Conformal Floquet dynamics with a continuous drive protocol
CConformal Floquet dynamics with a continuous drive protocol
Diptarka Das (1) , Roopayan Ghosh (2) , and K. Sengupta (2) ∗ (1) Department of Physics, Indian Institute of Technology Kanpur, Kanpur 208016, India (2)
School of Physical Sciences, Indian Association for the Cultivation of Science,2A and 2B Raja S. C. Mullick Road, Jadavpur, Kolkata 700032, India (Dated: January 13, 2021)We study the properties of a conformal field theory (CFT) driven periodically with a continuousprotocol characterized by a frequency ω D . Such a drive, in contrast to its discrete counterparts(such as square pulses or periodic kicks), does not admit exact analytical solution for the evolutionoperator U . In this work, we develop a Floquet perturbation theory which provides an analytic,albeit perturbative, result for U that matches exact numerics in the large drive amplitude limit.We find that the drive yields the well-known heating (hyperbolic) and non-heating (elliptic) phasesseparated by transition lines (parabolic phase boundary). Using this and starting from a primarystate of the CFT, we compute the return probability ( P n ), equal ( C n ) and unequal ( G n ) timetwo-point primary correlators, energy density( E n ), and the m th Renyi entropy ( S mn ) after n drivecycles. Our results show that below a crossover stroboscopic time scale n c , P n , E n and G n exhibitsuniversal power law behavior as the transition is approached either from the heating or the non-heating phase; this crossover scale diverges at the transition. We also study the emergent spatialstructure of C n , G n and E n for the continuous protocol and find emergence of spatial divergencesof C n and G n in both the heating and non-heating phases. We express our results for S mn and C n in terms of conformal blocks and provide analytic expressions for these quantities in several limitingcases. Finally we relate our results to those obtained from exact numerics of a driven lattice model. I. INTRODUCTION
The study of driven quantum systems have attracteda lot of attention in recent years . The theoreti-cal interest in this area stemmed from the fact thatsuch systems provide access to a gamut of phenomenathat have no analog in their equilibrium counterparts.Some of these phenomena, for periodically driven sys-tems, include dynamical phase transitions , dynamicalfreezing , realization of time crystals , and the pos-sibility of tuning ergodicity of the driven system with thedrive frequency . Moreover, periodically driven systemscan lead to novel steady states which has no counterpartin equilibrium quantum systems . The study of thesephenomena has also received significant impetus from thepossibility of realization of closed quantum systems usingultracold atom platforms; indeed, such platforms haverecently been used to experimentally probe several non-equilibrium phenomena .A large set properties of such periodically driven sys-tems can be inferred from studying its Floquet Hamil-tonin H F which is related to the evolution operator U of the system via the relation U ( T,
0) = exp[ − iH F T / (cid:126) ] .The computation of H F for a generic many-body sys-tem provides a significant challenge; indeed, an exactanalytic computation of H F is generally possible for in-tegrable models driven by discrete stepwise protocols(such a square pulse or kicks) . This has led to develop-ment of several perturbative schemes for computation of H F ; some of these include Magnus expansion , adiabatic-impulse approximation , the Hamilton flow method ,and the Floquet perturbation theory (FPT) . Out ofthese methods, the Floquet perturbation theory has theadvantage of being easily applicable to a wide class of systems as well as being accurate over a large range ofdrive frequencies .More recently, several theoretical studies concentratedon the effect of Floquet dynamics on conformal fieldtheories . Usually driving a CFT is expected to gen-erate an additional scale in the problem which drives thesystem away from its conformally invariant fixed point.However, it was recently realized that there is a classof models where drive protocols need not break theconformal symmetry . It is found that for CFTs with asine square deformation (SSD) such dynamics can be ini-tiated by a Hamiltonian whose holomorphic part is givenby H ( t ) = 2 πL (cid:20) f ( t ) L + 12 f ( t )( L + L − ) (cid:21) , (1)where L n for each integer n denotes a holomorphic gen-erator of the Virasoro algebra,[ L m , L n ] = ( m − n ) L m − n + c m ( m − δ m + n, , (2)where c denotes the central charge. These holomorphicgenerators are related to the stress tensor T µν of the CFTby L = L π (cid:90) L dxT ( w ) L ± = L π (cid:90) L dxe ± πw/L T ( w ) (3)where w = τ + ix , x is the spatial coordinate and τ isthe Euclidean time. The anti-holomorphic ones have asimilar expression in terms of the complex conjugates. a r X i v : . [ h e p - t h ] J a n It is well-known that the class of Hamiltonians given byEq. 1 are valued in su (1 , U ( T,
0) = T e − i (cid:126) T (cid:82) H ( t ) dt = (cid:18) a bc d (cid:19) , (4)valued in the group SU (1 ,
1) with ad − bc = 1. Usingthe isomorphism of SU (1 ,
1) and SL (2 , R ) we shall alsohave occasion to write the evolution operator in imagi-nary time in the generic form U ( T = − iτ,
0) = (cid:18) ˜ a ˜ b ˜ c ˜ d (cid:19) ∈ SL (2 , R ) . (5)Its action on the complex plane C is given by the M¨obiustransformation z → z (cid:48) = ˜ az + ˜ b ˜ cz + ˜ d , z ∈ C (6)where ˜ a, ˜ b, ˜ c, ˜ d ∈ R and ˜ a ˜ d − ˜ b ˜ c =1.The exact solution for U ( T,
0) for Hamiltonians val-ued in su (1 ,
1) and driven by discrete protocols werediscussed earlier for periodic kicks and square pulseprotocols . It was found that such driven systemcould display two distinct phases depending on the driveparameters; these are termed as the heating (hyper-bolic) and the non-heating (elliptic) phase and are foundto be separated by a transition line (parabolic phaseboundary) . The presence of these phases can be shownto be a direct consequence of non-compact nature of theSU(1,1) group. However, such studies have not been ex-tended to continuous drive protocols where exact ana-lytic results are not available ; in particular, the phasediagram of such a driven system for continuous drive pro-tocols has not been studied so far.It was noted in Ref. 23 that the action of the evolu-tion operator U on a primary operator of the CFT canbe understood, in the Heisenberg picture, in terms of aM¨obius transformation of it’s coordinates leading to U † ( T, O ( z, ¯ z ) U ( T,
0) = (cid:18) ∂z (cid:48) ∂z (cid:19) h (cid:18) ∂ ¯ z (cid:48) ∂ ¯ z (cid:19) ¯ h O ( z (cid:48) , ¯ z (cid:48) ) , (7)where z (cid:48) is defined in Eq. 6 and the bar designatesanti-holomorphic variables throughout. Using this rela-tion, and a straightforward mapping from cylindrical orstrip geometries to the complex plane following standardprescription , several results were obtained on the en-ergy density, correlation function and entanglement en-tropy of the driven system . It was shown that inthe heating phase, such a drive leads to emergent spa-tial structure of the energy density . Moreover, it wasfound that the time evolution of the entanglement en-tropy shows linear growth with n (in the large n or long-time limit) in the heating phase. In contrast, it shows anoscillatory behavior in the non-heating phase and a log-arithmic growth on the parabolic phase boundary. All of these studies focussed on evolution starting from theCFT vacuum on a strip geometry ; the dynamics ofthe system starting from asymptotic states correspond-ing to primary operators of the theory, which necessi-tates computation of four-point correlation functions ofthe primary fields of the driven CFT, has not been stud-ied so far.In this work, we study dynamics of conformal field the-ories subjected to a continuous drive protocol. The pro-tocol we use corresponds to the Hamiltonian given by Eq.1 with f ( t ) = 1 and f ( t ) = f cos( ω D t ) + δf (8)where f is the drive amplitude and δf is a constant pa-rameter. In this study we shall focus on the large driveamplitude regime which corresponds to f (cid:29) δf,
1. Inthis regime for ω D ≥ δf,
1, the FPT is expected to be ac-curate and we expect this to provide us with an analytic,albeit perturbative, understanding of the properties ofthe driven system.The central results that we obtain from such a studyare as follows. First, we chart out the phase diagram ofthe driven system as a function of δf and ω D using exactnumerics. Our results show re-entrant heating and non-heating phases separated by a parabolic phase boundariesas a function of δf and T . Second, we provide analyticexpressions of the evolution operator U ( T,
0) using FPT.The phase diagram obtained from the perturbative re-sult provides a near-exact match with its exact numer-ical counterpart over a wide range of frequencies. Ourperturbative results indicate the existence of a parame-ter α , given within first order FPT by (where we haveput (cid:126) = 1) α = ∞ (cid:88) n = −∞ J n (cid:18) f πLω D (cid:19) T ( nπ + πδf T /L ) , (9)which indicates proximity of the system to the phaseboundary. The system stays in the non-heating phasefor α < α >
1; the phaseboundary between these two phases is given by α = ± U within FPT, to obtain ana-lytic expressions for energy density, equal and unequaltime two-point correlation functions, entanglement en-tropy, and return probability of generic sine-square de-formed CFT starting from a primary state. Our resultsexpresses these quantities in terms of α and allows us tocharacterize their behavior near the transition from theheating phase. For example, we find that the return prob-ability of any primary state after n drive cycles shows anuniversal behavior below a crossover time n c both in theheating and the non-heating phases near the transitionline; for n > n c , the probability decays exponentially forthe heating phase and remains an oscillatory function inthe non-heating phase. We provide analytic estimate of n c as a function of α . Fourth, we discuss the nature of theemergent spatial structure of the energy density and thecorrelation function of primary operators for the driveprotocol. We analytically show that the energy densityof any primary state in the heating phase displays peakswhich shifts from L/ L/ L/ C n and half-chain m th Renyi entropy S mn of the driven CFT after n drive cycles starting from aninitial primary state in terms of conformal blocks V p . Ingeneral, these blocks do not have analytical expression forarbitrary CFTs; here, we provide their analytical formsin several asymptotic limits. We discuss the applicabilityof these limits to the driven CFT and discuss the proper-ties of C n and S mn in these asymptotic regimes. Finally,we relate some of our results to those obtained by exactnumerical study of the SSD model on a 1D lattice .The plan of the rest of the paper is as follows. In Sec.II, we derive expressions of U and provide the phase dia-gram for our drive protocol. This is followed by Sec. IIIwhere we compute energy density, correlation functionsand return probabilities starting from a primary state.Next, in Sec. IV, we relate our results to those obtainedfrom numerical study of driven SSD Hamiltonian on a 1Dlattice. Finally, we discuss our main results and concludein Sec. V. Some details of the perturbative FPT calcu-lations and representation independent derivation of theMobius transformation are presented in the Appendices. II. PHASE DIAGRAM
