Floquet Conformal Field Theory

MIT-CTP/4999 Floquet conformal ﬁeld theory

Xueda Wen1, ∗ and Jie-Qiang Wu2 1Department of Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA 2Center for Theoretical Physics, Massachusetts Institute of Technology, Cambridge MA, 02138 USA (Dated: May 23, 2018) Given a generic two-dimensional conformal field theory (CFT), we propose an analytically solvable setup to study the Floquet dynamics of the CFT, i.e., the dynamics of a CFT subject to a periodic driving. A complete phase diagram in the parameter space can be analytically obtained within our setup. We find two phases: the heating phase and the non-heating phase. In the heating phase, the entanglement entropy keeps growing linearly in time, indicating that the system keeps absorbing energy; in the non-heating phase, the entanglement entropy oscillates periodically in time, i.e., the system is not heated. At the phase transition, the entanglement entropy grows logarithmically in time in a universal way. Furthermore, we can obtain the critical exponent by studying the entanglement evolution near the phase transition. Mathematically, different phases (and phase transition) in a Floquet CFT correspond to different types of Möbius transformations.

Introduction The dynamics of periodically driven (Flo- be obtained if we approach the phase transition from the heat- quet) many-body systems has received extensive attentions re- ing phase, by studying the slope of the linear growth of en- cently. It sheds light on fundamental issues in condensed mat- tanglement entropy. At the phase transition, in the long time ter physics and statistical physics such as the phase structures limit, the entanglement entropy grows logarithmically in time c and thermalization. Striking examples include Floquet topo- as 3 log t, where c is the central charge of CFT. We confirm logical insulators,1–10 Floquet symmetry protected/enriched our CFT result with a numerical simulation based on a free topological phases,11–15 Floquet time crystals,16–21 and Flo- fermion lattice model. We also find an elegant mathematical quet thermodynamics.22–26 structure underlying the phase diagram. The heating phase, In this work, we are interested in the Floquet dynamics of a non-heating phase and phase transition in the Floquet CFT (1+1) dimensional quantum critical point which is described correspond to three kinds of Mobius¨ transformations, i.e., hy- by a conformal field theory (CFT). To our knowledge, little perbolic, elliptic, and parabolic transformations, respectively. attention has been paid in this direction. In Ref.27, the Flo- Our setup applies to a family of periodically driven CFTs. quet dynamics of a boundary driven quantum critical point Our setup Now we consider a generic (1+1) dimensional was studied. It was found that, depending on the driving CFT defined on a finite space of length L, with conformally frequency, there are multiple dynamics regimes, including a invariant boundary conditions imposed at x = 0 and x = L, heating regime and several other non-heating regimes. Since respectively.32 The initial state is prepared as the ground state the energy injected (from the boundary) per cycle is not exten- |Gi of Hamiltonian H0, and then we drive the system in the sive in system size, it is still an open question on the Floquet following way dynamics of a bulk-driven quantum critical point. It is well known that CFTs after a quantum quench have brought to us H1 much insight in the non-equilibrium dynamics of many-body ... 28–30 H0 systems. Now, for a periodically bulk-driven CFT, it is time desirable to understand its Floquet dynamics. However, an analytically solvable setup is still lacking. Here H0 denotes a uniform Hamiltonian of the form We fill this gap by proposing an analytically solvable setup for a bulk-driven Floquet CFT. Both the correlation functions Z L dx H = T (x), (0.1) and the entanglement entropy can be analytically obtained in 0 2π ττ the whole parameter space within our setup. We find two dif- 0

ferent phases depending on the driving frequency, namely the where Tττ (x) is the ‘time-time’ component of the stress ten- 31 arXiv:1805.00031v2 [cond-mat.str-el] 21 May 2018 heating and non-heating phases. In the heating phase, the sor. For later convenience, we have deﬁned our theory in Eu- entanglement entropy keeps growing linearly in time, which clidean space with w = τ + ix, so that Tττ = Tww + Tw¯w¯ =: indicates that the system keeps absorbing energy; in the non- T + T . The other Hamiltonian H1, among a family of 61 heating phase, the entanglement entropy keeps oscillating in candidates, is constructed by deforming H0 as follows time, indicating that the system is not heated. In particular, in the high frequency driving regime of the non-heating 1 H = H − (H + H ) , (0.2) phase, the oscillation period of entanglement entropy is in- 1 0 2 + − dependent of the driving frequency. In addition, as we ap- R L dx ±2πw/L ∓2πw/L¯ proach the phase transition, the oscillation period of entan- where H± = 0 2π e T (w) + e T (w ¯) . H1 glement entropy diverges, based on which we can extract the itself describes a sine-square deformed CFT which was ex- critical exponent ζ = 1/2. The same critical exponent can tensively studied recently.33–51 It was found that a CFT with 2

O(x=l) 2.5 H0 H1 2.0 ......

Lorentzian time Lorentzian 1.5 Heating T1 /L τ= -∞ +∞ 1.0

Euclidean time 0.5 Non- heating FIG. 1. Path integral representation of the single-point correlation 0.0 function hψ(t)|O(x)|ψ(t)i in w-plane, with w = τ + ix. 0.0 0.5 1.0 1.5 2.0 2.5 3.0 T0 /L

FIG. 2. (Part of) Phase diagram for a Floquet CFT, plotted according Hamiltonian H1 has a continuous Virasoro algebra that results in a continuous energy spectrum,45,46 which is in contrast with to Eq.(0.7). The solid dots are obtained from a numerical simulation 52 based on a free fermion chain with L = 500 and T0/L < 2. H0 that has discrete energy spacing ∝ 1/L. In short, starting from the ground state |Gi of H0, we drive the system with H1 for a time interval T , and then with H for a time interval T , 1 0 0 where we have associated to the Mobius¨ transformation z1 = πτ0 and repeat this driving procedure. To characterize the Floquet πτ1 L f(z) a matrix H, and defined a := (1 + L ) · e , b := dynamics of the system, we study the correlation functions −πτ0 πτ0 −πτ0 πτ1 πτ1 πτ1 − · e L , c := · e L , and d := 1 − · e L . and entanglement entropy evolution at time t = n(T0 + T1), L L L with n = 0, 1, 2, ··· . As will be seen shortly, the Mobius¨ transformation in Eq.(0.4) The wavefunction after n cycles of driving can be written determines the Floquet dynamics of our periodically driven CFT. Note that the Mobius¨ transformation has been normal- as |ψ(t)i = e−iH0T0 e−iH1T1 ··· e−iH0T0 e−iH1T1 |Gi, based on which we can evaluate the multi-point correlation func- ized so that ad − bc = 1. By defining the trace square of H as 2 tions. For simplicity, now let us consider the single point cor- σ(H) := Tr(H) , it is known that the value of σ(H) classi- relation function hψ(t)|O|ψ(t)i. Its path integral representa- fies different types Mobius¨ transformations. After analytical tion in w-plane is shown in Fig.1, with τ ∈ (−∞, ∞) and continuation τ0 → iT0 and τ1 → iT1, one can find that x ∈ [0,L], i.e., the path integral is defined on a strip. Note σ(H) = 4(1 − ∆), (0.5) there is Lorentz (real) time evolution introduced by the driving. To evaluate hOi, we go to the Euclidean space by writing with |ψ(t)i as |ψ(τ)i = e−H0τ0 e−H1τ1 ··· e−H0τ0 e−H1τ1 |Gi, and 2 h πT1 i 2 πT0 πT1 2πT0 do the analytical continuation τ → iT , τ → iT in the final ∆ = 1 − sin + · sin . (0.6) 0 0 1 1 L L L L step. One-cycle driving Before studying the effect of n-cycle Depending on the values of ∆, there are three types of Mobius¨ driving, it is helpful to check how a primary operator O transformations as follows:54 evolves under one-cycle driving. First, with a conformal map-  0 ≤ σ < 4, 0 < ∆ ≤ 1, Elliptic, 2πw  ping z = e L , we map the strip in w-plane to the complex  σ = 4, ∆ = 0, Parabolic, (0.7) z-plane, where the boundaries along x = 0 and x = L in w-  plane are mapped to the slit along the half real axis Re(z) ≥ 0  σ > 4, ∆ < 0, Hyperbolic. in z-plane. Based on the study in Ref.51, one can find that un- The elliptic, parabolic, and hyperbolic transformations are der one-cycle driving, the operator O in z-plane evolves from conjugate to the operation of rotation, translation, and dilation (z, z¯) to (z1, z¯1) as follows in z-plane, respectively. As we will see, ∆ = 0 determines the phase transition in a Floquet CFT (see Fig.2), and the elliptic h h¯ † ∂z1 ∂z¯1 and hyperbolic transformations correspond to non-heating and U O(z, z¯) Ueff = O(z1, z¯1), (0.3) eff ∂z ∂z¯ heating phases in the phase diagram, respectively. n-cycle driving To have a more intuitive picture on the where we have defined the time evolution operator Ueff := effect of different Mobius¨ transformations in Eq.(0.7), let us −H0τ0 −H1τ1 53 ¯ e e , and h (h) is the conformal dimension of consider the n-cycle driving. It can be found that the operator ¨ O. In particular, z1 (z¯1) is related to z (z¯) by a Mobius O in z-plane, after n cycles of driving, is driven from (z, z¯) to transformation:51,54 (zn, z¯n), with az + b a b zn − γ1 n z − γ1 z1 = f(z) = → H = , (0.4) = η · , (0.8) cz + d c d zn − γ2 z − γ2 3

