Exact Floquet Quantum Many-Body Scars under Rydberg Blockade

Quantum many-body scars have attracted much interest as a violation of the eigenstate thermalization hypothesis (ETH) due to recent experimental observation in Rydberg atoms and related theoretical studies. In this paper, we construct a model hosting exact Floquet quantum many-body scars, which violate the Floquet version of ETH (Floquet-ETH). We consider two uniformly-driven static Hamiltonians prohibiting neighboring up spins (Rydberg blockade) like the PXP model, and construct a binary drive composed of them. We show that there exists a four-dimensional subspace which completely avoids thermalization to infinite temperature and that any other states, including some special scar states reported in the static PXP model, are vulnerable to heating and relax to infinite temperature. We also construct a more generalized periodic drive composed of time-dependent PXP-type Hamiltonians showing exact Floquet quantum many-body scars and discuss possible experimental realization of the model in Rydberg atoms.


I. INTRODUCTION
Thermalization in closed quantum systems has been vigorously studied to understand the relationship between quantum physics and statistical physics.With the recent progress in numerical and experimental studies [1][2][3][4], generic nonintegrable systems have been believed to satisfy so-called eigenstate thermalization hypothesis (ETH).ETH dictates that all the eigenstates cannot be distinguished from thermal equilibrium states as long as only macroscopic observables are considered.Since ETH is a sufficient condition for thermalization to take place, ETH is believed to be a key to understand thermalization.
However, with the development of Rydberg atoms [4], it has been revealed that there exists a violation of ETH called quantum many-body scars [5].To be precise, there are several non-thermal eigenstates, which are eigenstates distinguishable from thermal equilibrium states (called exact scar eigenstates), while other states out of their subspace experience thermalization as with usual nonintegrable models.The PXP model, a typical model showing scars, has been realized on Rydberg atoms where adjacent atoms in Rydberg states are prohibited (Rydberg blockade) [6][7][8][9].A non-thermalizing oscillation of domain wall density, which seems to be related to exact scar eigenstates [10], has been observed.
In contrast, in periodically driven (Floquet) cases, nonintegrable systems are believed to satisfy the Floquet version of ETH (Floquet-ETH), which says that all the eigenstates of the time evolution operator for one period cannot be distinguished from a trivial infinite temperature state [11,12].This is a sufficient condition for any initial state to be thermalized to infinite temperature, which can be interpreted as a consequence from the absence of energy conservation.While many-body * mizuta.kaoru.65u@st.kyoto-u.ac.jp physics causes various attractive phenomena in static systems, interacting Floquet systems often become trivial due to Floquet-ETH in the thermodynamic limit, except for a few examples such as Floquet many-body localization [13,14] and Floquet time crystals [15][16][17][18][19][20][21].Thus, quantum many-body scars in Floquet systems (Floquet quantum many-body scars) are also of great interest as a violation of Floquet-ETH.
Some recent studies have tackled the realization of Floquet quantum many-body scars [22][23][24][25].The former references [22,23] consider a system dominated by random unitary matrices preserving charges and dipole moments, and numerically (Ref.[22]) and analytically (Ref.[23]) show the existence of states immune to thermalization.The latter one [25] rigorously constructs Floquet quantum many-body scars realized by quasienergydegeneracy modulo 2π (Floquet-intrinsic scars).Here, we demonstrate a new systematic construction of exact Floquet quantum many-body scars realized by a binary drive composed of uniformly-driven PXP-type static models.In our construction, the instantaneous Hamiltonians share a subspace immune to thermalization, though they have different non-thermal scar eigenstates.Using these properties, we can realize exact Floquet quantum many-body scars showing persistent dynamics both stroboscopically and microscopically without fine-tuning of the switching time.This also enables us to construct a generic periodic drive hosting exact Floquet quantum many-body scars.These results will shed a new light on understanding of nonequilibrium dynamics in Floquet systems.

