Absence of localization in weakly interacting Floquet circuits

We present a family of Floquet circuits that can interpolate between non-interacting qubits, free propagation, generic interacting, and dual-unitary dynamics. We identify the operator entanglement entropy of the two-qubit gate as a good quantitative measure of the interaction strength. We test the persistence of localization in the vicinity of the non-interacting point by probing spectral statistics, decay of autocorrelators


I. INTRODUCTION
Thermalization of quantum many-body systems has been an active topic of research in recent years [1][2][3].It is now known that generic interacting systems reach an equilibrium state that can be described by a statistical mechanics ensemble [4][5][6].Such systems show quantum chaotic dynamics, which encompasses several dynamical and static traits [7,8].There are two non-generic cases where the dynamics is not expected to be chaotic: (i) non-interacting systems can show free propagation or get localized in the presence of disorder, the latest being called Anderson localization [9].(ii) Interacting integrable systems whose extensive number of conserved quantities constrained their dynamics, giving rise to a special type of thermalization [10].When integrability is perturbed in an extensive manner, quantum chaos is expected to set in the long run and the system thermalizes [8,[11][12][13][14].However, in recent years, it was shown that strong disorder potentials may cause the break down of thermalization, the so-called many-body localization (MBL) [15][16][17].This is described as an ergodicitybreaking transition at finite interaction strength and disorder.In this regime, quasi-local conserved quantities [18,19] are stabilized, leading to emergent integrability out of an initially quantum chaotic system.There are lots of evidence of MBL in systems of moderate size [20,21], but recently, the fate of MBL in the thermodynamic limit has been put into question [22][23][24][25] and spark debates on the scaling of the critical disorder and possible de-localization mechanisms [26][27][28][29][30][31][32].
An interesting question arises at the intersection of the dynamical regimes exposed above: how stable are Anderson localized systems to small interactions?On the one hand, any small interactions are expected to drive the system towards thermal equilibrium, on the other hand, disorder may overcome the interactions and the system becomes MBL.In Ref. [33] the authors studied a model where the MBL persists at small enough interactions.The critical value of the localization length is the one expected from the avalanche mechanism [34,35].In Ref. [36] it is shown that small interactions are not effectively perturbing the Anderson localized orbitals when the disorder strength is large.In this work, we study the localization and delocalization transition in a maximally localized system subjected to small and medium-strength interactions.
Most of the studies that address the strong disorder and weak interaction limit focus on Hamiltonian systems so far.Here, we present a family of Floquet circuits that can interpolate between non-interacting qubits and strongly interacting systems.In this work, we carefully investigate the finite-size scaling of the putative localization-delocalization transition by analyzing various spectral and dynamical quantities [17,18,21,[37][38][39][40][41][42].Our results suggest that there exists no MBL phase for our system in the thermodynamic limit.Instead, our numerics suggest a finite-size crossover, reminiscent of integrability-breaking phenomena [43] in clean systems.
The structure of the paper is as follows: In Sec.II, we introduce our Floquet model and the quantities we use to study the onset of thermalization.In Sec.III, we present our results for various commonly studied quantities in the field, together with a finite-size scaling analysis.Finally in Sec.IV we discuss the implications and perspectives of our work.

II. MODEL AND OBSERVABLES
Before we present our results, we provide technical details about our model and different quantities to detect signatures of the MBL regime.
In the following work, we study a family of Floquet models U (c) shown in Fig. 1, defined by the three parameters c = (c 1 , c 2 , c 3 ).
The single-qubit unitaries u i and v i .are independently drawn from the Haar measure and introduce the spatial disorder of the model.The two-qubit gates G(c) are U (c) = u1 u2 u3 u4 u5 u6 u7 u8 w1 w2 w3 w4 w5 w6 w7 w8 1. Diagrammatic notation of the unitary defined in Eq. ( 3).The single-qubit gates ui and wi introduce spatial disorder to the system, while the interaction between neighboring qubit is mediated by the two-qubit gate G(c).
determined by three parameters c = (c 1 , c 2 , c 3 ) and are fixed for the entire circuit.The two-qubit unitary G(c) is defined in Sec.II A and it is shown there how it naturally appears in a classification of two-qubit gates.
After that, the family of Floquet models is introduced in Sec.II B, together with a description of special parameter choices c into the literature.Furthermore, we present a possible characterization for the "interaction strength" of two-qubit gates in terms of the gate operator entanglement in Sec.II C.This section is closed with a description of the various measures used in this work to characterize the finite-size scaling of the localizationdelocalization transition in Sec.II D and Sec.II E.

