Localised Dynamics in the Floquet Quantum East Model

We introduce and study the discrete-time version of the Quantum East model, an interacting quantum spin chain inspired by simple kinetically constrained models of classical glasses. Previous work has established that its continuous-time counterpart displays a disorder-free localisation transition signalled by the appearance of an exponentially large (in the volume) family of non-thermal, localised eigenstates. Here we combine analytical and numerical approaches to show that: i) The transition persists for discrete times, in fact, it is present for any finite value of the time step apart from a zero measure set; ii) It is directly detected by following the non-equilibrium dynamics of the fully polarised state. Our findings imply that the transition is currently observable in state-of-the-art platforms for digital quantum simulation.

Introduction.-Establishing the precise conditions for real space localisation in interacting systems, even in one dimension, turns out to be extremely challenging.Despite intense efforts to crack it [1][2][3][4][5][6][7][8][9], it currently remains a major unsolved problem in theoretical physics.It has been argued that a form of many-body localisation should emerge as a consequence of an external quenched disorder, which, under some conditions, might defeat interactions.Whether this mechanism can lead to a stable phase of matter remains an actively debated topic [7][8][9].A fundamental problem is that localisation studies are either limited to small systems accessible to numerical or experimental simulation or uncontrolled perturbative approximations.Nevertheless, for many-body localisation to be established as a phase of matter it has to exist in the thermodynamic limit: it should not (only) be a property of eigenstates, but (also) of dynamics.
Recently, it has been suggested that, other than by disorder, real space localisation can also be triggered by kinetic constraints which render transport a higher-order process.An advantage of this approach is its immunity to fluctuating rare events such as ergodic bubbles.A minimal example of this mechanism is realised in the socalled Quantum East model [10][11][12][13][14][15] (and its bosonic version [16]) where a localisation transition in the quantum Hamiltonian is in one-to-one correspondence to a first order activity-inactivity transition in the corresponding classical stochastic glass model.In agreement with this picture, Ref. [17] observed an eigenstate localisation transition in the Quantum East for an exponentially large family of eigenstates.
In this work, we take a fundamental step further and look for the possibility of dynamical localisation in a Floquet, or trotterised, version of the Quantum East model, where localisation is challenged by a steady pumping of energy into the system [18][19][20].This setting can be seen as the kinetically-constrained analogue of Floquet many-body localisation [21][22][23].We replace the continuous Hamiltonian dynamics by a discrete sequence of con-ditional unitary gate operations -a quantum circuitthat can be conveniently implemented on platforms for digital quantum simulation such as trapped ions [24][25][26][27] and superconducting circuits [28][29][30][31][32][33].Using timedependent perturbation theory, we argue that the model displays a localisation transition by tuning the parameters of the model.We demonstrate that in the dynamically localised phase the model can be efficiently simulated by time-dependent matrix product methods (i.e.TEBD algorithm) [34][35][36] to an arbitrary precision, showing very good quantitative agreement with the perturbative prediction.Moreover, we find qualitative agreement between the dynamical picture of localisation in the infinite system and the localisation of eigenstates in the finite system.
The model.-Our starting point is the Quantum East model [10] defined by the following Hamiltonian operator (in arbitrary energy units) Here a is the dimensionless coupling constant, 2L is the system size, {X j , Y j , Z j } are Pauli matrices acting nontrivially at site j, I is the identity operator, and P j = (I + Z j )/2.We are interested in discrete sequences of unitary operations U(a, τ ) that reproduce the dynamics generated by Eq. (1) in a special scaling limit.Namely where t is the number of discrete time steps, while t plays the role of physical time.This procedure is known as Trotter-Suzuki decomposition [37,38] and does not uniquely specify the unitary operator: there are many choices of U(a, τ ) fulfilling Eq. ( 2).Here we consider one that is local in space, i.e., it has the following brickwork structure (see Fig. 1) with where we use a standard notation O x,y for an operator O acting non-trivially only at the sites x, y, and define a local conditional gate U (a, τ ) = e −iτ (aP ⊗X−P ⊗I) .The (Trotter) time step τ , usually referred to as the Trotter step, sets the strength of the unitary operation (3).It is easy to verify that (2) holds for the evolution operator defined in Eq. (3).Note that the discrete-time dynamics generated by Eq. ( 3) is equivalent to a continuous-time dynamics in the presence of a periodic drive.