To find U ( T,
0) corresponding to the Hamiltoniangiven by Eq. 1 with f ( t ) given by Eq. 8, we first notethat the holomorphic generators (Eq. 2) for n = − , , su (1 ,
1) subalgebra[ L , L − ] = L − , [ L , L ] = − L , [ L − , L ] = − L . (10)A representation of this algebra is furnished in terms ofthe Pauli matrices as L = σ z , L ∓ = ± σ ∓ , σ ± = 12 ( σ x ± iσ y ) , (11)where σ α for α = x, y, z are standard Pauli matrices. Inthis representation (Eq. 11), the holomorphic part of theHamiltonian H ( t ) becomes H ( t ) = πL ( f ( t ) σ z − iσ y ) (12)which corresponds to a Zeeman Hamiltonian of a singlespin-half particle with a time dependent magnetic fieldalong ˆ z and an imaginary constant magnetic field along y . The latter feature is a consequence of the SU(1,1)group structure and is crucial in realization of the heatingand the non-heating phases . The expression for U ( T, and square-pulse protocols.In contrast, for the continuous protocol given by Eq.8, an exact analytic expression for U ( T,
0) does not ex-ist. However, the numerical result for U can be ob-tained in a straightforward manner. Such a numer-ics is carried out by dividing the time period T into N Trotter steps with width δt i ∼ T /N . The max-imal allowed width of each of these steps depend onsystem energy scale and the drive frequency; they arechosen so that H does not vary appreciably within anystep. This allows one to write U (cid:39) (cid:81) i =1 ,N U i , where U j = U ( t j − + δt j , t j − ) = exp[ − i (cid:82) t j − + δt j t j − dtH ( t ) / (cid:126) ].This procedure leads to U ( T,
0) = exp[ − iH F T / (cid:126) ]; wefind numerically that the Floquet Hamiltonian is H F = p ( T ) σ z + iq ( T ) σ y (13)We note that σ x does not appear in H F . The con-dition for the different phases are thus obtained by | Tr U ( T, | > < δf or ω D .In particular, we note that large (cid:126) ω D (cid:29) δf,
1, the phasetransition between the heating and non-heating phase oc-curs at δf = 1.The absence of σ x in H F can be shown to be the con-sequence of an emergent dynamic symmetry of the evo-lution operator at stroboscopic times which follows fromthe periodicity of H ( t ). To see this we note that since H ( t ) = H ( T − t ), one has U j = U N − j +1 for all j in theTrotter product. Further, we note for any of these Trot-ter steps, one has σ x U j σ x = U − j . Thus one can write σ x U ( T, σ x = σ x (cid:89) j =1 ,N U j σ x (14)= (cid:89) j =1 ,N U − j = (cid:89) j =1 ,N U j − = U − ( T, U − j = U − N − j . Eq. 14clearly implies that if U = exp[ − iH F T ] then σ x H F σ x = − H F . This forbids presence of terms ∼ σ x in H F . Itis to be noted that this symmetry is absent for t (cid:54) = nT where n is an integer.Next, we develop an perturbative analytic expressionof U in the limit when f is the largest scale in theproblem. Here the diagonal term in H ( t ) is treatedexactly and the effect of the off-diagonal term is takeninto account within standard time-dependent perturba-tion theory . In this scheme, the first term for U and H F is given by U ( t,
0) = e − i ( π/L )( f sin ω D t/ω D + δft ) σ z H (0) F = sσ z /T (15)where s = arccos[cos( δf T π/L )]. We note that H (0) F re-tain the periodic structure of U .To obtain the first order correction to U ( T, U ( T,
0) = U ( T, I − U (cid:48) ( T, I denotes the 2 × U (cid:48) denotesthe first-order correction to U in the interaction picturegiven by U (cid:48) ( T,
0) = − i (cid:90) T dtU † ( t, H U ( t,
0) (17)where H = − πL iσ y is the perturbative term of theHamiltonian. A simple calculation detailed in App. Ayields U (cid:48) ( T,
0) = iαe is sin( s ) σ y U ( T,
0) = (cid:18) e − is − iα sin siα sin s e is (cid:19) (18)where α is given by Eq. 9. We note that U ( T,
0) isnot unitary; this is a well-known issue with perturbationtheory for U . In what follows we unitarize U ( T,
0) asfollows. We note that this can be done by first writ-ing ( I − U (cid:48) ( T, (cid:39) exp[ − iH (cid:48) F T ], where the matrix H (cid:48) F = ( i/T ) U (cid:48) . The evolution operator is then given by U ( T,
0) exp[ − iH (cid:48) F T ] which is unitary. However, in thepresent case, we find that evolution operator obtained bythis method retains terms ∼ σ x . This is clearly incon-sistent with the dynamic symmetry discussed in Eq. 14.Thus we chose to unitarize U ( T,
0) directly since it doesnot have any term σ x . These two alternative routes toobtaining H F coincides with each other in the large ω D limit, where s → U ( T,
0) given by Eq. 18 can beachieved by writing U ( T,
0) = exp[ − iθ ( σ z n z + σ y n y )] , θ = s (cid:112) − α n z = 1 √ − α , n y = iα √ − α (19)Comparing Eqs. 13 and 19, we find that the first orderFPT result provides approximate analytic expressions for p ( T ) and q ( T ) given by q/p (cid:39) α, p (cid:39) s/T (20)In what follows we shall use Eq. 20 to compare betweenexact numeric and approximate analytic results. The evolution operator obtained in Eq. 19 indicatesseveral interesting features. First, we find that n y isimaginary; this is a direct consequence of the SU(1,1)group structure. Second, we note that Tr[ U ( T, θ ); thus the condition | Tr[ U ( T, | > ( < )2 for re-alization of heating (non-heating) phases translates to θ being imaginary(real). This happens when α > ( < )1which leads to our identification of α as the parameterwhose value determines the phase the system. Third,the parabolic phase boundary between these two phasesis given by | Tr[ U ( T, | = 2 which leads to θ = 0. Thisis realized for α = ± α inEq. 9, we find that for ω D (cid:29) f , the condition α = 1is satisfied for δf = 1. This can be verified directlyfrom the exact phase diagram in the left panel of Fig. 1.Fourth, the re-entrance of the heating and non-heatingphases as a function of δf shown in the right panel ofFig. 1 displaying the phase diagram obtained from firstorder FPT, can be easily explained by periodic natureof | cos( θ ) | since the phases must repeat for θ → θ + π .We note the two phase boundaries in each of the lobes ofthis phase diagram correspond to α = 1 (lower branch ofthe lobes) and α = − α → ∞ which occursalong the lines δf /ω D = n / n ∈ Z . This can bedirectly seen from Eq. 9 where the term in the sum cor-responding to n = n diverges in this limit. Finally, acomparison between the left and the right panels of Fig.1 shows that the two phase diagrams match qualitativelyfor (cid:126) ω D / ( π/L ) >
1. This allows us to justify the use ofthe analytical method for a qualitative understanding ofthe dynamics within first order FPT. A computation ofthe second-order results of FPT is carried out in App. A;we find that it retains all features of the first order theoryand provides a near-identical phase diagram. In App. B,we chart out a representation independent derivation ofthe first order perturbation results in the high frequencylimit which matches the results obtained here using theSU(1,1) representation of the Virasoro generators.Before ending this section , we obtain the elements of U n = U ( nT,
0) (where n ∈ Z is an integer). Using Eqs.4 and 13, we find the elements of U j for any integer j obtained using exact numerics in the non-heating phase(where p > q ) to be a j = cos( (cid:112) p − q jT ) − i p (cid:112) p − q sin( (cid:112) p − q jT ) b j = iq (cid:112) p − q sin( (cid:112) p − q jT ) (21)where d j = a ∗ j and c j = b ∗ j . For the heating phase, forwhich p < q , one obtains a (cid:48) j = cosh( (cid:112) q − p jT ) − i p (cid:112) q − p sinh( (cid:112) q − p jT ) b (cid:48) j = iq (cid:112) q − p sinh( (cid:112) q − p jT ) (22) FIG. 1: (Color online) Left Panel: Plot of the phase dia-gram, obtained from | Tr U ( T, | plotted as a function of theamplitude δf and frequency ω D as obtained from exact nu-merics. The red regions indicate heating phases while the vi-olet ones show the non-heating phases. The re-entrant transi-tion between these phases are shown by black lines indicatingparabolic transition lines. Right Panel: Similar phase dia-gram obtained from first order FPT. For both plots, we haveset π/L to unity and set f = 10. For the transition line where p = q , a careful evaluationof the limit shows a ” j = 1 − ipjT, b ” j = − ipjT (23)The corresponding analytic, first order FPT, expressionsof elements of U ( jT,
0) in terms of α and s can be directlyread off from these equations by using the correspondenceexpressed in Eq. 20. For example for the transition line,the first order FPT result is a ” j = 1 − ijs and b ” j = − ijs . Also, the corresponding elements ˜ a j , ˜ b j , ˜ c j and˜ d j of the evolution operator in Euclidean time can beobtained for each of these phase from Eqs. 21, 22, and 23via standard analytic continuation T → − iτ . We shalluse these expressions in the next section computation ofreturn probability, energy density, correlation functionsand the entanglement entropy. III. RESULTS FOR DRIVEN CFT
