Stretched-Exponential Melting of a Dynamically Frozen State Under Imprinted Phase Noise in the Ising Chain in a Transverse Field

The concept of dynamical freezing is a phenomenon where a suitable set of local observables freezes under a strong periodic drive in a quantum many-body system. This happens because of the emergence of approximate but perpetual conservation laws when the drive is strong enough. In this work, we probe the resilience of dynamical freezing to random perturbations added to the relative phases between the interfering states (elements of a natural basis) in the time-evolving wave function after each drive cycle. We study this in an integrable Ising chain in a time-periodic transverse field. Our key finding is, that the imprinted phase noise melts the dynamically frozen state, but the decay is"slow": a stretched-exponential decay rather than an exponential one. Stretched-exponential decays (also known as Kohlrausch relaxation) are usually expected in complex systems with time-scale hierarchies due to strong disorders or other inhomogeneities resulting in jamming, glassiness, or localization. Here we observe this in a simple translationally invariant system dynamically frozen under a periodic drive. Moreover, the melting here does not obliterate the entire memory of the initial state but leaves behind a steady remnant that depends on the initial conditions. This underscores the stability of dynamically frozen states.


Introduction:
The study of non-equilibrium phenomena in driven quantum systems has sparked significant interest [1][2][3][4][5][6][7], uncovering phenomena far from the realm of the notions and concepts of equilibrium physics or its intuitive extensions even in ostensibly simple tractable systems.In this work we focus on one such example -the phenomenon of dynamical freezing (DF) [2,6,[8][9][10][11][12][13][14][15][16][17], in particular, an incarnation of it in an integrable spin chain.In DF, a strongly driven closed quantum system (including quantum-chaotic interacting many-body systems) can evade the intuitive Floquet thermalization to a locally infinite-temperature-like state [18,19], when the drive amplitude crosses a threshold [11].This happens due to the appearance of exact (in systems mappable to non-interacting fermions or hard-core bosons) [20] and emergent (in generic interacting systems) [8,11,12] stroboscopic conservation laws that are not present in the undriven system.Here we investigate the robustness of the phenomenon in an integrable spin chain, against adding after each drive cycle a random component to the relative phases between the amplitudes of the members of a particular (natural) basis in the wave function.Our study reveals a rich pallet of interesting regimes of stability and decay.Previously, DF was argued to result from repeated quantum interference of the relative phases mentioned above [6,10].In particular, it was shown earlier by Amit Dutta and collaborators that if one considers the dynamics discounting the phases altogether, i.e., consider only the transition probabilities for each cycle ignoring the interference between the transition amplitudes, then an infinitetemperature-like scenario (within the integrable structure) is approached exponentially rapidly with the number of drive cycles [21].
Here we examine the robustness of DF by targeting those phases and corrupting them randomly in a controlled manner.The phenomenology we uncover here is expected to be realizable in experimental setups similar to that described in Ref. 10.
The Model and The Drive Protocol: The model under investigation is specified by the Hamiltonian where the Pauli matrices satisfy periodic boundary condition σ α L = σ α 0 for α ∈ {x, z}, and admits a ferromagnetic (paramagnetic) phase for |h| < 1 (|h| > 1) and critical points at |h| = 1, and we set J = 1.In our periodic driving protocol, the transverse field is time-dependent, quenched from +h 0 to −h 0 (h 0 > 0), and back periodically over a period of T such that We follow the system's dynamics stroboscopically at instants t = nT, where T is the period of one drive cycle, and n is a non-negative integer.This sets the final time t f = N c T (n = 0, 1, . . ., N c ).The system is prepared to be at the ground state of the Hamiltonian at t = n = 0, then allowed to evolve under the protocol mentioned above.Measurements of the relevant observables are recorded after N c number of full cycles.The standard steps to diagonalize the Hamiltonian at any instant are as follows [22].First, the Pauli spin operators are expressed in terms of (spinless) Jordan-Wigner fermions; the resultant fermionic Hamiltonian reads This is a quadratic problem in terms of the fermions, and invoking translation invariance (under periodic boundary condi-tion) can easily be solved in the Fourier space following the Fourier transform c k = 1/ √ L j c j e −ikj , where k denotes the Fourier space quantum numbers with k ∈ (−π, π).A particular advantage is that the Hamiltonian in the Fourier space is a non-interacting ensemble of two-level systems The manifold in which the ground state and states reachable from it via the Schrödinger dynamics under H k (t) lie, is spanned by the set of basis states |0 k , 0 −k ⟩ and | + k, −k⟩ that represent, respectively, the unoccupied and the doubly occupied states of the ±k fermions.