A. Classification of two-qubit gates
For any two-qubit gate U 2 , there exist general singlequbit gates u 1 , u 2 , v 1 and v 2 and a two-qubit gate of the form (with c = (c 1 , c 2 , c 3 )) such that it can be decomposed as [44] Here σ α with α = x, y, z are the Pauli operators acting on each qubit.
Restricting to the subset 0.5 ≥ c 1 ≥ c 2 ≥ c 3 ≥ 0, this allows for a classification of all two-qubit gates in terms of the vector c: Two gates which are described by the same vector c are equivalent to each other up to singlequbit gates [45,46].
The space enclosed by the independent set of c is known as the Weyl chamber [47][48][49] and indicated by the purple tetrahedron in Fig. 2.There are a few special points and lines [45](see Fig. 2): (iii) c 1 = c 2 = 0.5 and c 3 ≥ 0 denote the family of dualunitary gates (shown as an orange line in Fig. 2).
(iv) c 1 = 0.5 and c 2 = c 3 = 0 is the CNOT gate up to single qubit rotations.
Since any two-qubit gate corresponds to a specific point of the Weyl chamber, the entanglement properties of each gate Eq. ( 2) are uniquely determined by the vector c = (c 1 , c 2 , c 3 ) [45,46,50], as will be illustrated in Sec.II C.

B. Floquet circuit model
The characterization of two-qubit unitaries by means of the gate G(c) allows us to introduce the following family of Floquet circuits: A diagrammatic representation of this circuit is shown in Fig. 1: The gates u i and w i are single qubit rotations drawn from the Haar measure.They act as a spatial disorder.The gate G(c) is defined in equation Eq. (1).All bonds have the same set of parameters c = (c 1 , c 2 , c 3 ).The Floquet dynamics is given by repeatedly applying U Eq. ( 3), thus the Floquet circuit ensemble is determined by the vector c.
A few special cases of this model have been already studied in the literature: The case of space-time dual unitary circuits c 1 = c 2 = 0.5, c 3 ≥ 0 (see orange dashed line in Fig. 2) has been extensively studied [51,52].These circuits are quantum chaotic despite being exactly solvable, which makes them special in the study of thermalization [53][54][55][56][57][58][59][60].For instance, they are shown to saturate bounds on information scrambling [57,[61][62][63].A random version of this circuit -different single qubit gates at each time step -was studied in Ref. [64], the authors found that the fastest scrambler quantum circuit, i.e., highest entanglement rate production, are random circuits with c 1 = c 2 = 0.5 and c 3 ≥ 0.
The vicinity of the non-interacting point c 1 = c 2 = c 3 = 0 is less explored.A natural question is whether the system gets many-body localized [2,65] for small finite values of the coefficients (c 1 , c 2 , c 3 ).In principle, there are multiple possible choices for c that can be studied starting from the origin.We focus on three lines: Fig 2).It interpolates between single qubit rotations (c 1 = 0) and the SWAP gate (c 1 = 0.5).
In Ref. [67] the authors report MBL for small coefficients on the SWAP line.Besides the three lines mentioned above, we study a subset c = (c 1 , c 2 , c 3 ) with c 1 , c 2 ∈ (0.02, 0.28) and c 3 ∈ (0.0, 0.18) in order to test the generality of the chosen lines and possible links between long-time dynamics and two-qubit gate invariants.