It was pointed out in Ref. 23 that the operation of theevolution operator U on any primary operator of the CFTcan be understood in terms of a M¨obius transformationof its coordinates and is therefore given by Eq. 7. Thisis further discussed in App. B. In this section, we shalluse this result to study the properties of the driven CFTwith a cylindrical geometry which corresponds to peri-odic boundary condition of a driven 1D chain of length L . In our setup, the coordinate of the cylinder is givenby w = τ + ix , where x is the spatial coordinate and τ is the Euclidean time; we use w = ( L/ (2 π )) ln z formapping such a cylinder to the complex plane. Thus forany primary operator O ( w, ¯ w ) on the cylinder, we can write U n † O ( w, ¯ w ) U n = (cid:18) ∂w∂z (cid:19) − h (cid:18) ∂z n ∂z (cid:19) h (cid:18) ∂ ¯ w∂ ¯ z (cid:19) − ¯ h × (cid:18) ∂ ¯ z n ∂ ¯ z (cid:19) ¯ h O ( z n , ¯ z n ) (24)where z n is the transformed coordinates given by Eq. 6and ˜ a n , ˜ b n , ˜ c n , and ˜ d n are the elements of U n in Eu-clidean time which can be obtained from analytic con-tinuation of T → − iτ in Eqs. 21, 22 and 23. In allcomputations, we shall work in Euclidean time and carryout the analytic continuation to real time using τ = iT at the end of the calculation.Apart from the primary operators we shall also usetransformation properties of the stress tensor. The holo-morphic part of the stress tensor denoted by T ( w ) trans-forms, under a general coordinate transformation, as T ( w ) → T (cid:48) ( w (cid:48) ) = ( ∂w (cid:48) /∂w ) − ( T ( w ) + S w (cid:48) ,w ), where S is Schwarzian given by S w (cid:48) ,w = c (cid:34)(cid:18) ∂ w (cid:48) ∂w (cid:19) (cid:18) ∂w (cid:48) ∂w (cid:19) − − (cid:32)(cid:18) ∂ w (cid:48) ∂w (cid:19) (cid:18) ∂w (cid:48) ∂w (cid:19) − (cid:33) (25)Here we note that S z n ,z = 0 since S vanishes for anyM¨obius transformation and S z,w = c / (24 z ) for trans-formation from a cylinder to the complex plane. Thetransformation for the antiholomorphic part of the stresstensor can be obtained in the similar manner by replacing( w (cid:48) , w ) → ( ¯ w (cid:48) , ¯ w ).In what follows, we shall use these general results toobtain the stroboscopic time ( n ) dependence of the re-turn probability, energy density, equal and unequal timecorrelation functions, and the entanglement entropy ofthe driven CFT. The initial state for this purpose is cho-sen to be an asymptotic in-state of the CFT denoted by | h, ¯ h (cid:105) = lim w, ¯ w →−∞ φ ( w, ¯ w ) | (cid:105) (26)where | (cid:105) denotes the CFT vacuum, φ is a primary fieldwith dimension ( h, ¯ h ) and we note w, ¯ w → −∞ corre-sponds to τ → −∞ which in turn implies z, ¯ z → (cid:104) h, ¯ h | = (cid:104) | lim w, ¯ w →∞ φ ( w, ¯ w ) (27)which corresponds to τ → ∞ and hence z, ¯ z → ∞ on thecomplex plane.The reason for choosing these primary states are asfollows. First, we note that since L and L ± annihilatesthe CFT vacuum, the vacuum state does not evolve. Thisrenders all equal time correlation functions of primaryoperators, entanglement entropy and energy density tobe fixed at their equilibrium values under action of U ;only the unequal-time correlation function shows non-trivial dynamics. This feature of driven CFT usuallycompels one to work with the strip geometry or usedifferent M¨obius transformation where one can obtainnon-trivial time evolution starting from the ground state.Here we use an alternative approach by using the cylin-drical geometry but starting from the primary states ofthe CFT ( | h, ¯ h (cid:105) ) as initial states; these are the simpleststates of the model which have non-trivial dynamics un-der action of U . A. Return probability and energy density
The return probability amplitude A n after n cycles ofthe drive is given by A n = (cid:104) h, ¯ h | U ( nT, | h, ¯ h (cid:105)(cid:104) h, ¯ h | h, ¯ h (cid:105)| (28)where the denominator corresponds to the normalizationof the asymptotic states. To this end, we first look atthe contribution of the holomorphic part of the returnprobability amplitude given by A hol n = lim w → ,w →∞ (cid:104) | φ ( w ) U ( nT, φ ( w ) | (cid:105) lim w → ,w →∞ (cid:104) | φ ( w ) φ ( w ) | (cid:105) = ND (29)To compute A hol n , we take w i = τ i + ix i for i = 1 , τ → −∞ and τ → ∞ at the endof the calculation. Using the fact that (cid:104) | U † = (cid:104) | , wecan rewrite as N = (cid:104) | U n † φ ( w , ¯ w ) U n φ ( w , ¯ w ) | (cid:105) . Wethen use Eq. 24 to obtain N = lim z →∞ ,z → (cid:18) πL (cid:19) h (cid:18) z z (˜ c n z + ˜ d n ) ( z n − z ) (cid:19) h D = lim z →∞ ,z → (cid:18) πL (cid:19) h (cid:18) z z ( z − z ) (cid:19) h (30)where z n = (˜ a n z + ˜ b n ) / (˜ c n z + ˜ d n ) is the transformedcoordinate, ˜ a n , ˜ b n , ˜ c n and ˜ d n are obtained from Eqs. 21,22 ,23 after analytic continuation T = − iτ and we haveused ∂w/∂z = L/ (2 πz ) and ∂z n /∂z = (˜ c n z + ˜ d n ) − .After a careful evaluation of the limits, we finally obtain A hol n = ˜ a − hn . A similar computation shows the contri-bution from the anti-holomorphic part to be A ant − hol n =˜ a − hn . Thus one obtains the return probability P n , afteranalytic continuation to real time, to be P n = | A hol n A ant − hol n | = | a n | − h +¯ h ) (31)The exact numerical value of P n can thus be obtained byusing Eqs. 21, 22, 23. Here and for all quantities in therest of this section, we shall provide expressions for theprediction of FPT; the numerical result in terms of p and q can be directly read from these expressions by using Eq. - - - - - -
50 2 4 6 8 10 l n ( P n ) / ( h + h ) ω - - D ω = ω = ω = - - - - - l n ( P n ) / ( h + h ) n FIG. 2: (Color online) Left Panel: Plot of ln P n / (4( h + ¯ h ))as a function of frequency ω D for after n = 50 cycles of thedrive. The inset shows the variation of ln P n with ω D in thenon-heating phase. Right Panel: Plot of ln P n / (4( h + ¯ h ))as a function of n for three representative drive frequenciescorresponding to the heating ( ω D = 3 .
26 orange line) andnon-heating phases ( ω D = 3 .
28, green line) and the transitionline ( ω D (cid:39) . π/L has been setto unity, f = 10 and δf = 0 .
5. See text for details. i.e. with the mapping α → p/q and s → pT ). Thisprocedure yields P n = (cid:18) α (1 + cosh nθ ) − α − (cid:19) − h +¯ h ) , heating= (cid:18) − α (1 + cos nθ )2(1 − α ) (cid:19) − h +¯ h ) , non − heating= (1 + n s ) − h +¯ h ) , transition line (32)The behavior of ln P n is shown in the left panel of Fig.2 as a function of drive frequency after n = 50 cycles ofthe drive and for δf = 0 .
5. We find that ln P n shows dipsat large n in the heating phases; in contrast, it shows os-cillatory behavior near the transition in the non-heatingphase as can be seen from the inset of the left panel ofFig. 2. We also note that the return probability shows anexponential decay for large n in the heating phase and anoscillatory behavior in the non-heating phase accordingto standard expectation. On the transition line, it showsa power law decay ∼ n − h +¯ h ) for large n . These qualita-tively different behaviors of P n can be clearly seen in theright panel of Fig. 2 where ln P n is plotted as a functionof n for three values of drive frequencies correspondingto three different phases. Here we also note that near thephase transition line, in either phase, the behavior of P n is identical to that on the critical line below a crossovertimescale n ≤ n c . This is most easily seen by expand-ing the cosh( nθ ) term in powers of nθ in Eq. 32. Thisprocedure yields, in the heating phase,ln P n (cid:39) − h + ¯ h ) ln (cid:20) αsn ) + ( αsn ) α − α + ... (cid:21) n c (cid:39) √ / ( s (cid:112) α −
1) (33)where the ellipsis denote terms higher order in n . Wenote that n c here is estimated as n for which the con-tribution of the term ∼ n becomes equal to the leadingterm ∼ n . We find that n c diverges at the transition as1 / √ α − δ f n c FIG. 3: (Color online) Left Panel: Plot of ln P n / (4( h + ¯ h ))as a function of frequency n for δf = 1 . . ω D = 100. The red dashed line is aplot of − ln(1 + s n ). Right panel: Plot of n c as a functionof δf for ω D = 100 showing a sharp peak at the transition.For all plots π/L has been set to unity, and f = 10. See textfor details. drive frequency or amplitude since s ∼ δf T . For n (cid:28) n c ,the behavior P n is identical to that on the transition line.A similar result can be obtained if the transition line isapproached from the non-heating phase. Thus we findthat characteristic behavior the return probability on thetransition line can be observed upon approaching the linebelow a crossover timescale n c which diverges at the tran-sition; P n has a universal dependence on s for n (cid:28) n c .This behavior is shown in Fig. 3 in the high drive fre-quency regime ( (cid:126) ω D ) / ( π/L ) = 100 where δf = 1 givesthe position of the transition line. We clearly see thatln P n remains indistinguishable for n ≤ n c (cid:39)
50. The di-vergence of n c at the transition line is shown in the rightpanel of Fig. 3.Next, we study the behavior of the energy density of the driven system given by E n = (cid:104) h, ¯ h | U † n ( T ( w ) + ¯ T ( ¯ w )) U n | h, ¯ h (cid:105)(cid:104) h, ¯ h | h, ¯ h (cid:105) (34)To evaluate this, we first consider the holomorphic contri-bution to the energy density. To this end, we first movefrom the cylinder to the complex plane so that one canwrite T ( w ) = (cid:18) πzL (cid:19) T ( z ) + c (35)where the constant term c = (2 π/L ) c /
24 comes fromthe Schwarzian given by Eq. 25 and c denotes the centralcharge. In what follows we shall ignore this term sinceit does not change upon driving the system. Using thisand Eqs. 24 we find (cid:104) h | U † n T ( w ) U n | h (cid:105) = lim z →∞ ,z → (cid:18) πL (cid:19) h ) z h z h z × (˜ c n z + ˜ d n ) − (cid:104) | φ ( z ) T ( z n ) φ ( z ) | (cid:105) (36)where we have used w = ix so that z = exp[2 πix/L ] and w i = τ i for i = 1 , τ → −∞ , τ → ∞ leading tothe limits z → ∞ and z → (cid:104) φT φ (cid:105) correlator in the CFTvacuum state. This can be done in a straightforwardmanner using standard identity and yields (cid:104) | φ ( z ) T ( z n ) φ ( z ) | (cid:105) = h (cid:2) ( z n − z ) − ( z − z ) − h + ( z n − z ) − ( z − z ) − h − z n − z ) − ( z − z ) − (2 h +1) − z n − z ) − ( z − z ) − (2 h +1) (cid:105) (37)We substitute Eq. 37 in Eq. 36 and find that in the limit z → ∞ only the second term in the right side of Eq. 37contribute. Taking the z → E hol n = (cid:18) πL (cid:19) h ( R n ( x ) R n ( x )) (38) R n ( x ) = [˜ a n (˜ c n ) e πix/L + ˜ b n ( ˜ d n ) e − πix/L ] (39)The antiholomorphic contribution can be simply read offfrom this to be E hol n ( x ) = E ant − hol n ( − x ). Thus the totalenergy density which is the sum of the holomorphic andanti-holomorphic parts are given by E n ( x ) = (cid:18) πL (cid:19) h (cid:0) [ R n ( x ) R n ( x )] − +[ R n ( − x ) R n ( − x )] − (cid:1) (40) We note that the expressions of the energy density of theasymptotic states is identical to that obtained for thevacuum state in the strip geometry in Ref. 26. The onlydifference appears in the prefactor; the asymptotic stateshave a prefactor h while the vacuum in the strip geom-etry has a prefactor of c /
32. Thus we expect similaremergent spatial structure for E n ( x ) as discussed in Ref.26. Below, we analyze this phenomenon for the continu-ous drive protocol. To this end, we analytically continueto real time, substitute Eq. 21 in Eq. 40 and, using Eq. n = = = x / L E ( x / L ) δ f δ x FIG. 4: Left Panel: Plot of E n ( x ) as a function of x for severalrepresentative values of n in the heating phase with ω D = 3and δf = 0 .
