Slow heating in a quantum coupled kicked rotors system
Simone Notarnicola Alessandro Silva Rosario Fazio Angelo Russomanno










Dipartimento di Fisica ed Astronomia “Galileo Galilei” & INFN, via Marzolo 8, I-35131 Padova, Italy.











 SISSA, Via Bonomea 265, I-34136 Trieste, Italy





 Abdus Salam ICTP, Strada Costiera 11, I-34151 Trieste, Italy 
Dipartimento di Fisica, Università di Napoli “Federico II", Monte S. Angelo, I-80126 Napoli, Italy











 Abdus Salam ICTP, Strada Costiera 11, I-34151 Trieste, Italy 
 Max-Planck-Institut für Physik Komplexer Systeme, Nöthnitzer Strasse 38, D-01187, Dresden, Germany 





             



We consider a finite-size periodically driven quantum system of coupled kicked rotors which exhibits two distinct regimes in parameter space: a dynamically-localized one with kinetic-energy saturation in time and a chaotic one with unbounded energy absorption (dynamical delocalization). We provide numerical evidence that the kinetic energy grows subdiffusively in time in a parameter region close to the boundary of the chaotic dynamically-delocalized regime. We map the different regimes of the model via a spectral analysis of the Floquet operator and investigate the properties of the Floquet states in the subdiffusive regime. We observe an anomalous scaling of the average inverse participation ratio (IPR) analogous to the one observed at the critical point of the Anderson transition in a disordered system. We interpret the behavior of the IPR and the behavior of the asymptotic-time energy as a mark of the breaking of the eigenstate thermalization in the subdiffusive regime. Then we study the distribution of the kinetic-energy-operator off-diagonal matrix elements. We find that in presence of energy subdiffusion they are not Gaussian and we propose an anomalous random matrix model to describe them.




Valid PACS appear here
                             

                              
Slow heating in a quantum coupled kicked rotors system
    Angelo Russomanno 
    December 30, 2023
======================================================






§ INTRODUCTION




Chaos and energy absorption are intimately related. Efficient energy absorption occurs when the driving is resonant with some natural frequencies of the system and chaos develops around resonances <cit.>. If the driving amplitude is large enough, all the phase space is chaotic and the energy diffusively increases up to the so-called infinite-temperature value. For smaller values of the amplitude, the KAM theorem states that only a part of the phase space is chaotic <cit.>. Nevertheless, in the many-body case, this fact gives rise to a chaotic web uniformly spread in the phase space. Along this web, diffusion in phase space can occur giving rise to a slow energy increase up to the T=∞ value, possibly after a prethermal behavior <cit.>.

In the quantum case, the route leading to chaos, energy absorption and T=∞ thermalization is different. In this case a central role is played by the properties of the Floquet states, the eigenstates of the stroboscopic periodically-driven dynamics <cit.>. There is thermalization when these states are strongly entangled and locally equivalent to the thermal T=∞ density matrix. In this case the local observables asymptotically relax to the T=∞ thermal ensemble with fluctuations vanishing in the thermodynamic limit <cit.>. This is a form of the so-called eigenstate thermalization <cit.>, where thermal behavior appears because of the properties of the eigenstates of the dynamics. There is a chaotic and thermalizing behavior when the eigenstates of the dynamics behave as the eigenstates of a random matrix [This is rigorously true in the case of a driven system. In the autonomous case, the eigenstates of the Hamiltonian behave as the eigenstates of a random banded matrix <cit.>, in order to ensure energy conservation.] <cit.>.





Usually, there is correspondence between classical and quantum chaos. If the classical dynamics is chaotic and thermalizing, the quantum eigenstates obey eigenstate thermalization and the Hamiltonian (or the Floquet Hamiltonian in the driven case – see later) behaves as a random matrix in physical bases <cit.>.