To probe the localisation properties of the quantum circuit (3) we consider a local quantum quench.Namely, we prepare the circuit in the initial state which is an eigenstate of the bulk evolution due to the identity U (a, τ ) |↓↓⟩ = |↓↓⟩, but is not stationary at the left boundary.As a consequence, only the sites within a light cone spreading from the left boundary undergo a non-trivial evolution, see Fig. 1.Intuitively, one can think of our local quench protocol as creating a localised disturbance in (x, t) = (0, 0) in a state that is otherwise stationary.Importantly, this quench problem is also a caricature of local-operator spreading in a generic quantum many-body system after an operator-to-state mapping.Here, |↓⟩ [and (5)] represents the identity and |↑⟩ stands for any other traceless local operator that starts growing from the left edge.The question of localization now translates to that of the existence of a conserved local operator.
A simple measure of how the disturbance created by the local quench spreads through the system is given by the partial norms Since W (x, t) ≥ 0 and x W (x, t) = 1, the partial norms can be thought of as the probability of having the rightmost up spin at position x.Specifically, whenever the disturbance remains localised at the boundary we have W (x, t) ≈ 0 for x ≫ x 0 = O(t 0 ), while when it spreads through the light cone the partial norms attain non-zero values for all x ≤ 2t.We emphasise that due to the light cone W (x > 2t, t) = 0.The factor 2 is a direct consequence of the brickwork structure of (3), as each time step propagates information for up to two sites to the right.
In fact, to facilitate our numerical analysis we consider slightly modified quantities that bear the same physical information as those in Eq. ( 6): Instead of the partial norms of the state U(a, τ ) t |↓ • • • ↓⟩, we look at those of the state along the diagonal cut in the right panel of Fig. 1.The latter quantities are where we introduced the "ladder evolution operator" (cf.Fig 1) (8) which is related to U(a, τ ) by a similarity transformation [39].The quantities in Eq. ( 7) are more convenient than those in Eq. ( 6) because with the same computational effort one can access times that are twice as long.
Infinite system at finite times.-Letus begin considering the time evolution of the partial norms N (x, t) in the thermodynamic limit L → ∞.In this case, the main qualitative features of their evolution can be understood by performing a simple perturbative analysis (the same can be done for W (x, t) [39]).We begin by introducing the interaction representation of the time evolution operator where we defined Ũk,k+1 (a, τ ), (10) with Ũ (a, τ ) = e −iaP ⊗Xe −iτ Z .We now fix x, t, τ and expand (6) in powers of a. Looking at the local gate in the interaction picture, i.e., Ũ1,2 (a, τ we have that N (x) is at least of order a 2x .Indeed, due to the structure of Eq. ( 3), to get a spin up at position x we need to at least flip all the spins on its left.This also tells us where ≃ denotes equality at the leading order in a.A simple combinatorial calculation then allows us to express it in terms of q-deformed binomial coefficients [39] N where we set q = e iτ .Interestingly, the perturbative analysis commutes with the limit (2).Indeed lim τ →0 N ′ 1 (x, t/τ ) = (2a sin(t/2)) 2x /(x!) 2 , coincides with the leading order of (7) if one replaces (9) with its Trotter limit [39].
Let us now move on to analyse the localisation properties of the perturbative solution.To this end, we assume that N ′ 1 (x, t) gives the only relevant contribution to the partial norm.The first key feature of N ′ 1 (x, t) is that its localisation properties depend on whether or not τ is a rational multiple of 2π.Namely, whether it can be written as 2πc/d for some coprime integers c and d.When true, the q-Lucas theorem [40] connects the behaviour of q-deformed and regular binomials where ⌊x⌋ is the largest integer smaller than x and mod(c, d) is the remainder of the division of c ∈ N by d ∈ N. Using the Stirling approximation we find that Eq. ( 13) has a maximum at x = (aτ ) 2d t/(1 + (aτ ) 2d ).Therefore the support of N ′ 1 (x, t) grows in time ruling out localisation.
On the other hand, whenever τ is not a rational multiple of 2π the deformed binomial coefficients are bounded in time.Namely we have In the last step we used that, since {mod(τ p, 2π)} t p=1 covers [0, 2π) uniformly in the large t limit, we have Therefore the O(t) in Eq. ( 15) cancels, and we are left with an O(t 0 ) term.Plugging the bound Eq. ( 15) into Eq.( 13) we find that N ′ 1 (x, t) is localised within a distance x 0 = −1/(2 log aτ ) from the left boundary for all times.