04. Right panel: Plot of δx/L , as a function of δf for ω D = 100. The peaks of E n ( x ) occur at L/ ± δx . For allplots π/L is set to unity and f = 10. See text for details.
20, obtain for the heating phase E n ( x ) = (cid:18) πL (cid:19) h ( α − Q ( x ) + Q ( x )( Q ( x ) − Q ( x )) Q ( x ) = α cos (2 πx/L ) (cosh[2 nθ ] − α cosh[2 nθ ] − Q ( x ) = α (cid:112) α − πx/L ) sinh[2 nθ ] (41)We note from Eq. 41 that for a generic x , E n ( x ) de-cays exponentially with n with the large n limit: E n ( x ) ∼ e − nθ . This behavior can be in Fig. 4 where E n ( x ) is plot-ted for several n as a function of x . The plot shows that E n ( x ), at large n , decays at all positions except at twoplaces for which the leading terms of Q ( x ) and Q ( x ) inthe large n limit cancels each other. A straightforwardcalculation shows that x ± c = L ± δx, δx = L π arccos(1 /α ) (42)within first order FPT. Thus the position of the peaksof E n ( x ) in the large n limit moves from L/ L/ L/ ± δx/L ) as a functionof δf for (cid:126) ω D = 100( π/L ).In contrast for the non-heating phase we find using Eq.22, E n ( x ), in real time, is given by Eq. 41 with Q ( x ) → Q (cid:48) ( x ) and Q ( x ) → Q (cid:48) ( x ), where Q (cid:48) ( x ) = α cos (2 πx/L ) (1 − cos[2 nθ ]) + 1 − α cos[2 nθ ] Q (cid:48) ( x ) = α (cid:112) α − πx/L ) sin[2 nθ ] (43)Thus E n ( x ) is an oscillatory function of n as shown in theleft panel of Fig. 5 where E n ( x ) is plotted as a functionof x of several n .Finally on the transition line, using Eq. 23, we find E n ( x ) = (cid:18) πL (cid:19) h Q ” ( x ) + Q ” ( x ))( Q ” ( x ) − Q ” ( x )) Q ” ( x ) = 1 + 2 n s (1 + cos(2 πx/L )) Q ” ( x ) = 2 sn sin(2 πx/L ) (44) n = = = / L E ( x / L ) n = = = - x / L E ( x / L ) FIG. 5: Left Panel: Plot of E n ( x ) as a function of x for severalrepresentative values of n in the non-heating phase ith ω D = 3and δf = 0 .
4. Right panel: Same for ω D = 3 and δf = 0 . x = L/ n For allplots π/L is set to unity and f = 10. See text for details. ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆● δ f = ■ δ f = ◆ δ f = E ( L ) ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆ ◆◆◆● δ f = ■ δ f = ◆ δ f = E ( L / ) FIG. 6: (Color online) Left Panel: Plot of E n ( L ) as a func-tion of n in the heating ( δf = 0 .
7, blue points), non-heating( δf = 1 .
1, green points) phase, and on the transition line( δf = 1, orange points). The solid lines shows the scalinglaw1 / (1 + µn ) . The inset shows results obtained using firstorder FPT. Right panel: Plot of E n ( L/
2) as a function of n showing different behavior for E n ( L/
2) in the heating phase( δf = 0 .
7, blue points), non-heating phase ( δf = 1 .
05, yellowpoints) and on the transition line ( δf = 1, yellow points).The solid lines shows the scaling law 1 / (1 + µ (cid:48) n ) for eachcase. For the heating phase the color of the solid line is red forenhanced visibility. Here we have used f = 10 ω D = 100 and π/L has been set to unity for all plots. See text for details. where we have analytically continued to real time. Thuswe find the peak of E n ( x ) in the large n limit occursat x = L/ Q ” ( x ) = 1 and Q ” ( x ) = 0. Incontrast, for x = 0 , L , we have Q ” ( x ) = 1 + 4 n s and Q ” ( x ) = 0, so that E n ( x ) ∼ / (1 + 4 n s ) for all n .This behavior is consistent with the fact that n (cid:48) c divergesat the transition. A plot of E n ( x ) as a function of x forseveral n , shown in the right panel of Fig. 5, confirmsthis behavior.We also note that E n ( x ) takes particulary simple formsat the center and end of the chains where it is given by E n (0) = E n ( L ) = (cid:18) πL (cid:19) α − h ( α cosh[2 nθ ] − E n ( L/
2) = (cid:18) πL (cid:19) α + 1) h ( α cosh[2 nθ ] + 1) (45)From Eq. 45, we once again find the existence of acrossover timescale n (cid:48) c = n c (cid:112) (1 + α ) / (1 + 4 α ) belowwhich E n (0) decays as 1 / (1 + µn ) where µ = 2 s α (1 + α ). This behavior is verified in the left panel of Fig. 6which shows the behavior of E n (0) (with E n (0) = E n ( L ))as a function of n ; we find that the curves for the heatingand non-heating phase become identical to the transitionline for n (cid:54) = n (cid:48) c . For n (cid:29) n (cid:48) c , the decay becomes expo-nential in the heating phase. In contrast, for E n ( L/ E n in the heating phase andon the transition line. This can be attributed to the factthat x = L/ E n ( L/
2) decays in the heatingphase.
B. Correlation functions
In this subsection, we present results for both equal-time and unequal-time correlation functions of primaryoperators and the stress tensor.
1. Equal time correlation function
We begin with the analysis of the equal-time correla-tion function starting from the asymptotic state | h, ¯ h (cid:105) given by C n ( x , x ) = (cid:104) h, ¯ h | U † n φ ( w , ¯ w ) φ ( w , ¯ w ) U n | h, ¯ h (cid:105)(cid:104) h, ¯ h | h, ¯ h (cid:105) (46)where w j ( ¯ w j ) = +( − ) ix j for j = 1 , C n ( x , x )which is given by C hol n ( x , x ) = (cid:104) h | U n φ ( w ) φ ( w ) U n | h, (cid:105)(cid:104) h | h (cid:105) = lim z →∞ ,z → (cid:89) j =1 , (cid:18) ∂w j ∂z j (cid:19) − h (cid:89) j =1 , (cid:18) ∂z jn ∂z j (cid:19) h × z h (cid:104) | φ ( z ) φ ( z n ) φ ( z n ) φ ( z ) | (cid:105) (47)where z jn = (˜ a n z j + ˜ b n ) / (˜ c n z j + ˜ d n ). Here we choose thedimension of the primary fields φ to be same that of thestate | h (cid:105) . The anti-holomorphic part an be written sim-ilarly in terms of ¯ z and ¯ z jn . To compute the four-pointcorrelators of the primary fields, we use the standardresult which expresses these in terms of the cross ratioof the complex coordinates z i . To this end, we define z ij = z i − z j with z k = z kn for k = 1 ,
2. Using this notation, we define the cross ratio η = z z z z = ( z − z n )( z n − z )( z − z n )( z n − z ) (48)We note that for z → ∞ and z →
0, we have η → y n ( x , x ) = z n z n (49)= (˜ a n e iπx /L + ˜ b n e − iπx /L )(˜ c n e iπx /L + ˜ d n e − iπx /L )(˜ a n e iπx /L + ˜ b n e − iπx /L )(˜ c n e iπx /L + ˜ d n e − iπx /L )A similar result for ¯ y n can be obtained for the anti-holomorphic part by replacing x i → − x i .To compute the four-point correlator, we first definethe quantity C n = lim z, ¯ z →∞ z h ¯ z h (cid:104) | φ ( z, ¯ z ) φ ( z n , ¯ z n ) × φ ( z n , ¯ z n ) φ (0 , | (cid:105) (50)To evaluate this, we first note that in the non-heatingphase and on the transition line, | − z n /z n | , | − ¯ z n / ¯ z n | → n in Euclidean time. To takeadvantage of this limit, we make the conformal transfor-mation (where z ≡ z n and z ≡ z n ) z i → z (cid:48) i = − z ( z i − z n )( z n − z i ) z n (51)so that in the new coordinate z (cid:48) n = 0. A sim-ilar transformation is done for ¯ z i . Then using thestandard transformation rule of operators φ ( z, ¯ z ) → ( ∂z (cid:48) i /∂z i ) − h ( ∂z (cid:48) i /∂z i ) − h φ ( z (cid:48) i ), one can write, after somestraightforward algebra C n = z − h n ¯ z − h n F (1 − y n ; 1 − ¯ y n ) (52)where y n = z n /z n and ¯ y n = ¯ z n / ¯ z n . The advantageof using this form for F which admits conformal blockdecomposition is that one can write down a perturbativeexpansion of the blocks around | − y n | , | − ¯ y n | = 0.This allows us to obtain an analytic, albeit perturbativeexpression of C n ( x , x ) for arbitrary h, ¯ h . For the drivenproblem, | − y n | , | − ¯ y n | (cid:28) x , x in the large n limit. Also, for all phases, this limit holdsfor all n only for | x − x | /L (cid:28)
1. The perturbativeresults that we chart out next is expected to be accuratein these limits. For the present case, one obtains F = (cid:88) p C hhp V p ( y n , h ) ¯ V ¯ p (¯ y n , ¯ h ) V p ( y n , h ) = (1 − y n ) h p − h (cid:88) k F k (1 − y n ) k , ¯ V ¯ p (¯ y n , ¯ h ) = (1 − ¯ y n ) ¯ h p − h (cid:88) k F k (1 − ¯ y n ) k (cid:88) k F k x k = 1 + h p x + h p ( h p ( h p ( c + 8 h p + 8) + 2( c + 4 h − c + 8 h (2 h − h h p ( c + 8 h p −
5) + 4 c x + O ( x ) (53)
0. 0.5 1.0.00.10.20.30.40.50.60.7 x / L n ( x / L , x ' / L )
0. 0.5 1.0.00.51.01.5 x / L n ( x / L , x ' / L )
0. 0.5 1.0.00.51.01.52.0 x x x x FIG. 7: (Color online) Top left Panel: Plot of J n ( x , x ) =( C n ( x , x ) univ /C ( x , x ) − / (2 h /c ) as a function of x /L ≡ x/L in the non-heating phase ( δf = 1 . ω D = 100)for x (cid:48) /L = x/L + 0 .