Despite these expectations, there are exceptions. The best known is the quantum kicked rotor <cit.>. This non-integrable model describes a free rotor perturbed by a time-periodic  kick with strength K. In the classical case, when K is below some threshold, the system is quasi-integrable according to the KAM theory; here there is no energy absorption. On the opposite, when K is very large, the phase space is fully chaotic and the system absorbs energy which increases linearly and diffusively in time. The quantum behavior is completely different: quantum interference hinders energy absorption for all values of K and the kinetic energy linearly increases until a saturation value is reached <cit.>. This phenomenon is called dynamical localization and the reason behind it is that the wave function of the rotor is localized in the angular momentum representation. This form of localization has been found <cit.> to be equivalent to the Anderson localization of a particle hopping in a static, one dimensional disordered lattice <cit.>.

This is a remarkable result and many efforts have been devoted to see if dynamical localization survives when many interacting rotors are considered. When many interacting classical rotors are considered, chaos has a stronger effect on energy absorption. As stated above, even a partly chaotic phase space is enough to have Arnold diffusion, and this fact translates in a diffusive long-term dynamics, possibly after a transient <cit.>. In the quantum case the behavior is different and dynamical localization can persist for a finite number of rotors <cit.>. Nevertheless, it disappears in the thermodynamic limit, when the number of rotors tends to infinity <cit.> (although it can persist in the thermodynamic limit in other systems <cit.>). 

Even in absence of localization, quantum effects can alter the energy absorption dynamics. A way in which this can happen is the induction of a sub-diffusive energy increase, a regime where the  kinetic energy does not increase linearly and diffusively but as a power law with exponent smaller than 1. This regime up to now has only been found in  mean-field studies <cit.> and one might be tempted to think that it can appear only in the peculiar conditions where mean field is valid (thermodynamic limit, infinite-range interactions).

A hint that this might not be true comes from studies of localization in space. In Anderson models with mean-field interactions a breaking Anderson localization and consequent subdiffusion in space have been observed <cit.> even away from the mean-field limit. Indeed, subdiffusion in space for ergodic disordered models near the MBL transition has been theoretically predicted in <cit.> and experimentally observed in <cit.>.

  
  

 
Motivated by the emergence of subdiffusive behaviors in disordered systems, in the boundary region between localized and ergodic regimes, we want to study the case of a periodically-driven system exhibiting two separated regimes, a dynamically-localized one and an ergodic one.
We focus on the case of L=3 interacting quantum kicked rotors: indeed, this is the minimal case in which a transition from a dynamically-localized regime to a delocalized one can be observed <cit.>.
As it appears in other periodically-driven  <cit.> as well as time-independent, few-body models <cit.>, we show that in our system subdiffusion occurs not in the space  but in the momentum domain, manifesting itself in the energy absorption process. In particular, we observe a parameter region where quantum effects determine a dynamics different from the classical one even in the absence of localization: indeed, a subdiffusive power-law heating is observed in contrast with the linear, diffusive classical one.
 We show that the subdiffusion manifests in a region of the parameters space where the system is delocalized situated close to the boundary with the localized region, in analogy with the results relative to systems undergoing the MBL transition.

An example of this subdiffusion is shown in Fig. <ref> where we compare it with the corresponding classical dynamics. We see that after a transient, the quantum and the classical evolutions are completely different,  the first being  subdiffusive and the second  diffusive (as appropriate for an essentially chaotic classical dynamics). Subdiffusion is a genuinely quantum phenomenon originating from interference, with no classical counterpart.
It is therefore important to connect this subdiffusion to the quantum properties of the model, especially to its Floquet states which are the eigenstates of the stroboscopic dynamics.

We find that, in the subdiffusive regime, the kinetic-energy matrix elements in the Floquet basis show anomalous distributions, different from the pure Gaussians <cit.> of the fully chaotic and thermalizing case. This behavior has already been observed in ergodic disordered systems, near the transition to many-body localization, where two-time correlators show a subdiffusive behavior in space <cit.>. We can provide an interpretation of the anomalous distribution of the matrix elements. Indeed we see the same distributions if we consider a random-matrix model with fluctuations of the matrix elements decaying as a power law with the distance from the diagonal. 