C. Two-qubit gate entanglement
As mentioned in previous sections, a possible quantity to characterize the "interaction strength" in the introduced class of Floquet models is by means of their operator Schmidt decomposition [50]: where A l and B l are orthonormal operators for the corresponding single-qubit spaces.This is analogous to the Schmidt decomposition of states.The Schmidt coefficients λ l are normalized, i.e.  2) that are different only in single-qubit rotations will have the same set of Schmidt coefficients [50].Henceforth, the set (c 1 , c 2 , c 3 ) uniquely determines any function of the Schmidt coefficients.In the following, We use the second Renyi entropy [45,50] which allows us to define the operator entanglement per gate as: There are a few remarks with respect to the operator entanglement for some regions in the Weyl chamber.First, the line c 1 = c 2 = 0.5, c 3 < 0.5 has the largest possible s(c) = 2 log 2 [50].They are the building block of so-called dual-unitary circuits [62,68,69], which exhibit the fastest scrambling in the family of random circuits [63,64].Recently, it was shown that after perturbing the dual unitary point the gate operator entanglement plays a crucial role in recovering the more generic quantum chaotic behavior [70].This motivates our choice to study dynamical signatures of the localization-delocalization transition as a function of the operator entanglement of the two-qubit gate they consist of.
The operator entanglement at any point of the Weyl chamber is given by [50]: with

D. Level statistics and eigenstate entanglement entropy
The long-time dynamics of Floquet circuits can be probed by the spectral properties of the Floquet operator [8].In particular, we are interested in the eigenphases and eigenstates U |n⟩ = e iθn |n⟩.The gaps between consecutive eigenphases are defined as δ i = θ i+1 − θ i , the ratio between two consecutive gaps is denoted as: The average gap ratio r is known to serve as an order parameter for ergodicity breaking transitions [21,71]: When the Floquet dynamics leads to thermalization, the mean gap ratio is described by GUE random matrix ensemble r ≈ 0.60 [72].In contrast, when the Floquet dynamics gets localized, i.e.MBL, the gap ratio statistics is Poissonian such that r = 2 log 2 − 1 ≈ 0.386 [8,17].If our Floquet model undergoes a MBL transition it should show up in the behavior of r as a function of the Schmidt coefficients.
A second diagnostic for the transition is the structure of eigenstates of the time evolution operator: We introduce the reduced density matrix over half of the system as ρ A = Tr L/2 (|n⟩⟨n|).We probe the transition at the level of eigenstates using the half-chain entanglement entropy In the thermal phase, the eigenstates are expected to be essentially random vectors in Hilbert space; thus their entanglement entropy is proportional to the chain length S Page = (L log 2 − 1)/2, the so-called Page value [73].On the other hand, in the localized phase the eigenstates only exhibit short-range entanglement, resulting in an area law for the entanglement entropy S ≈ const [1].Hence the average entanglement entropy S signals the MBL transition [21,74,75].
These quantities are obtained using exact diagonalization.For small system sizes L = 8, 10, 12 the whole eigenspectrum is computed, while for larger system sizes L ≥ 14 polynomial filtered diagonalization [76] is used for extracting 100 eigenpairs of the Floquet unitary.All quantities are averaged over 3000-6000 disorder realizations (except for L = 20 where only 500-1000 realizations are used) and all available eigenstates.