03 and n = 100. Top Right panel: Sim-ilar plot for the heating phase ( δf = 1 . δf = 1). Bottom left panel: Plot of J n ( x , x ) as a functionof x and x in the non-heating phase for n = 100 showingemergence of spatial structure. Bottom right panel: Similarto the top left panel but for δf = 1 where the system is onthe transition line. Here we have used f = 10 ω D = 100 and π/L has been set to unity for all plots. See text for details. where (cid:80) p denotes sum over primaries and V p denotesthe p -th conformal block with dimensions h p , ¯ h p and thecoefficients C hhp that depend on the details of the CFT.We note here that the identity block corresponds to h p =0 and for this block C hhI = 1.Substituting Eq. 53 in Eq. 47 (and its correspondinganti-holomorphic part), one obtains C n ( x , x ) C ( x , x ) = (cid:88) p C hhp (1 − y n ) h p (1 − ¯ y n ) ¯ h p (54) C ( x , x ) = (cid:18) πL (cid:19) h +¯ h ) ( z z ) h ( z − z ) h (¯ z ¯ z ) ¯ h (¯ z − ¯ z ) h where only the leading term of the (cid:80) k F k x k (cid:39) C hhI = 1 = C ¯ h ¯ hI . For this block, h p = ¯ h p = 0. We note that if we retain only the firstorder term, we find from Eq. 54 that C n ( x , x ) becomesindependent of n . The first non-trivial contribution of theidentity block arises from the x term in the expansionof (cid:80) k F k x k . This universal contribution is given by C n ( x , x ) univ C ( x , x ) = 1 + 2 h c Re[(1 − y n ) ] (55)Thus in this limit, the identity block contribution to thedeviation of C n from its equilibrium value provides ameasure of the central charge. A plot of J n ( x , x ) =( C n ( x , x ) univ /C ( x , x ) − / (2 h /c ) (after analyticcontinuation to real time) is shown in the top panels ofFig. 7 as a function x /L = x/L for x /L = x /L + 0 . n = 100 cycles of the drive. For these plots wehave chosen (cid:126) ω D = 100 π/L . The top left panel showsthe behavior of C n ( x , x ) univ in the non-heating phase( δf = 1 .
5) displaying a broad oscillatory structure. Incontrast, in the top right panel, for the heating phase( δf = 0 .
9) it displays two sharp peaks consistent withthe behavior of E n ( x ). The position of these peaks shiftto L/ C n ( x , x ) univ for large n = 100 inthe non-heating phase (left panel) and on the transitionline (right panel) where the perturbative expansion of F is expected to be accurate. We find clear emergence ofspatial pattern in the non-heating phase in contrast tothe behavior of E n . We shall discuss this behavior inmore details in the context of unequal-time correlationfunction.In the large central charge c limit, with the conformaldimensions held fixed, the Virasoro conformal blocks re-duce to global conformal blocks which are given in termsof hypergeometric functions, by , V p ( x ) = x h p − h F ( h p , h p ; 2 h p ; x ) + O (1 /c ) . (56)Hence the correlator is given by C n ( x , x ) C ( x , x ) (cid:39) (cid:88) p (cid:54) = I C hhp (1 − y n ) h p (1 − ¯ y n ) ¯ h p (57) × F ( h p , h p ; 2 h p ; 1 − y n ) F (cid:0) ¯ h p , ¯ h p ; 2¯ h p ; 1 − ¯ y n (cid:1) . where C ( x , x ) is defined in Eq. 54. Note that sincethe Virasoro blocks reduce to global block in this limit,there is no identity block, i.e., h p (cid:54) = 0.1Finally, we note for CFTs with large central charge c (cid:29)
1, when the asymptotic states have large con-formal dimension H (cid:29) h , with h/c and H/c both held fixed, a closed form answer is availablevia the monodromy methods. Therefore we com-pute the equal-time correlation function C (cid:48) n ( x , x ) = (cid:104) H, ¯ H | φ ( w , ¯ w ) φ ( w , ¯ w ) | H, ¯ H (cid:105) / (cid:104) H, ¯ H | H, ¯ H (cid:105) . This canbe done exactly in the same way as charted out above;the only difference is that one needs to keep track of twooperator dimension H and h . A straightforward calcula-tion shows that in this case one has V p ( z n , z n , h ) = (cid:18) a ( z n z n ) ( a − / z a n − z a n (cid:19) h × (cid:32) − y a / n ) a (1 + y a / n ) (cid:33) h p (58)with a = (cid:112) − H/c . A similar expression can beobtained for ¯ V ¯ p by substituting z jn → ¯ z jn and h, h p → ¯ h, ¯ h p . One can then write C n in terms of V p and ¯ V p as C (cid:48) n ( x , x ) = (cid:18) πL (cid:19) h +¯ h ) (cid:88) p C hhp V p ( z n , z n , h ) × ¯ V ¯ p (¯ z n , ¯ z n , ¯ h ) ( z z ) h (¯ z ¯ z ) ¯ h . (59)A motivation for studying such large c CFTs comes fromthe AdS/CFT correspondence. These are CFTs whichare expected to have semiclassical gravity duals. Theglobal block answer for light correlators is reproducedin bulk AdS by the geodesic Witten diagrams . TheVirasoro vacuum block is non-trivial in two dimensionsunlike its higher dimensional versions as it contains thestress tensor and its descendants. In the large c limit,the dynamics of the Virasoro vacuum block matches withresults from semiclassical gravity . In CFT , since onehas Virasoro, one can use the geodesic Witten diagramsto interpolate between the global and the semiclassicalmonodromy answer (when pair of operator conformal di-mensions scale with the central charge) by taking intoaccount backreaction due to the heavy geodesics . The time-dependent drive of an inhomogeneous metric willhave implications for the physics of black holes in thedual gravitational theory. We are not going to explorethis issue further here.
2. Unequal time correlation function
In this subsection, we compute the unequal-time cor-relation function of the primary fields G n ( x , x ). Wenote that unequal-time correlation functions of the vac-uum state, unlike-their equal-time counterparts, displaynon-trivial dynamics . In what follows, we chart out theresult for these correlation functions for the CFT vacuumgiven by G n ( x , x ) = (cid:104) | φ ( w , ¯ w ) U † n φ ( w , ¯ w ) U n | (cid:105) (60)The holomorphic part of this correlator yields after map-ping to the complex plane, G hol n ( x , x ) = (cid:89) j =1 , (cid:18) ∂w j ∂z j (cid:19) − h (cid:18) ∂z n ∂z (cid:19) h z n − z ) h = (cid:18) πL (cid:19) h ( z z ) h (cid:16) ˜ a n z + ˜ b n − ˜ c n z z − ˜ d n z (cid:17) h (61)The antiholomorphic part can be computed in a similarmanner with z i → ¯ z i . Thus one finally gets for h = ¯ hG n ( x , x ) = (cid:18) πL (cid:19) h | ˜ a n z + ˜ b n − ˜ c n z z − ˜ d n z | h (62)where z i = exp[2 πix i /L ]. Defining the dimensionlesscenter of mass and relative coordinates as x cm = π ( x + x ) /L and x rel = π ( x − x ) /L , analytically continuingto real time, and substituting Eqs. 21, 22 and 23 in Eq.62, one obtains G n ( x cm , x rel ) = (cid:16) π L (cid:17) h (cid:104) ( α cos( x cm ) + cos( x rel )) sinh( nθ ) + (cid:112) α − nθ ) sin( x rel ) (cid:105) − h heating= (cid:16) π L (cid:17) h (cid:104) ( α cos( x cm ) + cos( x rel )) sin( nθ ) + (cid:112) − α cos( nθ ) sin( x rel ) (cid:105) − h non − heating (63)= (cid:16) π L (cid:17) h (cid:12)(cid:12)(cid:12) ns [cos( x cm ) + cos( x rel )] (cid:12)(cid:12)(cid:12) − h transition linewhere we have assumed h = ¯ h .A straightforward analysis of G n ( x cm , x rel ) in the heat-ing phase reveals that it will display peaks, in the large n limit, whencos( x cm ) = − sin( x rel + arcsin(1 /α )) (64)2
0. 0.5 1.1.0.50. x x G /( h ) FIG. 8: (Color online) Left Panel: Plot of G after n = 10drive cycles for the heating phase( ω D = 40, δf = 0 .
98) asa function of x /L and x /L showing the position of thepeaks of G n in the of x − x plane. Right panel: Plot of f ( x , x ) = α cos x cm + √ x rel + π/
4) as a function of x /L and x /L for the same parameter values showing theposition of zeros(darker shades) almost coinciding with theposition of peaks in the left panel. For all plots, f = 10. Seetext for details.