The paper is organized as follows. In Section <ref> we describe the model and the details of the numerical exact-diagonalization analysis we perform on it. In Section <ref> we show the power-law behavior in time of the kinetic energy. We map 
the different regimes we observe

in 
Fig. <ref>: We find a dynamically-localized regime and a delocalized one where numerics suggests unbounded energy absorption. Subdiffusion occurs close to the boundary of the dynamically-delocalized region in parameter space. We focus on the subdiffusive regime  and show that in this regime the eigenstate thermalization is broken. We can see this fact by studying the properties of the Floquet states: They show 

 long tails in the Inverse Participation Ratio (IPR) distributions and large IPR fluctuations. The breaking of eigenstate thermalization is reflected also in the asymptotic value of the energy with a truncated Hilbert space.
We also study the
off-diagonal matrix elements of the kinetic-energy operator in the Floquet basis: We find that the distributions of these matrix elements have long tails and are different from a Gaussian, marking the fact that the dynamics is not perfectly chaotic (similar distributions appear in cases of anomalous thermalization in many-body systems <cit.>).  In Section <ref> we interpret the distributions of the off-diagonal matrix elements of the kinetic-energy operator: We show that they can be derived from a model based on a random matrix with fluctuations of the matrix elements decaying as a power law with the distance from the diagonal.  In Section <ref> we draw our conclusions.







§ THE MODEL
 

The model we consider is described by the following Hamiltonian:

    Ĥ(t) = Ĥ_0+∑_n=-∞^+∞δ(t-n T ) V̂(θ) 
    
    
    
       Ĥ_0 = ∑_j=1^Lp̂_j^2/2 I_ m. i. ,  
    
    
    
    V̂(θ) = ∑_j=1^L[K cosθ̂_j+Jcos(θ̂_j-θ̂_j+1)] ,

with a time-independent kinetic term and an angle-dependent periodic perturbation; in particular, K is the strength of the kick acting on each single rotor and J is the amplitude of a nearest-neighbor coupling term.
With a proper redefinition <cit.> of the operators and the constants K and J in the Hamiltonian in Eq. (<ref>) we set the time period T and the inertia momentum I_ m. i. to 1.
We choose periodic boundary conditions such that L+1≡ 1;


we impose the canonical commutation relations to the angle and angular momentum operators, namely [p̂_j, p̂_k]=[θ̂_j, θ̂_k]=0 and [θ̂_j, p̂_k]=i δ_j k, where =ħ  T/I_ m. i.. 
 The Hilbert space in which the system evolves is 
ℋ=⊗_j=1^L ℋ_j where ℋ_j is the Hilbert space of a single rotor. The basis vectors in the angular momentum representation are indicated with  | p⟩=|p_1, ... p_L⟩, where p̂_j is the angular momentum of the j-th rotor. The spectrum of the operators p̂_j is unbounded; moreover, due to the 2 π-periodicity of the wave function with respect to each of the θ̂_j operators, it is discrete with eigenvalues p_j= m_j with m_j∈ℤ.


In general, the stroboscopic dynamics of periodically-driven quantum systems is studied by introducing the Floquet operator Û_F, defined as the time propagator over one period. In our case, we have

    Û_F =
    ^-iĤ_0/(2)^- i V(θ̂)/^- iĤ_0/(2) ,

where we set the initial time in the middle of the free evolution in order to make the time inversion symmetry explicit. We can numerically diagonalize this unitary operator in the form


    Û_F|ψ_β⟩=^-iμ_β|ψ_β⟩ ,


where the eigenstates |ψ_β⟩ are the so-called Floquet states and the eigenvalue phases μ_β the corresponding quasienergies <cit.>. 

