Classification of S L 2 deformed Floquet Conformal Field Theories
CClassification of SL deformed Floquet Conformal Field Theories Bo Han and Xueda Wen Theory of Condensed Matter Group, Cavendish Laboratory, University of CambridgeJ. J. Thomson Avenue, Cambridge CB3 0HE, United Kingdom Department of Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
Classification of the non-equilibrium quantum many-body dynamics is a challenging problem incondensed matter physics and statistical mechanics. In this work, we study the basic question thatwhether a (1+1) dimensional conformal field theory (CFT) is stable or not under a periodic drivingwith N non-commuting Hamiltonians. Previous works showed that a Floquet (or periodically driven)CFT driven by certain SL deformed Hamiltonians exhibit both non-heating (stable) and heating(unstable) phases. In this work, we show that the phase diagram depends on the types of drivingHamiltonians. In general, the heating phase is generic, but the non-heating phase may be absentin the phase diagram. For the existence of the non-heating phases, we give sufficient and necessaryconditions for N = 2, and sufficient conditions for N >
2. These conditions are composed of N layers of data, with each layer determined by the types of driving Hamiltonians. Our results alsoapply to the single quantum quench problem with N = 1. I. INTRODUCTION
Non-equilibrium many-body dynamics have receivedextensive attention recently because they show exoticproperties that are missing in static systems and alsothey can be realized in experiments such as optical lat-tice and cold atomic systems. For example, a periodicdrive creates novel systems that may not have an equi-librium analog, such as Floquet topological phases and time crystals . Moreover, a periodic drive is alsoone of the basic protocols to study non-equilibrium phe-nomena, such as localization-thermalization transitions,prethermalization, dynamical Casimir effect, etc.
Although there are rich properties and applications inthe time-dependent driving physics in quantum many-body systems, exactly solvable setups are very rare. Ingeneral, we need to resort to numerical methods thatare limited to a small-system size. In this work, we areinterested in a quantum (1 + 1) dimensional conformalfield theory (CFT), which may be viewed as the low en-ergy effective field theory of a many-body system at thecritical point. For (1 + 1)D CFTs, the property of con-formal invariance can be exploited to constrain the op-erator content of the critical theory, which makesit tractable for the study of non-equilibrium dynamics,such as the quantum quench problems.
For a time-dependent driven CFT, however, relatively little is knownon the analytical properties of the non-equilibrium dy-namcis.Recent study along this direction was initialized inRef. 38 and 39, where two non-commuting Hamiltoniansare used to drive the CFT periodically in time. One of thedriving Hamiltonians is chosen as the uniform one, andthe other is chosen by deforming the uniform one with asine-square deformation (SSD), which we will in-troduce in detail shortly. In this periodically driven CFT(or Floquet CFT), it is found there are both heating andnon-heating phases, separated by a critical (phase tran-sition) line. One of the ‘order parameters’ characterizingthe phase diagram is the time evolution of entanglement entropy. It was found that the entanglement entropygrows linearly in time in the heating phase, oscillates inthe non-heating phase, and grows logarithmically in timeat the phase transition. Later in Ref. 53, further interesting features were foundin the same setup. For example, the total energy growsexponentially in time in the heating phase, oscillates inthe non-heating phase, and grows polynomially in time atthe phase transition. In particular, in the heating phase,it was found there are interesting emergent spatial struc-tures during the driving: The (chiral and anti-chiral)energy-momentum densities form an array of ‘peaks’ inthe real space (See also Ref. 54 for the study of the energydensity distribution). In Ref. 53, it was also found thatthe entanglement pattern in a Floquet CFT is closelyrelated to the energy-momentum density distributions.The main contribution of quantum entanglement in theFloquet CFT comes from those between nearby energy-momentum density peaks of the same chirality.Most recently, the types of driving sequences are gen-eralized from periodic to quasi-periodic and randomones. On the one hand, it was found that the heat-ing phase is generic in all these three types of drivingsequences. In particular, the features as found in theperiodic driving in Ref. 53 turn out to be also genericin the other two types of drivings. On the other hand,there are some new features in the phase diagrams of thequasi-periodic and random drivings. For example, in aquasi-periodically driven CFT with Fibonacci sequence,the non-heating phases form a Cantor set of measurezero, and the heating rates in the heating phases ex-hibit self-similarity structures. In the random driving,the driven CFT is generally in the heating phase, butwith some isolated exceptional points (See Ref. 57 formore details). The mechanism of the heating phase in arandomly driven CFT is analogous to the Anderson lo-calization in (1+1)d disordered system. In short, thephase diagrams of time-dependent driven CFTs dependon the types of driving sequences.In this work, by fixing the driving sequence to be a a r X i v : . [ c ond - m a t . s t a t - m ec h ] A ug periodic one, we are interested in how the types of driv-ing Hamiltonians affect the phase diagrams of a FloquetCFT. Let us specify the meaning of “Hamiltonian types”first. Considering a CFT defined on a circle of length L ,the driving Hamiltonians we consider are of the followingform: H CFT = H chiral + H anti-chiral , (1.1)i.e., one can decompose the total Hamiltonian as the sumof chiral and anti-chiral parts. In terms of the energy-momentum tensor T ( x ), we have H chiral = 12 π Z L f ( x ) T ( x ) dx, (1.2)where f ( x ) is an envelope function which deforms thechiral energy-momentum density, and it is similar for theanti-chiral part, where the anti-chiral energy-momentumtensor is denoted as T ( x ). Here T ( x ) and T ( x ) are relatedto the energy density T and momentum density T ( x )as T = π ( T + T ) and T = π ( T − T ). For differentdriving Hamiltonians, we can choose different envelopefunctions f ( x ). In this work, we are interested in thedeformation of a single wavelength, with f ( x ) = σ + σ + cos 2 πqxL + σ − sin 2 πqxL , q ∈ Z , (1.3)where σ and σ ± are real numbers which characterize thedeformation. With this deformation, the Hamiltoniancan be written as H chiral = 2 πL (cid:0) σ L + σ + L q, + + σ − L q, − (cid:1) − πc L , (1.4)where we have defined L q, + := ( L q + L − q ) and L q, − := i ( L q − L − q ), with L n := c δ n, + L π R L dx π e i πnL x T ( x )being the generators of Virasoro algebra, and c isthe central charge. One can find that H chiral in(1.4) is composed of three generators that generate theSL ( q ) (2 , R ) group, which is isomorphic to the q -fold coverof SL(2 , R ) . For this reason, we call the periodicallydriven CFT with the driving Hamiltonians in (1.1)-(1.3)as the SL deformed Floquet CFT. In general, the Hamil-tonians in (1.4) can be classified into three types basedon the quadratic Casimir: c (2) := − ( σ ) + ( σ + ) + ( σ − ) . (1.5)Different types of Hamiltonians can be classified as fol-lows: Quadratic Casimir c (2) < c (2) = 0 c (2) > The reason we use these terminologies will be clearwhen we discuss operator evolutions in Sec. II. Therein,we will see that the operator evolutions governed by dif-ferent Hamiltonian types correspond to the Möbius trans-formations of hyperbolic/parabolic/elliptic types. An in-tuitive picture to understand the difference of these three c (2) < , elliptic c (2) = 0 , parabolic c (2) > , hyperbolic σ + σ σ − FIG. 1. Different types of manifolds determined by Eq.(1.5)with different quadratic Casimir c (2) . Each single point onthe manifold specifies a deformed Hamiltonian through (1.2)and (1.3). Any point on the manifold is SL(2 , R ) equivalentto arbitrary points on the same manifold. Hamiltonian types is to consider the manifold determinedby (1.5) with different signs of c (2) , as shown in Fig. 1.Recently, the properties of energy spectrum for differ-ent types of Hamtiltonians have been studied in litera-ture. As a remark, the specific choice of q = 1, σ = − σ + = 1 / σ − = 0 in (1.3) corresponds tothe CFT with SSD, as used in the simplest setup of aFloquet CFT in Ref. 39.Our motivation in this work is to classify the non-equilibrium dynamics in a Floquet CFT with all possiblechoices of Hamiltonian types in (1.6). More specifically,during the periodic driving, we choose N ( N ≥
2) non-commuting Hamiltonians, with each Hamiltonian speci-fied by a certain ( σ , σ + , σ − ) in (1.4). What we meanby ‘classification’ is to determine whether there are heat-ing phases/non-heating phases/phase transitions in thephase diagram. The main results we found can be sum-marized as follows. For arbitrary choices of Hamiltoniantypes, there are always heating phases in the FloquetCFT, i.e., the heating phase is generic. The non-heatingphases, however, do not always exist in the phase dia-gram. When there are N = 2 driving Hamiltonians, wegive the sufficient and necessary conditions for the ex-istence of non-heating phases in the phase diagram, asshown in Table. I. For N >
2, we give sufficient condi-tions for the existence of non-heating phases, as presentedin Sec. III D. These conditions can be summarized as fol-lows: There are in total N layers of data. At each layer n , we pick arbitrary n driving Hamiltonians by keep-ing the time order of driving. If there is at least oneelliptic Hamiltonian in { H i , H i , · · · , H i n } , then theremust exist non-heating phases. If all the Hamiltoniansin { H i , H i , · · · , H i n } are non-elliptic (either parabolicor hyperbolic), then the following condition ensures theexistence of non-heating phases: ∃ η n < , n = 1 , , · · · , N (1.7)where η n is the indicator as defined in (3.25). It is notedthat η n is only determined by the vectors ( σ i , σ + i , σ − i )that characterize these driving Hamiltonians. One canfind there are in total 2 N − N drivingHamiltonians. If at least one of these conditions is sat-isfied, then there must exist non-heating phases in thephase diagram. We suspect that these conditions arealso necessary, i.e., if there exists a non-heating phase inthe phase diagram, then at least one of the 2 N − N > deformations. Then we study theoperator evolution corresponding to different Hamilto-nian types, based on which the entanglement/energy-momentum evolution can be obtained. In Sec. III, westudy how the phase diagrams depend on the types of N driving Hamiltonians in a Floquet CFT. We give suffi-cient conditions for the existence of non-heating phasesfor arbitrary N , and illustrate these conditions with ex-amples N = 1, 2, and 3. We give some discussions andconclude in Sec. IV. There are also several appendicesfocusing on the detailed features of the phase diagramswith different types of driving Hamiltonians. II. SL DEFORMED FLOQUET CFTA. Setup