1 8 T=10 0 T=40 7 T=250 −1 1 0 T=450 −1 0 5 10 15 20 25 30 6 CFT

5 (t) A S 1 4 0

−1 1 3 2 0 −1 0 5 10 15 20 25 30 1 n 2 3 4 10 10

z z n 1 FIG. 3. Trajectory of n in -plane as a function of in the non- 0 2000 4000 6000 8000 10000 heating phase (top) and the heating phase (bottom). We choose T0 = t T1 = L/10 in the non-heating phase and T0 = T1 = L/2 in the heating phase. The subsystem length is chosen as l = L/2. FIG. 4. Comparison of entanglement entropy evolution between CFT calculations and numerical simulations in heating phase, non- heating phase, and at the phase transition (inset). We choose T0 = and similarly for z¯n. Here γ1 and γ2 are called ‘fixed points’ of the Mobius¨ transformation and are determined by f(γ) = γ T1 = T , L = 500 and l = L/2. The phase transition happens at T ∗/L ' 0.416 in CFT prediction and at T ∗/L ' 0.418 in the in Eq.(0.4), with the explicit expression γ1,2 = a − d ∓ numerical simulation for T < L. p(a − d)2 + 4bc/2c. The multiplier η shows qualitatively different behaviors depending on the types of Mobius¨ transformations (after analytical continuation): (0+1)-d quantum Mathieu’s harmonic oscillator. These two systems have similar phase diagrams due to the underlying  i2φ e , Elliptic, algebra structures which are isomorphic to each other.56  η = 1, Parabolic, (0.9) Single-point function and Entanglement entropy To char-  2φ0 acterize the Floquet dynamics, now let us focus on two phys- e , Hyperbolic. ical quantities, i.e., single-point correlation functions and 0 entanglement entropy for a subsystem A = [0, l]. For φ and φ are real functions of driving periods T0 and T1, and the explicit expressions will be given in the following discus- a primary operator O, one has hψ(t)|O(w, w¯)|ψ(t)i = h h¯ h h¯ ∂z ∂z¯ ∂zn ∂z¯n sions on entanglement entropy. Several remarks here: First, ∂w ∂w¯ ∂z ∂z¯ hO(zn, z¯n)iz, where h· · · iz as shown in Fig.3, for ∆ > 0, i.e., the Mobius¨ transformation represents the correlation function of “··· ” in z-plane, and b 1 1 1 h 2i 2h b is elliptic, η is a phase, and one can find that the trajectory hO(zn, z¯n)iz = A · ( √ √ ) · ( √ √ ) . Here A O 4 zn z¯n zn− z¯n O of zn in the complex z-plane keeps oscillating as a function is an amplitude depending on the selected boundary condi- 55 of n. On the other hand, for ∆ < 0, i.e., the Mobius¨ trans- tion |bi, and is a UV cutoff which may be interpreted as the formation is hyperbolic, zn will converge to one of the fixed lattice constant in a lattice model. Then the α-th Renyi en- 0 0 points γ1,2 (depending on φ > 0 or φ < 0) exponentially tanglement entropy S(α)(t) for A = [0, l] is directly related in n, and will not come back to its initial value. This dif- A to the single point correlation function of twist operator Tα, ference will result in different behaviors of correlation func- which is itself a primary operator with conformal dimension tions and entanglement entropy evolution. Second, for ∆ = 0, h = h¯ = c α − 1 , where c is the central charge and α is i.e., the Mobius¨ transformation is parabolic, one can find that 24 α the Renyi index. Explicitly, one has57,58 η = 1 and the two fixed points merge into a single one, namely γ1 = γ2 = γ = (a − d)/2c. In this case, one cannot use 1 S(α)(t) = loghψ(t)|T |ψ(t)i, (0.11) Eq.(0.8) to determine the trajectory of zn. It can be found that A 1 − α α zn is now determined by where Tα is inserted at w = 0 + il in the w-plane. In later 1 1 = + n · β, (0.10) discussion, we will use the von Neumann entropy defined by zn − γ z − γ (α) SA(t) := limα→1 SA (t). Based on Eq.(0.11), one can infer (α) where β = c [see the expression of c below Eq.(0.4)] is the the behavior of single-point correlation function from SA (t) 59 so-called ‘translation length’. Then, zn converges to the fixed (and vice versa), and therefore we will mainly focus on the 1 point γ in the way zn − γ ∝ n for large n, in contrast to the entanglement entropy hereafter. In 59, we have obtained the exponential convergence in the hyperbolic case. As a remark, analytical expression of SA(t) for A = [0, l] with l ∈ (0,L) it is interesting to compare our (1+1)-d Floquet CFT with the under arbitrary driving periods T0 and T1. Since the expres- 4

Mobius Transformation Conjugate to Multipliers Entanglement growth Single-point function Phases in Floquet CFT Elliptic Rotation η = ei2φ Oscillating Oscillating Non-heating Parabolic Translation η = 1 Logarithmic Power-law decay Phase transition 0 Hyperbolic Dilation η = e2φ Linear Exponential decay Heating

TABLE I. Summary of correspondence between M¨obius transformations and different phases (and phase transition) in a Floquet CFT. sion is quite involved in general, as an illustration, we will mainly focus on l = L/2, for which the entanglement entropy 16 CFT 12 has an elegant expression. Numerics

Non-heating phase In the non-heating phase, both the cor- / L 8 E relation functions and the entanglement entropy keep oscillat- T ing in time. This phase corresponds to the case ∆ > 0 in 4 ¨ 0 Eq.(0.5) and the Mobius transformation is elliptic. One can 0 0.1 0.2 0.3 0.4 ﬁnd the entanglement entropy of subsystem A = [0, L/2] as T / L

c h L R cos(2nφ + ϕ) − K i 4 S (t) ' log · , for ∆ > 0, 10 A 6 π R cos ϕ − K E

T Numerics (0.12) CFT ∝ −1/2 where the driving time is deﬁned as t := n(T0 + T1). Here y (T*−T) 3 10 (and in the following) we use “'” instead of “=” because 0 1 10 10 we only keep the leading term of SA(t). The subleading T*−T constant term that depends on the boundary condition is of order O(1) and is neglected hereafter. The parameters in FIG. 5. (Top) Oscillation period TE of the entanglement entropy Eq.(0.12) depend on the driving periods T0 and T1 as follows: for A = [0, L/2] as a function of the driving period T . (We choose i2φ πT L πT e := (Q−iP )(Q+iP ), with Q := sin 0 − cos 0 , T = T = T .) (Bottom) Scaling behavior of T near the phase √ L πT1 L 0 1 E and P := L ∆. Reiϕ := W cos πT0 + 1 + iP sin πT0 and transition. πT1 L L K := W cos πT0 + W 2, with W := cos πT0 + L sin πT0 . L L πT1 L 1 For n = 0, i.e., there is no driving, one has SA(t = 0) = HF = 2 (H0 + H1), which has been studied in in Ref.51. c πL 6 log , which is the entanglement entropy in the ground Therein it was found that the entanglement entropy√ evolution 51 state of H0, as expected. indeed displays oscillations with period TE = 2L/ 3. (See There are several remarkable features for SA(t) in also 59 for more discussions.) Eq.(0.12): (i) SA(t) oscillates as a function of driving cy- Heating phase In contrast to the non-heating phase, the cles n all the way (and so does the single-point correlation entanglement entropy keeps growing in time in the heating function59), indicating that the system is not heated. The os- phase, and the single-point function decays exponentially in cillation period of entanglement entropy in time t is time. This phase corresponds to the case ∆ < 0 in Eq.(0.5) and the Mobius transformation is hyperbolic. The entangle- TE = π · (T0 + T1)/|φ|. (0.13) ment entropy for A = [0, L/2] has the expression

(ii) In the high-frequency driving limit T0,T1 L, SA(t) c h L R0 cosh(2nφ0 + ϕ0) − K i only depends on the ratio of T and T .59 Here let us take SA(t) ' log · , for ∆ < 0. 1 0 6 π R0 cosh ϕ0 − K T0 = T1 = T , then one can ﬁnd that in the limit T L, S (t) has a simple form (0.15) A As before, here we neglect the subleading constant term. 0 c L c √ The parameters in Eq.(0.15) are as follows. e2φ := (Q + S (t) ' log + log 2 − cos( 3πt/L), (0.14) A 6 π 6 P )/(Q − P ), where Q has the same expression as the non- heating case, i.e., Q := sin πT0 − L cos πT0 , but P is L πT1 L where t := n(T + T ) = 2nT . Then the oscillation pe- √ 0 0 1 √ now expressed as P := L −∆. R0eϕ := W cos πT0 + πT1 L riod of SA(t) is TE = 2L/ 3, which is proportional to the 0 πT0 0 −ϕ πT0 πT0 length of the system. In other words, in the high frequency 1 − P sin L , R e := W cos L + 1 + P sin L , and K := W cos πT0 + W 2, with W := cos πT0 + L sin πT0 . limit, TE is independent of the driving frequency, as shown L L πT1 L in Fig.5. To further understand the result in Eq.(0.14), we Compared to SA(t) for the non-heating phase in Eq.(0.12), all note that in the high frequency limit T L, one may con- the parameters are deﬁned similarly except that ∆ → −∆ and sider the approximation e−H0T e−H1T ' e−(H0+H1)T . Then therefore P → iP . The ‘cos’ term in Eq.(0.12) now becomes the high-frequency driving limit of Floquet dynamics cor- a ‘cosh’ term, which may be intuitively viewed as a transition responds to a single quench with the effective Hamiltonian from ‘real time’ to ‘imaginary time’. Again, for n = 0, SA(t) 5

x 10−3 1 Different from SA(t) in Eq.(0.18), now SA(t) decreases ﬁrst and then grows in time. Again, in the large n limit, one has 0.8 c SA(t) ' 3 log t. A typical plot for the entanglement entropy 0.6 at the phase transitions can be found in Fig.12. E