Starting from the ground state, the wavefunction of the system at time t is given by |ψ(t)⟩ = k |ψ k (t)⟩ where |ψ k (t)⟩ = u k (t)|0 k , 0 −k ⟩+v k (t)|+k, −k⟩ for some complex amplitudes u k and v k .The phases of these complex numbers are of pivotal importance in the physics we explore through this work.The allowed values of the quantum number k are set by the fermion parity or equivalently the eigenvalue of the operator P = L−1 j=0 σ z j .We place the system in the evenparity sector (where the ground state is found) by considering L to be even which fixes the quantization of the k values as Physical observables that are local in c k and c † k are functions of the amplitudes u k (t) and v k (t).For instance, timedependent transverse magnetization is In addition, of particular interest will be the long-time average of m z (t) [8]: which is a measure of the degree of symmetry breaking in the system.Since each k-mode undergoes the time evolution independently, it renders the time-evolution operator accounting for the full dynamics a simple form T = k T k .In other words, with the initial state specified in a given k-sector, the time-evolved state is |ψ k (t)⟩ = T k |ψ k (t = 0)⟩ where T k is the relevant part of the full time-evolution operator T when projected onto the subspace spanned by |0 +k , 0 −k ⟩ and | + k, −k⟩.With this, we can readily compute T , and hence m z (t).The repeated interference between the amplitudes u k , v k leads to the remarkable phenomenon of DF.In the following section, we provide a brief account of this and highlight some notable features that will constitute the backbone of subsequent investigations.Dynamical Freezing: For the specific protocol of periodic driving we use, the Hamiltonian remains constant over each half cycle, however, the transverse field flips in the consecutive half cycles.Therefore the time-evolution operator T k is simply (with ℏ = 1), and the evolution operator after N full cycles i.e., after time t Let us consider the initial state (at t = 0) to be the ground state of H k (h 0 ) at each k.It was shown in Ref. 8 and 9 that depending on the driving frequency ω = 2π/T , the memory of the Z 2 symmetry breaking in the initial state can get imprinted in the response, i.e., the transverse magnetization m z (t) [and so in Q(t)] through all later time, though the driving respects this symmetry after each full cycle (over period T ).This effect is especially prominent when the dimensionless quantity p = 2h 0 /ω assumes nonzero integer values -a condition referred to as the freezing condition, and the corresponding ω or the p values, the freezing peaks.
The profile of m z (t = nT ) as a function of n as well as its long-time average Q(t = N c T ) (denoted as Q| NcT ) as a function of p is shown in Fig. 1.The peaks of Q occur at integer values of p, aligning with the initial condition m z (t = 0) ≈ 1 (set by h 0 = 20), indicating a robust freezing behavior.However, slight deviations from the respective initial values are noticeable at larger integer values of p (increasing with p) corresponding to lower-frequency drives which tend towards an approximately adiabatic regime (ω → 0 as p → +∞).This suggests that the phenomenon of DF is genuinely rooted in nonadiabatic quantum dynamics.The plot of Q| 500T in Fig. 1 FIG. 2. Shown is the peak-valley structure in the behavior of Q|1000T as a function of p = 2h0/ω, in the presence of the random phase noise averaged over 1000 random realizations (error bars showing the standard deviations) when we vary the strength of the randomness in the added phases λ.The red dashed line is the plot for λ = 0 (also shown in Fig. 1).The plots on the (right) are for mz as a function of time along the two vertical dashed lines: the top is at a representative integer value of p (a peak) and the bottom is for a non-integer value (a valley).The red-dashed lines in these again correspond to λ = 0.The results are for the chain with L = 100.also reveals intriguing valley structures near half-integer values of p.These points denote instances where m z (t) exhibits the largest deviation from its initial values between two consecutive peaks.Remarkably, the magnitude of such deviations diminishes with higher values of the valley locations along the p-axis indicating that the valleys at smaller frequencies emulate the freezing behavior of the integer p values at larger frequencies.