The second key feature is the τ dependence of a cthe critical a for localisation -in the case of irrational τ /(2π).In our setup this amounts to ask for what range of a we expect the perturbative result to apply (at least qualitatively).From Eq. ( 13) we see that for finite τ the parameter that has to be small to ensure the validity of the perturbative approach is aτ .Instead, in the limit τ → 0 the perturbative solution requires a itself to be small [39].This suggests that a c should be of the form for some α, β ∈ R.
Remarkably, by computing N (x, t) and N ′ (x, t) via a simple version of the TEBD algorithm [39] we find that all these qualitative features persist away from the perturbative regime.Some representative examples of our numerical results are presented in Figs. 2 and 3, where, as a further indicator of localisation, we also report the entanglement entropy S(x, t) between the x leftmost sites and the rest of the system at time t.
For small enough a we see that disturbance created by the local quench remains localised only for irrational values of τ /(2π).This is clearly shown in the insets of Fig. 2: While for rational τ /(2π) we see the peak (and its position) of the entanglement entropy growing linearly in time, for irrational τ /(2π) we see it saturating (additional corroborating plots of the (spatio) temporal behaviour of S(x, t), W (x, t) and entanglement spectra are found in [39]).Note that we observe this stark difference between rational and irrational τ /(2π) also for times that are significantly out of the perturbative regime (t ≫ 1/a) and at which Eq. ( 13) is not quantitatively accurate: see the comparison in the main panel of Fig. 2. As a result of this localised behaviour, for irrational τ /(2π) we are able to run our TEBD simulations with essentially no truncation error for hundreds of time steps.
On the other hand, for a larger than a certain critical value a c (τ ) the system transitions to the ergodic regime also for irrational τ /(2π), see Fig. 3.In this case the perturbative result does not describe the system's behaviour even at the qualitative level: the support of the partial norms grows linearly in time signalling a delocalisation of the disturbance caused by the impurity.Concerning the τ dependence of a c (τ ), our numerical results are compatible with the functional form in Eq. ( 17) [39].Namely, the critical a appears approximately τ -independent for small τ , while it starts to decay as τ −1 when τ is increased beyond a threshold value.
Finite systems at infinite times.-Interestingly, the phenomenology observed above in the thermodynamic limit is also observed for finite volumes.Here we again look at a quench from the initial state in Eq. ( 5) but keep L finite while taking t → ∞.A convenient indicator of the localisation transition is then the time averaged square of Loschmidt Echo (LE) this quantity is the same for brick-wall U and ladder propagators U ℓ ].
Assuming that there are no degeneracies in the spectrum of U ℓ , the LE can be written as lim t→∞ where the sum over i, j goes over all eigenstates.I |↓•••↓⟩ is the inverse participation ratio (IPR), which measures the spreading of the initial state in the eigenbasis of the time-evolution operator.It can be interpreted as the purity of the probability distribution {P i }, with For random eigenstates |E i ⟩ the probability distribution is flat, i.e., P i = 2 −L , which gives In contrast, in the localised phase, we expect that up to exponential corrections, the initial state spreads up to a finite distance k, so We computed I |↓•••↓⟩ numerically for several values of a, τ , and L: our main numerical results are summarised in Fig. 4. The behaviour of the IPR aligns remarkably well with the phenomenology of finite-time data.The For the first two τ s the transition occurs around a = 1 and a = 0.3.The third τ is a rational multiple of 2π, and its transition occurs at a much smaller a.For a detailed analysis of this plot, see [39].
bottom panel of the figure shows the IPR versus L for different values a and three choices of τ .Identifying the localisation transition as the transition between constant and exponentially decaying IPR, we can estimate a c (τ ).