k The correspondence between different phases and Mobius¨ 0.4 transformations et al. is summarized in Table I. CFT 0.2 Numerics Near the phase transition Now let us check the entanglement entropy evolution near the phase transition. First, as 0 0.4 0.5 0.6 0.7 0.8 0.9 1 we approach the phase transition from the non-heating phase, T / L as shown in Fig.5, one can find that the oscillation period of S (t) diverges. By taking T = k · T with arbitrary k > 0, k S (t) T A 1 0 FIG. 6. The slope E of linear growth in A as a function of one can find that59 in the heating phase. We choose T0 = T1 = T and L = 500 in ∗ the numerical simulations. The phase transitions happen at T /L ' ∗ −1/2 0.416 and T ∗/L = 1 for T ≤ 1. TE ∝ |T0 − T0 | . (0.19) The critical exponent is independent of k. If we approach the phase transition from the heating phase, as shown in Fig.6, reduces to the ground state entropy. As n grows, SA(t) can be the slope kE of the linear growth will vanish. In other words, approximated as 59 1/kE will diverge, and we find that 0 c L c |φ | ∗ −1/2 SA(t) ' log + · ·t, t = n(T0+T1). (0.16) 1/kE ∝ |T0 − T0 | . (0.20) 6 π 3 T0 + T1 In short, by approaching the phase transition from both sides, I.e., the entanglement entropy grows linearly in time t [see one can obtain the critical exponent ζ = 1/2. Fig.4 for a typical plot based on Eq.(0.15)], with slope Comparison with numerics We compare our CFT cal- 0 culation with the numerical simulations based on a free ∂SA(t) c |φ | kE := = · . (0.17) fermion lattice which has finite sites L with open bound- ∂t 3 T + T 0 1 ary conditions. We prepare the initial state as the ground 1 PL−1 † We emphasize that here SA(t) keeps growing all the way since state of H0 = 2 i=1 ci ci+1 + h.c. with half fill- there are infinite number of degrees of freedom and the energy ing. The sine-square deformed Hamiltonian has the form 60 PL−1 2 π(i+1/2) † † spectrum goes to infinity with no upper bound in CFT. In a H1 = i=1 sin L ci ci+1 + h.c., where ci (ci ) are lattice model, however, the entanglement entropy will finally fermionic operators, which satisfy the anticommutation rela- saturate because of the UV cutoff introduced by the lattice † † † 59 tions {ci, cj} = {ci , cj} = 0, and {ci, cj} = δij. We com- constant. As shown in Fig.6, we plot the slope of the linear pare our field theory result with the numerical simulations in i.e. k T T = T = growth, , E, as a function of (by choosing 0 1 Figs.2,4,5 and6, respectively. The agreement in the non- T ). It can be found that the slope goes to zero as we approach heating phase is remarkable. In the heating phase, the numer- the phase transitions, which indicates that the linear-growth ical results deviate from the CFT results as t grows. This is behavior disappears at the phase transitions, as expected. as expected, since the lattice system can no longer be well de- Phase Transition The phase transition between heating and scribed by a CFT as it keeps absorbing energy. (Recall that non-heating phases happens at ∆ = 0, where the entangle- only the low energy limit can be well described by a CFT.) ment entropy grows logarithmically in time, and the single- Discussion and Conclusion We have proposed an ana- point function decays in a power-law in time. There are two lytically solvable setup to study the Floquet dynamics of a sets of solutions for ∆ = 0 [see Eq.(0.6)]. One is T0 = mL, generic CFT. The phase diagram, entanglement entropy and with m = 1, 2, 3, ··· , as depicted in the vertical lines in Fig.2. correlation functions can be analytically obtained. There are The entanglement entropy for A = [0, L/2] has a simple ex- many future problems, and we mention a few of them: (i) The pression Hamiltonians H0 and H1 considered in this work are com- 2 posed of three generators of sl(2,R) algebra, which is a sub- c n L h 2 πT1 io algebra of the Virasoro algebra in a two dimensional CFT. Our SA(t) ' log · 1 + 4n · , (0.18) 6 π L setup applies to the general case with H(t) = H(t + T ), as long as H(t) is a combination of the three generators of where t := n(T0 + T1) = n(mL + T1). In the large n limit, sl(2,R) algebra, i.e., the Virasoro generators L0, Ln and L−n c one has SA(t) ' 3 log t. Another set of solutions for ∆ = 0 (and the anti-holomorphic parts), as will be discussed in more h 2 i πT1 πT0 πT1 πT0 detail in Ref.61. On the other hand, it is an open question if the are determined by 1− L sin L +2· L ·cos L = 0. H(t) In this case, one has Hamiltonian is a combination of generators of the Vira- soro algebra, which is infinite dimensional. (ii) It is also de- ( 4 2 ) sirable to study the multi-point correlation functions (although c L 4 πT1 4 πT1 L 2 L quite involved) in our setup. As discussed in 59, our system SA(t) ' log 2 n − 2 n+1 . 6 π πT1 πT1 1 + L 1 + L with periodic driving is not uniformly heated. It is our future 6 work to use two-point correlation functions to measure the of a CFT in the collaboration in 48. We thank for helpful local ‘temperature’ of the Floquet CFT. (iii) Our setup also conversations and discussions with Zhen Bi, Po-Yao Chang, works for non-periodic driving schemes, such as the quasi- Yingfei Gu, Max Metlitski, Xiao-Liang Qi, Yang Qi, Cecile periodic driving and random driving CFTs, which deserve fu- Repellin, Shinsei Ryu, and Xiao-Gang Wen, and thank Liujun ture studies. (iv) Since our setup applies to a generic CFT Zou for many helpful comments on various aspects of our re- including the large-c CFT, it would also be interesting to con- sults. We also thank for the helpful comments and questions sider the holographic description of our setup62–64, which may during the seminar talk at MIT. XW is supported by the Gor- shed new lights on the Floquet dynamics in AdS/CFT.65–67 don and Betty Moore Foundation’s EPiQS initiative through Grant No. GBMF4303 at MIT. JQW is supported by Mas- Acknowledgement XW thanks Shinsei Ryu and Andreas sachusetts Institute of Technology and the Simons foundation W. W. Ludwig for introducing to him the concept of SSD it from qubit collaboration.

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the system keeps absorbing energy in a quantum field theory with essential here. We can also consider the Hamiltonian H1 = H0 − tanh(2θ) infinite degrees of freedom. 2 (H+ + H−) with θ ≥ 0. For finite θ, H1 has discrete 32 We can also consider a system with periodic boundary condition energy spacing ∝ 1/L cosh(2θ). One can find similar physics in if the Hamiltonian is composed of three generators of Virasoro the Floquet CFT in this case.61. 53 algebra L0, Ln and L−n with n > 1. It is noted that studying Ueff is equivalent to studying the property 33 Andrej Gendiar, Roman Krcmar, and Tomotoshi Nishino, “Spher- of the Floquet Hamiltonian HF , which is defined through Ueff = ical deformation for one-dimensional quantum systems,” Progress e−HF (τ0+τ1). Aside from the types of Möbius transformations of Theoretical Physics 122, 953–967 (2009). in Eq.(0.4), one can alternatively use the Floquet Hamiltonian to 34 Andrej Gendiar, Roman Krcmar, and Tomotoshi Nishino, “Spher- characterize/classify different phases. A detailed discussion on the ical deformation for one-dimensional quantum systems,” Progress Floquet Hamiltonian and its spectrum in a Floquet CFT will be of Theoretical Physics 123, 393 (2010). given in 61. 35 Toshiya Hikihara and Tomotoshi Nishino, “Connecting distant 54 See, e.g., https://en.wikipedia.org/wiki/Mobius_ ends of one-dimensional critical systems by a sine-square defor- transformation for more details on Mobius transformation. mation,” Phys. Rev. B 83, 060414 (2011). 55 It is noted that although the trajectory is plotted in a continuous 36 A. Gendiar, M. Daniska,ˇ Y. Lee, and T. Nishino, “Suppression of way, it is only well defined at discrete n. It is the same in the finite-size effects in one-dimensional correlated systems,” Phys. following plots for entanglement entropy evolution SA(t), where Rev. A 83, 052118 (2011). t is defined at discrete values t = n(T0 + T1). 37 Naokazu Shibata and Chisa Hotta, “Boundary effects in the 56 In our setup for the (1+1)-d Floquet CFT, the trajectory of density-matrix renormalization group calculation,” Phys. Rev. B O(zn, z¯n) in z-plane displays three kinds of behaviors depending 84, 115116 (2011). on the types of Möbius transformations. This is similar to certain 38 Chisa Hotta and Naokazu Shibata, “Grand canonical finite-size classical Floquet dynamics such as the classical Mathieu’s har- numerical approaches: A route to measuring bulk properties in an monic oscillator (see, e.g., Ref.68), where the harmonic oscillator applied field,” Phys. Rev. B 86, 041108 (2012). displays three kinds of trajectories in the phase space depending 39 Chisa Hotta, Satoshi Nishimoto, and Naokazu Shibata, “Grand on the elliptic, parabolic or hyperbolic transformations between T T canonical finite size numerical approaches in one and two dimen- (xn, pn) and (xn+1, pn+1) . xn and pn are the position and sions: Real space energy renormalization and edge state genera- momentum of the harmonic oscillator after n cycles of driving. tion,” Phys. Rev. B 87, 115128 (2013). Depending on the trajectories of the harmonic oscillator, there 40 Hosho Katsura, “Exact ground state of the sine-square deformed are stable and non-stable regions separated by a boundary, sim- xy spin chain,” Journal of Physics A: Mathematical and Theoreti- ilar to the non-heating and heating phases with a phase transition cal 44, 252001 (2011). in our Floquet CFTs. (For the non-stable region in Mathieu’s har- 41 Hosho Katsura, “Sine-square deformation of solvable spin chains monic oscillator, the amplitude of oscillator keeps increasing by and conformal field theories,” Journal of Physics A: Mathematical absorbing energy from the external driving. This is similar to the and Theoretical 45, 115003 (2012). heating phase of our Floquet CFTs, where the entanglement en- 42 Isao Maruyama, Hosho Katsura, and Toshiya Hikihara, “Sine- tropy keeps growing in time.) In addition, the ‘phase diagram’ of square deformation of free fermion systems in one and higher di- Mathieu’s oscillator also shows periodic structure as the driving mensions,” Phys. Rev. B 84, 165132 (2011). period increases, which results from higher order resonances. In 43 Tsukasa Tada, “Sine-square deformation and its relevance to fact, there is a deep reason on the similarity between our (1+1)-d string theory,” Modern Physics Letters A 30, 1550092 (2015). Floquet CFTs and the (0+1)-d quantum Mathieu’s harmonic os- 44 Kouichi Okunishi and Hosho Katsura, “Sine-square deformation cillators. The Hamiltonians in our Floquet CFTs are composed and supersymmetric quantum mechanics,” Journal of Physics A: of three generators of sl(2,R) algebra, while the Hamiltonians Mathematical and Theoretical 48, 445208 (2015). in quantum Mathieu’s harmonic oscillators are composed of three 45 Nobuyuki Ishibashi and Tsukasa Tada, “Infinite circumference generators of su(1, 1) algebra.69 The similarity on the ‘phase di- limit of conformal field theory,” Journal of Physics A: Mathemat- agram’ of the two systems originates from the algebraic struc- ical and Theoretical 48, 315402 (2015). ture sl(2,R) =∼ su(1, 1), where ‘=∼’ represents ‘is isomorphic 46 Nobuyuki Ishibashi and Tsukasa Tada, “Dipolar quantization and to’. From this point of view, we may say that within our setup a the infinite circumference limit of two-dimensional conformal (1+1)-d Floquet CFT =∼ a (0+1)-d quantum Mathieu’s harmonic field theories,” International Journal of Modern Physics A 31, oscillator. 1650170 (2016). 57 Pasquale Calabrese and John Cardy, “Entanglement entropy and 47 Kouichi Okunishi, “Sine-square deformation and mbius quanti- quantum field theory,” Journal of Statistical Mechanics: Theory zation of 2d conformal field theory,” Progress of Theoretical and and Experiment 2004, P06002 (2004). Experimental Physics 2016, 063A02 (2016). 58 Pasquale Calabrese and John Cardy, “Entanglement entropy and 48 Xueda Wen, Shinsei Ryu, and Andreas W. W. Ludwig, “Evo- conformal field theory,” Journal of Physics A: Mathematical and lution operators in conformal field theories and conformal map- Theoretical 42, 504005 (2009). pings: Entanglement hamiltonian, the sine-square deformation, 59 See supplementary materials. 60 and others,” Phys. Rev. B 93, 235119 (2016). It is noted that the UV cutoff in SA(t) is introduced only at the 49 Shota Tamura and Hosho Katsura, “Zero-energy states in confor- entanglement cut. In the bulk of the subsystem A, there are al- mal field theory with sine-square deformation,” Progress of The- ways infinite degrees of freedom in a quantum field theory. And oretical and Experimental Physics 2017, 113A01 (2017). the energy spectrum (in a CFT) goes to infinity without an upper 50 Tada Tsukasa, “Conformal quantum mechanics and sine-square bound. Then the system can keep absorbing energy. Similar things deformation,” arXiv:1712.09823. also appear in the entanglement entropy in a CFT with finite tem- 51 Xueda Wen and Jie-Qiang Wu, “Quantum dynamics in sine- perature. In the high temperature limit, the entanglement entropy c l c πl square deformed conformal field theory: Quench from uniform for a finite subsystem of length l is SA(β) ' 3 log + 3 · β , to non-uniform cfts,” arXiv:1802.07765. where is the UV cutoff introduced at the entanglement cut.70 The 52 It is noted that the feature of continuous spectrum for H1 is not 8