This phenomena of DF cannot be explained in terms of counting the probabilities after each half cycle and the resultant recurrence relation [10,23].For a specific k-mode, the probability of finding it in the excited state in the second half of the n-th full cycle reads where we have used the notation v k (n + 1/2) ≡ v k (t = (n + 1/2)T ) and P ex (k) is the probability that the state ends up at |+k, −k⟩, following a flip in the transverse field, starting from the same state before the flip.The solution to this relation predicts |v k (n + 1/2)| 2 → 1/2 independent of k, T, h 0 as n → +∞ suggesting the transverse magnetization (m z in Eq. 5) diminishes over time regardless of p.However, the presence of a distinct peak-valley structure in Fig. 1 presents a contrasting scenario; an explanation for DF, therefore, must take into account the amplitudes and their interference rather than solely relying on the probability count outlined above.
The robustness of the peak-valley structure in this simple integrable model was tested against disorder in the spin-spin interaction as well as in the transverse field [15].Remarkably, it was observed that the decay of the transverse magnetization m z (t), exponential in time with a characteristic scale τ (similar to the undriven case), is significantly suppressed at the peak points when plotted against the driving frequency ω.Curiously, this characteristic time scale τ (ω) displays an exponential growth with ω (or inversely with p) which warrants that the stability of the peaks intensifies at larger frequencies due to adequate nonadiabatic dynamics induced by the periodic drive even in the presence of disorder.
Quite intriguingly, the peak-valley structure survives even in scenarios where integrability is compromised.In Ref. 12, the integrability was broken by introducing higher-neighbor spin-spin interactions, and longitudinal magnetization m x (t) was measured at late times (in the diagonal ensemble) with the driving field aligning along the ordering direction.When plotted against the driving frequency ω, a peak-valley structure, akin to the integrable counterpart, emerges in the late-time profile of m x with robust peaks and valleys gaining prominence at larger values of ω.
These deliberations lay the groundwork for our present inquiry, prompting the question: how does random phase noise in the amplitudes u k and v k , and their interference, influence the stability of the peak-valley structure within the integrable framework?An extension to the realm of nonintegrable models will be discussed elsewhere.
Adding Random Component to The Phases: To answer this, we set the following protocol of inserting a random phase difference between u k and v k after each full cycle and for all k by drawing independent and identically distributed random numbers uniformly over the range ϕ k (n) ∈ [0, 1] (the phenomena we describe here do not alter if the mean of the noise is 0 i.e., ϕ k (n) ∈ [−0.5, 0.5]).We then introduce a parameter λ as the effective strength of these random phases by modifying ϕ k (n) → λϕ k (n).Such a protocol yields the full time-evolution operator for a specific k-mode over a period of N full cycles as ) and λ is treated as a control parameter to vary the strength of the random phases as discussed above.In this setting, we embark on analyzing the effect of the interference between such random phases accumulated by the amplitudes after many consecutive cycles on the phenomenon of DF.We aim to elucidate the influence of this interference on both the transverse magnetization m z (t) and its late-time average Q after a sufficiently large number of complete cycles by scrutinizing the fate of the peak-valley structure.