0. 0.5 1.1.0.50. x x G /( h ) G n / ( h ) ( . , . ) FIG. 9: (Color online)Left Panel: Plot of G n ( x , x ) after n = 10 drive cycles for the non- heating phase ( ω D = 100, δf =1 . G n ( x = 0 . L, x = 0) as a functionof n for ω D = 100 and δf = 1 . n as predicted by Eq. 65. For all points f = 10 and π/L is set to unity. See text for details. For x rel = 0, we find that the peak position coincide withthose of E n ( x ). In general, the position of the peakstrace a curve in the ( x , x ) space and can be tuned bychanging α ; for α → ∞ , the peaks only occurs when x cm = x rel + π/
2. Thus this phenomenon constitutes an-other example of emergence of spatial structure in drivenCFTs which was initially found by analysis of E n ( x ) .For all pairs of values ( x , x ) which do not satisfy Eq.64, G n decays exponentially with n in the large n limit: G n ∼ exp[ − nh ]. This behavior is clearly seen in theleft panel of Fig. 8 where G n ( x , x ) is plotted as a func-tion of x /L and x /L . The position of the divergence of G n ( x , x ) coincides with the solution of Eq. 64 as canbe seen from the right panel of Fig. 8.In contrast, the peaks of G n ( x , x ) in the non-heatingphase do not occur at an fixed positions independent of n in the large n limit. Here the divergences for occurwhen nθ = mπ (for integer n and m ) if x = x ; for all x (cid:54) = x , they occur when n , x , and x satisfies α cos( x cm ) + cos( x rel ) √ − α sin( x rel ) = − cot( nθ ) (65)
0. 0.5 1.1.0.50. x x G /( h ) G n / ( h ) ( . , . ) FIG. 10: (Color online)Left Panel: Plot of G / hn ( x , x ) after n = 10 drive cycles when the system is on the transitionline ( ω = 100, δf = 1). Right panel: Plot of G / hn ( x =0 . L, x = 0) as a function of n for ω D = 100 and δf = 1showing linear decay with n . The blue dots indicate values of G / hn while the red line is the linear fit. For all points f = 10and π/L is set to unity. See text for details. These divergences constitute emergent spatial singulari-ties of the unequal time correlation function in the non-heating phases and have no analog in energy density ofthe system. Such divergences, which also showed up forequal-time correlation function in the botom left panel ofFig. 8, can be clearly seen in the left panel of Fig. 9 where[ G n ( x , x )] / h is plotted as a function of x and x for n = 30. The right panel shows multiple divergences of G h +¯ h ) n for x = 0 . x = 0 as a function of n . Fi-nally, we note that on the transition line G n ( x , x ) candiverge if cos( x cm ) = − cos( x rel ), i.e. , for x = L/ x = L/
2. On the transition line, for large n one finds a1 /n h +¯ h ) decay of the correlator for all ( x , x ) exceptwhen x = L/ x = L/ G n ( x , x ) on the transition line as afunction of x and x is shown in the left panel of Fig. 10showing line of divergences at x = L/ x = L/ x , x (cid:54) = L/ G / hn decays linearly with n as shownin the right panel of Fig. 10 for x /L = 0 .
99 and x = 0.These features, obtained from exact numerics, confirmsthe analytic prediction of the first order FPT. C. Entanglement
In this subsection, we consider the evolution of the m th Renyi entropy S mn after n drive cycles starting from | h, ¯ h (cid:105) states. We note that the ground state has S mn = S m sincethe state does not change under the drive in the cylindri-cal geometry. This entanglement can be computed mostsimply by considering the correlation of the twist oper-ators T ( w, ¯ w ) . Here we shall concentrate on the m th Renyi entropy which is given, after n cycles of the drive,in terms of the twist operator T m ( w, ¯ w ) as S mn ( (cid:96) ) = 11 − m ln α mn ( (cid:96) ) , α mn ( (cid:96) ) = Tr ρ mn ( (cid:96) ) α mn ( (cid:96) ) = (cid:104) h, ¯ h | U n † T ( w , ¯ w ) T ( w , ¯ w ) U n | h, ¯ h (cid:105)(cid:104) h, ¯ h | h, ¯ h (cid:105) (66)3where ρ n ( (cid:96) ) is the reduced density matrix of state after n cycles of the drive corresponding to an initial state | h, ¯ h (cid:105) , (cid:96) is the spatial dimension of the subsystem, w i ( ¯ w i ) =+( − ) ix i , we choose x = 0 and x = (cid:96) , and the twistoperator T m represents a primary field with dimension h m = c ( m − /m ) / . In what follows, we shall focus onthe half-chain entanglement entropy which correspondsto (cid:96) = L/ (cid:96) .As before we evaluate the holomorphic part of the α mn ( (cid:96) = L/ α mn = lim z →∞ ,z → (cid:18) πL (cid:19) h m C m hol4 n e πih m (cid:16) ˜ d n − ˜ c n (cid:17) h m C m hol4 n = lim z →∞ z h (cid:104) | φ ( z ) T m ( z n ) T m ( z n ) φ (0) | (cid:105) (67) The computation of C m n = C m hol4 n C m anti − hol4 n and hence S mn thus reduce to a problem similar to that worked outfor the correlation function. However, here the opera-tor dimension of T m are different from h , and hence allthe expressions obtained in Sec. III B can not be directlyused. Nevertheless, the computation procedure is simi-lar, and we present the main results here. We find that C m n can once again be written in terms of sum of contri-bution over conformal blocks. In the perturbative limit,where | − y n | , | − ¯ y n | (cid:28)
1, one has C m n = z − h − h m n ¯ z − ¯ h − ¯ h m n F (cid:48) (1 − y n ; 1 − ¯ y n ) , F (cid:48) = (cid:88) p C hhp C h m h m p V p ¯ V ¯ p (68) V p ( y n , h, h m ) = (1 − y n ) h p − h − h m (cid:88) k F k (1 − y n ) k , ¯ V ¯ p (¯ y n , ¯ h, ¯ h m ) = (1 − ¯ y n ) ¯ h p − ¯ h − ¯ h m (cid:88) k F k (1 − ¯ y n ) k , (cid:88) k F k x k = 1 + h p x + h p ( h p ( h p ( c + 8 h p + 8) + 2( c + 2 h − c − h ) + 4 h m ( h p ( h p + 4 h −
1) + 2 h )8 h p ( c + 8 h p −
5) + 4 c x + O ( x )We note that this perturbative result is expected to beaccurate for large n and in the non-heating phases oron the transition line as discussed earlier. Also in thesecases since | − y n | , | − ¯ y n | (cid:28)
1, only the leading term in the sum may be retained. Substituting Eq. 68 in Eq.67 and after some straightforward algebra one obtains,assuming h = ¯ h and using h m = ¯ h m α mn ( L/ (cid:39) (cid:18) πL (cid:19) h m (cid:88) p C hhp C h m h m p h p +¯ h p − h m − h (cid:32) ˜ a n + ˜ b n c n + d n (cid:33) ¯ h p − h p | ˜ c n − ˜ d n | h − h m ) ( ˜ d n − ˜ c n ) − h p (˜ a n − ˜ b n ) − ¯ h p (69)If we retain only the identity contribution which is uni-versal, then using C aaI = 1 we can simplify, α mn ( L/ univ (cid:39) (cid:18) πL (cid:19) h m − h m − h ×| ˜ c n − ˜ d n | h − h m ) (70)This universal contribution, after analytic continuation to real time, yields a simple expression for the evolutionof δS mn = S mn − S mn =0 , given by δS mn ( L/ univ (cid:39) h − h m )1 − m ln | c n − d n | (71)In the non-heating phase and on the transition line, thisyields, after analytic continuation to real time,4 δS mn ( L/ univ (cid:39) h − h m − m ln (cid:104) (cid:0) α cos 2 nθ (cid:1) / (cid:112) − α (cid:105) , non − heating (cid:39) h − h m − m ln(1 + 4 n s ) , transition line (72)Thus the oscillatory behavior of δS n in the non-heatingphase and its logarithmic growth on the transition lineare expected qualitative features that are reproduced inthis perturbative approach. We note that the lineargrowth of the entanglement in the hyperbolic phase isalso reproduced by this procedure; however, here we can not ascertain the accuracy of this result since the pertur-bation expansion of F over conformal blocks are uncon-trolled.In the large central charge limit, for states with fixedconformal dimensions, we can use the global block ap-proximation to express (assuming h m = ¯ h m and h = ¯ h ) α mn ( L/ (cid:39) (cid:18) πL (cid:19) h m (cid:88) p (cid:54) = I C hhp C h m h m p h p +¯ h p − h m − h (cid:32) ˜ a n + ˜ b n ˜ c n + ˜ d n (cid:33) ¯ h p − h p ( ˜ d n − ˜ c n ) − h p (˜ a n − ˜ b n ) − ¯ h p ×| ˜ c n − ˜ d n | h − h m ) × F ( h p − h + h m , h p − h + h m ; 2 h p ; 1 − y n ) F (cid:0) ¯ h p − h + h m , ¯ h p − h + h m ; 2¯ h p ; 1 − ¯ y n (cid:1) . (73)Defining α m n ( L/
2) = (2 π/L ) h m − ( h m + h ) and δS mn = ln[ α mn ( L/ /α m n ( L/ / (1 − m ), we find, using Eq. 73, δS mn (cid:39) − m ln (cid:104) (cid:88) p (cid:54) = I C hhp C h m h m p h p +¯ h p (cid:32) ˜ a n + ˜ b n ˜ c n + ˜ d n (cid:33) ¯ h p − h p ( ˜ d n − ˜ c n ) − h p (˜ a n − ˜ b n ) − ¯ h p (74) ×| ˜ c n − ˜ d n | h − h m ) × F ( h p − h + h m , h p − h + h m ; 2 h p ; 1 − y n ) F (cid:0) ¯ h p − h + h m , ¯ h p − h + h m ; 2¯ h p ; 1 − ¯ y n (cid:1) . (cid:105) This provides the drive induced contribution to the half-chain m th Renyi entropy after n drive cycles.Finally we consider the case for large c CFTs suchthat h/c (cid:29) h m /c with h/c and h m /c held fixed. Here once again, it is possible to obtain analytic expression of α mn ( L ) using the monodromy block Eq. 58. The universalcontribution (from the identity block) non-perturbativein n is given by, α mn ( L/ univ = (cid:18) πa L (cid:19) h m | c n − d n | − h m ( a +1) | a n − b n | h m ( a − (cid:12)(cid:12)(cid:12) (cid:18) a n + b n c n + d n (cid:19) a − (cid:18) a n − b n c n − d n (cid:19) a (cid:12)(cid:12)(cid:12) − h m (75) δS mn ( L/ univ = 11 − m (cid:16) h m ln (cid:12)(cid:12)(cid:12) (cid:18) a n + b n c n + d n (cid:19) a − (cid:18) a n − b n c n − d n (cid:19) a (cid:12)(cid:12)(cid:12) + 2 h m ( a + 1) ln | c n − d n | + 2 h m (1 − a ) ln | a n − b n | . (cid:17) where we have analytically continued to real time. Wenote here that the exact conformal block contributioncan also be determined numerically using the Zamolod-chikov recursion relations ; however, we are not going to address this in this work.5 IV. RELATION TO LATTICE MODELS