II. MODELS AND OUTLINE
First, we identify the protocol of driving and show the outline of this paper.Floquet systems are described by a time-periodic Hamiltonian H(t) which satisfies H(t) = H(t + T ) (T : period).Here, we focus on a binary drive as the simplest protocol, described by and then the Floquet operator U f (the time evolution operator for one period T ) is given by What distinguishes Floquet systems from static systems is the non-commutative property of Hamiltonians at different time, and hence we assume To construct Floquet quantum many-body scars, we make use of two different static Hamiltonians H 1 and H 2 hosting quantum many-body scars.With choosing proper uniform PXP-type Hamiltonians, defined on a Hilbert space prohibiting adjacent up spins (Rydberg blockade), we rigorously show the existence of a fourdimensional subspace immune to relaxation to infinite temperature.
Following the above strategy, this paper is organized as follows.In Section III, we introduce two static models hosting quantum many-body scars.One of them is a well-known model called the PXP model.We construct the PY 4 P model as another model inequivalent to the PXP model.In Section IV, we construct a binary drive which shows Floquet quantum many-body scars using these static models, and numerically examine its nonintegrability to demonstrate that it can be a nontrivial example for the violation of Floquet-ETH.Then, we rigorously show the existence of an embedded subspace, which is a subspace completely immune to thermalization.We also demonstrate the real-time dynamics and show thermalization dependent on whether the initial states belong to the embedded subspace (Section V).Finally, we discuss generalization of our binary drive to a generic timeperiodic drive (Section VI) and end up with discussing a possible experimental realization in Rydberg atoms and concluding this paper (Section VII).

A. PXP model and PY4P model
In this section, we introduce the PXP model as a typical static model showing scars, and construct another model called the PY 4 P model with its analogy.Throughout the paper, we consider a one-dimensional Ising chain under open boundary conditions (OBC).Assume that the number of the sites L is a multiple of four.We consider a Hilbert space prohibiting states which include neighboring up spins, and then the dimension of the Hilbert space D L is given by D L = F L+2 (F n : the Fibonacci sequence) [26].The PXP model under OBC is described by the following Hamiltonian [4,6]: Here, we denote Pauli operators on the i-th site by I i ,X i ,Y i and Z i .P i represents the projection to a down spin state on the i-th site.Such a Hamiltonian on the constrained Hilbert space is realizable in Rydberg atoms by quite strong repulsive interactions between adjacent atoms in Rydberg states [4].The PXP model is known to violate ETH since the following four eigenstates |Γ x αβ (α, β = 1, 2) are not thermal [10]: and they satisfy For convenience, we define the b-th block by a pair of neighboring spins at the (2b − 1)-th site and 2b-th site.Then, each block can take three states ↑↓, ↓↓, and ↓↑, which we denote as s = L, O, R respectively.Then, the states |Γ x αβ are rewritten as follows: where the number of the blocks is L b = L/2.The derivation of these relations is given in Ref. [10].
Next, we construct another static model, the PY 4 P model, described by which possesses quadruple lattice-periodicity.We define four states |Γ y αβ by In a similar way to the derivation of the PXP model, we can show that these four states are exact scar eigenstates of the PY 4 P model as follows: The PY 4 P Hamiltonian H Y is related to the PXP Hamiltonian H X by a unitary transformation as follows: Thus, properties of the PXP model, such as the nonintegrability and the violation of ETH, are inherited to the PY 4 P model.

B. Exact Scar eigenstates of PY4P model
In this section, we derive the four exact scar eigenstates of the PY 4 P model, Eq. ( 17).This calculation is necessary to obtain the physical interpretation of the absence of thermalization in our periodically-driven model (Section IV) and generalize our binary drive to generic timeperiodic drives (Section VI).This part can be skipped if the reader is only interested in the existence of exact Floquet quantum many-body scar eigenstates and other thermal behaviors in the model.
Proof.-We consider the PY 4 P Hamiltonian H Y based on blocks of neighboring two spins [27].We denote a state of the b-th block composed of the (2b − 1)-th and are obtained.The boundary terms of H Y are given by Using the properties Thus, only the single-body terms have a nonzero contribution: To calculate this, we consider superblocks corresponding to pairs of neighboring blocks.Here, we define a u-th superblock by a pair of (2u − 1)-th and 2u-th blocks.Each superblock has seven degrees of freedom t u = (s (2u−1) , s 2u ) = OO, LL, RR, OL, OR, LO, RO, while RL and LR are not included in |Γ y αβ because of A R A L = A L A R = 0.Then, the states |Γ y αβ are rewritten in the following form, The matrices Ãt are given by Ãt = A s A r , t = (s, r), The action of h y,(1) When we define Y = σ y / √ 2, the matrices Ãt and F t are related as follows: Finally, we obtain In the last equality, we have used Eq. ( 16) indicating that v 1,2 are eigenvectors of Y .Thus, we obtain Eqs.(17).

