Sensitivity of the spectral form factor to short-range level statistics
SSensitivity of the spectral form factor to short-range level statistics
Wouter Buijsman, ∗ Vadim Cheianov, and Vladimir Gritsev
1, 3 Institute for Theoretical Physics Amsterdam and Delta Institute for Theoretical Physics,University of Amsterdam, P.O. Box 94485, 1090 GL Amsterdam, The Netherlands Instituut-Lorentz and Delta Institute for Theoretical Physics,Universiteit Leiden, P.O. Box 9506, 2300 RA Leiden, The Netherlands Russian Quantum Center, Skolkovo, Moscow 143025, Russia (Dated: October 6, 2020)The spectral form factor is a dynamical probe for level statistics of quantum systems. The early-time behaviour is commonly interpreted as a characterization of two-point correlations at largeseparation. We argue that this interpretation can be too restrictive by indicating that the self-correlation imposes a constraint on the spectral form factor integrated over time. More generally,we indicate that each expansion coefficient of the two-point correlation function imposes a constrainton the properly weighted time-integrated spectral form factor. We discuss how these constraintscan affect the interpretation of the spectral form factor as a probe for ergodicity. We propose a newprobe, which eliminates the effect of the constraint imposed by the self-correlation. The use of thisprobe is demonstrated for a model of randomly incomplete spectra and a Floquet model supportingmany-body localization.
I. INTRODUCTION
Level statistics play an unambiguously important rolein studies on quantum ergodicity [1, 2], thanks to theuniversal properties as described by random matrix the-ory [3, 4]. The applicability of random matrix theory todescribe the correlations between energy levels is quan-tified by the Thouless energy [5], which gives the rangeover which the random matrix theory description for fullyergodic systems holds. For diffusive mesoscopic systems,this range is intimately related to numerous quantities,such as the conductance and the time required for elec-trons to diffuse over the full sample [6, 7].The dependence of the Thouless energy on the param-eters of the system allows one to study the onset of er-godicity. In the spirit of studies on disordered mesoscopicsystems conducted in the 90’s [8–10], recent years show arevival of interest for this quantity from several directions[11–19]. The correlations within the spectra are typi-cally studied through the spectral form factor [20, 21], atime-dependent probe for level statistics on ranges longcompared to the mean level spacing.In this work, we show that the self-correlation imposesa constraint on the spectral form factor integrated overtime. More generally, it is shown that each expansion co-efficient of the two-point correlation function imposes aconstraint on the properly weighted time-integrated spec-tral form factor. As the lower order expansion coefficientscharacterize correlations at small separation, the con-straints are effectively determined by short-range levelstatistics. We argue that these constraints can affect theinterpretation of the spectral form factor as a probe forlong-range level statistics, as well as the usability of thespectral form factor as a tool to study the scaling of theThouless energy. ∗ [email protected] We propose a new probe for ergodicity which elim-inates the constraint imposed by the self-correlation,thereby providing a more transparent quantification ofergodicity than the spectral form factor. For this probe,quantifying ergodicity does not involve a comparisonwith the evaluation for fully ergodic systems, giving it theadditional benefit that it is applicable even for systemsthat obey intermediate level statistics. We demonstratethe use of this probe for an ensemble of random incom-plete spectra [22, 23] and a Floquet model supportingmany-body localization [24].The outline is as follows. Sec. II discusses the spectralform factor and the conventional procedure that is usedto study the evolution of the Thouless energy. Sec. III de-rives the constraints imposed by short-range level statis-tics. Sec. IV illustrates the potential consequences ofthese constraints with physically relevant examples fromrandom matrix theory. Sec. V introduces the new probe.Sec. VI demonstrates the use of the probe for two modelsof intermediate level statistics. A discussion and outlookis provided in Sec. VII.