Our analysis is based on the full exact diagonalization of the Floquet operator, through which we infer its spectral properties and we compute the exact dynamics of the system. 
Since the local Hilbert space ℋ_j relative to each site has infinite dimension, a local truncation is necessary in order to write the Floquet operator matrix: we fix a maximum value for the angular momentum on each site m_ max∈ℕ and consider only angular momentum eigenstates with eigenvalue |p_j|≤ m_ max so that the dimension of the whole Hilbert space is M^L, with M=2 m_ max+1. We restrict our analysis to the subspace invariant under the following symmetry transformations: spatial translation (j'=j+1), spatial inversion (j'=L-j) and global momentum parity (p'_j=-p_j ∀ j), so that the dimension of the Hilbert subspace we deal with is D=M^L/(4L).




§ SUBDIFFUSION AND BREAKING OF EIGENSTATE THERMALIZATION
 


In this section we study the subdiffusion behavior of the energy (Sec. <ref>) and relate it with the properties of the Floquet states and of the off-diagonal matrix elements of the kinetic energy operator in the Floquet basis. We discuss the last point in Sec. <ref>. We see that the distributions are not Gaussian, exactly as occurs in cases of anomalous thermalization in many-body systems <cit.>. In Sec. <ref> we study the properties of the Floquet levels and we see that subdiffusion is associated to an average level spacing ratio very near to the ergodic value. Nevertheless, the Floquet states do not obey the eigenstate thermalization, as we see in Sec. <ref> by studying the properties of the IPR distribution of the Floquet states. Moreover, for the local Hilbert space truncations M we can reach, the system does not thermalize to T=∞ when there is subdiffusion. This suggests that in our case subdiffusion appears in association with breaking of eigenstate thermalization. This is different from the known many-body cases, where subdiffusion is associated to thermalization in the thermodynamic limit <cit.>. Honestly, due to numerical limitations, we do not know if a thermalization behavior is attained in the limit M→∞ and if this would imply a breaking of subdiffusion in the long time.




 §.§ Energy evolution
 
We start studying the kinetic-energy dynamics, showing examples of power-law increase in time of this quantity.
The stroboscopic dynamics of  a given initial state |ψ_0⟩ is given by
|ψ(n)⟩ = (Û_F)^n|ψ_0⟩. The observable on which we focus is the kinetic energy of the system per rotor defined as E(n) =⟨ψ(n) | Ĥ_0 | ψ(n)  ⟩/L, that can be re-written as


    E(n)=∑_β,γℋ^ 0_β γ e^-in(μ_β-μ_γ) ψ_βψ_γ ,


where ℋ^ 0_β γ=⟨ψ_β|Ĥ_0|ψ_β⟩/L and ψ_β=⟨ψ_β|ψ(0)⟩. 


We analyze the dynamics of the system focusing on the kinetic energy operator defined in Eq. (<ref>). We choose as initial state the momentum eigenstate |0⟩ in which all the rotors have zero angular momentum. Let us consider Fig. <ref> in which we plot the growth of the kinetic energy for a choice of the values of  and  and different values of M. In this figure we can distinguish two regimes. The first occurs at intermediate times: the dynamics of the system is independent on the truncation, while its duration increases with M. The second occurs at long times dynamics: the energy tends to an asymptotic value which depends on the truncation M. In the following we separately analyze these two regimes.   We postpone the analysis of the asymptotic regime to Sec. <ref> and here we focus on the intermediate-time dynamics.