IV. FLOQUET QUANTUM MANY-BODY SCARS
In this section, we construct a binary drive showing Floquet quantum many-body scars.To confirm that our model becomes a nontrivial example violating Floquet-ETH, we numerically demonstrate nonintegrability of our model, and rigorously proves the existence of exact Floquet scar eigenstates, which are distinguishable from infinite temperature states.

A. Model
Assume that the system is a one-dimensional Ising chain where pairs of neighboring up spins are prohibited.Then, we consider a binary drive composed of the static Hamiltonians in the previous section: and then its Floquet operator is written by Here, for Floquet quantum many-body scars to take place, T 1 and T 2 are arbitrary except for the case when either one of them is zero (there is no need for fine-tuning of them).We note the symmetries underlying in this model.First, it possesses an inversion symmetry I which maps each i-th site to (L − i + 1)-th site, and the Floquet operator U f is invariant under the inversion.Second, a nonlocal chiral symmetry C, designated by is also respected.This chiral symmetry makes the spectrum of quasienergy {ε} (the eigenvalues of −i log U f ) symmetric to ε = 0.In addition, if T 1 = T 2 is satisfied, the model also respects a time-reversal symmetry (TRS), described by where we use the relation Eq. ( 18).

B. Nonintegrability
To confirm that the model can be a nontrivial example of violation of Floquet-ETH, we begin with analyzing nonintegrability of the model.Considering the nonintegrability of the PXP model, that of the PY 4 P model [6,28], and the noncommutability [H X , H Y ] = 0, the model is also expected to be nonintegrable.Here, we demonstrate this by calculating level statistics.Using the n-th quasienergy ε n (replaced by eigenenergy E n in static cases) and its gap ∆ n = ε n+1 −ε n , let us define the level spacing ratio r n by r n = min(∆ n /∆ n+1 , ∆ n+1 /∆ n ) and denote their spectrally averaged value by r ≡ r n .When the model is nonintegrable with the increasing system size, r approaches a Gaussian orthogonal ensemble (GOE) value close to 0.53 if it is time-reversal symmetric, or approaches a Gaussian unitary ensemble (GUE) value close to 0.6 otherwise [28][29][30].On the other hand, in integrable systems, it approaches a value close to 0.39, that of Poisson statistics.
Figure 1 (a) shows the numerical result for r calculated by exact diagonalization (ED).Considering the inversion symmetry I, we limit the Hilbert space to the inversion -plus sector, the subspace whose eigenvalue of I is +1.The red upper solid line (blue middle solid line) represents the case of T 1 = 9.5, T 2 = 0.5 without TRS (that of T 1 = T 2 = 5 with TRS) respectively.In each case, r flows to a value close to 0.6 (a value close to 0.53) as the system size grows, and hence we can conclude the nonintegrability of the model.
We also demonstrate the nonintegrablity in terms of entanglement entropy of each Floquet eigenstate.Entanglement entropy of a given state |ψ is defined by where the subsystems A (B) represents the left (right) half of the system respectively.A state |ψ indistinguishable from the infinite temperature state is expected to possess entanglement entropy equal to that of infinite temperature S ∞ , and hence it obeys a volume law in the thermodynamic limit as follows [26]: volume at infinite temperature, while the blue lines represent the averaged entanglement entropy per volume of all the Floquet eigenstates.As the system size L increases, Floquet eigenstates become featureless, with its entanglement entropy approaching the one at infinite temperature.On the other hand, there exist four anomalous low-entangled Floquet eigenstates designated by the four marked points in the figure.As discussed in the following section, these four states are nothing but exact Floquet scar eigenstates, which are distinguishable from the infinite temperature state.This result elucidates both the nonintegrability and the existence of nontrivial Floquet quantum many-body scars.

