (b), (d), (f)] panels,respectively. In Figure 2a,b, the value of ξ leads to the GOE-like level spacing distribution; in Figure 2c,d thedistribution is GUE-like, and in Figure 2e,f is GSE-like.The main result of Figure 2 is the fact that for experimental sizes (few dozens of sites), the correlation hole emergesat times ( t < ) and values of the number operator ( n ,L/ ( t ) > − ) that are experimentally reachable. Thecorrelation hole is the dip below the saturation point of the dynamics. In all panels of Figure 2, the saturation of thedynamics is marked with a red horizontal dashed line for the smallest and the largest system sizes. The correlation holecorresponds to the values of the numerical curves that are below this dashed line. The difference between saturationand minimum of the hole is most evident for the GSE-like spectrum in Figure 2e.One can write the number operator in terms of the energy eigenstates and eigenvalues as n ,L/ ( t ) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:88) α | C (1 ,L/ α | e − iE α t (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:90) E max E min (cid:32)(cid:88) α | C (1 ,L/ α | δ ( E − E α ) (cid:33) e − iEt dE (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (10)where E min is the lower bound of the spectrum and E max is the upper bound. In the equation above, the sum in -3 -2 -1 n t -2 -1 n L / t t (b)(a) (c)(d) (f)(e) FIG. 2. Evolution of the number operator for a particle initially placed on Site 1 ( a , c , e ) and for one initially placed on Site L/ b , d , f ) respectively, for spectra with GOE- ( a , b ), GUE- ( c , d ), and GSE-like ( e , f ) level spacing distributions. On each panel,the size L of the chain increases from top to bottom, L = 20 , , , , , , , /t for ( a , c , e ) and 1 /t for ( b , d , f ). The horizontal dashed lines mark the saturation values for the smallestand largest L ’s. In all panels: averages over 10 random realizations. parenthesis, ρ ,L/ = (cid:88) α | C (1 ,L/ α | δ ( E − E α ) , (11)is the energy distribution of the initial state, often known as local density of states (LDOS) or strength function [59–62]. The number operator in Equation (10) is the square of the Fourier transform of the LDOS. We denote thevariance of the LDOS by σ ,L/ = (cid:88) j (cid:54) = j |(cid:104) ϕ j | H | ϕ j (cid:105)| . (12)The envelope of the LDOS for the Hamiltonian with GOE-, GUE-, and GSE-like level spacing distributions isanalogous to the shape obtained for the clean model [33]. It is a semicircle when j = 1 and it has a U-shapewhen j L/ = 1, as shown in Figure 3 for the GSE-like spectrum and three system sizes increasing from left to right, L = 20 , , L = 20, one sees approximately L/ L increases, the curves become smoother.The Fourier transform of the semicircle gives n ( t ) = [ J (2 σ t )] σ t , where σ = 1 , (13)and J is the Bessel function of the first kind. For the U-shaped LDOS, we get n L/ ( t ) = [ J (2 σ L/ t )] , where σ L/ = 2 . (14) r -2 -1 0 1 2 E r L / -2 -1 0 1 2 E -2 -1 0 1 2 E (b)(a) (c)(d) (f)(e) FIG. 3. Energy distribution of the initial state (LDOS) for a particle initially on Site 1 ( a , c , e ) and on Site L/ b , d , f ) forspectra with GSE-like level spacing distribution. The sizes of the chain increase from left to right: L = 20 ( a , b ); L = 80 ( c , d ), L = 400 ( e , f ). Average over 10 random realizations. The equations above imply that the initial decay of n ( t < σ ) ≈ − t is slower than for n L/ ( t < σ L/ ) ≈ − t ,which is noticeable by comparing the top and bottom panels of Figure 2 for t <
1. This is expected, since the particleon Site 1 can only hop to Site 2, while Site L/ t > σ ,L/ , the picture changes and the dynamics become faster for n ( t ) than for n L/ ( t ). The quadratic decayis succeeded by a power-law decay that envelops the oscillations of the Bessel functions. This non-algebraic decay ∝ /t γ is caused by the bounds in the spectrum [63, 64]. The exponent is γ = 3 for n ( t ) [65] and γ = 1 for n L/ ( t )[66].The power-law decay is followed by a plateau that is below the saturation value, n ,L/ = (cid:88) α | C (1 ,L/ α | , (15)of the number operator. This saturation point is marked with dashed horizontal lines in Figure 2. The plateau belowthis point corresponds to the correlation hole. It is related to the level number variance [11, 37], which explains why itgets deeper as we move from the GOE- to the GUE- and to the GSE-like spectrum (compare Figure 1c and Figure 2).The hole does not develop in integrable models where the level spacing distribution is Poissonian and the eigenvaluesare uncorrelated. But it does emerge in integrable models with a picket-fence spectrum.By checking where the curve of the power-law decay first crosses the plateau below n ,L/ , we estimate numerically,the time t hole for the minimum of the correlation hole. As shown in Figure 4, we find that t hole ∝ L / for n ( t ) and t hole ∝ L / for n L/ ( t ). The first estimate can be derived from the fact that the power-law decay is ∝ /t and theminimum value of n ( t ) at the plateau is ∝ /L . The estimate for the t hole for n L/ ( t ) comes from the power-lawdecay ∝ /t and the minimum value of n L/ ( t ) at the plateau, which is ∝ /L / . Both times should be reachable bycurrent experiments with cold atoms realized with few dozens of sites.The correlation hole holds up to the revival of the dynamics, which first happens at t rev ∼ L for n ( t ) and at t rev ∼ L/ n L/ ( t ), as seen in Figure 2. The revival is followed by another decay and a possible correlation hole,but at higher values. This behavior is better seen for the GSE-like spectrum in Figure 2f, where the correlation isdeep. The revival repeats itself at t rev ∼ L for n ( t ) and at t rev ∼ L for n L/ ( t ) with an yet larger value of thecorrelation hole. This second revival is better seen for larger L ’s. We may expect subsequent revivals to become visibleto even larger system sizes, although they should eventually become indistinguishable of the temporal fluctuations atthe saturation point. L L t h o l e (a) (b) FIG. 4. Log–log plots for the time to reach the correlation hole for a particle initially placed on Site 1 ( a ) and a particle initiallyplaced on Site L/ b ) versus the system size for spectra with GSE-like level spacing distributions. ( a ) t hole ∝ L / and ( b ) t hole ∝ L / . Average over 10 random realizations. IV. CONCLUSIONS
This work shows that the effects of level repulsion can be directly observed by studying the evolution of the numberoperator in the finite one-particle 1D Aubry-Andr´e model. Level repulsion is manifested in the form of the so-calledcorrelation hole. The number operator, the Aubry-Andr´e model, the system sizes, and timescales studied here areaccessible to experiments with cold atoms.
ACKNOWLEDGMENTS
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