entanglement entropy grows linearly with the temperature 1/β all 1. Phase transition I 16 the way. This is not the case in a lattice, where there are always a 2. Phase transition II 16 finite number of degrees of freedom in a finite subsystem, and the D. On single-point correlation function 17 bandwidth of energy spectrum is finite. The entanglement entropy in a lattice system will finally saturate as 1/β increases. III. A lattice model on critical fermion chain 17 61 Xueda Wen, “A family of analytically solvable floquet conformal A. Ground state 18 field theory,” In preparation. B. Quantum quench and Floquet case 18 62 Shinsei Ryu and Tadashi Takayanagi, “Holographic derivation of entanglement entropy from the anti–de sitter space/conformal field theory correspondence,” Physical review letters 96, 181602 (2006). I. ON THE SETUP AND ENTANGLEMENT ENTROPY 63 Shinsei Ryu and Tadashi Takayanagi, “Aspects of holographic entanglement entropy,” Journal of High Energy Physics 2006, 045 In the supplementary materials, we present more details on (2006). the calculations, as well as analysis and discussions on the 64 Veronika E Hubeny, Mukund Rangamani, and Tadashi results in the main text. Takayanagi, “A covariant holographic entanglement entropy pro- posal,” Journal of High Energy Physics 2007, 062 (2007). 65 Roberto Auzzi, Shmuel Elitzur, Sven Bjarke Gudnason, and A. More about the setup Eliezer Rabinovici, “On periodically driven ads/cft,” Journal of High Energy Physics 2013, 16 (2013). 66 Mukund Rangamani, Moshe Rozali, and Anson Wong, “Driven The system is driven by two different Hamiltonians period- holographic cfts,” Journal of High Energy Physics 2015, 93 ically, with (2015). 67  Z L A. Biasi, P. Carracedo, J. Mas, D. Musso, and A. Serantes, “Flo-  quet Scalar Dynamics in Global AdS,” ArXiv e-prints (2017),  H0 = h(x)dx,  0 arXiv:1712.07637 [hep-th]. L (1.1) 68 Z Ryoichi Kawai, Katja Lindenberg, and Christian Van den Broeck,  2 πx  H1 = sin h(x)dx, “Parametrically modulated oscillator dimer: an analytic solution,” 0 L Physica A: Statistical Mechanics and its Applications 312, 119– 140 (2002). where h(x) is the Hamiltonian density which is uniform in 69 Askol’d Mikhailovich Perelomov and Vladimir Stepanovich space. We start from the ground state of H0, and drive the Popov, “Group-theoretical aspects of the variable frequency oscil- system with H1 and H0 periodically (see the main text). In the lator problem,” Theoretical and Mathematical Physics 1, 275–285 CFT calculation, it is convenient to rewrite the Hamiltonian in (1969). terms of stress energy tensors, i.e., 70 John Cardy and Erik Tonni, “Entanglement hamiltonians in two- dimensional conformal field theory,” Journal of Statistical Me-  Z L  dx chanics: Theory and Experiment 2016, 123103 (2016).  H0 = Tττ (x), 71 Ingo Peschel, “Calculation of reduced density matrices from cor- 0 2π (1.2) relation functions,” Journal of Physics A: Mathematical and Gen-  1  H1 =H0 − (H+ + H−) , eral 36, L205 (2003). 2 ¯ Floquet CFT: with Tττ = T (w) + T (w ¯), and Supplemental Materials Z L dx ±2πw/L ∓2πw/L¯ H± = e T (w) + e T¯(w ¯) . (1.3) 0 2π L CONTENTS The CFT lives on a finite space of length , with conformal boundary condition imposed. In path integral, the partition function is defined on a strip I. On the setup and entanglement entropy8 A. More about the setup8 w = τ + ix, (1.4) B. Expression for entanglement entropy9 where τ is the imaginary time, and x is the space, with II. Entanglement entropy evolution 11 A. Non-heating phase 11 τ ∈ (−∞, +∞), x ∈ (0,L). (1.5) 1. High frequency limit 12 2. Comparison with a single-quench 12 The wavefunction after n cycles of driving can be written as 3. Near the phase transition 13 −iH T −iH T −iH T −iH T |ψ(t)i = e 0 0 e 1 1 ··· e 0 0 e 1 1 |Gi. (1.6) B. Heating phase 13 1. Entanglement entropy evolution 13 We can evaluate the n-point (equal time) correlation function 2. Near the phase transition 15 in state |ψ(t)i as follows: 3. Long time limit in a lattice model 15 C. Phase transitions 16 hψ(t)|O1O2 ···On|ψ(t)i. (1.7) 9

To obtain the correlation functions of operators at different That is, we have defined a, b, c and d as follows: time, we simply need to insert these operators at different  πτ1 πτ0 time slices. Shown in Fig.1 is the path integral representa- a :=(1 + ) · e L ,  L tion of single-point correlation function hψ(t)|O(x)|ψ(t)i. In   πτ1 −πτ0  L the calculation, we will consider the Euclidean space, i.e.,  b := − · e , L (1.15) πτ1 πτ0 |ψ(τ)i = e−H0τ0 e−H1τ1 ··· e−H0τ0 e−H1τ1 |Gi, (1.8)  c := · e L ,  L   πτ1 −πτ0 and take analytical continuation τ → iT and τ → iT in  d := 1 − · e L , 0 0 1 1 L the final step. We map the strip w-plane to z-plane as follows, which satisfies ad − bc = 1. Note that τ0 and τ1 are real numbers, and z¯1 has the form w z πτ0 −πτ0 =L πτ1 L πτ1 L x (1 + L ) · e · z¯ − L · e z¯1 = . (1.16) πτ πτ0 πτ −πτ0 1 L 1 L x L · e · z¯ + 1 − L · e τ This Mobius¨ transformation (before analytical continuation) x=0 forms a SL(2,R) group. For later convenience, we write the Mobius¨ transformation in Eq.(1.14) in the normal form: by using the conformal transformation z − γ z − γ 1 1 = η · 1 , (1.17) 2πw z2 − γ2 z − γ2 z = e L . (1.9) where γ1 and γ2 are called ‘fixed points’, and η is called ‘mul- −H τ −H τ Instead of studying how e 0 0 and e 1 1 act on the ground tiplier’ in a Mobius¨ transformation. γ1, γ2 and η have the state, we consider the Heisenberg picture here. I.e., we study explicit form how the operator evolves during the periodic driving. For  p 2 the operator O(z, z¯) in z-plane, it is found that the effect of a − d − (a − d) + 4bc  γ1 = , Hamiltonian Hi, with i = 0, 1, is to evolve the operator in the  2c  following way  a − d + p(a − d)2 + 4bc γ2 = , (1.18) h h  2c ∂zi ∂z¯i  p eτHi O(z, z¯)eτHi = new new O(zi , z¯i ).  (a + d) + (a − d)2 + 4bc new new  η = . ∂z ∂z¯  p 2 (1.10) (a + d) − (a − d) + 4bc 51 To be concrete, we have Note that one can take an inverse on both sides of Eq.(1.17), −1  0 2πτ0 so that γ1 → γ2, γ2 → γ1 and η → η . z =e L z,  new Now we repeat the above procedure for n cycles of driving, πτ1 πτ1 (1.11) 1 (1 + )z − and denote the coordinate of O as zn and z¯n. Then one has  z = L L ,  new πτ1 z + 1 − πτ1 L L h h¯ ∂zn ∂z¯n hψ(τ)|O(z, z¯)|ψ(τ)i = O(zn, z¯n), Then, after one-cycle driving, one can obtain ∂z ∂z¯ (1.19) h h¯ where † ∂z1 ∂z¯1 Ueff O(z, z¯) Ueff = O(z1, z¯1), (1.12)  ∂z ∂z¯ zn − γ1 n z − γ1  =η · ,  zn − γ2 z − γ2 where we have defined the time evolution operator Ueff := (1.20) z¯n − γ1 z¯ − γ1 −H0τ0 −H1τ1 n e e . z1 has the explicit expression  =η · .  z¯n − γ2 z¯ − γ2