C. Exact Floquet Scar eigenstates
We rigorously show a violation of Floquet-ETH in our model.To be precise, we prove that there exist four exact Floquet scar eigenstates, which can be distinguished from infinite temperature states.We define S by the fourdimensional subspace spanned by {|Γ x αβ } α,β=1,2 .Within the subspace S, thermalization does not take place under exp(−iH X T 1 ) by its definition.On the other hand, using Eqs.(11)  ( Since this transformation is invertible, the subspace S is identical to the one spanned by {|Γ y αβ } α,β=1,2 .Thus, thermalization does not take place in the subspace S also under exp(−iH Y T 2 ), and hence we can conclude the absence of thermalization in the subspace S under the Floquet operator U f .This can be understood also from local conserved quantities in the embedded subspace S. From Eq. ( 25) and similar relations of the PXP model, the interaction terms of the Hamiltonian disappear within S, and this results in the Floquet operator on S as follows: Therefore, a set of local observables defined by ) exp(−iT 1 h x,( ))} b composes a macroscopic number of local conserved quantities of the system, and this implies the absence of thermalization in S. The existence of the four-dimensional embedded subspace ensures the existence of four exact Floquet scar eigenstates.Two of them given by have quasienergy 0 and are invariant under the chiral symmetry operation C modulo constant.The other two eigenstates are related to each other by C and hence they have quasienergies with the opposite signs, though their analytical forms are too complicated to be shown here.
Since C becomes an onsite symmetry within S due to Eqs. ( 25) and (39), they have the same entanglement entropy, while other pairs outside of S do not.These four states appear in Fig. 1 (b)-(d) as the four marked lowentangled states.Since they are represented by matrix product states with at-most bond dimension 8, their entanglement entropy per length decays with O(1/L), corresponding to the numerical result.Finally, to confirm the violation of Floquet ETH, we examine whether Floquet eigenstates within the subspace S can be distinguished from the infinite temperature state.Here, we focus on a domain wall density defined by Then, the expectation value of D b is obtained as lim for any state |ψ ∈ S, ψ|ψ = 1 including the renormalized eigenstates of U f within S [See Appendix A1 C].On the other hand, the expectation value of domain wall density at infinite temperature in the thermodynamic limit is 2/ √ 5φ = 0.542 . . .= 2/3 (φ: the golden ratio) under OBC [See Appendix A2 B].Therefore, it can be concluded that the model Eq.(37) violates Floquet-ETH.

V. REAL-TIME DYNAMICS
We discuss the real-time dynamics in and out of the embedded subspace S, which possibly leads to experimental detection of Floquet quantum many-body scars.
First, let us consider the real-time dynamics within the embedded subspace S. We define renormalized scar eigenstates by | Γ x αβ ≡ |Γ  [31].With this basis, the Floquet operator U f | S is represented by and stroboscopic dynamics of any observable is determined by its matrix representation.For example, the total magnetization in the x direction, lim L→∞ i X i , is given by diag(0, √ 2, − √ 2, 0) and shows a persistent oscillation in general, while the local Pauli operator Z i and the domain wall density D b remain constant since they are proportional to identity in S [31].On the other hand, concerning the microscopic dynamics, generic initial states in S, different from |Γ 0 , show some persistent motion since |Γ 0 is the unique simultaneous eigenstate of H X and H Y in S. We show typical real-time dynamics in Fig. 2 (a).
Next, we demonstrate the behavior outside of the embedded subspace S. Following the nonintegrability of the model, generic initial states are expected to relax to infinite temperature, and we numerically confirm it by ED [See Fig. 2  √ 2 and denote the projection to the constrained Hilbert space prohibiting adjacent up spins as P.They have an exponentially small overlap with S in terms of the system size, and |ψ 1 (|ψ 2 ) is an infinite-temperature state (a state with finite temperature β eff ) under the PXP Hamiltonian H X because of Here, the temperature of a state |ψ is determined by solving the energy conservation in terms of β. Figure 2 (b) and (c) show the dynamics at T 1 = 9.5 and T 2 = 0.5, which we choose so that pre-equilibration under an effective static Hamiltonian in the high-frequency regime can be avoided [32][33][34][35].The model shows thermalization to infinite temperature regardless of initial states outside of the embedded subspace in contrast to the static PXP model and the PY 4 P model, where the system relaxes to thermal states with a certain temperature depending on its initial states.
In the static PXP model, there also exist some special states such as |Z 2 ≡ |↑↓↑↓ . . .which show nonthermalizing behaviors of observables (e.g.domain wall density), though their overlap with the embedded subspace is small enough [4,6,8,10].We demonstrate the existence of such special states in the periodically-driven model in Appendix A3 [36].The numerical result says that such a special state is thermalized to infinite temperature as well as other generic initial states out of the embedded subspace.In the static PXP model and the PY 4 P model, three types of dynamics-complete absence of thermalization within the embedded subspace, seemingly non-thermalizing behavior of some special states, and thermalization of other generic states, are observed.By contrast, we conclude that the periodically-driven model only shows complete absence of thermalization to infinite temperature within the embedded subspace or thermalization of other generic states.