The setup we consider is based on a (1+1) dimen-sional CFT on a circle of length L with periodic bound-ary conditions. The driving Hamiltonians we choose areof the form in Eqs. (1.1)-(1.3). Then we study a time-dependent driving with a discrete and periodic sequence: | Ψ n i = (cid:0) U N · · · U · U (cid:1) m | Ψ i , with U j = e − iH j T j , (2.1)where n = N m , and T j is the time duration of driv-ing with Hamiltonian H j . Within each period, there arein general N (possibly) different driving Hamiltonians.Here the initial state | Ψ i may be taken as the groundstate of a uniform CFT Hamiltonian H = 12 π Z L T ( x ) dx + anti-chiral part . (2.2)It is noted that the initial state can also be chosen as anexcited state or even a thermal ensemble at finite tem-perature. Each driving Hamiltonian H i in (2.1) can bechosen with different envelope functions f i ( x ) in (1.3),which are characterized by a triple ( σ i , σ + i , σ − i ). Our goalis to classify the non-equilibrium dynamics and phase di-agrams with respect to the types of H i ( i = 1 , · · · , N ) inthe N dimensional parameter space spanned by n(cid:16) T l , · · · , T N l (cid:17) (cid:12)(cid:12)(cid:12) < T i l < ∞ , i = 1 , · · · , N o , (2.3) where l := L/q is the wavelength of deformation in f ( x )[see Eq.(1.3)]. As a remark, in the above discussions, we only specifythe deformation of the chiral part of the Hamiltonian,which are characterized by the vector ( σ i , σ + i , σ − i ). Thedeformation of the anti-chiral parts are characterized byanother independent vector ( σ i , σ + i , σ − i ), because wechoose periodic boundary conditions and the chiral andanti-chiral modes are decoupled from each other. In thefollowing study, without loss of generality, we will focuson the deformation of the chiral parts. The analysis ofthe anti-chiral parts can be performed in the same way. B. Hamiltonian types and operator evolution
To understand the non-equilibrium dynamics underthe driving in Eq. (2.1), we first study the operatorevolution, based on which the time evolution of corre-lation functions such as the entanglement entropy andthe energy-momentum density can be obtained .Let us start from driving the CFT with a single Hamil-tonian H with a time duration t . We consider themethod as used in Ref. 39, which we briefly sketchas follows. In the Euclidean spacetime, the correla-tion function h Ψ( t ) |O ( x ) | Ψ( t ) i = h G | e Hτ O ( x ) e − Hτ | G i = h G |O ( x, τ ) | G i , where τ = it , can be considered as thepath integral on a w -cylinder with the operator O in-serted at ( x, τ ), as depicted in Fig. 2. This cylinder canbe mapped to a q -sheet Riemann surface with a confor-mal map z = e i πqwL . Then the Hamiltonian in Eqs.(1.1)-(1.3) can be written as H = H ( z ) + H ( z ) , where H ( z ) = 2 πl I dz πi h σ z + 12 ( σ + − iσ − ) z + 12 ( σ + + iσ − ) i T ( z ) − σ πc l . (2.4)One can further perform a Möbius transformation z =( a e z + b ) / ( c e z + d ), where a = − i , b = σ + + iσ − √ c (2) , c = −√ c (2) + iσ σ + + iσ − ) , and d = − σ + i √ c (2) √ c (2) . Then the Hamiltoniandefined on the q -sheet e z Riemann surface is of the simpleform: H ( e z ) = − πi √ c (2) l I d e z πi e z T ( e z ) − σ πc l . (2.5)Several remarks here. First, from (2.5), one can al-ready see the difference for positive and negative c (2) .The choice of branch cut of √· · · will not affect the re-sults of operator evolution as presented later in Eqs.(2.8),(2.9), and (2.10). Second, for c (2) = 0, the expression in(2.5) is not well defined. To obtain the operator evolu-tion, one can do the calculation by keeping nonzero c (2) and take the limit c (2) → c (2) > c (2) < On the q -sheet e z Riemann surface, the operator evo-lution becomes a dilatation: e H ( e z ) τ O ( e z, e z ) e − H ( e z ) τ = xτ w •O ( x, τ ) x = 0 x = L FIG. 2. Path integral representation of the correlation func-tion h G |O ( x, τ ) | G i in a CFT with periodical boundary con-ditions. Here x = 0 and x = L are identified. λ h O ( λ e z, e z ), where λ = e − πi √ c (2) l τ , and h is the con-formal dimension of the operator O . Then by mappingback to the z -surface, one can find the operator evolvesas e H ( z ) τ O ( z, z ) e − H ( z ) τ = (cid:18) ∂z ∂z (cid:19) h O ( z , z ) . (2.6)By doing an analytical continuation τ = it , where t thetime duration of driving, one has z = αz + ββ ∗ z + α ∗ =: M · z, M = (cid:18) α ββ ∗ α ∗ (cid:19) ∈ SU(1 , . (2.7)Note that SU(1 , ∼ = SL(2 , R ), which is as expected sincethe three generators L and L q, ± in Eq. (1.4) generatethe SL ( q ) (2 , R ) group. Depending on the types of drivingHamiltonians in (1.6), the SU(1 ,
1) matrices in Eq.(2.7)have different expressions as follows:1. Elliptic ( c (2) < α = cos (cid:18) π C tl (cid:19) + i σ C sin (cid:18) π C tl (cid:19) ,β = i σ + + iσ − C sin (cid:18) π C tl (cid:19) . (2.8)2. Parabolic ( c (2) = 0): α = 1 + i σ πtl ,β = i ( σ + + iσ − ) πtl . (2.9)3. Hyperbolic ( c (2) > α = cosh (cid:18) π C tl (cid:19) + i σ C sinh (cid:18) π C tl (cid:19) ,β = i σ + + iσ − C sinh (cid:18) π C tl (cid:19) . (2.10)In all these three cases, we have defined the real number: C := q(cid:12)(cid:12) − ( σ ) + ( σ + ) + ( σ − ) (cid:12)(cid:12) . (2.11) One can find that for different types of Hamiltonians, thecorresponding Möbius transformations are qualitativelydifferent. It is well known that there are in total threetypes of SU(1 ,
1) matrices (See, e.g., Ref. 61) dependingon the value of their traces: For | Tr( M ) | <
2, = 2, and >
2, the corresponding SU(1 ,
1) matrices are called ellip-tic, parabolic, and hyperbolic matrices, respectively. It isstraightforward to check that for a general time duration t , where t >