E. Quench dynamics
Another direct way to detect localization in our system is using transport properties.A common tool is the autocorrelator [20,[77][78][79][80]: Here O is a normalized observable with vanishing mean (Tr Ô = 0, Tr Ô2 = 2 L ), and the time evolution is generated by the circuit introduced in Eq. ( 3), i.e.Û (t) = U (c) t .Since our goal is to probe scrambling caused by the entangling gates, we choose an operator Ô such that C(t) = 1 for c 1 = c 2 = c 3 = 0. To achieve this, consider the product of the single-site operators at i = ⌊L/2⌋: Since R is unitary, there exists a diagonal matrix D and an unitary V such that By choosing we obtain an autocorrelator C(t) with the desired properties.It is important to note that the choice of V depends on the specific disorder configuration.When the system thermalizes, the disorder-averaged autocorrelator vanishes in the long-time limit [1].In contrast, C(t) is expected to converge to a non-zero value in the MBL regime [1].In summary, we obtain Another way to probe transport properties in the study of MBL is the entanglement entropy production starting from a product state |ψ(0)⟩ = |ψ 0 ⟩ [41,81].Analog to Sec.II D, we compute the half-chain entanglement entropy S(t) [cf.Eq.( 9)], but now for the time-evolved state |ψ(t)⟩ instead.The entanglement growth rate depends on the overall dynamics: quantum chaotic systems show linear growth in time, while MBL systems exhibit logarithmic growth of entanglement [37,41].The latest signals the existence of quasi-local integrals of motion [18,19].

A. Gap ratio and eigenstate entanglement entropy
As a first check, we probe the operator entanglement per gate s(c) as a unifying parameter for the interaction strength.To do so, we show the gap ratio and the eigenstate entanglement entropy as a function of the gate entanglement entropy s(c) in Fig. 3  originate from the specific choice of the vector c.On the CNOT line, two Schmidt coefficients λ l [cf.Eq. ( 8)] are zero [50], which is not the case for any other choice of c in the Weyl chamber.Therefore the finite size behavior visible in r and S/S Page may be affected by this choice of c (see Appendix A for a more quantitative comparison).
For our further studies, we choose three parameter lines that differ in the number of non-vanishing coefficients c i : the SWAP line, the iSWAP line, and the CNOT line (cf Sec.II B).The gap ratio and the half-chain entanglement entropy on these lines of the Weyl chamber and systems sizes are shown in Fig. 4. In all three cases, we see a crossover between an MBL regime (indicated by a small eigenstate entanglement entropy and an rvalue close to the Poissonian case) for small interactions towards a thermal regime at large interactions.
The curves of different system sizes intersect.Analogously to Ref. [17,21,26], we then compare the different crossings between consecutive system sizes to check the stability of the phase in the thermodynamic limit [71].In Fig. 5, the crossings of gap ratio and entanglement entropy for consecutive sizes L and L + 2 are shown.The trend of the crossing suggests a scaling ∝ 1/L, at least for the accessible system sizes.Remarkably, finite size effects for the half-chain entanglement entropy are less pronounced in comparison to the r-value.The trend of the data suggests that the crossings are shifting towards zero in the limit L → ∞, thus ergodicity is restored at any finite interaction.However, given the smallness of the accessible system sizes, we can not rule out a change in the trend at larger sizes.
Finally, we present an analysis of the entanglement entropy fluctuations of the Floquet eigenstates.It is known that fluctuations around the mean eigenstate entanglement entropy S, probed by σ 2 S = S 2 − S 2 , peak at the transition point [21] and thus are a good indicator to identify the delocalization-localization transition [82,83].Such fluctuations are shown in Fig. 6.We see that the peak of the fluctuations is moving towards smaller s(c), as has been reported in other models where MBL might be stable [75].Importantly, the entanglement entropy fluctuations also seem to be sensitive only to the operator entanglement s(c) of the gate rather than the specific choice (c 1 , c 2 , c 3 ) in the Weyl chamber.