II. SPECTRAL FORM FACTOR
We consider an ensemble of spectra { λ n } Nn =1 with N (cid:29) (cid:104)·(cid:105) denoting an ensemble average, the den-sity ρ (1) ( x ) and two-point correlation function ρ (2) ( λ, λ (cid:48) )are given by ρ (1) ( λ ) = (cid:28) (cid:88) n δ ( λ − λ n ) (cid:29) , (1) ρ (2) ( λ, λ (cid:48) ) = (cid:28) (cid:88) m (cid:54) = n δ ( λ − λ n ) δ ( λ (cid:48) − λ m ) (cid:29) . (2) a r X i v : . [ c ond - m a t . d i s - nn ] O c t Because of the unfolding, one finds ρ (1) ( λ ) = 1 over thefull range of support. The two-point correlations areassumed to be translationally invariant, meaning that ρ (2) ( λ, λ (cid:48) ) = ρ (2) (0 , λ (cid:48) − λ ). The spectral form factor K ( t ) [3] is defined as the Fourier transform of the clus-ter function ρ (2) (0 , λ ) − ρ (1) (0) ρ (1) ( λ ) accompanied by anoffset, K ( t ) = 1 + (cid:90) (cid:18) ρ (2) (0 , λ ) − (cid:19) e iλt dλ. (3)Because of the finite range of support for λ , the timeis a discrete variable taking values 2 πn/N for n ∈ Z .The translational invariance of the correlations allows oneto replace exp( iλt ) by the real-valued cos( λt ). Utilizingthe translational invariance of the correlations again, thespectral form factor can be evaluated at relatively lowcomputational costs as K ( t ) = (cid:42) N (cid:18) (cid:88) n,m e i ( λ n − λ m ) t (cid:19)(cid:43) − N δ ( t ) (4)= (cid:42) N (cid:12)(cid:12)(cid:12)(cid:12) (cid:88) n e iλ n t (cid:12)(cid:12)(cid:12)(cid:12) (cid:43) − N δ ( t ) . (5)Ensemble averaging is required as the spectral form fac-tor is not a self-averaging quantity [25]. Sec. VI coversthe interpretation of the wavenumber t as a time.The large- λ behaviour of ρ (2) (0 , λ ) is equivalent to thesmall- t behaviour of the spectral form factor (restrictingthe focus to t ≥ (cid:90) λ − n e iλt dλ = π ( − n (2 n − | t | n − (6)with n ∈ N [26, 27]. It is suggestive to associate thebehaviour of the spectral form factor at time t with thetwo-point correlator acting over a distance proportionalto 1 /t . Then, there is a lowest time from which onwardsthe spectral form factor matches the evaluation for fullyergodic systems. In the literature, this time known as theThouless [11, 13–19], ergodic [8–10] or ramp [12] time. Itis inversely proportional to the Thouless energy, givingthe range over which the spectra obey the correlationsas for fully ergodic systems [5]. We remark that each ofRefs. [11–19] appeared in recent years. III. CONSTRAINTS IMPOSED BYSHORT-RANGE LEVEL STATISTICS
A key distinction between ergodic and non-ergodic sys-tems is the occurence of level repulsion [3]. Unfoldedspectra obey ρ (2) ( λ, λ ) = (cid:40) , . (7) Integrating the expression for the spectral form factoras given in Eq. (3) over time shows that the value ofthe self-correlation ρ (2) ( λ, λ ) imposes a constraint on thetime-integrated spectral form factor, (cid:90) ∞ (cid:18) − K ( t ) (cid:19) dt = π (cid:18) − ρ (2) (0 , (cid:19) . (8)For spectra with and without level repulsion, the integralevaluates to respectively π and zero. An important con-sequence appears when determining the Thouless time.Namely, positive (negative) differences between the spec-tral form factor and the evaluation for fully ergodic sys-tems at earlier times have to be compensated by negative(positive) differences at later times. As such, one couldexpect the estimated Thouless time to deviate signifi-cantly from the value that would have been obtained byusing probes that are not sensitive to constraints.The constraint imposed by Eq. (8) is a specific examplefrom a more general set of constraints when ρ (2) (0 , λ ) canbe expanded in powers of λ . Examples are given in Sec.IV. Consider the inverse Fourier transform ρ (2) (0 , λ ) − π (cid:90) (cid:18) K ( t ) − (cid:19) e − iλt dt. (9)On the left-hand side, we expand ρ (2) (0 , λ ) in powers of λ as, ρ (2) (0 , λ ) = c + c λ + c λ + c λ + . . . . (10)Next, we take the 2 n -th derivative with respect to λ onboth sides, and set λ = 0 to obtain (cid:90) ∞ (cid:18) − K ( t ) (cid:19) t n dt = π ( − n (2 n )! c n . (11)This equation establishes a relation between the 2 n -thderivative of ρ (2) (0 , λ ) at λ = 0 and the time-integratedspectral form factor weighted by a factor t n . By a sim-ilar argument as above, these constraints can make theestimated Thouless time to deviate from the value thatwould have been obtained by unconstrained probes. IV. ILLUSTRATIONS