Let us first note a recurrence relation due to the successive application of the unitary matrix T k (−h 0 )T k (+h 0 ) (no random phase added) considering this unitary matrix admits a generic form a b c d (a, b, c, d are complex functions of k, h 0 , T ).In the absence of the cross term in the square parentheses, we readily identify this relation with Eq. 8, the recurrence relation for the probability count, with |d| 2 ≡ P ex (k).The contribution of the interference term is, at this point, explicit in giving rise to DF.While the full expressions of a, b, c, d are too complicated to place here, a simplified version can be obtained by inspecting the large h 0 limit which also offers additional insights into the peak-valley structure.We find, in this limit, the offdiagonal element of the unitary time-evolution matrix is therefore, when h 0 T = 2πh 0 /ω is integer multiples of π, namely, 2h 0 /ω = pπ -the freezing condition, the offdiagonal terms of the matrix T k (−h 0 )T k (+h 0 ) vanish leading to an arrest of the magnetization dynamics regardless of k.As a result, Eq. 10 renders a peak-valley profile to m z (t) when plotted against h 0 at a fixed frequency of the drive [9].When random phases are included by setting u k (n) → e −2iπλϕ k (n) u k (n) and v k (n) → e 2iπλϕ k (n) v k (n) at the end of each (n-th) full cycle, Eq. 9 modifies to From the results of the large h 0 limit, one would then infer that this alters the freezing condition to depend on k.More specifically, for a given p satisfying pπ = h 0 T (in the absence of the random phases), a k-dependent random shift ∆ k is introduced for which p → p ′ = p + ∆ k where ∆ k = 2λϕ k .The parameter p ′ is not in general an integer anymore (unless ∆ k is so at some special values of λ and ϕ k ) and the off-diagonal element in Eq. 10 is also modified to In what follows, we numerically investigate the effect of the resultant interference due to these random phases on the resilience of the overall freezing phenomenon by studying the behavior of m z (t) and Q| NcT (after N c number of full cycles) which are the integrated response of all k-modes.We do so by varying the parameter λ which, as defined earlier, serves as the effective strength of the random phases ϕ k (n).
For the numerical analysis, we fix L = 100 as no prominent L-dependence is observed at larger L-values.For λ ̸ = 0, an average is performed in computing the response over 1000 random realizations for each k which turns out to be sufficient in the present context.
At the Freezing Peaks: As seen in Fig. 2, the qualitative form of the peak-valley structure remains although the difference in the values of Q| NcT at the peaks and the nearby valleys gets larger with the effective strength λ of the random phases.This implies that even when there is random interference between the amplitudes after consecutive cycles, special frequencies do appear at which the deviation of Q| NcT from the initial magnetization remains almost frozen.The peak locations are well described by the clean case λ = 0.The values of Q| NcT at these peaks eventually decay with time, however, the rate is much slower compared to that of any noninteger values of p.
It turns out one can model the decay of m z (t) for any λ ̸ = 0 by a stretched-exponential form where the remnant magnetization m rm (which can be readily identified with Q in Eq. 6), the steering exponent α, and the time scale τ all are frequency-dependent, and the initial magnetization, denoted as m 0 , is set by h = h 0 at t = 0.A nonzero remnant magnetization indicates a perpetually sustained memory of the initial value of m z (perpetual persistence of the shards of the emergent conservation law).We demonstrate this in Fig. 3.For small integer values of p, equivalently, for large frequencies of the drive, the profile of m z (t) is fairly exponential attributed to strong nonadiabaticity in the dynamics.The stretched-exponential form gradually develops at slower frequencies as evident in Fig. 4. We further find the timescale (τ ) is suppressed with λ leading to a faster approach of m z (t) to its remnant value m rm at the presence of strong phase noise (Fig. 4) and, for a fixed λ, it grows with faster frequencies.The parameters here are extracted from an average value of m z computed over 1000 random realizations of the phase noise in each cycle.We also trace the behavior of Q| NcT by varying the parameter λ which features prominent oscillations with a decaying envelope, shown in Fig. 5.The simple expression in Eq. 12 at large driving amplitudes is speculative of this feature; because of the modified phase factor, the integrated response is expected to fall as λ −1 in the leading order: Q(p)| NcT = Q 0 (p) + A(p)/λ, regardless of p.For a fixed λ, a plot of Q| NcT as a function p exhibits a similar pattern since the term with λ renders an additive correction to the term h 0 T as noted in Eq. 12.
Away From The Freezing Peaks: Away from the frequencies that respect the freezing condition at λ = 0 maximally (i.e., the peaks in Fig. 2), we observe a prominent stretchedexponential decay of m z (t) with the time scale τ much smaller than those at the freezing points or peaks, as evinced from Fig. 3. Other features of the valleys are qualitatively similar to those at the freezing peaks (cf.Fig. 4), such as the non-monotonic behavior of Q| NcT with a decaying envelope as a function of both λ and p and the decay of Q| NcT in the form λ −1 at large driving amplitudes (Fig. 5).The values of Q| NcT at large non-integer values of p approach those at the peaks signaling the freezing to occur at small frequencies even in the presence of interference due to random phase noise, however, the scale is enhanced by the effective strength (λ) of the noise.