2πτ0 πτ1 L πτ1 (1 + L ) · e · z − L z1 = , (1.13) πτ 2πτ0 πτ 1 L 1 B. Expression for entanglement entropy L · e · z + 1 − L

Written in a normalized form of Mobius¨ transformation, z1 The entanglement entropy of subsystem A = [0, l] with has the expression: 0 < l < L can be obtained by calculating the single-point correlation function of a twist operator Tα. The entanglement πτ πτ0 πτ −πτ0 (1 + 1 ) · e L · z − 1 · e L az + b measure we use is the so-called Renyi entropy z = L L =: . 1 πτ0 −πτ0 πτ1 πτ1 · e L · z + 1 − · e L cz + d 1 L L S(α)(t) = log tr [ρα (t)] , (1.21) (1.14) A 1 − α A 10 where α is the Renyi index, and the von Neumann entropy One can find that h (α) h h " 2 # ∂zn ∂z¯n (ad − bc) SA(t) = lim SA (t). (1.22) = α→1 ∂z ∂z¯ 2 2πl 2 2 (c + 2cd · cos L + d ) α (α) (1.30) The term tr(ρA) in SA (t) is related with the single-point correlation function of twist operator as follows:57,58 We also need to evaluate the single-point correlation function (z) hTα (zn, z¯n)i in Eq.(1.26). For convenience, we write zn as tr(ρα ) = hψ(t)|T (x = l)|ψ(t)i, (1.23) A α E + iF iϕn zn = =: Rn · e , (1.31) where Tα is a primary operator with conformal dimension G where c 1 h = h¯ = α − , (1.24)  2πl 24 α  E =ac + (ad + bc) cos + bd,  L  In the following, we will evaluate the correlation func-  2πl F =(ad − bc) sin , (1.32) tion in Eq.(1.23) with path integral method. Pictorially, L  hψ(t)|Tα(x = l)|ψ(t)i is shown in Fig.1 by setting O(x) =  2πl  G =c2 + 2cd · cos + d2. Tα(x). Note that there are both Euclidean time and Lorentzian L time in the path integral. As shown in the following, we will Then, we have do calculation in the Euclidean space by setting it = τ, and (z) analytically continue back to Lorentzian time in the final step. hTα (zn, z¯n)i Let us start by evaluating the single-point correlation func- 1 1 b 1 − 2 − 2 h 2i 2h tion: =Aα · ( zn z¯n ) · ( 1 1 ) , 4 2 2 zn − z¯n hψ(t)|T (w, w¯)|ψ(t)i α 1 1 h −42 h h h h h b ∂z ∂z¯ ∂z ∂z¯ =Aα · · √ · √ , n n 4 znz¯n zn +z ¯n − 2 znz¯n (1.33) = hTα(zn, z¯n)iz, ∂w ∂w¯ ∂z ∂z¯ h 2 h b 1 1 −4 (1.25) =Aα · · · 4 Rn 2Rn · cos ϕn − 2Rn ¯ h where we have considered the fact h = h for the twist op- 2 1 erator. Note that hT (z , z¯ )i is the single point correlation =Ab · · . α n n z α 2 R2 · (1 − cos ϕ ) function in the ground state in z-plane, with a slit lying along n n the half real-axis [0, ∞). Conformal boundary conditions are Note that imposed along the slit. Then one has  E2 + F 2 1/2   Rn := 2 , 1 − 1 − 1 2i  G b 2 2 h 2h (1.34) hTα(zn, z¯n)iz = Aα · ( zn z¯n ) · ( 1 1 ) , (1.26) E/G 4 2 2 zn − z¯n  cos ϕn := .  p(E2 + F 2)/G2 where Ab is an amplitude depending on the selected boundary α Collecting all these terms, one can obtain [see Eq.(1.25)] condition |bi as well as the Renyi index α. It will affect the entanglement entropy by an order ∼ O(1) term. is a UV hψ(t)|Tα(w, w¯)|ψ(t)i cut-off, which may be considered as the lattice spacing in a h h h h ∂z ∂z¯ ∂z ∂z¯ lattice model. In Eq.(1.25), one has = n n hO(z)(z , z¯ )i ∂w ∂w¯ ∂z ∂z¯ n n h h 2h h ∂z ∂z¯ 2π 2h " 2 # = . (1.27) 2π (ad − bc) ∂w ∂w¯ L = · L 2 2πl 2 2 (c + 2cd · cos L + d ) To evaluate other terms in Eq.(1.25), first we rewrite Eq.(1.20) h 2 1 as b · Aα · · 2 2 Rn · (1 − cos ϕn) n n (γ1 − η γ2)z − (1 − η )γ1γ2 az + b h 2h " 2 2 # zn = n n =: , (1.28) 2π (ad − bc) (1 − η )z − (γ2 − η γ1) cz + d b =Aα · · · √ √ . L 2 E2 + F 2 − G2 E E2 + F 2 where we have defined G (1.35)  a :=γ − ηnγ ,  1 2 Then, based on Eqs.(1.21) and (1.23), the α-th Renyi entropy  n  b := − (1 − η )γ1γ2, is related with hψ(t)|T (w, w¯)|ψ(t)i as follows (1.29) α c :=1 − ηn,  (α) 1  n S = loghψ(t)|Tα(w, w¯)|ψ(t)i. (1.36) d := − (γ2 − η γ1). A 1 − α 11

1.6 with t := n(T0 + T1) and l=0.5L  0.2L E =R cos(2nφ + ϕ) − K, 1.5  0.1L (2.2) 2 2πl CFT  F =P sin , 1.4 L (t) A

S where we have defined 1.3  ei2φ :=(Q − iP )(Q + iP ),  1.2  πT L πT  0 0  Q := sin − cos ,  L πT1 L 1.1  √ 0 1000 2000 3000 4000  L t  P := ∆,  πT1 πT 2πl (2.3) FIG. 7. Comparison between numerical simulations and CFT cal- 0 2  K :=W cos − W cos , culations for entanglement evolution in the non-heating phase with  L L  c = 1. Here we choose L = 500, and T0 = T1 = T = 20.  πT0 L πT0  W := cos + sin ,  L πT1 L   πT0 2πl πT0  Reiϕ :=W cos − cos + iP sin . One can find the entanglement entropy as L L L √ √ One can find that in the non-heating phase, the entanglement c L c E2 + F 2 − G2 E E2 + F 2 S (t) ' log + log G , entropy oscillates in time, with the period A 6 π 12 2(ad − bc)2 π (1.37) TE = · (T0 + T1). (2.4) where we only keep the leading term, and the subleading term |φ| of order O(1) has been neglected. In the following parts, we Now let us do a self-consistent check. For n = 0, i.e., the need to evaluate Eqs.(1.35) and (1.37), by making analytical system is not driven at all and stays in the ground state, one continuation τ → iT and τ → iT . 0 0 1 1 can find that 2πl E =R cos ϕ − K = (W 2 − 1) cos L (2.5) II. ENTANGLEMENT ENTROPY EVOLUTION 2πl =P 2 cos , L As discussed in the main text, the behavior of the entanglement entropy evolution is determined by the sign of ∆ in where we have considered the fact that W 2 − P 2 = 1. Then Eq.(0.6). For ∆ > 0, we have the non-heating phase. Both the one can obtain entanglement entropy and the single-point correlation func- c L πl tion oscillate in time; For ∆ < 0, we have the heating phase. SA(t = 0) = log sin , (2.6) The entanglement entropy keeps growing linearly in time, and 6 π L the single-point correlation function decays exponentially in which is the entanglement entropy in the ground state, as ex- time; For ∆ = 0, there is a phase transition. The entangle- pected. ment entropy grows logarithmically in time, and the single For a generic l, a typical plot deep in the non-heating phase point correlation function shows a power-law decay in time. is shown in Fig.7. It can be found that SA(t) for different l os- In the following, we give the explicit expressions of SA(t) cillate with the same period, as can be also straightforwardly in Eq.(1.37) for different cases, by doing analytical continua- observed in Eqs. (2.1)∼(2.3). tion τ0 → iT0 and τ1 → iT1. The procedure is tedious but Now let us check the specific case l = L/2 as discussed in quite straightforward, and here we list the main results and the main text. In this case, one has E = R cos(2nφ + ϕ) − K, give some discussions. and F = 0. Moreover, one can find that

|R| − K < 0, E < 0, (2.7)

2 A. Non-heating phase 2 πT0 2 based on the following facts: R = W + cos L , K = 2 W 2 · W + cos πT0 , W 2 = P 2 + 1 > 1, and K > 0. Note The non-heating phase corresponds to ∆ > 0 in Eq.(0.6). L that K > 0 because K = W 2 +W cos πT0 > W 2 −W 2 = 0. After doing analytical continuation, one can ﬁnd L Therefore, the entanglement entropy in Eq.(2.1) becomes √ c L c E2 + F 2 − E E2 + F 2 c L c K − R cos(2nφ + ϕ) S (t) ' log + log , (2.1) S (n) = log + log . (2.8) A 6 π 12 2P 4 A 6 π 6 P 2 12

2 c L πl Note also that K − R cos ϕ = P . Therefore, one has the expression SA(t = 0) = 6 log π · sin L , which is the entanglement entropy in the ground state, as expected. From c h L K − R cos(2nφ + ϕ)i Eqs.(2.1) and (2.14), one can ﬁnd that the oscillation period SA(n) = log · , (2.9) 6 π K − R cos ϕ of entanglement entropy is which is Eq.(0.12) in the main text. 2L TE = √ , (2.15) 3 which is observed in the numerical simulation in Fig.4 and 1. High frequency limit Fig.7. In particular, for l = L/2, SA(t) can be further simpli- ﬁed as Now let us look at the behavior of SA(t) in the high fre- " √ !# quency limit T ,T L. We will show that the result only c L c 3π 0 1 S (t) ' log + log 2 − cos t , (2.16) depends on the ratio A 6 π 6 L

σ := T0/T1. (2.10) which is Eq.(0.14) in the main text.