0, the Möbius transformations in Eqs. (2.8),(2.9), and (2.10) are elliptic, parabolic, and hyperbolic,respectively. For this reason, we denote the correspond-ing Hamiltonian types in (1.6).Now let us consider an SL deformed Floquet CFTwith N driving Hamiltonians. Then the operator evolu-tion after each period is determined by z N = Π N · z (andsimilarly for the anti-holomorphic part), whereΠ N = M · · · M N =: (cid:18) α N β N β ∗ N α ∗ N (cid:19) ∈ SU(1 , . (2.12)The operator evolution after m periods of driving is de-termined by Π N as z n = (Π N ) m · z, n = mN. (2.13)Then the phase diagram of the Floquet CFT is deter-mined by | Tr(Π N ) | as follows: | Tr(Π N ) | < , non-heating phase , | Tr(Π N ) | = 2 , phase transition , | Tr(Π N ) | > , heating phase . (2.14)As studied in Ref. 39, the value of | Tr(Π N ) | in (2.14)determines the trajectories of operator evolutions. For | Tr(Π N ) | <
2, the operator will rotate along the circle allthe way; for | Tr(Π N ) | = 2, the operator will approach afixed point polynomially fast in time; for | Tr(Π N ) | > C. Entanglement and energy-momentum evolution
Once we know the operator evolution in (2.7), we canstudy the time evolution of correlation functions. The‘order parameters’ we use to distinguish different dy-namical phases are the entanglement entropy and energy-momentum evolutions, which can be viewed as the corre-lation functions of twist operators and energy-momentumtensor, respectively. Some of related details can also befound in Refs. 38, 39, 53, and 55.Let us consider the time evolution of entanglemententropy first. With the twist-operator approach, one can find the α -th Renyi entropy of the subsystem A = ( x , x ) as S ( α ) A = 11 − α log h Ψ n |T ( w , w ) T ( w , w ) | Ψ n i , (2.15)where | Ψ n i is the wavefunction in (2.1) by choosing theinitial state | Ψ i as the ground state of the uniformHamiltonian H in (2.2), and T ( T ) are twist operatorswith conformal dimensions h = h = c ( α − /α ). Asstudied in the previous subsection, to evaluate the corre-lation function of the twist operators, we first map the w -cylinder to the q -sheet Riemann surface z by z = e i πqwL ,where the evolution of z ( z ) z ( z ) are governed byMöbius transformations in Eq. (2.13). Next, we mapthe z -Riemann surface to a complex plane ζ by ζ = z /q .One can find that h Ψ n |T ( w , w ) T ( w , w ) | Ψ n i = Y i =1 , (cid:18) ∂ζ i ∂w i (cid:19) h Y i =1 , (cid:18) ∂ζ i ∂w i (cid:19) h hT ( ζ , ζ ) T ( ζ , ζ ) i ζ , (2.16)where we have w i = x i + iτ = x i . For a general choice ofthe subsystem A , the expression of S ( α ) A is complicated. Here, for simplicity we choose the subsystem A as a unitcell with ( x , x ) = ( kl, ( k + 1) l ) where k ∈ Z . Then itis straightforward to find that S A ( n ) − S A (0) = c (cid:16) log (cid:12)(cid:12) α n + β n (cid:12)(cid:12) +log (cid:12)(cid:12) α n + β n (cid:12)(cid:12)(cid:17) , (2.17)where α n ( β n ) corresponds to the driving effect inthe anti-chiral parts and we have considered S A =lim α → S ( α ) A . Here α n and β n are the matrix elementsin (Π N ) m in (2.13), i.e., (cid:18) α n β n β ∗ n α ∗ n (cid:19) = (cid:18) α N β N β ∗ N α ∗ N (cid:19) m , n = mN. (2.18)As a remark, if one studies the entanglement entropyof A by shifting a half unit cell, i.e., A = [( k + ) l, ( k + ) l ]where k ∈ Z , then one can find that S A ( n ) − S A (0) = c (cid:16) log | α n − β n | +log | α n − β n | (cid:17) . (2.19)The difference between Eq. (2.17) and Eq. (2.19) reflectsthe fact that the system is driven in a non-uniform way.Next, let us consider the energy-momentum densityevolution. With the operator evolution in (2.13), onehas U † T ( z ) U = (cid:18) ∂z ∂z (cid:19) T (cid:0) z (cid:1) + c
12 Sch { z , z } , (2.20)where U = (cid:0) U N · · · U · U (cid:1) m [see Eq. (2.1)], and thesecond term represents the Schwarzian derivative. One can obtain the expectation value of the chiral energy-momentum tensor density as follows π h T ( x, n ) i = − q πc L + πc L · ( q − · | α n · z + β n | (2.21)where z = e πiqxL , and α n and β n are those defined inEq.(2.18). By integrating over the energy-momentumdensity, one can obtain the total energy as E ( n ) = 12 π Z L h T ( x, n ) + T ( x, n ) i dx = − q πc L + πc L ( q − · ( | α n | + | β n | + | α n | + | β n | )(2.22) Now let us comment on how different types of Möbiustransformations in Eqs. (2.8), (2.9), and (2.10) resultin different behaviors of entanglement/energy evolution.Based on the analysis in Refs. 39 and 55, the norm | α n | ( | β n | ) will grow exponentially/polynomically/oscillatein time when the corresponding Möbius transforma-tions within one period is hyperbolic/parabolic/elliptic.Based on the expressions of entanglement and energy-momentum density evolution in Eqs. (2.17) and (2.22),one can find that in general the entanglement entropywill grow linearly/grow logarithmically/oscillate in time,and the total energy will grow exponentially/grow poly-nomially/oscillate in time accordingly. We will see anillustrating example later in Sec. III A. III. CLASSIFYING THE SL DEFORMEDFLOQUET CFT
Now we come to the main section of this work: westudy the conditions for the existence of heating and non-heating phases in the phase diagrams when there are N driving Hamiltonians. It is found that there are alwaysheating phases in the phase diagram. For the non-heatingphases, we will give conditions for their existence in thephase diagram. Our conditions are both sufficient andnecessary for N = 1 and N = 2, and are sufficient for N >