In order to analyze this regime we measure the power-law exponent of the kinetic energy growth for different values of  and . We observe a region in the parameter space in which the heating process in the classical system is very different from the quantum one, as in the first case heating is linear, while in the second we find E(n)∼ n^α with α<1 (see Fig. <ref>).  
The power-law heating is not due to the local truncation, as we checked by computing the dynamics for several values of M, as it is shown  in Fig. <ref>. Although we cannot exclude a priori that different regimes may arise at longer times, the available time scales and the truncation values are enough to claim the existence of a genuine quantum regime, different from the classical one, as it is evident from Fig. <ref>.
We repeat the same procedure for a grid of values of  and  and  we compute the power law coefficient. In Fig. <ref> we plot α as a function of  for different values of ; the lower boundary in the interval of  values  is that of the dynamically-delocalized regime, while the upper one is determined by numerical limitations. Even though a clear dependence of α from the parameters K and J is missing we notice that α seems to increase as J is increased. 




 §.§ Distribution of the off-diagonal matrix elements


Subdiffusion corresponds to the breaking of the perfect chaoticity of the dynamics. From one side this can be already seen from the fact that a perfectly chaotic classical dynamics leads to diffusion and not subdiffusion. From the quantum perspective this can be seen by observing the properties of the Floquet states and noticing that they have properties different from perfect eigenstate thermalization. In order to do that, we start considering that the diffusion dynamics is given by the off-diagonal matrix elements of the energy operator Ĥ_0 in the Floquet basis (see Eq. (<ref>)). This suggests to inquire the behavior of the distribution of the off-diagonal elements ℋ^ 0_βγ with γ≠β, in order to interpret the power-law increase behavior of the energy (a similar analysis was performed to interpret an anomalous thermalization behavior in <cit.> possibly associated with subdiffusion behavior in space <cit.>). If we had a perfect chaotic behavior, the operators expressed in the basis of the Floquet states should behave as a perfect random matrix <cit.>, therefore the matrix elements ℋ^ 0_βγ should be distributed according to a Gaussian.

We plot the distribution of ℋ^ 0_βγ/Σ (where Σ is the variance of the distribution of the ℋ^ 0_βγ) for many subdiffusive cases in the upper panels of Fig. <ref>. We find indeed a significant deviation from a Gaussian behavior, as it should have been expected being the corresponding behavior of E(n) different from the perfectly chaotic diffusion. 



(One of the distributions of Fig. <ref>–right panel corresponds to the subdiffusion depicted in Fig. <ref>.) In order to do a comparison, we plot in the lower panel of Fig. <ref> the distribution for a case which is fully thermalizing without any subdiffusion (at least for the truncations we have access to). We see that it is an (almost) perfect Gaussian, in agreement with the expectations from Random Matrix Theory. From this comparison we  see that the behavior of the distribution of the off-diagonal elements and the time behavior of the energy are intimately connected. 






 §.§ Level-spacing ratio analysis and dynamical regimes
 


Here we consider the properties of the Floquet quasienergies. In particular, in order to understand if the dynamics is chaotic, we study the average level spacing ratio <cit.>. It is defined as


    ⟨ r ⟩ = 1/D-2∑_β=1^D-2min{λ_β, λ_β+1}/max{λ_β, λ_β+1} ,


where λ_β=μ_β+1-μ_β.

If the dynamics is chaotic, the Hamiltonian should behave as a random matrix in the angular momentum basis:


The level spacings λ_β obey the Circular Orthogonal Ensemble (COE) distribution and the average level spacing ratio is ⟨ r ⟩≃0.5269 (the Floquet operator belongs to the Circular Orthogonal Ensemble of symmetric unitary matrices <cit.>).  


On the opposite, a regular non-thermalizing behavior generically corresponds to a Poisson distribution <cit.> of the λ_β; in this case the average level spacing ratio is ⟨ r ⟩≃0.386. These considerations are important for the energy absorption. As we extensively analyzed in <cit.>, a chaotic behavior corresponds to Floquet states delocalized in the angular momentum basis and to energy absorption. On the opposite, a regular behavior corresponds to localized Floquet states and then to dynamical localization.




In Fig. <ref>  we plot ⟨ r ⟩ as a function of  for different values of . In Fig. <ref> we map the 
regimes we observe in our model into the parameter space.