In this limit, one can ﬁnd that the parameters in Eq.(2.3) can be 2 πT0 2. Comparison with a single-quench approximated by (keeping the leading order) ∆ ' L + r 2 √ πT1 2πT0 , P ' T0 + 2T0 = σ2 + 2σ, Q ' − L , L L T1 T1 πT1 As mentioned in the main text, in the high-frequency 2 driving limit, one can consider the approximation T0 T0 T0 2πl W ' 1 + , K ' 1 + − 1 + cos , φ ' −H0T −H1T −(H0+H1)T T1 T1 T1 L e e ' e , and then the Floquet dy- r 2 namics can be effectively described by a single quench with πT1 · T0 + 2T0 , and Reiϕ ' 1 + T0 − cos 2πl . Then E, 1 L T1 T1 T1 L the effective Hamiltonian HF = 2 (H0 + H1). Here let us F and P in Eq.(2.1) are approximated by check this approximation explicitly. In Ref.51, we have considered a single quench starting from  2πl πT1 p the ground state of H0, and switch the Hamiltonian to HMo¨b  E' 1 + σ − cos · cos 2n · · σ2 + 2σ  suddenly, with  L L   2 2πl tanh(2θ)  − (1 + σ) + (1 + σ) cos , H (θ) = H − (H + H ), (2.17) L Mo¨b 0 2 + −  2 2πl  F'(σ + 2σ) · sin , where  L   L L  p Z dx Z dx  P ' σ2 + 2σ.  H = T (x) = T (w) + T¯(w ¯) ,  0 ττ (2.11) 0 2π 0 2π Z L From Eqs.(2.1) and (2.11), one can obtain the oscillation pe-  dx ±2πw/L ∓2πw/L¯ ¯ riod of entanglement entropy as  H± = e T (w) + e T (w ¯) . 0 2π (2.18) 1 + σ 51 TE = √ L. (2.12) Then the entanglement entropy evolution has the form: σ2 + 2σ c L c f(t)2 + f(t) · h(t) S (t) ' log + log (2.19) In particular, for l = L/2, the entanglement entropy has a A 6 π 12 2 simple expression where √ σ2+2σ 2πt r 1 + σ − cos · 2 2πl c L c 1+σ L f(t) = h(t)2 + sin , (2.20) SA(t) ' log + log . L 6 π 6 σ (2.13) and Now let us consider the simple case T = T = T . Then one 0 1 πt πt 2πl has h(t) = − sin2 · cosh(4θ) + cos2 cos Leff Leff L  √ ! 2πl 3π 2πl πt  + sin2 · sinh(4θ),  E' 2 − cos · cos t − 2 + 4 cos ,  L L L Leff  (2.21) 2πl  F'3 sin ,  L with Leff = L cosh(2θ). Now we consider the high frequency  √  P ' 3, limit of the Floquet CFT with T0 = T1 = T . Then one has H = 1 (H + H ) = H (θ) (2.14) F 2 0 1 Mo¨b with where t = 2T . As a self-consistent check, one can ﬁnd that 1 2πl tanh(2θ) = . (2.22) for t = 0, one has E' 3 cos L . Then SA(t) in Eq.(2.1) has 2 13

After some straightforward algebra, one can obtain 3 √ 2.5 T=200 1/2 ∗ 1/2 CFT |φ| ' ∆ ' κδ = κ · (T0 − T0 ) , (2.26) 2 p 1.5 where κ := 2π2km/L. Therefore, near the phase transition ∗ 1 in Eq.(2.24), the oscillation period T depends on (T − T ) 0 5000 10000 15000 E 0 0 as follows 3 T=206 2.5 CFT (1 + k)mL 1 TE ' · . (2.27) (t) p ∗ 1/2 A 2 2km/L (T0 − T ) S 0 1.5

1 In particular, for T0 = T1 = T , i.e., k = 1, one has 0 5000 10000 15000

3 √ 1 T ' 2m · L3/2 · . (2.28) 2.5 E (T − T ∗)1/2 2 T=209.1 1.5 CFT The other set of solutions for the phase transition are deter- h 2 i 1 πT1 πT0 πT1 πT0 3 4 1 − sin + 2 · · cos = 0 10 10 mined by L L L L . By t ∗ ∗ choosing (n−1)L < T0 < nL, T0 = T0 −δ, and T1 = k·T0, one can find that FIG. 8. Comparison between numerics and CFT calculation for 0 2 entanglement evolution in the non-heating phase near the phase tran- ∆ ' κ · δ + O(δ ), (2.29) sition. The driving period in CFT calculation is T0 = T1 = T = ∗ n 2 ∗ ∗ ∗ 199.5, 205.1, and 207.94, respectively. Here we choose L = 500. 0 πT0 π T1 πT0 π πT0 where κ = sin L 2· L2 ·sin L ·(1+k)− L ·cos L · ∗ 2 o πT1 √ 1 + 2k − . Then one can obtain 4θ L Then√ one has e = 3, cosh(2θ) = 2/ 3, and sinh(2θ) = 1/ 3. It is straightforward to check that the entanglement en- ∗ ∗ ∗ ∗ ∗ πT1 πT0 πT0 π(T0 + T1 ) 1 tropy evolution in Eq.(2.19) is the same as the high-frequency TE ' sin − cos · √ · . 0 ∗ 1/2 L L L κ (T0 − T0) limit of a Floquet CFT in Eqs.(2.1) and (2.14). (2.30) ∗ 1/2 That is, TE ∝ 1/(T0 − T0) . In a short summary, for the non-heating phase near the 3. Near the phase transition phase transitions, one always has ∆ ∝ δ, and |φ| ∝ δ1/2, ∗ −1/2 based on which we can obtain TE ∝ |T0 − T0 | . As we approach the phase transition from the side of non- heating phase, one can find that the oscillation period of entanglement entropy diverges, as shown in Fig.5 in the main text. [See also Fig.8.] B. Heating phase Now let us check the behavior of TE in Eq.(2.4), i.e., TE = π(T0 + T1)/|φ|, near the phase transition explicitly. Since 1. Entanglement entropy evolution there are two sets of solutions for the phase transition [see the main text], here we consider them separately. We approach The heating phase corresponds to ∆ < 0 in Eq.(0.6). After the phase transition along doing analytical continuation, one can find

T1 = k · T0, for arbitrary k > 0. (2.23) √ c L c E2 + F 2 − E E2 + F 2 SA(t) ' log + log , One set of solution are T0 = mL, with m = 1, 2, 3, ··· 6 π 12 2P 4 [see the vertical lines in Fig.2]. Let us take (2.31) with t = n(T0 + T1) and ∗ ∗ ∗ T0 = mL, T1 = kT0 . (2.24)  πT0 2πl  E = − cosh(2nφ0) · W · cos − cos Since we approach the phase transition from the non-heating  L L ∗ ∗  phase, then we make T0 = T0 + δ and T1 = T1 + kδ, with  ∗ 0 πT0 δ T0 . Expanding to the ﬁrst order in δ, one has + sinh(2nφ ) · P · sin + K, (2.32)  L  2π2km  2πl ∆ ' δ. (2.25)  F =P 2 sin , L L 14

where we have deﬁned 8  0 πT0 πT0  R0eϕ =W cos + 1 − P sin ,  L L 7 Numerics (2.36) CFT 0 πT πT  R0e−ϕ =W cos 0 + 1 + P sin 0 . L L 6 Then the entanglement entropy can be expressed as 5 c h L R0 cosh(2nφ0 + ϕ0) − K i (t)

A SA(t) ' log · . (2.37) S 6 π P 2 4 At n = 0, one can check that 3 R0 cosh ϕ0 − K = 1 − W 2 = P 2. (2.38)

2 c L Therefore, one has SA(t = 0) = 6 log π , which is the entanglement entropy in the ground state. For generic t, one has 1 0 2000 4000 6000 8000 10000 12000 c h L R0 cosh(2nφ0 + ϕ0) − K i t S (t) ' log · , (2.39) A 6 π R0 cosh ϕ0 − K

FIG. 9. Comparison between numerics and CFT calculation for which is Eq.(0.15) in the main text. A typical plot of SA(t) entanglement evolution in the heating phase. From top to bottom: for different driving periods is shown in Fig.9. It is noted that T0 = T1 = T = 400, 450, 480, and 495. Here we choose L = as n grows, SA(t) grows linearly in time as follows 500. c L c S (t) ' log + · |φ0| · n. (2.40) A 6 π 3 where Noting that t := n(T0 + T1), one has