The last two τ are similar in size, but they are respectively irrational and rational multiples of 2π.We see that the difference between these two cases is stark also in this setting: the irrational τ /(2π) shows a transition at sizeable a, while rational τ /(2π) shows ergodic behaviour for the same choice of a.In the phase diagram, the rational τ /(2π) generate some irregular behaviour reminiscent of Arnold tongues [41].Some further discussion and additional finite-volume data is reported in the SM [39].Discussion and Outlook.-Weintroduced a discretetime version of the Quantum East model [10] and analysed its localisation properties in real-time.Combining a perturbative analysis with exact numerics we identified a localisation transition taking place in this system despite the periodic drive: for couplings smaller than a critical value a c (τ ) the effect of a boundary perturbation remains localised in space, while it spreads ballistically for a > a c (τ ).This is also shown by a stark difference in the entanglement scaling (linear vs bounded), which is more marked than what is reported in other accounts of localisation in Floquet settings [21,42].Interestingly, this transition has a non-analytic dependence on the Trot-ter time τ and takes place only when the latter is an irrational multiple of 2π.In fact, irrational Trotter step and dynamical constraints can be identified as the two key mechanisms for the onset of localisation.To understand this one can imagine expanding the state of the system at time t in the computational basis.Because of the dynamical constraints, there will be far fewer states appearing in this sum than those allowed by the locality of the interactions.Moreover, all configurations are dampened by a factor (τ a) x , where x is the position of the up spin, that is further from the left boundary.This, however, is not enough to ensure that the configurations with large x are suppressed -i.e.localisation -because each configuration can be reached in many different ways, i.e., by many different "trajectories".This means that in the expansion of the state at time t each configuration is multiplied by a "combinatorial weight", which can in principle overcome the dampening.An irrational Trotter step avoids this by introducing destructive interference between the different trajectories, and hence, ensuring that the combinatorial weight never overcomes the exponential dampening.In the continuous time limit, τ → 0, the dampening factor goes to 0 and the combinatorial weight diverges, therefore one has to combine the two effects.The outcome suggested by our analysis is that in this limit the system is localised for any a < 1, in agreement with Ref. [17].This is not in contrast with the statement that the Floquet Quantum East is localised only for irrational τ /(2π) as in this case the limits of τ → 0 and t → ∞ do not commute [43].
A natural question is what are the initial states for which localisation occurs.We note that our analysis can be repeated for all states written as tensor products of arbitrary finite-block states with an infinite block of down spins on the right.This is consistent with the local-operator spreading analogy discussed earlier, as such states are those corresponding to local operators.States not fitting this form evade our treatment, leaving their analysis for future research.Our expectations is that those states will not show localisation as in the continuous-time setting [17].Indeed, they lack the first ingredient of the localisation mechanism we identified, i.e., the presence of dynamical constraints.A key future direction is the rigorous characterisation of the observed transition within the convenient discrete space-time setting introduced here.3 ∼ 2.0944 (green) for system sizes L = 8, 12, 14 (cross, triangle, square).Even thought these τ are close, the transition of the irrational one is close to a = 0.3 versus the rational one shows IPR shrinking with system size for much smaller a.

FIG. 4 .
FIG. 4. (Top)Logarithm of the IPR at L = 13 as a function of τ and a.We see the transition for smaller (bigger) τ at a = 1 (a ∝ 1/τ ).The dashed lines denote the three τ s considered in the bottom panel.(Bottom) Logarithm of the IPR versus L for three values of τ and several values of a (0.5, 0.7, . . ., 1.5 top to bottom for the first plot and 0.05, 0.15, . . ., 0.55 for the second and third).The solid grey line corresponds to random eigenstates.For the first two τ s the transition occurs around a = 1 and a = 0.3.The third τ is a rational multiple of 2π, and its transition occurs at a much smaller a.For a detailed analysis of this plot, see[39].
FIG. S8. (Left)Logarithm of the IPR at L = 13 as a function of τ and a.We see that for small τ the transition is at a = 1, whereas for bigger τ the transition is at a ∝ 1/τ .With the white dashed lines we show the three values of τ considered in the bottom panel.(Right) Logarithm of the IPR versus L for three values of τ and several values of a (0.5, 0.7, . . ., 1.5 top to bottom for the first plot and 0.05, 0.15, . . ., 0.55 for the second and third).The grey solid line corresponds to fastest decay 2 −L for random eigenstates.For the first two values of τ , we can estimate the transition between constant and exponential decay at a = 1 and a = 0.3.The third τ is a rational multiple of 2π, and its transition occurs at a much smaller a.
1. (Left) State (5) after t = 3.5 time steps of discrete dynamics.White bullets denote the state |↓⟩ and the blue circles the activation part of the conditional gate U (a, τ ).The brick-wall Floquet propagator U(a, τ ) is highlighted in yellow.(Right)Explicitsimplification of the dynamics out of the light cone.Dashed lines indicate the cut for the ladder evolution in Eq. (7).The ladder propagator U ℓ (a, τ ) (cf. (8)) is highlighted in orange.