Universal spectral correlations in interacting chaotic few-body quantum systems

The emergence of random matrix spectral correlations in interacting quantum systems is a defining feature of quantum chaos. We study such correlations in terms of the spectral form factor and its moments in interacting chaotic few- and many-body systems, modeled by suitable random-matrix ensembles. We obtain the spectral form factor exactly for large Hilbert space dimension. Extrapolating those results to finite Hilbert space dimension we find a universal transition from the non-interacting to the strongly interacting case, which can be described as a simple combination of these two limits. This transition is governed by a single scaling parameter. In the bipartite case we derive similar results also for all moments of the spectral form factor. We confirm our results by extensive numerical studies and demonstrate that they apply to more realistic systems given by a pair of quantized kicked rotors as well. Ultimately we complement our analysis by a perturbative approach covering the small coupling regime.

We study the SFF in a similar random matrix model, which is built from large independent chaotic subsystems subject to an all-to-all, and hence spatially non-local, interaction of tunable strength.For the bipartite case of just two subsystems our setting reduces to the so-called random matrix transition ensembles (RMTE) introduced in Ref. [38].We therefore refer to our setting as the extended RMTE henceforth.The bipartite RMTE models a universal transition from an uncorrelated Poissonian spectrum with exponentially distributed level spacings in the non-interacting case to a correlated spectrum whose spacings follow Wigner Dyson statistics at strong interaction [38].This universal transition has been observed subsequently also in the average eigenstate entanglement [39][40][41] and in the entanglement generation after a quench [42,43].
In the extended RMTE we describe the full transition of the SFF from a simple product structure in the noninteracting case towards the full random matrix result at strong interaction as a simple convex combination of these two extreme cases.This transition is universally governed by a single scaling parameter, which combines the dependence of the SFF on all parameters of the system, namely the interaction strength as well as the size and the number of subsystems, into a single number.The SFF signals an intricate interplay between different time (and associated energy) scales, such as the Heisenberg time of the subsystems and the full system, and most notably a non-trivial Thouless time.We confirm our prediction by extensive numerical studies in few-body systems but expect our results to hold even in the many-body case.For the minimal setting of a bipartite system, we obtain a similar description also for the moments of the spectral form factor, which characterize its distribution and indicate correlations between multiple levels.Moreover we go beyond random matrix models and demonstrate that the above results equally well apply in quantized dynamical systems, i.e., a pair of coupled kicked rotors.Ultimately, we complement our results by a perturbative treatment of the interaction.
Extended random matrix transition ensemble.-Tomodel interacting few-or many-body systems, we generalize the RMTE introduced in Ref. [38] by allowing for an arbitrary number L of subsystems.We consider Floquet systems which evolve in discrete time with unitary time evolution operator given by Each of the U i is an independent, N -dimensional Haar random unitary drawn from the circular unitary ensemble, CUE(N ), and models a chaotic subsystem.By considering the CUE we restrict ourselves to systems without anti-unitary symmetries, e.g., time-reversal invariance.The interaction is introduced by the N Ldimensional diagonal unitary matrix U c ( ), where controls the strength of the interaction with = 0 corre-sponding to the non-interacting situation, U c (0) = 1.In the canonical product basis the interaction reads Here the phases ξ i1•••i L are i.i.d.random variables with zero mean and variance σ 2 , which gives rise to an effective interaction strength σ .For numerical simulations we use phases uniformly distributed in [−π, π].Note, that imposing a spatial locality structure on U c recovers the random-phase circuit of Ref. [26].Spectral form factor.-The SFF indicates correlations between the eigenphases (or quasi-energies) φ i defined by the eigenvalue equation U |i = exp (iφ i ) |i .For chaotic Floquet systems the spectral density is a constant, but its two-point correlation function r 2 (ω) yields the probability of finding two eigenphases with a distance ω and hence encodes spectral correlations.The SFF K(t) is then given by the Fourier transform of the connected part of r 2 (ω) and depends on a time variable t conjugate to the quasi-energy difference ω.The SFF has a simple representation in terms of the time evolution operator U as Here, the brackets denote an ensemble average over the subsystems, i.e., over the L independent CUE(N ), as well as an average over the random phases ξ i1•••i L .For numerical simulation we average over at least 1000 realizations.This averaging procedure is necessary as the SFF is not self averaging [44] and fluctuates wildly for an individual realization.