0  2φ0 Q + P c L c |φ |  e := , SA(t) ' log + · ·t, t = n(T0+T1). (2.41)  Q − P 6 π 3 T + T  0 1  πT L πT  Q := sin 0 − cos 0 , In the above result, the entanglement entropy keeps growing   L πT1 L linearly in time without saturation. This is because in the con-  L √ P := −∆, (2.33) formal field theory, the energy spectrum goes to infinity with-  πT1 out an upper bound, and there are infinite degrees of freedom.   πT0 L πT0 On a lattice, however, we always have a finite number of de-  W := cos + sin ,  L πT L grees of freedom for a finite subsystem and the bandwidth of  1  πT 2πl energy spectrum is finite. Therefore the entanglement entropy  K :=W · cos 0 − W 2 · cos . L L will finally saturate, as will be discussed shortly. (See Fig.11) Now let us check the entanglement entropy for an arbitrary It is helpful to compare the parameters above with those in subsystem A = [0, l] with 0 < l < L. Based on Eqs.(2.31) Eq.(2.3) for the non-heating phase. As a self-consistent check, and (2.32), it looks that for a generic l the entanglement en- now let us look at the case with n = 0, i.e., t = 0. Then one tropy will always grow linearly in time in the large n limit. 2 2πl 2 2πl However, this is not the case. Let us rewrite E in Eq.(2.32) as has E = (1 − W ) cos L = P cos L , where we have used 2 2 follows the fact that W + P = 1. Then SA(t) in Eq.(2.31) can be 0 simplified as e2nφ πT 2πl πT E = −W cos 0 + cos + P sin 0 2 L L L c L πl 0 S (t = 0) ' log sin , (2.34) e−2nφ πT 2πl πT A 6 π L + −W cos 0 + cos − P sin 0 + K. 2 L L L (2.42) which is the entanglement entropy in the ground state, as expected. The entanglement entropy will grow linearly in time for large 0 πT0 2πl Now let us check the specific case with l = L/2, so that n when satisfying: (i) φ > 0 and −W cos L + cos L + πT 0 πT 2πl SA(t) in Eq.(2.31) can be further simplified. One can find 0 0 P sin L < 0, or (ii) φ < 0 and −W cos L + cos L − πT0 2 that F = 0, K = W · cos + W , and πT0 L P sin L < 0. One can check that for l = L/2, one of the above situations must be satisfied, and therefore the entangle- E = −R0 cosh(2nφ0 + ϕ0) + K, (2.35) ment entropy will always grow linearly in time for large n. 15

0.35 Based on the deﬁnitions in Eq.(2.33), one can obtain 0.3 √ 0 1/2 ∗ 1/2 |φ | ' −∆ ' κδ = κ · (T0 − T0) , (2.45) 0.25

l* / L p 0.2 where κ := 2π2km/L. Therefore, near the phase transi- 0.15 tions at T1 = mL with m = 1, 2, 3 ··· , 1/kE for the linear growth of entanglement entropy depends on (T − T ∗) as fol- 0.1 0 0 0 0.5 1 1.5 2 2.5 3 lows T / L 1 1 3(T + T ) 1 3(T ∗ + T ∗) 1 = 0 1 · ' 0 1 · . ∗ 0 p 2 ∗ 1/2 FIG. 10. A typical plot of l as a function of driving period T1 in kE c |φ | c 2π km/L (T0 − T0) the heating phase. Here we choose T0/L = 0.9. (2.46) The other set of solutions for the phase transition are deter- h 2 i 1 − πT1 sin πT0 + 2 · πT1 · cos πT0 = 0 However, one can find there exists a length l∗, so that for mined by L L L L . By ∗ ∗ ∗ l < l , neither condition (i) nor (ii) is satisfied. That is, for choosing (n−1)L < T0 < nL, T0 = T0 +δ, and T1 = k·T0, A = [0, l] or A = [L − l, L] with l < l∗, the entanglement one can find that n entropy will not grow linearly in time even for large . In 0 other words, the region with l < l∗ is not ‘heated’, and only ∆ ' −κ · δ, (2.47) ∗ ∗ ∗ the region in (l ,L−l ) is ‘heated’. A typical plot of l in the πT ∗ n π2T ∗ πT ∗ πT ∗ ∗ 0 0 1 0 π 0 heating phase is shown in Fig.10. For l < l , one can find that where κ = sin L 2· L2 ·sin L ·(1+k)− L ·cos L · as n grows, S (t) will not grow linearly in time, but evolve to πT ∗ 2 o A 1 + 2k − 1 . Then one can obtain a stable value with L 1 πT ∗ πT ∗ πT ∗ 3(T ∗ + T ∗) 1 0 c L 2πl ∗ ' 1 sin 0 − cos 0 · 0 √ 1 · . SA(nφ 1) ' log · sin , l < l . (2.43) 0 ∗ 1/2 6 2π L kE L L L c · κ (T0 − T0 ) (2.48) As a remark, this is a typical feature in a quantum quench In short, for the heating phase near the phase transitions, one 0 1/2 by quenching the ground state of H0 [see Eq.(0.1)] with a always has ∆ ∝ −δ and |φ | ∝ δ , based on which we can h ∗ −1/2 obtain 1/kE ∝ |T0 − T | . new Hamiltonian H1 = H0 − 2 (H+ + H−) where h > 1 0 R L dx ±2πw/L ∓2πw/L¯ As a short summary, by approaching the phase transitions and H± = 0 2π e T (w) + e T (w ¯) . More details will be presented in 61. from both the non-heating phase and the heating phase, one ζ = 1/2 In the lattice model under a periodic driving, we did not ob- can obtain the critical exponent from the entangle- serve this stable behavior in Eq.(2.43). For arbitrary l < L in ment entropy evolution. a lattice model, we always observed a linear growth in SA(t) before saturation [see Fig.11 for example]. This disagreement 3. Long time limit in a lattice model may result from the lattice effect, which we leave as a future problem. As seen from Eqs.(0.15) and (0.16) in the main text, the entanglement entropy for A = [0, L/2] grows linearly in time 2. Near the phase transition all the way, without saturation. As we already mentioned, this is because there are infinite number of degrees of freedom in- side the subsystem A and the energy spectrum goes to infinity As shown in Fig.6, the slope kE of linear growth of the entanglement entropy will vanish near the phase transitions. without an upper bound, so that the system can absorb energy all the way. In a lattice model, however, the degrees of free- In other words, 1/kE diverges near the phase transitions. In the following, we will show that as we approach the phase dom in a finite subsystem are finite. The bandwidth of energy spectrum is also finite. It is expected that the entanglement transition along T1 = k · T0, where k is an arbitrary positive ∗ −1/2 entropy will saturate in the long time limit. real number, 1/kE always diverges as 1/kE ∝ |T0 −T0 | . The critical exponent ζ = 1/2 is the same as that obtained As shown in Fig.11, we calculate the entanglement evolu- from the side of non-heating phase. tion in the long time limit on a free fermion lattice (See Sec.III The analysis is similar to that in the non-heating phase in for the lattice model.). The entanglement entropy grows lin- Sec.II A 3. There are two sets of solutions for the phase tran- early in time first, and then saturates, as expected. At the current stage, in the field theory approach, it is an sitions and let us discuss them separately. First, for T0 = mL, with m = 1, 2, 3 ··· , [see the vertical lines in Fig.2]. Let us open question for us to introduce the saturation in the entan- ∗ ∗ ∗ glement evolution in the heating phase. (Note that this is dif- take T0 = mL, T1 = kT0 . Since we approach the phase ∗ ferent from the case of a global quench in CFTs where the transition from the heating phase, then we make T0 = T0 − δ, and T = T ∗ − kδ. Expanding to the first order in δ, one has saturation in entanglement evolution is introduced by the finite 1 1 energy density in the initial state.29 In the Floquet CFT, since 2π2km we drive the system periodically, the system can absorb en- ∆ ' − δ. (2.44) L ergy all the way if there are infinite degrees of freedom and the 16

35 1. Phase transition I l=0.5L 30 l=0.3L 25 l=0.2L One set of solutions for the phase transitions are T0 = mL, l=0.1L with m = 1, 2, 3 ··· , which correspond to the vertical lines in 20

(t) l=0.05L

A Fig.2. Here we denote this set of solutions as phase transition S 15 I. In this case, one can find that the entanglement entropy has 10 the expression in Eq.(2.51) with 5  πT 2 πl 2πl 0  2 1 2 0 200 400 600 800  E = − 4n · sin + cos , t / L L L L (2.52)  2πl  F = sin . FIG. 11. Numerical simulation for the entanglement entropy L evolution in the long time limit in the heating phase. We choose T1 = T2 = T = 0.9L, with L = 500. One can check that for t = 0, i.e., n = 0, one has c L πl S (t = 0) = · log · sin , (2.53) A 6 π L energy spectrum goes to infinity.) Similar problems also appear in the entanglement entropy in a CFT with finite temper- which is the entanglement entropy in the ground state, as ex- ature. In the high temperature limit, the entanglement entropy pected. For l = L/2, the entanglement entropy can be simpli- c l c πl for a finite subsystem of length l is SA(β) ' 3 log + 3 · β , fied as where is the UV cutoff introduced at the entanglement cut.70 " 2# The entanglement entropy grows linearly with the tempera- c L c 2 πT1 SA(n) = log + log 1 + 4n · , (2.54) ture 1/β all the way. In a lattice system, however, the entan- 6 π 6 L glement entropy will finally saturate as 1/β increases, since there are a finite number of degrees of freedom in a finite sub- which is Eq.(0.18) in the main text. A typical plot is shown in system and the bandwidth (of the energy spectrum) is finite. Fig.12 (top). As a remark, it is interesting that SA(n) at phase transition I has the same form as that after a single quench in Ref.51. In Ref.51, we start from the ground state of H0, and evolve it C. Phase transitions with the new Hamiltonian H1. Then the entanglement entropy evolution has the expression in Eq.(2.51) with The phase transitions happen at ∆ = 0 [see Eq.(0.6)]. To  2 study the entanglement entropy at the phase transition, we  πt 2 πl 2πl  E = − 4 sin + cos , cannot use the formula in Eq.(1.37) directly. It is because after L L L (2.55) analytical continuation, one has η = 1 and γ1 = γ2. There-  2πl  F = sin . fore, a = b = c = d = 0 in Eq.(1.29), and then Eq.(1.37) is L not well defined. In this case, zn is related to z in Eq.(0.10), i.e., By making t = nT1, Eqs.(2.52) and (2.55) are the same. This is not a coincidence. In the case of Floquet CFTs, for 1 1 T0 = mL with m = 1, 2, 3 ··· , the state ‘revives’ after a time = + n · β. (2.49) evolution of T with H . Effectively, the state only evolves ac- zn − γ z − γ 0 0 cording to H1, corresponding to the single-quench case. This where γ = (a − d)/2c and β = c, with a, b, c, d given in can be easily seen based on Eq.(0.4). After one cycle of driv- Eq.(1.15). Then Eq.(2.49) can be rewritten as ing, one has (after analytical continuation)