VI. GENERALIZATION
We generalize our binary drive to generic time-periodic drives.For this purpose, we first introduce another static model, the PZ 4 P model defined by Then, the PZ 4 P model possesses the following eigenstates: where {|Γ z αβ } is given by This derivation is given in Appendix [31].Since the Hamiltonian H Z commutes with Z i for every i, the PZ 4 P model is integrable and does not host nontrivial phenomena by itself.However, by combining the PXP Hamiltonian and the PY 4 P Hamiltonian, we can construct nontrivial models described by with a = (sin θ cos ϕ, sin θ sin ϕ, cos θ).This generalized model is no longer unitarily equivalent to the PXP model or PY 4 P model like Eq. ( 18).We can compose four exact scar eigenstates of this Hamiltonian, given by Using Eq. ( 35) and similar relations for the PXP model [9,10] and the PZ 4 P model [31] results in with Σ = (σ x , σ y , σ z )/ √ 2, and then we confirm this by the fact that the vectors u a α are eigenvectors of a • Σ with eigenvalues (−1) α−1 / √ 2. Now, we are ready to construct a generalized version of the binary drive Eq. (37), represented by Since |Γ a αβ is represented by a linear combination of {|Γ x αβ } for arbitrary a, the dynamics under H(t) is closed within the subspace spanned by {|Γ x αβ }.Thus, this model always has a four-dimensional embedded subspace S. Using the fact that { u a α } α=1,2 is a complete orthonormal basis of C 2 , which is independent of the choice of a, is an eigenstate of H a whose eigenvalue is 0. Thus, we always provide one of the four exact Floquet scar eigenstates of H(t) with |Γ 0 , whose quasienergy is 0.

VII. DISCUSSION AND CONCLUSIONS
Before concluding the paper, let us briefly discuss how to realize the Hamiltonian (37) showing Floquet quantum many-body scars.The PXP model, which hosts static quantum many-body scars, is experimentally realized in Rydberg atoms [4].Each atom can occupy the ground state ↓ and the Rydberg state ↑, which is an excited state with a large quantum number.Since the repulsive interactions between neighboring atoms in the Rydberg state are quite large, neighboring ↑↑ pairs are prohibited.The Rabi oscillation in this limited subspace results in the PXP Hamiltonian H X .
Once the PXP Hamiltonian H X is realized, our model is also realizable.We consider a potential with quadruple periodicity of the lattice, where the coefficients c i are given by Eq. ( 14).Then, using the unitary equivalence Eq. ( 18), the Floquet operator U f is = e −iZ4(π/4) e −iH X T2 e −i(−Z4)(π/4) e −iH X T1 .(65) Thus, our model can be realized by switching the PXP Hamiltonian and the quadruple-periodic potential alternately in the Rydberg atoms.
In summary, we have constructed a nonintegrable model which hosts Floquet quantum many-body scars, driven by uniformly-imposed Hamiltonians on the constrained Hilbert space prohibiting adjacent pairs of up spins.We have rigorously shown that the model violates Floquet-ETH with the fact that instantaneous Hamiltonians share a subspace immune to thermalization although the scar eigenstates do not correspond to one another.The entanglement spectrum of Floquet eigenstates and the real-time dynamics of the model indicate that any initial state outside of the embedded subspace is thermalized to infinite temperature.We have also discussed a possible experimental realization of the model in Rydberg atoms, and thereby our result would contribute to understanding how closed Floquet systems equilibrate to infinite temperature.This is nothing but the definition of Fibonacci sequence, and hence we obtain We evaluate the entanglement entropy at infinite temperature when we split the system in half.Denoting the left half (the right half) by A (B), the reduced density operator of the infinite temperature state ρ ∞ = I D L /D L for the subsystem A is and in the thermodynamic limit, the entanglement entropy per volume becomes lim

B. Observables
The expectation value of a certain observable O at infinite temperature is given by where K L represents a set of classical spin configurations of an L-site chain which includes no adjacent up spins.We here discuss the expectation values of the Pauli operators X i , Z i , and the domain wall density D b .Since the operator X i has only off-diagonal elements in the basis {| σ } σ∈K L , we obtain Next, we consider the Pauli z operator Z i .When we fix the i-th spin by ↑ (↓), the number of possible spin configurations is . Therefore, we obtain the expectation value for finite-size and infinitesize systems as follows: In a similar way, we obtain the domain wall density as follows:

A3. DYNAMICS OF SPECIAL SCAR STATES
In this section, we discuss the relationship between the model showing exact Floquet many-body scars and a nonthermalizing oscillation of observables in the static PXP model [4,6].In the PXP model, the embedded subspace spanned by the exact scar eigenstates is perfectly immune to thermalization.However, there also exist some special scar states seemingly immune to thermalization although they are not included in the embedded subspace.In fact, an extremely long-term oscillation of the domain wall density is observed under the preparation of special initial states such as |Z 2 = |↑↓↑↓ . . .and |Z 3 = |↑↓↓↑↓↓ . . . in Rydberg atoms.Our periodically-driven model is composed of the PXP Hamiltonian H X and the PY 4 P Hamiltonian H Y , and each of them shows non-thermalizing oscillations of observables under the specific initial-state preparation (See Fig. A1 (a) for the non-thermal behavior of the PY 4 P model).Here, we numerically examine whether a non-thermalizing oscillation appears also in the driven cases, and discuss its origin.
Figure A1 shows the real-time dynamics when we begin with the Z 2 ordered state |Z 2 [See (b) and (c)].Figure A1 (c) indicates that the model for the relatively-small period T = T 1 + T 2 = 1 shows non-thermalizing oscillations of the domain wall density D L/2 and the Pauli operator Z L/2 .These non-thermalizing behaviors are expected to originate from pre-equilibration of Floquet systems in the high-frequency regime [32][33][34][35].When the local energy scale of the Hamiltonian is small enough compared to the frequency, its stroboscopic dynamics is well described by a static effective Hamiltonian given by the Floquet-Magnus expansion.Up to the lowest order in T , the static effective Hamiltonian for our model is given by the time-averaged one over one period, Thus, through Fig. A1 (c), we observe non-thermalizing behaviors caused by static quantum many-body scars in the periodically-driven model.On the other hand, when the local energy scale is comparable to the frequency or larger than it, we expect that effective static behaviors do not appear.Figure A1 (b) shows the dynamics for such a Floquet intrinsic regime, where the local energy scale 1 is larger than the frequency 2π/T = π/5.This result represents that the domain wall density D L/2 and the Pauli operator Z L/2 quickly relax to the values of infinite temperature states, and that the non-thermalizing oscillations disappear by the periodic drive in spite of the instantaneous Hamiltonians H X and H Y .We focus on the origin of this behavior below.
We expect that this can be explained in terms of forward scattering approximation (FSA) [6,7].By means of FSA, we can obtain an approximately-closed subspace of the dynamics under a certain initial state, and write down the effective Hamiltonian within this subspace.In the case of the PXP model, the Hamiltonian H X can be divided into two terms as follows: under open boundary conditions.When we begin with |Z 2 , we define a set of orthonormal states by When we denote the Hamming distance (the smallest number of spin flips required to convert two states each other) from |Z 2 by H.D., the term H + X (H − X ) increases (decreases) H.D. by one.Thus, the state |v n becomes a superposition of states whose spin configurations satisfy H.D. = n, and thereby we obtain |v L = | Z2 and |v L+1 =0.From the numerical calculation up to L = 32 [7], it is known that the dynamics from the initial state |Z 2 under H X is approximately closed within the subspace R X , spanned by {|v n } L n=0 [See Fig. A2 (a)].This is one of the possible explanations for long-term non-thermalizing oscillations in the PXP model [6,7].
In a similar way, we apply FSA to the PY 4 P model.The Hamiltonian is written as is n as well.However, in the case of the PY 4 P model, a spin flip gives additional phases +i or −i to each state depending on the flipped-spin's site due to the signs of c i and the coefficients in Eq. (A38) [See Fig. A2 (b)].Then, we can understand the thermalization to infinite temperature under the Floquet drive with the initial state |Z 2 from the difference between the closed subspaces R X and R Y .We immediately obtain |v 0 = |w 0 , |v L = |w L , and v i |w j = 0 for different i, j by their definitions.First, let us consider the overlap between |v 1 and |w 1 .These states are equally-weighted superpositions of states where one of the odd sites are flipped from |Z 2 .Since the number of the states with an additional phase +i is equal to that of the states with an additional phase −i, we obtain depends on the parity of p. Choosing odd p flipped sites with the blue squares is equivalent to choosing even n − p flipped sites with orange circles.Assume that the system size L is large enough, and then we can neglect the effect of the boundaries.With considering the symmetry of the blue squares and the orange circles in the bulk, the total contributions with odd p is equal to that for even p.Thus, we obtain lim L→∞ v n |w n = 0, for odd n (A40) because N H.D.=n grows with increasing L. These macroscopic number (at least L/2) of orthogonality relations represent that a generic state in R X (or R Y ) flows out of R X (or R Y ) under the time evolution by the Hamiltonian H Y (or H X ).Finally, we conclude that, under the periodic switching of H X and H Y , the dynamics from |Z 2 is no longer closed within the original subspaces R X or R Y , and hence thermalization to infinite temperature is observed without showing non-thermalizing oscillations.