We recognize 

the light-blue region on the right where r acquires the COE value and the energy increases up to a value scaling with the truncation. This is the chaotic dynamically delocalized region where the system heats up without a bound. This heating does not always correspond to full chaoticity and diffusive energy behavior: subdiffusion occurs in 
the delocalized regime, close to its boundary.

This is strictly reminiscent of the subdiffusion in space domain occurring in the delocalized phase near the transition to MBL <cit.>. Going on with our description of the 

diagram of Fig. <ref>, we find on the left a red region which we define “dynamically localized” because we have numerically verified that the energy saturates after a transient to a value independent of the truncation [see an example in Fig. <ref>(b)]. This fact marks the presence of dynamical localization. Here r takes the Poisson value and the dynamics is regular-like. In between the localized and the delocalized region there is an intermediate regime where r has a value in between Poisson and COE. Here it is not easy to characterize the energy dynamics and we postpone the analysis of this regime to a future publication. 






 §.§ Inverse participation ratio
 
 
 

In order to explore the eigenstate thermalization breaking related to subdiffusion we have to consider the localization properties of the Floquet states. This can be realized by looking at the average Inverse Participation Ratio (IPR) <cit.> in the angular momentum basis, defined as


    IPR=1/D∑_β∑_ m |⟨ψ_β| m⟩|^2q ,


where q=2 is an exponent whose meaning will be clarified later. Given a state |ψ⟩ which is uniformly delocalized  over all the states of the angular momentum basis, one finds that its IPR satisfies IPR_ψ∼ 1/D. 

Let us focus on the delocalized region of the parameter space [see Fig. <ref>]. We remind that we are in the truncated Hilbert space whose dimension is D=M^L/(4L). If we suppose that there is eigenstate thermalization, then all the Floquet states are locally equivalent to the T=∞ density matrix. They behave as random states and are fully delocalized, we can therefore infer that they should satisfy IPR∼ 1/D as well. We find, instead, IPR∼ 1/D^δ with δ<1. This fact occurs in a wide range of parameters inside the delocalized region, as we can see in Fig. <ref>. In Fig. <ref> (a) we show some example of the scaling of the IPR with M at a fixed value of  and some values of  chosen in the dynamically-delocalized regime,
 
 while in panel (b) the corresponding exponent δ is plotted versus  (dark line). From a physical point of view this fact marks the breaking of eigenstate thermalization near the boundary of the delocalized region which we have already discussed from a different point of view in Sec. <ref>. 



Breaking of eigenstate thermalization implies that the Floquet states are not all equivalent, so we expect large fluctuations from one state to the other. In order to estimate the fluctuations in the IPR, we consider the scaling of the logarithmic average of the IPR, defined as 


    IPR _ log = exp⟨ln IPR⟩ ,


where (…)=1/D∑_β(…)_β. We find a scaling of the form IPR _ log∼ 1/D^η; we plot the scaling exponent η versus  in Fig. <ref> (b) (lighter curve). We see that the exponents δ and η are similar and near 1 for high values of . Here all the Floquet states are similar (small fluctuations) and obey eigenstate thermalization (full delocalization). For smaller , on the opposite, the exponents are different and smaller than 1. Here eigenstate thermalization is broken and large fluctuations in the IPRs of the Floquet states emerge. We would like to emphasize that the  eigenstate thermalization breaking witnessed by δ ,η<1 in Fig. <ref> and the subdiffusion with α<1 occur in the same parameter range (see Fig. <ref>): Further research will be devoted to investigate the relation between these two results.



