2. We will illustrate these conditions by considering N = 1, 2, and 3, and then give the general results forarbitrary N . A. N = 1 As a warm up, let us first consider the simplest casewith N = 1, i.e., there is only one driving Hamiltonian H . This case corresponds to a single quantum quenchrather than a Floquet CFT. Here we consider this simplecase to illustrate how different Hamiltonian types in (1.6)determine different behaviors of entanglement evolution.It is noted that in Ref. 38, the single quantum quenchwas studied for some specific Hamiltonians H with c (2) = 0 and c (2) <
0, respectively. It was found thatthe entanglement entropy grows logarithmically in timefor c (2) = 0, and simply oscillates in time for c (2) < A = [ kl, ( k + 1) l ],where k ∈ Z , it can be found that in the long time drivinglimit (i.e., C t/l (cid:29) t/l (cid:29) c (2) > c (2) = 0 (parabolic case), andoscillate in time for c (2) < S A ( t ) − S A (0) ’ πc · C tl , c (2) > ,c πtl , c (2) = 0 ,c (cid:12)(cid:12)(cid:12) a + b sin (cid:0) π C tl + φ (cid:1)(cid:12)(cid:12)(cid:12) , c (2) < , (3.1) where C is defined in Eq. (2.11). The real numbers a , b , and φ depend on the parameters ( σ , σ + , σ − ). Onecan refer to Appendix. A for a complete expression ofthe entanglement entropy evolution with arbitrary timeduration t > N = 1 isonly determined by the Hamiltonian types in (1.6). B. N = 2 Now we consider the properties of the phase dia-gram in the case of N = 2, i.e., there are two non-commuting Hamiltonians. Since there are three possi-bilities of Hamiltonian types for H ( H ), we have intotal six different unordered pairings of H and H (SeeTable I).For later use, let us define the Casimir vector that char-acterizes the SL deformed Hamiltonian in (1.3): C i = ( σ i , σ + i , σ − i ) , (3.2)as well as the product of two Casimir vectors C i · C j := − σ i σ j + σ + i σ + j + σ − i σ − j . (3.3)Our strategy to determine the phase diagram in the pa-rameter space spanned by { ( T /l, T /l ) | < T /l, T /l < ∞} can be briefly summarized as follows. The effect ofdriving with Hamtiltonian H ( H ) for a time duration T ( T ) are represented by a SU(1 ,
1) matrix M ( M ).Depending on the types of H ( H ), the SU(1 ,
1) ma-trix M ( M ) takes the form in one of Eqs. (2.8), (2.9), FIG. 3. Phase diagrams of a Floquet CFT with the bothnon-heating (in blue) and heating (in red) phases for the sixkinds of pairings with N = 2 in Table.I. The parameters are(from left to right, and then top to bottom): elliptic-ellipticwith C = (1 , ,
0) and C = (1 , . , C = (1 , ,
0) and C = (1 , , C =(1 , ,
0) and C = (0 , . , C =(1 , ,
0) and C = (1 , , C =(1 , ,
0) and C = (1 , . , C =(1 , . ,
0) and C = (1 , , . and (2.10). Then the phase diagram is determined by | Tr( M · M ) | = | Tr( M · M ) | based on Eq. (2.14).Fig. 3 is a sample plot of the phase diagrams for the sixdifferent pairings of H and H in Table I. The param-eters in C and C are chosen such that there are bothheating and non-heating phases in the phase diagram.For arbitrary choices of C and C , the heating phasesare generic, but the non-heating phases may be absent.In Table I, we give the sufficient and necessary conditionsfor the existence of non-heating phases in the phase dia-gram. The details of derivations of these conditions canbe found in Appendix B. H H Elliptic Parabolic HyperbolicElliptic √ √ √
Parabolic √ C · C < C · C < √ C · C < C · C C C < N = 2. “ √ ” means the non-heating phases exist for arbitrary choices of non-commutingHamiltonians H and H of the corresponding types. Let us give several remarks on how to obtain the con-ditions in Table I. If at least one of the driving Hamilto-nians is elliptic, then there must exist non-heating phasesin the phase diagram. This can be straightforwardly un-derstood as follows. Denoting the elliptic Hamiltonianas H , then the limit T /l = 0 and T /l = 0 ( j = i )corresponds to a single quench problem with N = 1.In this case we have | Tr( M · M ) | <
2. Now we turnon T /l . As long as T /l is small enough, we still have σ + σ σ − FIG. 4. C C = ( σ , σ +1 , σ − ) = (0 , ,
0) is fixed (the vector inblack). The normalized vectors C C that satisfy the conditionin Eq.(3.11) are in the region in green. | Tr( M · M ) | <
2, i.e., the Floquet CFT is in a non-heating phase. Essentially, this condition is a quasi-( N −
1) condition, since only ( N −
1) driving Hamil-tonians dominate while the left Hamiltonian plays littlerole. The five conditions labeled by ‘ √ ’ in Table I areall quasi-( N −
1) conditions, with N = 2. The left fourconditions are intrinsic- N conditions, which can be ex-pressed by one condition: η < . (3.4)Here the indicator η N =2 is constructed as follows. Foreach driving Hamiltonian H j , we arrange a matrix P j inthe following way. If H j is parabolic, then P j = (cid:18) i σ j i ( σ + j + iσ − j ) − i ( σ + j − iσ − j ) − i σ j (cid:19) . (3.5)If H j is hyperbolic, then the corresponding matrix is P j = i σ j C j i ( σ + j + iσ − j ) C j − i ( σ + j − iσ − j ) C j − i σ j C j . (3.6)Here P j are obtained based on the SU(1 ,
1) matrix in(2.7) as follows. For parabolic H j , one has P j = lim t/l →∞ "(cid:18) πtl (cid:19) − M , (3.7)with M in (2.9). For hyperbolic H j , one has P j = lim C t/l →∞ "(cid:18) cosh π C tl (cid:19) − M , (3.8)with M in (2.10). Then the indicator is defined as η := Tr( P · P ) . (3.9)One can check explicitly that if at least one of the non-elliptic Hamiltonians is parabolic, then the condition in(3.4) becomes C · C < , (3.10) where the product of two Casimir vectors is defined in(3.3). If both of the two driving Hamiltonians are hyper-bolic, then the condition in (3.4) becomes1 + C · C C C < , (3.11)The general principle of defining η in (3.9) can bestraightforwardly understood as follows. By writing theMöbius transformations in Eqs. (2.9) and (2.10) in termsof Pauli matrices, one can find that in the limit T i l → ∞ ( i = 1 ,