In this section, we relate our results obtain usingconformal field theory in the last section to those ob-tained by exact numerics on a specific lattice model.The model chosen is the sine-square deformed (SSD)fermionic model whose Hamiltonian is given by H SSD = [ H + ( H + + H − ) / / H = − (cid:88) j J ( c † j c j +1 + h . c) (76) H ± = − (cid:88) j J e ijδ ( c † j c j +1 + h . c . )where c j is the fermion annihilation operator on site j , δ = 2 π/L , L is the chain length, the lattice spacing is setto unity, we have assumed that the system to be at half-filling, and J is the hopping strength of the fermions.In what follows, we shall use periodic boundary condi-tion for this Hamiltonian. We note that due to the localphase factor exp[ ± iδj ], H ± are not Hermitian operators.However, their sum is still Hermitian and leads to H SSD = − (cid:88) j J Λ j ( c † j c j +1 + +h . c . ) = (cid:88) j h j Λ j = 1 + 2 J cos (cid:18) πL ( j − / (cid:19) /J (77)where J is the hopping strength of the fermions. Inwhat follows, we shall implement the drive via a time-dependent hopping J → J ( t ) = J + J cos( ω D t )) + δJ .It is well known that the low-energy sector of Hamil-tonian can be expressed as H = 2 πL ( J ( t ) L + J ( L + L − ) / − holomorphic part (78)so that one can identify δf = δJ + J and J = f .In what follows, we shall scale all energies by J andcompute the time evolution of the instantaneous energydensity of the system, E n ( x ) after n cycles of the driveas follows.To this end, we first compute U ( T,
0) = T t exp[ − i (cid:82) T dtH SSD ( t ) / (cid:126) ] numerically. The proce-dure for this identical to the one carried out in Sec.II and involves decomposition of U into N time stepsof width δt = T /N : U ( T,
0) = (cid:81) j =0 ..N − U j , where U j = U ( t j + δt, t j ). The width δt is chosen such that H SSD does not vary appreciably within this interval.One then diagonalizes H j and expresses U j in terms ofits eigenvalues and eigenvectors. The matrix U is thenconstructed by taking product over all U j s. Finally onediagonalizes U to obtain the Floquet eigenvalues (cid:15) F SSD m and eigenvectors | m SSD (cid:105) . In terms of these after n drivecycles, the wavefunctions of the driven chain can bewritten as | ψ n (cid:105) = (cid:88) m e − i(cid:15) F SSD m nT/ (cid:126) c m | m SSD (cid:105) (79) ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆ δ f = δ f = δ f = ● δ f = ■ δ f = ◆ δ f = × - × - × - × - × - × - - n E ( L ) ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆ δ f = δ f = δ f = ● δ f = ■ δ f = ◆ δ f = × - × - × - × - × - × - - n E ( L ) CFTlattice
0. 0.25 0.5 0.75 1.0.00.20.40.60.81.01.2 x / L E ( x / L ) δ f n c | δ f - | n c FIG. 11: (Color online)Top Left Panel: Plot of E n ( L ) = E n (0) as a function of n as obtained from lattice (solid lines)and CFT (dotted lines) calculations showing universal behav-ior for n ≤ n c (cid:39)
200 for several representative values of δf (in units of π/L ). Top Right panel: Plot of E ( L ) as a func-tion of n obtained from lattice (solid lines) and CFT (dottedlines) calculations showing deviation oscillatory and decay-ing behaviors in non-heating and heating phases and on thetransition line. Bottom left panel: Plot of E ( x ) as a func-tion of x after n = 1500 drive cycles and δf = 5 π/L (cid:39) . n c as a function of the δf (in units of π/L ) show-ing the divergence at the transition line. The inset showsplots of δn c vs δf on log scale. The red line corresponds to aplot of 1 / (cid:112) | δf − | . For all points ω D = 40 J / (cid:126) for which δf c (cid:39) π/L and we have chosen L = 202. See text for details. where c m = (cid:104) m SSD | ψ init (cid:105) denotes the overlap of the Flo-quet eigenstates with the initial state. The initial stateis chosen to be one of the primary CFT states; for thelattice model studied here, these states are tabulated inRef. 42. We note since f (cid:29)
1, the initial primary statescorresponding to H SSD is expected to be accurately de-scribed by those charted in Ref. 42. Here we choose thestate corresponding to h = ¯ h = 1 / | ψ init (cid:105) = c † k F + π/L c †− k F − π/L | FS (cid:105) (80)where | FS (cid:105) is the half-filled Fermi sea. Thus | ψ init (cid:105) cor-responds to two particles populating the lowest avail-able energy states over the half-filled Fermi sea . Onethen computes the instantaneous energy density at anygiven site as E n ( j ) = (cid:104) ψ init | h j ( n ) | ψ init (cid:105) , where h j ( n ) = U † ( nT, h j ( t = 0) U ( nT, E n ( j ) and relate it to the corre-sponding CFT results obtained in Sec. III A.The results obtained from this procedure is shown inFig. 11 for a chain of length L = 200. The top left panelshows a plot of E ( L ) as a function of n for ω D = 40 J / (cid:126) for several representative values of δf . At this frequency,the transition line is at δf c (cid:39)
1. One therefore finds thatbelow a threshold number of drive cycles, n c , E ( L ) as ob-tained from the lattice shows universal behavior for boththe phases and mimics the behavior of the system on the6transition line. Moreover, these results show an excellentmatch with the corresponding CFT results outlined inEqs. 41, 43, and 44 (with L in Eq. 1 identified to thechain length of the lattice Hamiltonian). The long timebehavior of E ( L ) as a function of n in all the phases isshown in the top right panel of Fig. 11. We find that thelattice model reflects a clear distinction, as predicted byCFT, between the behavior of E ( L ) in the heating andnon-heating phases at long times (large n ). The bottomleft panel shows the energy density E ( x ) as a function of x/L in the heating phase corresponding to ω D = 40 J / (cid:126) and δf = 5 π/L after n = 1500 cycles of the drive. Wefind that the peak positions predicted by CFT are cor-rectly captured by the lattice model; however, the peakamplitudes are lower which is due to lattice effects thatare expected to cause deviation of lattice dynamics fromthe CFT results in the heating phase at large n . Fi-nally, in the bottom right panel of Fig. 11, we find that n c diverges at the transition; this divergence seems tocoincide with the predicted 1 / (cid:112) | − α | behavior since α ∼ /δf for ω D π/L (cid:29) V. DISCUSSION
In this work, we have studied the dynamics of drivenCFTs using a continuous protocol. Our analysis showsthat such a drive protocol, characterized by amplitudes f , δf and frequency ω D yield heating and non-heatingphases separated by transition lines. Such phases wereobtained for discrete protocols earlier in Refs. 22 and23 where the evolution operator U ( T,
0) admits an exactanalytic solution. In contrast, for continuous drive proto-col, there is no exact analytic result for U . We thereforefirst present the phase diagram in the limit of large driveamplitude as a function of δf and ω D showing severalre-entrant transitions between the heating and the non-heating phases. We also develop an analytic, albeit per-turbative, approach to these driven systems using FPT.We find that for (cid:126) ω D ≥ δf, π/L ), the firstorder FPT results provide an excellent match with thenumerical results. Our analysis allows us to identify aparameter α as a function of f , δf , ω D whose valuesdetermine the phase of the system; for | α | < ( > )1, thesystem is in the non-heating (heating) phase. The tran-sition lines correspond to α = ± P n of a primarystate displays decaying (oscillatory) behavior in the heat-ing (non-heating) phase as expected. On the transitionline P n shows a power-law decay with n . Our analy-sis identifies a crossover stroboscopic timescale n c ; for n ≤ n c , the return probability shows a universal behav-ior analogous to that of a system on the transition line. We show that n c ∼ / (cid:112) | − α | and can thus be tunedby changing both δ and ω D .For energy density of a primary state, we find emer-gence of spatial structure as identified earlier in Ref. 25.The peaks of the energy density for large n occurs at L/ L/ | α | (cid:29) L/ α = ± n in the large n limit; here E n ( x ) showsan oscillatory behavior as a function of x for all n . Forsmall n , we find that E n ( x ) obeys universal behavior sim-ilar to that when the system is on the transition line andidentify a crossover time scale till which this behaviorpersists. However this phenomenon does not occur if x corresponds to the position of the peaks in the heatingphase or on the transition line.We have also computed the equal-time correlationfunctions of primary fields, C n ( x , x ) of the driven CFTstarting from a primary state. These correlation functionrequires evaluation of the four-point function of the CFTand are therefore expressed in terms of F which admitsdecomposition into Virasoro conformal blocks V p . Theanalytical expression of V p for arbitrary ( h, h p ) does notexist. Here we have identified several limiting case whereanalytical results may be presented. The first of these isthe case when the cross ratios that appear in the argu-ment of F ( | − y n | and | − ¯ y n | in our case) are small.This limit is applicable for large n if | x − x | (cid:28) L forall phases; it is also applicable for all x and x in thenon-heating phase and on the transition line provided n is large. Our analysis in this limits shows emergent peaksin the hyperbolic phase analogous to the energy density.In addition, we also find emergent spatial structure in-dicating a line of divergence in the non-heating phaseand on the transition line. We also provide analytic ex-pression using large c limit for the block with fixed h, ¯ h where the Virasoro blocks can be replaced by the globalconformal block. Finally, we note that using monodromymethods, it is possible to find analytic expressions of thecorrelation functions in the large c when the dimensionof the primary state H (cid:29) h with H/c and h/c heldfixed. We point out that this regime may be relevant forlarge c CFTs used in AdS/CFT correspondence.The structure of the unequal time correlator G n ( x , x ) in the presence of the drive provides anotherexample of emergent spatial structure. We note that hereone can study the dynamics starting from the cylindervacuum state since the unequal-time correlation functionfor such an initial state, in contrast to its equal-timecounterpart, shows non-trivial evolution. Our analysisshows that G n , in the heating phase, diverges along acurve in the x , x plane; we provide an analytic expres-sion for this curve within first order FPT which showsreasonable match with exact numerics. We also find thatin contrast to E n ( x ), G n also shows divergences along acurve in the non-heating phase; the shape of this curvedepends on n through Eq. 65. This constitutes an ex-7ample of emergent spatial structure in the non-heatingphase which does not exists for E n ( x ).Next, we provide a computation of the half-chain en-tanglement entropy ( m th Renyi entropy) for the drivenCFT staring from a primary state. The computation of S mn is similar to that of the equal-time correlation func-tion since it can indeed be viewed as equal time corre-lation function of the twist operator T m with conformaldimension h m . We express S mn in terms of the conformalblocks and discuss limits in which their analytic expres-sions are available. Such a limit constitutes the case oflong-time ( n (cid:29)