h iπT0 i −iπT0 iπT1 L iπT1 L az + b (1 + L ) · e z − L · e zn = . (2.50) z1 = . (2.56) cz + d iπT iπT0 iπT −iπT0 1 L 1 L L · e z + 1 − L · e where a = 1+nβ·γ, b = −nβ·γ2, c = nβ, and d = 1−nβ·γ. One can ﬁnd that for T0 = mL with m = 1, 2, 3 ··· , z1 has Then, following the procedure in Sec.IB, one can obtain the the same form as that for T = 0. I.e., effectively, the state entanglement entropy at the phase transitions as follows: 0 only evolves with the Hamiltonian H1. √ c L c E2 + F 2 − E E2 + F 2 S (n) = log + log . A 6 π 12 2 2. Phase transition II (2.51) Note that there are two sets of solutions for ∆ = 0 at the The other set of solutions for phase transitions are deter- phase transitions, and the expressions of E and F are different h 2 i πT1 πT0 πT1 πT0 for these two sets of solutions, as discussed in the following. mined by 1 − L sin L + 2 · L · cos L = 0. 17

which is Eq.(0.19) in the main text. A typical plot of SA(t) is shown in Fig.12 (bottom). It is noted that for both phase transitions I and II, the entanglement entropy grows as SA(t) ' 2.5 Numerics c CFT 3 log t for large n. (t)

A 2 S D. On single-point correlation function 1.5

Since the entanglement entropy in this work is calculated 3 4 10 10 through the correlation function of twist operators which are t themselves primary operators, it is straightforward to ob-

tain the correlation functions from the results of entanglement entropy (and vice versa). This can be clearly seen in 2.5 Numerics Eqs.(1.35)∼(1.37). CFT 2 As an example, for a primary operator in the non-heating (t)

A phase, one can find that S 1.5 hψ(t)|O(x = l)|ψ(t)i 1 π2h 2P 4 h (2.60) b 3 4 =AO · · √ 10 10 L E2 + F 2 − E E2 + F 2 t where h is the conformal dimension, and P , E, and F are given in Eqs.(2.2) and (2.3). For l = L/2, one has FIG. 12. Entanglement entropy evolution at the phase transition. (Top) Phase transition at T0 = L, with T1 = 500, 150, 50, and 20 hψ(t)|O(x = L/2)|ψ(t)i (from top to bottom); and (Bottom) phase transition for T0 < L with π K − R cos ϕ 2h (2.61) T1 = 150, 80, 50 and 30 (from top to bottom). Here we choose =Ab · · . L = 500, and l = L/2. O L K − R cos(2nφ + ϕ) See Eq.(2.3) for definitions of variables. That is, the single- We denote this set of solution as phase transition II. It can point correlation function oscillates in time with t := n(T0 + be found that the entanglement entropy has the expression in T1). Similarly, one can find that the single-point correlation Eq.(2.51) but with E and F as follows function decays exponentially in time in the heating phase, and decays in a power-law in time at the phase transitions.  2 The behaviors of entanglement entropy and single-point h 2 2(1 − x ) 2 2πl  E = − n · − cos correlation functions and their correspondence with Mobius¨  x2(1 + x2) x2 L  transformations are summarized in Table I.  4 2πl i − n · − cos (2.57)  1 + x2 L   2πl III. A LATTICE MODEL ON CRITICAL FERMION  F = sin . L CHAIN where we have defined x := L . As a self-consistent check, πT1 Here we give some details on the calculation of entangle- 2πl for n = 0, one has E = cos L . Then one can find that ment entropy in a free fermion lattice under periodic driving. The essential part is to calculate the equal time two-point cor- c L πl relation functions. Then based on the method in 71, one can S = log · sin , (2.58) A 6 π L evaluate the entanglement entropy explicitly. We consider a free fermion chain with half filling. It has fi- which is the entanglement entropy in the ground state, as ex- nite sites L with open boundary conditions. The Hamiltonians 51 pected. For the specific case with l = L/2, one has H0 and H1 have the following form:  1 L−1 2 4 4  X † E = − n · − n · + 1 , (2.59)  H0 = ci ci+1 + h.c., x2(1 + x2) 1 + x2  2 i=1 (3.1) L−1 and F = 0. Then the entanglement entropy can be simplified  X 2 π(i + 1/2) †  H1 = sin c ci+1 + h.c. as  L i  i=1 ( 4 2 ) c L 4 πT1 4 πT1 † S (t) ' log L n2 − L n+1 . where ci (ci ) are fermionic operators, which satisfy the A 2 2 † † 6 π πT1 πT1 {c , c } = {c , c } = 0 1 + L 1 + L anticommutation relations i j i j , and 18

† 1 iH1t −iH1t −ij t {ci, cj} = δij. At t = 0, we prepare the initial state as the e βj e = e βj. Then one can ﬁnd that ground state |Gi of H0, and then evolve the state with H1 X † X † X for time T1, and H0 for time T0. Then we repeat this driving βi = (V )ijcj = (V )ij Ujkγk procedure in time. j j k For completeness, in the following we list the procedures of X X (3.9) = (V †U) γ =: W γ . calculating two-point correlation functions in various cases. ik k ik k k k That is, A. Ground state X βi = Wijγj. (3.10) Now we consider the ground state |Gi of H0, and evaluate j † hG|cmcn|Gi. With a unitary transformation where we have deﬁned W = V †U. Similarly, one has X X † cn = Uniγi, γi = (U )ijcj, (3.2) X † i j γk = (W )kiβi. (3.11) i one can diagonalize the Hamiltonian H0 as follows Then we can check that

L 1 X † iH1t −iH1t X −ii t X e cne |Gi = Vnie Wikγk|Gi. (3.12) H0 = iγi γi. (3.3) i=1 i k

† P † ∗ P † † † It is convenient to deﬁne Note also that cn = i γi Uni = i γi (U )in, and γi = P † ∗ † P † (t),1 1 (U ) c = c Uji. The ground state |Gi can be writ- −ii t j ij j j j Wik := e Wik, (3.13) ten as and then L/2 Y † |Gi = γ |vaci. (3.4) iH1t −iH1t X X (t),1 i e cne |Gi = Vni Wik γk|Gi i=1 i k (3.14) X (t),1 Then one can ﬁnd = V · W nkγk|Gi. i † X † hG|cmcn|Gi = Uni(U )im, (3.5) i∈occ. Then, it is straightforward to check that

iH1t † −iH1t iH1t −iH1t where ‘occ.’ denote the occupied modes. hG|e cme e cne |Gi † (t),1† X (t),1 =hG|γk0 V · W k0m [V · W ]nkγk|Gi k (3.15) B. Quantum quench and Floquet case X (t),1 (t),1† = [V · W ]nk · [V · W km. Now we consider a single quench by evolving the ground k∈occ. state |Gi with the Hamiltonian H1. Then the time dependent Let us move one step further to the ‘double quench’, and wavefunction has the form consider the state |ψ(t)i = e−iH0T0 e−iH1T1 |Gi, with t = T + T . We check the following quantity: |ψ(t)i = e−iH1t|Gi. (3.6) 0 1

iH1T1 iH0T0 −iH0T0 −iH1T1 † e e cne e |Gi Now we evaluate hψ(t)|c cn|ψ(t)i. We need another unitary m 0 1 X −i T0 X † −i T1 X (3.16) transformation to diagonalize H1, i.e., = Unie i (W )ije j Wjkγk|Gi. i j k X X † cn = Vniβi, βi = (V )ijcj, (3.7) i j By deﬁning

0 (T0),0 −i T † so that i 0 Wik = e W ik, (3.17) X 1 † H1 = i βi βi. (3.8) one has i iH1T1 iH0T0 −iH0T0 −iH1T1 e e cne e |Gi † P † ∗ P † † † h i Note also that cn = i βi Vni = i βi (V )in, βi = X (T0),0 (T1),1 (3.18) 1 = U · W · W γk|Gi. P † ∗ † P † iH1t † −iH1t ij t † nk j(V )ijcj = j cjVji, e βj e = e βj , and k∈occ. 19

Then it is straightforward to obtain where h i † X (T0),0 (T1),1 hψ(t)|cmcn|ψ(t)i = U · W · W · nk k∈occ. h i† U · W (T0),0 · W (T1),1 . km (T0),0 (T1),1 (T0),0 (T1),1 (3.19) W = U · [W · W ] ··· [W · W ]. (3.22)

Now we consider the Floquet case, with

|ψ(t)i = e−iH0T0 e−iH1T1 ··· e−iH0T0 e−iH1T1 |Gi, (3.20) where t = n(T0 + T1). Based on the above examples, it is In the above, we showed how to obtain the two-point correla- straightforward to obtain tion functions for various cases, based on which we can obtain the entanglement entropy evolution by using the method † X † hψ(t)|cmcn|ψ(t)i = Wnk · (W )km, (3.21) in Ref.71. k∈occ.