Eqs. (8) and (11) indicate dependencies between thespectral form factor evaluated at earlier and later times.A physically relevant illustration is provided by the bulkstatistics of the unitary (Dyson index β = 2) randommatrix ensemble [3, 28]. In the thermodynamic limit N → ∞ , unfolded spectra obey ρ (2) (0 , λ ) = 1 − (cid:18) sin( πx ) πλ (cid:19) (12)= − ∞ (cid:88) n =0 ( − n n π n (2 n )!(2 n + 3 n + 1) λ n . (13)The level statistics of the unitary random matrix en-semble apply to fully ergodic systems with broken time-reversal symmetry. The constraints given by Eqs. (8)and (11) impose (cid:90) ∞ (cid:18) − K ( t ) (cid:19) t n dt = − n π n +1 n + 3 n + 1 (14)for the n -th term of the expansion given in Eq. (13).These constraints are consistent with the evaluation ofthe spectral form factor [3], given by K ( t ) = (cid:40) | t | π if | t | ≤ π, | t | > π. (15)Various models for intermediate level statistics, such asthe short-range plasma model [29], are described by Eq.(12) on short ranges, i.e. up to lower orders. Due tothe constraints, the corresponding spectral form factorcould match Eq. (15) down to early times, despite dis-agreement of the two-point correlation function with Eq.(12) at large separations. Again, one might consequentlyexpect the estimated Thouless time to deviate from thevalue that would have been obtained when using alter-native probes that are not sensitive to constraints.A generalization of the above example is provided bythe Calogero-Sutherland model [30]. This model is di-rectly related to the random matrix models, see e.g Ref.[4]. It describes particles on a ring, interacting through apairwise potential with a magnitude controlled by a pa-rameter β ≥
1. The positions are indicated by an angle θ n ∈ [0 , π ), where n runs over 1 , , . . . , N with N thenumber of particles. The Hamiltonian H is given by H = − N (cid:88) n =1 ∂ ∂θ n + (cid:88) n 1. Eq. (11) imposes a con-straint on the time-integrated spectral form factor when β is even (i.e. β/ ∈ N ). In the thermodynamic limit N → ∞ , these coefficients have been obtained analyti-cally in terms of generalized factorials [31]. Noting thatthe Hamiltonian of the Calogero-Sutherland model is in-tegrable, we conjecture that these coefficients are relatedto the conserved charges. For β = 4 (symplectic randommatrix ensemble), the coefficients can alternatively beobtained from the evaluation of the spectral form factor[3], given by K ( t ) = | t | π − | t | π ln (cid:12)(cid:12)(cid:12)(cid:12) − | t | π (cid:12)(cid:12)(cid:12)(cid:12) if | t | ≤ π, | t | > π. (18) V. PROPOSAL FOR A NEW PROBE The evolution of the Thouless time as a function of theparameters of the system allows one to quantify the onsetof ergodicity through the spectral form factor. Because ofthe constraints outlined in Sec. III, the interpretation canhowever be fairly non-straightforward. Eq. (8) suggestsan alternative probe, defined as ρ (2) (0 , | t ) = 1 − π (cid:90) t (cid:18) − K ( t (cid:48) ) (cid:19) dt (cid:48) . (19)This probe gives the self-correlation as captured by thespectral form factor evaluated at times less than t . Pois-sonian level statistics are characterized by K ( t ) = 1 asthe levels are uncorrelated. With increasing t , the valueof ρ (2) (0 , | t ) thus tends to zero or one for respectivelyergodic and non-ergodic systems. It approximates theself-correlation in a controlled way, thereby serving as adiagnostic when evaluated at a fixed, late time.The diagnostic proposed here is advantageous com-pared to the Thouless time in at least two respects. First,the constraint on the time-integrated spectral form factorimposed by the self-correlation through Eq. (8) is elimi-nated, making it arguably more transparent. Second, thequantification of ergodicity is not based on a comparisonwith fully ergodic systems, thereby allowing one to studysystems exhibiting intermediate level statistics.Semiclassically, the Heisenberg time t H = 2 π is thelargest physically relevant time (see e.g. Refs. [32, 33]).In this setting, it can be natural to quantify the onsetof ergodicity by ρ (2) (0 , | t H ). Using the evaluations ofthe spectral form factors for the bulk statistics of theorthogonal ( β = 1), unitary ( β = 2) and symplectic ( β =4) random matrix ensembles as given in respectively Eqs.(22), (15), and (18), one finds ρ (2) (0 , | t H ) = − ln(3) if β = 1 , β = 2 , β = 4 , (20)with 1 − ln(3) ≈ . t H = 4 π for thesymplectic ensemble as the spectra contain 2 N elements[3]. A method to numerically evaluate the integral in Eq.