2) the sum of coefficients of the leading termsin Tr( M · M ) is nothing but η . If η <
0, then onehas Tr( M · M ) = −∞ in the limit T i l → ∞ ( i = 1 , T i l → i = 1 , M · M ) →
2. Therefore, as we change the value of T i l continuously, there must be a non-heating phase with | Tr( M · M ) | < Tr( M · M ) = 2 cosh (cid:16) π C T l − π C T l (cid:17) + 2 (cid:16) C · C C · C (cid:17) · sinh (cid:16) π C T l (cid:17) · sinh (cid:16) π C T l (cid:17) . (3.12) For 1+ C · C C ·C = 0, the Floquet CFT will stay at the phasetransition (or critical phase) along the line C T l = C T l . Away from this critical line, the driven CFT will alwaysbe in the heating phase. For 1 + C · C C ·C >
0, one alwayshas Tr( M · M ) > < T /l, T /l < ∞ ,and therefore the system is always in the heating phase.The non-heating phase can appear if and only if (3.11) issatisfied, which can be understood as follows. In the limit T /l, T /l →
0, one has Tr( M · M ) →
2. In the otherlimit T /l, T /l → ∞ , one has Tr( M · M ) → −∞ . Bycontinuously changing T /l and T /l , there must exista region where | Tr( M · M ) | <
2, which correspondsto the non-heating phase. As an intuitive picture, byfixing C C = ( σ , σ +1 , σ − ) = (0 , , C C thatsatisfies (3.11) is shown in Fig. 4.Before we leave this subsection, we hope to pointout one interesting feature in the phase diagram ofhyperbolic-hyberbolic driven Floquet CFT. As we ap-proach 1 + C · C C ·C = 0 from (3.11), it is found that thenon-heating phase does not vanish continuously. Whatwe observe is that the non-heating phase is composed ofan island connected to three lines (see Fig. 5). As weapproach 1 + C · C C ·C = 0, this island of non-heating phasedoes not vanish but simply moves to the infinity (SeeAppendix B for more details). C. N = 3 Now let us consider the case of N = 3, i.e., there arethree driving Hamiltonians H , H , and H in a driving FIG. 5. Phase diagram in a Floquet CFT with N = 2 drivingHamiltonians, both of which are of hyperbolic types. Thecorresponding Casimir vectors are C = (1 , a,
0) and C =(1 , , a ), where we choose a = 1 . . . a = √ a < √
2. For a > √
2, the condition in (3.11) is violated, andthe non-heating phase does not exist. period. Similar to the previous subsection, we determinethe phase diagram based on the value of | Tr( M · M · M ) | according to the criteria in (2.14).One can find there are in general three layers of con-ditions to ensure the existence of non-heating phases:1. Quasi- n (with n = 1) condition. In this case, onesimply needs to look at if there is an elliptic Hamil-tonian in the driving. If there is, then there mustexist a non-heating phase in the phase diagram.In particular, the non-heating phase can be real-ized when the elliptic Hamiltonian dominates in thedriving.2. Quasi- n (with n = 2) condition. One needs to con-sider all possible choices of pairings ( i, j ) in thedriving. If both Hamiltonians are non-elliptic, thenthe quasi- n ( n = 2) condition is η n =2 < . (3.13)One can find that these conditions are nothing butthose obtained in Table I. In particular, the non-heating phases are realized when H i and H j domi-nate in the driving.3. Intrinsic- N ( N = 3) condition. Now we needto consider all the three driving Hamiltonians to-gether. If all the three Hamiltonians are non-elliptic, then the intrinsic- N ( N = 3) conditionensuring the existence of non-heating phases is η N =3 < , (3.14)where we have defined η N =3 := Tr( P · P · P ) , (3.15)with the matrices P j ( j = 1 , ,
3) of the form inEqs. (3.5) or (3.6) depending on the types of Hamil-tonian H j . If at least one the above conditions is satisfied, thenthere must exist a non-heating phase.The condition in (3.14) is obtained in a similar way tothat in obtaining (3.4). That is, by tracking the behaviorof Tr( M · M · M ) with T i l ( i = 1 , ,
3) varied from 0 + to ∞ , one can find that the condition η N =3 < η N =3 in(3.15) (One can find more examples in Appendix C). Ifall the three Hamiltonians are parabolic, one has η N =3 = C ∗ C ∗ C , (3.16)where we have defined C ∗ C ∗ C := σ σ +2 σ − − σ σ − σ +3 + σ +1 σ − σ (3.17) − σ +1 σ σ − + σ − σ σ +3 − σ − σ +2 σ . If all the three Hamiltonians are of hyperbolic types, then η can be expressed as η N =3 = 1 + X i 0, then we will have Tr( M · M · M ) → −∞ . That is, as we increase πT i l , Tr( M · M · M ) changes from 2 to −∞ continuously. Apparently,there will be a non-heating phase (with | Tr( M · M · M ) | < 2) in the parameter space.In the second illustrating case, we consider three hy-perbolic Hamiltonians. Based on the Möbius transfor-mations in (2.10), one can find that Tr( M · M · M ) = 2 h cosh π C T l cosh π C T l cosh π C T l + X i 2. Next, by taking the limit C i T i l → ∞ ,one has cosh π C i T i l ’ sinh π C i T i l , and therefore Tr( M · M · M ) ’ − η N =3 (cosh π C T L cosh π C T l cosh π C T l ), where FIG. 6. Non-heating phases in a Floquet CFT with N = 3driving Hamiltonians, all of which are of hyperbolic types. Wechoose λ = 1 . η n =2 < λ = 3 (right) such that only thecondition η N =3 < η N =3 is defined in Eq. (3.18). If η N =3 < 0, we will haveTr( M · M · M ) → −∞ . Therefore, as we tune C i T i l from0 + to ∞ , the amplitude of Tr( M · M · M ) will changefrom − ∞ continuously. Apparently, there will be anon-heating phase (with | Tr( M · M · M ) | < 2) in theparameter space.To have an intuitive picture of the phase diagram, wegive a sample plot of the non-heating phases when thedriving Hamiltonians are all hyperbolic. We consider theCasimir vectors: C i = (1 , λ cos θ i , λ sin θ i ) , (3.21)where λ > θ = 0, θ = π ,and θ = π , one can check explicitly that for