Summary and Outlook: Our investigation focuses on probing the robustness of the phenomena of DF in periodically driven many-body systems against interference from random phase noise.In practical quantum devices designed for exploring out-of-equilibrium quantum dynamics, even the slightest incoherence in the driving protocol can naturally introduce such a noise.In earlier observations, within an Ising chain subjected to coherent periodic driving, specific values of the ratio between the drive frequency ω and amplitude h 0 led to a notable slowing down of transverse magnetization relaxation -a hallmark known as the freezing condition.This phenomenon manifested as a distinctive peak-valley structure in the late-time response (Q) of the system as a function of ω.
Our study delves into how random phase accumulation after each drive cycle affects the freezing at these specific frequencies and the overall peak-valley structure.We find that while such random phase interference maintains the qualitative form of the peak-valley structure and the locations of the peaks in the frequency domain, the transverse magnetization m z eventually decays over time.However, the decay time scale is significantly extended at the freezing peaks compared to frequencies away from these points.Moreover, we observe that increasing the strength (λ) of the phase noise enhances the disparity in relaxation between m z (t) at the peaks and the valleys; additionally, a notable stretched-exponential decay of m z (t) emerges as an exclusive feature of the nonadiabatic dynamics (earlier observed in complex systems in the presence of disorder including many-body localization [24][25][26]).Interestingly, as a function of the strength λ of the noise, the long-time-averaged response Q exhibits a non-monotonic behavior, indicating an intricate interplay between the repeated quantum interference and the stochastic process of random phase insertion.Extending our analysis to interacting systems presents a promising avenue for exploration.Recent advancements in cold-atom experiments [27][28][29] suggest the feasibility of experimentally observing the freezing phenomenon in periodically driven arrays of hard-core bosons in optical lattices.Such experiments hold promise for validating our predictions regarding the stability of many-body DF against the random phase noise considered.
Dedication: We dedicate this paper to our much-loved colleague, friend, and mentor, Late Amit Dutta, who left us all of a sudden in 2022.His untimely demise has left a vacancy in the heart of, we dare say, anyone who came close to him in any capacity.We are no exception.
Funding: No additional funding was received particularly to support this work.

FIG. 1 .
FIG. 1. Left: Temporal profile of the transverse magnetization mz(t) at various p = 2h0/ω with h0 = 20 plotted over Nc = 500 full cycles.The time-axis (t) is in the units of the full-cycle period T as t = nT where T = 2π/ω.DF is observed at special frequencies given by the integer values of p. Right: The long-time averaged magnetization Q after 500 complete cycles (denoted as Q|500T ) as a function of p reveals a peak-valley structure where the valleys are located almost halfway between the two consecutive peaks.The results are for the chain with L = 100.

FIG. 3 .
FIG. 3. The profile of mz(t) for various integer values of p = 2h0/ω (left) and non-integer values of p (right) at a fixed strength of the phase noise λ = 0.5.The plots fit best to the stretched-exponential of the form mz(t) = mrm + (m0 − mrm)Exp[−(t/τ ) α ] with a p-dependent remnant magnetization mrm, the steering exponent α, and time scale τ .The results are for L = 100 averaged over 1000 random realizations of the phase noise (error bars showing the standard deviations).

FIG. 4 .
FIG.4.The behavior of the remnant magnetization mrm, the steering exponent α, and the time scale τ as a function of the strength of the phase noise λ after Nc = 1000 full cycles (obtained from the fitting of the average mz(t) over 1000 random realizations of the phase noise).The first three figures from the left show results for various integer values of p and the next three show the same for various non-integer values of p.The stretched-exponential behavior gains prominence at larger integer values of p or equivalently at slower frequencies, where almost-adiabatic dynamics set in, and approaches the conventional exponential form at higher frequencies (see the right-most figure where the blue-filled circles correspond to the integer values of p or the peaks (freezing points) and the red ones correspond to the half-integers or the valleys).The time-scale τ is also seen to be significantly higher at the integer values of p (freezing points).The results are for L = 100.