It is instructive to begin with the SFF for a single CUE of dimension M , for which the SFF takes the simple form K M (t) = min{t, M }.The initial linear ramp ∼ t indicates correlations in the spectrum and substantially differs from the constant SFF of an uncorrelated Poissonian spectrum, characteristic for, e.g., integrable systems [11].Hence a linear ramp of the SFF indicates quantum chaos and ergodicity.In interacting physical models the linear ramp is usually approached after a non-universal time scale known as Thouless time t Th .It sets the energy scale ∼ 1/t Th below which the system exhibits random matrix like spectral correlations and hence indicates the onset of universal dynamics.In contrast for non-interacting systems modeled by the tensor product of independent CUE(N ) matrices, e.g., the extended RMTE at = 0, the SFF factorizes into a product L .In the extended RMTE we expect a transition from this factorized SFF to the full CUE N L SFF for increasing interaction strength.In the following we fully characterize this transition and demonstrate that it depends on a single scaling parameter only.To this end we adapt the large N expansion of the SFF for the random-phase circuit of Ref. [26] based on the Weingarten calculus for integration over unitary groups [45,46] to the extended RMTE.The average over the subsytems proceeds in the same fashion whereas the average over the phases simplifies; see Ref. [47] for a detailed derivation.Ultimately, as our first main result, we represent the SFF in the simple form of a time-dependent convex combination of the two extreme cases discussed above.It is given by (4) where χ( ) = exp (i ξ) ξ is the characteristic function of the distribution of the phases ξ i1•••i L .This result is exact in the limit N → ∞.For finite N it provides the leading contribution (in 1/N ) for times t < t SH = N , i.e., smaller than the subsystems' Heisenberg time t SH set by the mean level spacing 2π/N of the subsystems.For this times K N (t) = K N L (t) = t.It has a natural extension to larger times by including the plateaus of K N (t) = N for t > t SH and K N L (t) = N L for times t > t H = N L , i.e., larger than the full systems Heisenberg time t H .This extension is an approximation, which is in excellent agreement with numerical data as depicted in Fig 1 with possible deviations occurring around Heisenberg time and for small coupling.We emphasize, that requiring large N limits numerical studies to few-body systems, i.e., small L, while our arguments do not depend on L being small.We therefore expect our results to hold also in the many-body setting.However, before discussing the qualitative features of the SFF in the RMTE in more detail, we first point out its universal dependence on a single scaling parameter.
Universality.-Tocompare the SFF for different systems it is appropriate to measure both K(t) and time t in units of t H and to introduce the rescaled SFF κ(τ ) and the rescaled time τ via κ(τ ) = K(t)/N L and τ = t/N L . ( This results in a rescaled Heisenberg time τ H = 1 of the full system and τ SH = N −L+1 of the subsystems.Apart from the latter, the only N dependence is implicitly contained in |χ( )| 2t via t = N L τ .By applying the central limit theorem to the characteristic function the N dependence together with the dependence on the effective coupling strength σ can be converted into the dependence on a single scaling parameter Γ via Here we use the characteristic function exp −x 2 /2 of the standard normal distribution.Consequently, the SFF becomes independent from the concrete choice of the distribution of the phases ξ i1•••i L entering U c .Moreover, it depends only on Γ for times τ > τ SH .This universal dependence on a single scaling parameter constitutes our second main result.It is well confirmed in Fig. 1, where we depict the SFF for different combinations of N , L, and all leading to the same Γ and coinciding SFF for τ > τ SH .In the non-interacting case Γ = 0 and hence exp −Γ 2 τ = 1 the SFF initially grows as κ(τ ) = τ L up to τ = τ SH and subsequently is constant, κ(τ ) = 1 (not shown).For small Γ we still observe an initial growth of the SFF as κ(τ ) ∼ τ L , but after times larger than τ SH the SFF drops down to the linear ramp κ(τ ) ∼ τ because all other terms are exponentially suppressed as exp −Γ 2 τ .This indicates the Thouless time τ Th as the smallest time for which κ(τ ) ∼ τ .For intermediate Γ one has τ SH < τ Th < 1 and we obtain [47] t Th = N L τ Th = L ln(N ) 2| ln |χ( )|| , which scales linear with the number of subsystems.This is in contrast with, e.g.logarithmic scaling [20,26,28] for local interactions or even t Th = 0 in local dual-unitary quantum circuits [21,22].For large Γ the linear ramp is approached earlier than τ SH , as shown for Γ = 27.21 for N = 80 and L = 2. Ultimately for very large Γ all terms involving the characteristic function are almost immediately suppressed and the SFF reduces to the CUE N L result (not shown).
Higher moments.-As the SFF is defined via an average over the RMTE one might study its distribution via its moments of order m defined by For the CUE(M ) the SFF follows an exponential distribution, i.e., K M,m (t) = m!K M (t) m [48].To compute the moments in the extended RMTE for t < t SH we follow Ref. [29] to perform the average over the independent CUE(N ).The remaining average over the phases for initial times t < t SH .Here the combinatorical factors A k (t) are polynomials of degree m(L − 1) in t which can be obtained exactly only for the bipartite case L = 2.