(19) up to arbitrarily large times at low computationalcosts is mentioned in Sec. VI. VI. EXAMPLESA. Randomly incomplete spectra Randomly incomplete spectra provide an interpolationbetween Poissonian and Wigner-Dyson level statistics,for which the spectral form factor can be obtained an-alytically [22]. Following Ref. [23], we consider the bulkstatistics of the orthogonal ( β = 1) random matrix en-semble with a randomly selected fraction 1 − f of thelevels omitted. This ensemble was introduced originallyto study the effect of missing levels in experimental con-texts. After rescaling the levels to unit mean level spacing(unfolding), the spectral form factor K ( t ) is given by K ( t ) = 1 − f + f K (cid:48) ( f t ) , (21)where K (cid:48) ( t ) denotes the spectral form factor for the bulkstatistics of the orthogonal random matrix ensemble [3], K (cid:48) ( t ) = | t | π − | t | π ln (cid:18) | t | π (cid:19) if | t | ≤ π, − | t | π ln (cid:18) | t | /π +1 | t | /π − (cid:19) if | t | > π. (22)The ensemble interpolates between Poissonian ( f = 0)and Wigner-Dyson ( f = 1) level statistics. Level repul-sion can be observed for f > 0. For the intermediatevalue f = 1 / 2, the statistics are close to those of thesemi-Poisson model [34]. Evaluating ρ (2) (0 , | t ) at theHeisenberg time t = 2 π yields ρ (2) (0 , | t H ) = 1 + 52 ( f − f ) + (cid:18) − f (cid:19) ln(1 + 2 f ) . (23)Consistent with Eq. (20), this evaluates to respectivelyunity and 1 − ln(3) for f = 0 and f = 1.Eq. (21) gives the spectral form factor for a weightedsum of the complete spectra (factor f , combined witha scaling of the density) and uncorrelated levels (factor1 − f ). Superimposed spectra appear more frequently asmodels for intermediate level statistics [35]. Potentially,alternative interpolations can be obtained from superim-posed spectra. B. Many-body localization Quantum systems with time-periodic Hamiltonians areknown as Floquet systems [36]. The Floquet operator U F is given by the time-evolution operator of the Hamilto-nian H ( t ) acting over a single period T , U F = exp (cid:18) − i (cid:126) (cid:90) T H ( t ) dt (cid:19) . (24)Since U F is unitary, the eigenvalues can be parametrizedas e iθ with 0 ≤ θ < π . The set { θ n } gives the quasi-energy spectrum. The set of quasi-energy levels of the n -th power of U F , which is the time-evolution operatorfor n cycles, is given by { θ i n } = { x i t } . As t only entersin the expression for the spectral form factor as θt , it hasthe interpretation of a (discrete) time.We consider the Floquet model introduced in Ref. [24].It describes a spin-1 / . . . . . . t ρ ( ) ( , | t ) Γ = 0 . 3Γ = 0 . 5Γ = 0 . FIG. 1. Numerically obtained ρ (2) (0 , | t ) as a function of t atsystem size L = 12 for Γ = 0 . . . directions. The Floquet operator is given by U F = exp( − iτ H x ) exp( − iτ H z ) , (25) H x = L (cid:88) n =1 g Γ σ xn , (26) H z = L − (cid:88) n =1 σ zn σ zn +1 + L (cid:88) n =1 (cid:16) h + g (cid:112) − Γ G n (cid:17) σ zn . (27)The σ x,zn represent Pauli matrices acting on site n . Pe-riodic boundary conditions σ x,zL +1 = σ x,z are imposed.The G n represent disorder sampled independently from aGaussian distribution with mean zero and unit variance.The free parameters are taken as g = 0 . h = 0 . τ = 0 . 8. The spectral density is uniform. The re-sults below are obtained from statistics over at least 1000disorder realizations. The model exhibits many-body lo-calization [37–41]. At large L , it is indicated to be in alocalized phase for Γ (cid:46) . ρ (2) (0 , | t ) as a functionof t . In the ergodic phase (Γ = 0 . K ( t ) = 1 for Poissonian level statistics. Fig. 2shows the evolution of ρ (2) (0 , | ≈ . ρ (2) (0 , | t ) can be evaluated at relativelylow computational costs by using Eq. (4) and involving N (cid:88) n =0 cos( nx ) = 1sin( x/ 2) sin (cid:16) x N + 1) (cid:17) cos (cid:16) x N (cid:17) . (28) . . . . . . . . . . . ρ ( ) ( , | ) L = 8 L = 10 L = 12 FIG. 2. Numerically obtained ρ (2) (0 , | L = 8 (squares), L = 10 (circles), and L = 12 (pentagons). For Poissonian and Wigner-Dyson levelstatistics, one expects respectively ρ (2) (0 , | ρ (2) (0 , | ≈ . VII. DISCUSSION AND CONCLUSIONS In summary, we revisited the interpretation of thespectral form factor as a probe for ergodicity. Wehave shown that short-range level statistics imposes con-straints on the spectral form factor integrated over time, which could affect its interpretation as a probe for long-range level statistics, as well as the usability of the spec-tral form factor as a tool to study the scaling of the Thou-less energy. We have proposed a new probe, and arguedthat it is more transparent. We demonstrated the use ofthis probe for two models of intermediate level statistics.The Thouless energy can alternatively be determinedfrom the number variance [43]. Interestingly, this yieldsresults conflicting with the analysis of the spectral formfactor for several classes of systems [12]. Possibly, thesediscrepancies can be explained using the results of thispaper. Next, our probe could be relevant in the recentlyemerging debate on the stability of many-body localiza-tion [19, 44–46], in which the spectral form factor playsa prominent role. 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