In order to better understand the origin of the fluctuations of the Floquet IPRs, we have focused on their distributions. To this purpose we have considered a fixed value of  and computed the distributions for some values of , as shown in Fig. <ref>(a): The distributions exhibit a power-law tail for smaller values of  which tends to disappear as  is increased. This observation is in agreement with the behavior of the exponents δ and η discussed above: a long tail means large fluctuations and then δ≠η. Moreover, a long tail means coexistence of more localized and more delocalized states with states not completely localized. A last remark regards the behavior of the distributions with M, which is shown in Fig. <ref>(b). In the presence of fluctuations [δ≠η – left panel of Fig. <ref>(b)] a small fraction of localized states persists as M is increased, as it emerges from the right extreme of the power law-tail. Differently, in absence of fluctuations and with δ=1, all the states are delocalized and the whole distribution shifts to lower values of the IPR as M is increased [Fig. <ref>(b)].

 It is worth noting that similar scaling properties of the IPRs have been found
 
 at the Anderson transition point of  a three-dimensional disordered lattice <cit.> and across the MBL transition <cit.>: 
 
The eigenstates exhibit  an anomalous scaling of the probability distribution momenta <cit.> and they are said to have a multifractal structure. In our analysis,
 we focused on the averaged second momentum,

 namely the IPR. By computing the exponent δ for higher momenta (q>2),
 
 a set of fractal dimensions  can be obtained for characterizing a single state. Multifractal states  are also characterized through their probability distribution correlation functions and their level spacing distributions. The mapping existing between this rotors model and a disordered L-dimensional lattice <cit.> suggests the existence of multifractal properties also in our case. This multifractal analysis, which may help understanding the intermediate region in Fig. <ref>, is left for future work.
 











 §.§ Asymptotic behavior of the energy
 

The asymptotic behavior of the kinetic energy is governed by the diagonal matrix elements ℋ^ 0_ββ: At long times, the system reaches the infinite-time  averaged kinetic energy, defined as


    E_n→∞=lim_𝒯→∞1/𝒯∑_n=0^𝒯E(n)=∑_β |ψ_β|^2 ℋ^ 0_ββ .


The value of E_n→∞ is independent of M in the case of dynamical localization: The initial state evolves until it reaches an asymptotic state which is localized in the angular momentum space. On the other side, when the dynamics is ergodic the system is expected to reach the so called infinite-temperature state, defined as 


    ρ_T→∞=1/D∑_β |ψ_β⟩⟨ψ_β| .


This state is the equivalent, in the infinite-temperature case, of the equilibrium thermal state which is reached by time-independent ergodic systems at a given temperature T. The corresponding expectation value of the energy, obtained by taking E(T=∞)=Tr[Ĥ_0 ρ_T→∞]/L, is:


    E(T=∞) = ^2/2 LD∑_ m | m|^2 .


In the  case of delocalized dynamics the ratio I(M) defined as


    I(M)=E_n→∞/E(T=∞)


should be O(1) with respect to the truncation M. Differently, in the dynamically-localized regime
 
 the averaged, infinite-time energy is constant with respect to M, while E(T=∞)∼ M^2, so we have I(M)∼ M^-2 [This relation comes by observing that the infinite temperature energy of a single rotor can be re-written as ^2/2 M∑_m=-m_max^m_max m^2.].





 In Fig. <ref> we plot the ratio I(M) at a fixed value of  for some choices of  in the dynamically-delocalized regime
 
 (a) and in the localized one (b). In the latter case we find the expected power-law behavior of I(M); in the former case we find that I(M) does not change significantly with M in the interval we can access with simulations. Notice that in case of ergodicity one would expect, in particular, I(M)=1, and this occurs for the thermalizing case ==10. In the other subdiffusive cases, we observe that I(M) increases towards 1 as  is increased in the range of accessible values of . Nevertheless, due to  numerical limitations, we cannot distinguish whether I(M) tends to 1 for larger values of M, independently on .