1) limit of S mn in the non-heating phaseand on the transition line. Here we show that universalcontribution to F (and hence S mn ) from the identity blockhas oscillatory dependence of n in the non-heating phaseand a logarithmic growth on the transition line. We alsoprovide analytic expression for S mn in the large c limitwhere the Virasoro blocks can be replaced by global con-formal blocks. Finally for h (cid:29) h m , and c (cid:29) h/c and h m /c held fixed, we find analytic expressionfor S mn ( L/
2) using monodromy methods; our results heremay be relevant to CFTs used in AdS/CFT correspon-dence.Finally, we point out that our results show excellentmatch with exact numerics of a 1D lattice model offermions on a finite chain of length L with Hamiltonian H SSD (Eq. 77). In particular, we find that the emer-gent spatial structure of the energy density in the heat-ing phase at long times and its universal behavior belowa crossover scale n c is accurately reflected in such lat-tice dynamics. We note that we study the system in thepresence of a global drive. In a typical lattice systemwhich obeys Galilean invariance, such a drive does notusually lead to emergent spatial structure of correlationfunctions or energy densities. The fact that we find suchan emergent structure here clearly shows the necessityof a CFT based interpretation of such a dynamics wherespace and time are intertwined .In conclusion we have studied driven CFTs using acontinuous periodic protocol and have provided a phasediagram showing re-entrant transitions between heatingand non-heating phases. We have also studied the re-turn probability, energy density, correlation functionsand Renyi entropies of such a driven CFT starting fromprimary states. Our results indicate several features ofthese quantities such as the universal behavior of the re-turn probability and the energy density below a crossoverstroboscopic timescale and emergence of spatial structurein both heating and non-heating phases as found in thecorrelation function of primary fields. We discuss rela-tions of these results to a recently studied lattice modeland find excellent match between exact numerical latticemodel based results with analytic prediction of the CFT. VI. ACKNOWLEDGEMENT
The authors thank Shouvik Datta and espe-cially, Koushik Ray for several stimulating discus-sions. RG acknowledges CSIR SPM fellowship for sup-port. DD acknowledges supports provided by
SERB,MATRICS and Max Planck Partner Group grant,
MAX-PLA/PHY/2018577 . Appendix A: Floquet perturbation theory
In this appendix, we provide details of the Floquetperturbation theory used in the main text. We begin ouranalysis starting from Eq. 1 of the main text with f ( t )given by Eq. 8. We shall put π/L = (cid:126) = 1 in this section.In the limit of large f , U , the zeroth order term inthe perturbative expansion of U is given by , U ( t,
0) = e − i (cid:82) t H ( t (cid:48) ) dt (cid:48) = e − iσ z ( f ωD sin( ω D t )+ δft ) (A1)Since sin( ω D T ) = 0, where T = πω D is the time period ofthe drive, we find (restoring π/L and (cid:126) ) U ( T,
0) = e − iδfT σ z , H F = sσ z /T (A2)where s = arccos(cos( δf T )) (where π/L is set to unity)is defined in the main text. We note that H (0) F reflectsthe periodicity of U .The first order term for U in the perturbation expan-sion is given by U (cid:48) ( T,
0) = − i (cid:90) T dtU † H U (A3)Since H ∼ iσ y and U only depends on σ z (Eq. A2), astraightforward calculation yields U (cid:48) ( T,
0) = = − i ( I σ + + I ∗ σ − ) I = (cid:88) m J m (cid:18) f ω D (cid:19) (cid:90) T dte i ( mω D +2 δf ) t = (cid:88) m J m (cid:18) f ω D (cid:19) e is T sin smπ + δf T (A4)where J m ( x ) are m th Bessel functions. This leads toEq. 18 and then, following the unitarization procedurediscussed in the main text, to Eq. 19 for H (1) F .Next, we compute the second order term in the per-turbative expansion of UU (cid:48) ( T,
0) = ( − i ) (cid:90) T dt (cid:90) t dt U † ( t , H U ( t , × U † ( t , H U ( t ,
0) (A5)8
FIG. 12: (Color online) Left Panel: Plot of the phase diagramshowing | Tr U ( T, | as a function of the amplitude δf andfrequency ω D as obtained from second order FPT. Once again using the Pauli matrix dependence of H and U we find U (cid:48) = − ( σ + σ − I + σ − σ + I ∗ ) (A6) I = (cid:88) m,n J m ( x ) J n ( x ) (cid:90) T dt e i ( mω D + δf ) t × (cid:90) t dt e i ( nω D + δf ) t = (cid:88) m,n J m ( x ) J n ( x ) iT πm + δf T ) [ − δ mn + e is sin sπn + δf T ]where x = 2 f /ω D . The real and imaginary parts of I can be read off asRe[ I ] = (cid:88) m,n J m ( x ) J n ( x ) T πm + δf T )( πn + δf T ) sin s Im[ I ] = β = (cid:88) m,n J m ( x ) J n ( x ) T πm + δf T ) × [ δ mn − cos s sin s ( πn + δf T ) ] (A7)From Eq. A7 we note that U − U / I ]. Thus to second order in perturbation theory, U is given by U = U ( I + U (cid:48) + ( U (cid:48) − U (cid:48) / e − iH (2) F T = (cid:18) e − is (1 + iβ ) − iα sin yiα sin y e iy (1 − iβ ) (cid:19) (A8) where α is defined in Eq. 9 in the main text. We unitarize U following the same procedure as H (2) F = θ (2) (cid:16) n (2) z σ z + in (2) y σ y (cid:17) /T sin( θ (2) T ) = (cid:113) (sin s − β cos s ) − α sin sn (2) z sin( θ (2) T ) = sin s − β cos sn (2) y sin( θ (2) T ) = α sin s (A9)The phase diagram is obtained from Eq. A9 by imposingcondition on Tr[exp[ − iH (2) F T ]] as discussed in the maintext. This translates to the conditions | cos( θ (2) ) | > ( < )2for the heating (non-heating) phases and | cos( θ (2) ) | = 2on the transition line. The phase diagram, shown in Fig.12 as a function of δf and ω D turns out to be qualitativelysimilar to that obtained using first order FPT (Fig. 1 inthe main text). Appendix B: Mobius transformation
In this section, we discuss several aspects of the Mobiustransformation corresponding to continuous protocol dis-cussed in this work. To this end, we first relate to theMobius transformation used in Ref. 23. It was shownthat for an Hamiltonian H = H + tanh( θ )( H + + H − ) / z new = [(1 − λ ) cosh(2 θ ) − ( λ + 1)] z + ( λ −
1) sinh(2 θ )(1 − λ ) sinh(2 θ ) z + [( λ −
1) cosh(2 θ ) − ( λ + 1)](B1)where λ = exp[ δτ / cosh(2 θ )] and τ is the imaginary time.We show below that this relation is reproduced for us atevery Trotter steps.To this end, we note that the instantaneous Hamilto-nian at any time τ is of the form, H = aH − b H + + H − ) = a [ H − b a ( H + + H − )] (B2)where a and b depends on τ . This indicates that forthe non-heating phase one can write tanh 2 θ = b/a and λ = exp[ aδτ / cosh(2 θ )] = exp[ √ a − b δτ ], sincecosh 2 θ = a/ √ a − b and sinh 2 θ = b/ √ a − b . TheMobius transformation corresponding to such a Hamil-tonian is given by9 M = (cid:18) a b c d (cid:19) = (cid:32) s − s b √ a − b (exp[ √ a − b δτ ] − − b √ a − b (exp[ √ a − b δτ ] − s + s (cid:33) (B3)where s = (1 − exp[ √ a − b δτ ]) a/ √ a − b and s =(exp[ √ a − b δτ ]+1). Thus we seek a matrix of the form M = ( aσ z − ibσ y ) whose exponential gives the Mobiusmatrix M . Since M = e − ∆ τM = cosh (cid:0) √ a − b δτ (cid:1) − a sinh ( √ a − b δτ ) √ ( a − b )( a + b ) b sinh ( √ a − b δτ ) √ ( a − b )( a + b ) − b sinh ( √ a − b δτ ) √ ( a − b )( a + b ) cosh (cid:0) √ a − b δτ (cid:1) + a sinh ( √ a − b δτ ) √ ( a − b )( a + b ) (B4)we find M (1 ,
1) = a exp[ δτ √ a − b ]. Similar expres-sions can be seen for other elements M . Thus up toan overall irrelevant factor, our analysis reproduces thesame Mobius at every Trotter step as in Ref. 23. A simi-lar analysis can be easily carried out for the heating phaseand leads to similar results.Next, we show derive the form of H F using algebra ofthe Virasoro operators without resorting to their SU(1,1)representations. To this end, we begin from the time-dependent Hamiltonian given by Eq. 1. For f (cid:29) δf, U ( t,
0) = exp (cid:20) − iL πtL (cid:18) f ω D t sin( ω D t ) + δf (cid:19)(cid:21) (B5)This leads to U ( T,
0) = exp[ − πisL /L ] and H F (0) =2 πsL / ( LT ); these results coincide with Eq. 15 for L = σ z / U , we write U (cid:48) ( T,
0) = − i (cid:90) T U † ( t, πL ( L + L − ) U ( t, dt (B6)To evaluate this, we use the Baker-Campbell-Hausdorffrelations for L and L ± which states e iaL ( L + L − ) e − iaL = e ia L + e − ia L − (B7)Using Eq. B7 and identifying a = − π/L ( f sin( ω D t ) /ω D + δf t ) (Eq. B5), one canevaluate U (cid:48) ( T,
0) in a straightforward manner. Theintegrals involved are similar to those in App. A and oneobtains U (cid:48) ( T,
0) = iα [ i ( L + L − ) sin s +2 sin s/ L − L − ) (cid:3) (B8) where α is given by Eq. 9 of the main text. Thus theexpression of the evolution operator, up to first order inperturbation theory, is given by U ( T,
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