Computing the latter for L = 2 and fixed m allows for expressing the SFF as a time dependent convex combination between the full random matrix result K N 2 ,m (t) and the non-interacting result [K N,m (t)] 2 as well as additional terms involving products of lower moments.For instance for the second moment, m = 2, we find [47] and similar for m > 2. By explicitly including the plateaus for the moments of the CUE spectral form factors the above results again extends also to times t > N .Moreover, it reproduces the correct result for the noninteracting case = 0 for all m and for the interacting case implies K(t) ∼ K N 2 ,m (t), i.e., an exponential distribution, for t > t Th as all the terms involving |χ( )| 2t have decayed.Given this exponential distribution we define the rescaled moments via Repeating the argument invoking the central limit theorem, we again find that the rescaled moments of the SFF depend only on Γ for times τ > τ SH .Both Eq. ( 10) and its variants for m > 2, see [47], as well as the universal dependence on Γ for fixed L = 2 is confirmed in Fig. 2 for the second and third moment.Due to the rescaling (11) higher moments exhibit the same phenomenology as the SFF κ(τ ).They depend on both L and Γ even for times τ SH < τ τ Th while coinciding with the CUE(N L ) result afterwards.
Coupled kicked rotors.-Todemonstrate, that the RMTE describes actual physical systems, we apply our results to a quantized dynamical system given by two coupled kicked rotors [49].While individual kicked rotors [50] are a paradigmatic model for both classical and single particle quantum chaos, coupling two rotors provides an example for the corresponding two-body setting [38][39][40][51][52][53][54].We consider coupled kicked rotors with periodic boundary conditions, whose classical phase space is the four torus with canonical conjugate coordinates (q 1 , q 2 , p 1 , p 2 ).After quantization the effective Planck's constant h is constraint to integer values 1/h = N .The time evolution operator is a N 2 -dimensional unitary of the form (1) with [55][56][57][58][59] Here k 1 = 9.7 and k 2 = 10.5 governs the strength of the kicks end ensures chaotic classical dynamics.The coupling is introduced by with coupling strength γ and effective = γN/(2π).We choose boundary conditions for the quantum states which break time-reversal invariance and average over such boundary conditions in order to perform the average in the definition of the SFF and its moments.The resulting SFF and its second moment is depicted in Fig. 3 and shows qualitatively similar behavior as in the RMTE.However, initial fluctuations are more pronounced, which we attribute to short periodic orbits in the classical dynamics.In order to model the coupled kicked rotors with the bipartite RMTE we choose ξ ij = cos(η ij ) with i.i.d. and uniformly distributed η ij .This yields χ( ) = J 0 (γN/(2π)).The corresponding RMTE result is in good agreement with numerical data and again implies universal dependence on Γ for τ > τ SH ; see Fig. 3.
For scaling parameters Γ for which the Thouless time is given by Eq. ( 7) we note, that t Th does not coincide with the Ehrenfest time t E .The latter is the time it takes for an initially localized wave packet to spread over the system and hence indicates the time for which quantum follows classical dynamics.It is determined by the classical system's Lyuapunov exponents and for the coupled kicked rotors approximately reads t E ≈ ln(N )/(2 ln(k A k B /4)) [50].For chaotic subsystems t E is necessarily smaller than the subsystem's Heisenberg time t = N and is also much smaller than t Th even though both times scale logarithmic with N .Perturbative regime.-Forvery small scaling parameter extrapolating the exact result from t < N to larger times gives a less accurate description of the SFF.This is visible already for Γ = 2.27 in Fig. 1 around Heisenberg time τ ≈ τ H .A natural approach for Γ 1 is to extend the regularized Raleigh-Schrödinger perturbation theory introduced in Ref. [38] from the bipartite to the extended case of arbitrary L. Viewing U c ( ) as a perturbation to the non-interacting system the eigenphases φ i = φ i ( ) can be expanded in a perturbative series in which allows for computing K(t) = exp i ij φ i − φ j .While Eq. ( 9) still holds for τ < τ SH the perturbative approach yields [47] for τ > τ SH up to arbitrary large times.Again, this universally depends on the scaling parameter Γ only.The validity of the perturbative approach for very small Γ is depicted in Fig. 4. Summary and Outlook.-We have given a simple description of the SFF (and its moments for the bipartite case) for interacting chaotic subsystems as a convex combination of the results for the non-interacting and the strongly interacting case.We confirm this numerically for few-body systems and expect it to hold also for manybody systems at large N .Interestingly relatively small subsystem sizes, N ≈ 10, seem to be large enough for our description to apply.Our description additionally implies the universal dependence of the SFF on a single scaling parameter Γ and is insensitive to the detailed statistics of the phases ξ i1•••i L .However, using i.i.d.phases we ignore all correlations in the phases as they would be present for instance due to spatial locality of typical many-body systems.It therefore is an interesting open question, whether such a simple picture applies also for these situations.Moreover, our results for the RMTE are exact only for small times t < N whereas a derivation for larger times might be possible using field theoretical methods [60,61].For systems originating from the quantization of classically chaotic systems, e.g., the coupled kicked rotors, semiclassical periodic-orbit based techniques might shed further light on spectral correlations.The latter approaches, however, are left for future research.

FIG. 4 .
FIG.4.SFF κ(τ ) for small Γ for τ > τSH and different N, L. We smooth the SFF by an additional moving time average.The perturbative result (14) is depicted as black lines.Deviations from universality at Γ = 0.199 are of subleading order 1/N .