The diffusion and thermalization occurring for large kick parameters may suggest that in the fully ergodic regime each rotor evolves 
as if it interacts with an external bath. Hence some of the phenomenology we observe may have connections with the results in <cit.>




We remark the strong dependence of I(M) on , in opposition with the behavior of the subdiffusion exponent α (see Fig. <ref>). Nevertheless, we see that I(M) is significantly smaller than 1 in the same interval where there is subdiffusion and no eigenstate thermalization [see Figs. <ref> and <ref>(b)], marking the connection between these phenomena. 
































§ ANALYSIS OF THE RANDOM-MATRIX MODEL
 

We have seen in the previous section that the operator Ĥ_0 does not look like a random matrix in the Floquet basis, as the anomalous distributions of ℋ^ 0_βγ of Fig. <ref> testify. Here we build a model which can reproduce these distributions. The rationale is the following. In the chaotic case, the operators look like random matrices in the Floquet state basis <cit.>. On the opposite, in the localized case the Hamiltonian (Ĥ_0) looks like a banded random matrix <cit.> and one can argue that the operators showing localization appear as banded random matrices in the Floquet basis (otherwise they would be delocalized). We construct a model interpolating between these two conditions. It is a random matrix where the elements are Gaussian distributed, but the variance of this distribution depends on the position inside the matrix and gets smaller as the distance from the diagonal is increased. In particular, it depends as a power law on the distance from the diagonal, so we assume that ℋ^0_βγ for γ≠β is a Gaussian random variable with variance


    σ_β,γ=σ_|β-γ|=1/|β-γ|^b ,


with b some real non-negative number. In the limit b=0 we recover the standard random-matrix behavior, while in the limit b→∞ we move towards a banded random matrix behavior.

In order to show the validity of our model, we can use it to fit the distributions of ℋ^0_βγ/Σ (Fig. <ref>) obtained through exact diagonalization. To do the fit, we adjust the parameter b in order to numerically minimize the quantity


    d_ log(b)=∫_-∞^∞ x|log(P(x))-log(P_ model^(b)(x))| .


We perform some plots of d_ log(b) versus b. For small values of b, our numerics gives us distributions restricted to a too narrow interval, that's why we have a non-physical increase of d_ log(b) (we do not show this interval of b in Fig. <ref>). For larger values of b we find a physical minimum. 
The minimum we find is very shallow and we cannot clearly determine it, being overwhelmed by fluctuations (Fig. <ref>). Nevertheless, we find values of b for which  the agreement between the distributions resulting from ED and those from this model is good (see Fig. <ref>). Notice the very clear fluctuations at large deviations, giving rise to the wigglings of d_ log. From our results, we see that b is closer to 0 for large K. This is in agreement with the physical expectation that these cases are more chaotic and then closer to a pure random matrix condition.





































§ CONCLUSIONS
 

In conclusion we have studied the energy subdiffusion in an interacting quantum kicked rotors model. 
We have noticed that this is a purely quantum phenomenon and, through a numerical analysis, we have mapped 
the different dynamical regimes

in the parameter space.
 We have considered the subdiffusion and the asymptotic properties of the energy. 
 About the latter, the absence of full thermalization in the truncated Hilbert space is strictly related to an anomalous 
 behavior of the Floquet states, marking the breaking of eigenstate thermalization. 
 Subdiffusion is associated to anomalous non-Gaussian distributions of the off-diagonal matrix elements of the energy in the Floquet basis. 
 
 These distributions are well described by a model of anomalous random matrices. 
 
 Future directions of research include the application of the anomalous random matrix model to subdiffusion in ergodic systems near the MBL transition <cit.>. 
 It would be also worth investigating the nature of the intermediate region: by exploiting the mapping between the rotors model and the disordered Anderson one, a possible way would be  analyzing the possible multi-fractal structure of the Floquet states.



We acknowledge useful discussions with V. Kravtsov and F. Iemini.
SN  acknowledges SISSA and ICTP for supporting this work during his PhD; SN acknowledges partial support from the H2020 Project - QUANTUM FLAGSHIP - PASQUANS (2019-2022).

























  

  





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