i, j ∈ , , i = j , one always has η = 12 − 32 1 λ − , (3.22)and η = − − · λ − √ · λ ( λ − / . (3.23)Then by solving the conditions in (3.13) and (3.14), onecan obtain η < < λ < 2, and η < λ > 2. Typical plots of the phase diagrams for these twocases can be found in Fig. 6. D. General N Based on the discussions in the previous subsections,now we are ready to give the conditions for the existenceof non-heating phases in an SL deformed Floquet CFTwhen there are N driving Hamiltonians.Let us denote the N driving Hamiltonians as { H , H , · · · , H N } , which are arranged in time order ofdriving. That is, we drive the CFT with H for timeduration T , H for time duration T , and so on. These driving Hamiltonians are characterized by Casimir vec-tors { C , C , · · · , C N } as defined in (3.2).The sufficient conditions for the existence of non-heating phases are composed of N layers of condi-tions, which add constraints on the Casimir vectors { C , C , · · · , C N } . In layer n (1 ≤ n ≤ N ), we con-sider all possible choices of sets { H i , · · · , H i n } , wherethe Hamiltonians are again arranged in the time order ofdriving. Then the layer- n conditions are:1. If there is at least one elliptic Hamiltonian in theset { H i , · · · , H i n } , then there is no constraint onthe Casimir vectors { C i , C i , · · · , C i n } .2. If all the Hamiltonians in { H i , · · · , H i n } are non-elliptic (either parabolic or hyperbolic), then theconditions ensuring the existence of non-heatingphases are ∃ η n < , n = 1 , · · · , N. (3.24)Here the indicator η n is defined as η n := Tr( P i · P i · · · P i n ) , (3.25)with each matrix P j determined by the Casimir vec-tors C j = ( σ j , σ + j , σ − j ) as follows. If the drivingHamiltonian H j is parabolic, then P j has the form P j = (cid:18) i σ j i ( σ + j + iσ − j ) − i ( σ + j − iσ − j ) − i σ j (cid:19) . (3.26)If the driving Hamiltonian H j is hyperbolic, then P j has the form P j = i σ j C j i ( σ + j + iσ − j ) C j − i ( σ + j − iσ − j ) C j − i σ j C j . (3.27)By considering all possible 1 ≤ n ≤ N , there are intotal 2 N − n conditions:– The way to obtain condition (3.24) is similar to theexamples considered in Sec. III B and III C. That is, η n corresponds to the sum of coefficients of the leading-orderterms in Tr( M i · M i · · · M i n ) in the limit T i k /l → ∞ for all i k ∈ { i , i , · · · , i n } . Then the condition in(3.24) ensures that Tr( M i · M i · · · M i n ) will changefrom 2 to −∞ as we tune T i k /l from 0 to ∞ continu-ously. Then there must exist non-heating phases with | Tr( M i · M i · · · M i n ) | < C nN = N !( N − n )! n ! conditions in layer- n conditions. In addition, as we havementioned, if there exists at least one elliptic Hamiltonianin the chosen set { H i , · · · , H i n } , there is no constraint onthe corresponding Casimir vectors { C i , C i , · · · , C i n } .We hope to emphasize that this does not mean there is no0constraint on the layer- n condition when n < n , becauseit is totally possible there is no elliptic Hamiltonian in thesubset { H i , · · · , H i n } ⊂ { H i , · · · , H i n } .– In the specific case n = 1, the layer- n condition men-tioned above is simply reduced to the existence of anelliptic driving Hamiltonian.Finally, let us comment on where to find these non-heating phases in the N -dimensional parameter spacespanned by { T l , T l , · · · , T N l } . Suppose a certain layer- n condition is satisfied, then if there exists at least one ellip-tic Hamiltonian (which we denote as H i m ) in the subset { H i , · · · , H i n } , then the non-heating phase can be ob-tained by taking all T i /l → T i m /l ;if all the n Hamiltonians in { H i , · · · , H i n } are non-elliptic, then the non-heating phases can be found in the n -dimensional subspace spanned by { T i l , T i l , · · · , T in l } .One can simply take T i l = T i l = · · · = T in l := T ∗ l . Byincreasing T ∗ l from 0 to ∞ gradually, one will necessarilyfind a non-heating phase. For example, one can refer toFig. 5 for the case of N = n = 2 and the right plot inFig. 6 for the case of N = n = 3. IV. CONCLUSION AND DISCUSSION In this paper, we have studied how the types of drivingHamtiltonians affect the phase diagrams in an SL de-formed Floquet CFT. It is found that the heating phasesare generic, but the non-heating phases may be absentin the phase diagram. We give the N -layer conditions(with each layer of conditions expressed in (3.24) ) for theexistence of non-heating phases in an SL deformed Flo-quet CFT with N driving Hamiltonians. We showed thatthese conditions are sufficient and necessary for N = 2.For N > 2, we only showed that these conditions aresufficient. In fact, for small N with N > 2, we havescanned the parameter space numerically and did notfind any non-heating phases if the conditions in (3.24)are violated. We conjecture that our conditions in (3.24)are also necessary conditions. It is an interesting futureproblem to prove this conjecture.Besides the types of driving Hamiltonians, the con-crete driving sequences will also affect the phase diagram,such as quasi-periodic drivings and random drivings, asdiscussed in Refs. 55–57. One simple form of the quasi-periodic drivings is the Fibonacci quasi-periodic drivingas studied in Refs. 55 and 56 recently. One way to obtainthe phase diagram is to use a periodic driving to approachthe quasi-periodic driving, by taking larger and largerdriving period. Our conditions may be helpful to un-derstand how the phase diagram in a quasi-periodicallydriven CFT depends on the types of driving Hamiltoni-ans. In the random drivings, the dependence of phasediagrams on the types of driving Hamiltonians will