B. Single spin autocorrelation and entanglement entropy after a quench
From the previous section, we can conclude that the critical operator entanglement per gate s(c) is scaling roughly as 1/L.The largest system size for which we could extract eigenvalues and eigenvectors is L = 20.From Fig. 5, we estimate the crossover region for current system sizes to be around s(c) ∼ 1/20 = 0.05.In this section, we explore signatures for this crossover in quench dynamics for up to 10 5 cycles in the regime s(c) ≲ 0.1 and system sizes L ≥ 18.
In Fig. 7, we show the autocorrelation function introduced in Sec.II E on the SWAP line of the model.The circuit dynamics is simulated using Cirq [84] that allows to reach 10 5 cycles and up to L = 22 qubits.
For s(c) ≈ 0.09, 0.07 close to the crossover region, C(t) decays with a scale either logarithmically or stretched exponentially, in line with previous work on autocorrelation decay in prethermal systems [77].The long-time limit lim t→∞ C(t) decreases with system size, suggesting a trend towards thermalization.For smaller interactions s(c) ≈ 0.05, 0.03, our accessible time-scales are too short to draw conclusions about a drift in the long-time dynamics: we do not reach a steady state in our numerics.
Finally, we study the entanglement entropy growth for initial product states along the SWAP line.As discussed in Sec.II E, a signature of localization is logarithmic entanglement growth.As is visible in Fig. 8, the entanglement entropy S(t) is growing faster than logarithmically (black dashed lines) at time scales t ≳ 10 3 and interactions s(c) ≳ 0.07.In order to confirm this observation, we compute the derivative of S(t) with respect to log t (see Fig. 9).A logarithmic curve would be visible as a constant value.Instead, we see that the derivative keeps growing with system size even in the regime when the level statistics is Poissonian s(c) = 0.0369.We conclude that the entropy growth is faster than pure logarithmic growth even for the smallest interactions.The latest is in odds with the steady logarithmic growth in MBL regimes [37].

IV. DISCUSSION
In this work, we have introduced a generic Floquet circuit model that allows us to parametrically tune the interaction and keep the disorder maximal.We have identified the gate entanglement entropy s(c) as a quantitative measure for the interaction.In the limit s(c) = 0, our model reduces to a non-interacting system.Our results for various quantities suggest that the observed MBL regime for small interactions does not persist in the thermodynamic limit.
Our reachable system sizes and our investigated model are not sufficient to draw conclusions about the general fate of the MBL transition in the thermodynamic limit.Nevertheless, they suggest analyzing whether the results for other commonly studied models in the field of manybody localization are in alignment with an integrabilitybreaking transition [36,85] instead.
Moreover, it is an interesting question to establish connections between this model, where both direction and strength are subjected to disorder, and models where the direction of the single-qubit unitaries is fixed.The XXZ spin chain and its variants are part of the latter.Furthermore, the effect of spatial variations on the gate operator entanglement can give rise to "slow" and "fast" dynamical regions very much in the spirit of quantum avalanches proposed as delocalization mechanism [34,35].The investigation of the role of both types of disorder and the effects of spatial fluctuations in gate operator entanglement are both interesting venues for future research.
Apart from that, our model contains dual unitary circuits as another special case for a specific choice of parameters c.This model is thus a good starting point to study the effects of breaking dual-unitarity in more detail [70].
V. ACKNOWLEDGMENTS breaking phenomenon for finite system sizes.

FIG. 7 .
FIG. 7. Single spin autocorrelation C(t) as a function of time t, gates G(c) are chosen on the SWAP line for c = (0.05, 0.06, 0.07, 0.08) (the corresponding interaction strength s(c) is shown in each panel).Disorder average is taken over 1000-4000 disorder realizations.Errors bars are 68% confidence interval.

FIG. 8 .FIG. 9 .
FIG.8.Half chain entanglement entropy after a quench on the SWAP line for c = (0.05, 0.06, 0.07, 0.08) (same values but translated to s(c) are shown in each panel) and system sizes L = 18, 20, 22. Disorder average is taken over 1000-4000 disorder realizations, error bars denote 68% confidence interval.The black dashed lines denote a fit a log t + b for comparison, with a determined by the derivative of the L = 22 curve at t = 10 3 .For large interactions S(t) the logarithmic growth is proceeded by a faster entanglement entropy growth.
5is the equivalence class of the SWAP gate up to single qubit rotations.