Figure 1 (
Figure 1 (b)-(d) shows the numerical results at T 1 = 9.5 and T 2 = 0.5.The red lines are entanglement entropy per FIG. 2. (a) Real-time dynamics of total x spin i Xi under the initial states within S at T1 = 9.5, T2 = 0.5.The states |Γ x 12 and |Γ y 12 show a persistent oscillation.(b) Real-time dynamics under the initial state |ψ1 , which is at infinite temperature under HX .The red lines represent the values at infinite temperature.(c) Real-time dynamics under the initial state |ψ2 , which is at finite temperature β eff under HX .The lower solid yellow lines represent the finite-temperature-equilibrium values under HX [β eff is obtained by numerically solving Eq. (52)].The Floquet drive breaks such a feature of the initial state, and makes the observable approach the values at infinite temperature (the upper solid red lines).
(b), (c)].We consider two different initial states |ψ 1 ≡ |↓↓ . . .and |ψ 2 ≡ P |− ⊗L / √ D L , where we define |− by (|↑ −|↓ )/ ) Here, we define K ↑ (K ↓ ) by a set of configurations of L/2 spins, whose spin at the right edge is ↑ (↓).Using the equations |K ↑ | = D L/2−2 and |K ↓ | = D L/2−1 , we obtain the entanglement entropy at infinite temperature as follows: FIG. A1.Real-time dynamics under the special initial state |Z2 = |↑↓↑↓ . . .: (a) under the static PY4P Hamiltonian (b) under the binary drive at T1 = 9.5, T2 = 0.5 (c) under the binary drive at T1 = 0.95, T2 = 0.05.(a) Both of the domain wall density and the Pauli z operator show long-lasting oscillations without approaching their thermal equilibrium values at T = ∞.(b) The observables rapidly approach those of the infinite temperature due to the drive.(c) Non-thermalizing behaviors of the observables are observed as in the static case in spite of the existence of the drive.These are brought by pre-equilibration under an effective static Hamiltonian in the high-frequency regime of Floquet systems.

4 FIG
FIG. A2.Schematic pictures of approximately-closed subspaces obtained by FSA: (a) for the PXP model and (b) for the PY4P model at L = 4.Each layer denoted by |vn or |wn is composed of product states whose Hamming distance from |Z2 is fixed to n.In the PY4P model (b), blue squares and orange circles represent additional phases +i and −i obtained when the spins at their positions are flipped.The term "phase" at each state means its coefficient due to these additional phases.

v 1 |w 1 = 1 N
H.D.=1 {(−i) × L/4 + i × L/4} = 0, (A39) where N H.D.=n represents the number of possible spin configurations with H.D. = n under the constrained Hilbert space.Next, we consider the overlap v n |w n for generic n in the limit of L → ∞.Here, let p denote the number of flipped spins with an additional phase +i [the number of the blue squares flipped in Fig. A2 (b)].Then, each state in |w n has an additional phase given by (+i) p (−i) n−p = (−1) p (−i) n , and the overlap v n |w n is determined by its summation over states with H.D. = n.Let us consider the case where n is odd.The additional phase (−1) p (−i) n (15)15), we obtain + i P i+1 , H − Y = (H + Y ) † ,(A38)and then the dynamics under H Y is approximately closed within the subspace R Y spanned by|w n = (H + Y ) n |Z 2 /||(H + Y ) n |Z 2 || L n=0 .Astate |w n is a superposition of states whose Hamming distance from |Z 2 H + Y = i i:odd c i P i−1 S − i P i+1 − i i:even c i P i−1 S