alsoexhibit very interesting structures, as will be discussedin detail in Ref. 57. One interesting problem is to generalize the SL defor-mations to the more general deformations, by choosinga general real function f ( x ) in (1.2), where the under-lying group structure is the Virasoro group. Recently,in Ref. 64, the authors consider related problems in anopposite way. That is, one can start from a certain inter-esting conformal map on the complex z -plane, and mapit back to the physical spacetime to find out the corre-sponding deformation of the energy-momentum tensor.In general, this ‘mapping back’ procedure cannot be ana-lytically done, and one needs to perform numerical calcu-lations. In addition, we hope to emphasize that it is pos-sible that the envelope function f ( x ) [see Eq. (1.2)] gen-erated in this way may be not a real function, which mayresult in non-Hermitian deformed Hamiltonians. Never-theless, one may use this method to search for interestingconformal maps under which the driven CFT exhibit ex-otic features.Another interesting problem is on the characteriza-tion of the Floquet CFTs. Previous works character-ize the phase diagrams based on either entanglemententropy or energy evolution. More detailed fea-tures of the time-dependent driven CFT can be capturedby the entanglement Hamiltonian (and its spectrum),which was recently used to study the non-equilibrium dy-namics such as quantum quenches in (1+1)d CFTs bothanalytically and numerically . In the setup ofFloquet CFTs with SL deformation, we expect that theentanglement Hamiltonians in different phases of FloquetCFTs may be classified into three types (see (1.6)) up tocertain envelope functions. We will leave this problem toa future work. ACKNOWLEDGMENTS We thank B. Beri, R. Fan, Y. Gu, H. Shapourian, C.von Keyserlingk, A. Ludwig, I. Martin, S. Ryu, T. Tada,A. Vishwanath, A. Wall and J. Q. Wu for useful dis-cussions. X. W. also thanks D. Ageev, A. Bagrov, andA. Iliasov for communications on Ref. 64. B. H. is sup-ported by ERC Starting Grant No. 678795 TopInSy.X. W. is supported by Gordon and Betty Moore Foun-dation’s EPiQS initiative through Grant No. GBMF4303at MIT. Appendix A: N = 1 In this appendix, we give further detailed discussionson the time evolution of entanglement entropy after asingle quantum quench. Let us first consider the simplechoice of subsystem A = [ nl, ( n +1) l ] where l = L/n . Letus keep the anti-chiral part undeformed, and only focuson the effect of deformation in the chiral part. Thenbased on Eq. (2.17) and Eqs. (2.8), (2.9), and (2.10), onecan obtain1 S A ( t ) − S A (0) = c (cid:26)h cosh (cid:16) π C tl (cid:17) − σ − C sinh (cid:16) π C tl (cid:17)i + h σ + σ + C · sinh (cid:16) π C tl (cid:17)i (cid:27) , c (2) > ,c (cid:26)(cid:16) − πσ − tl (cid:17) + (cid:16) π ( σ + σ + ) tl (cid:17) (cid:27) , c (2) = 0 ,c (cid:26)h cos (cid:16) π C tl (cid:17) − σ − C sin (cid:16) π C tl (cid:17)i + h σ + σ + C · sin (cid:16) π C tl (cid:17)i (cid:27) , c (2) < . (A1) For general choices of ( σ , σ + , σ − ), one can find that inthe long time driving limit t/l (cid:29) 1, the entanglemententropy can be approximated by the formulas in (3.1) inthe main text. But there is one subtlety we hope to pointout. In the case of c (2) > 0, one can find that by choosing σ − = 0 and σ = σ + = 0, one has S A ( t ) − S A (0) = − c · π C tl . (A2)That is, the entanglement entropy decreases linearly intime. This phenomenon has been analyzed in Ref. 55.The reason is that the entanglement cut and the energy-momentum density peaks coincide with each other. In-tuitively, in the study of S A ( t ), one needs to introducea UV cutoff at the entanglement cuts. Since the energy-momentum density peaks are also located at the entan-glement cuts, during the driving, the degree of freedomthat carries the entanglement between A and its comple-ment will accumulate at the entanglement cut. Due tothe UV cut-off, these degrees of freedom cannot be de-tected by the entanglement entropy, which results in adecrease in the entanglement entropy. To see the lineargrowth of the entanglement entropy in this case, one sim-ply needs to shift the locations of entanglement cuts. Forexample, by choosing A = [( k + ) l, ( k + ) l ] where k ∈ Z ,the entanglement entropy is expressed in (2.19). In thiscase, the entanglement cuts and energy-momentum den-sity peaks do not coincide with each other. With thesame choice of σ − = 0 and σ = σ + = 0, one canfind that the entanglement entropy grows linearly in timenow. Appendix B: N = 2 In this appendix, we give a derivation of the re-sults in Table. I in the main text. That is, we con-sider six different pairings of H and H with theHamiltonian types in (1.6): (i) elliptic-elliptic, (ii)elliptic-parabolic, (iii) elliptic-hyperbolic, (iv) parabolic-parabolic, (v) parabolic-hyperbolic, and (vi) hyperbolic-hyperbolic. Some detailed features of the phase diagramwill also be discussed. 1. Features of phase diagram (i) Elliptic-elliptic If both the driving Hamiltonians are elliptic, thenbased on the Möbius transformations in Eqs. (2.8), (2.9),(2.10), we have Tr( M · M ) = 2 cos (cid:16) π C T l (cid:17) · cos (cid:16) π C T l (cid:17) + 2 C · C C · C · sin (cid:16) π C T l (cid:17) · sin (cid:16) π C T l (cid:17) . (B1) First, as proved in Appendix B 2, there always exists aheating phase along the lines in Eqs. (B18) and (B19) inthe parameter space.Second, let us prove there always exist non-heatingphases in the phase diagram. It is noted that when both H are H are elliptic, we always have (cid:12)(cid:12)(cid:12) C · C C C (cid:12)(cid:12)(cid:12) > 1, as dis-cussed in Appendix B 2. Let us consider the cases with C · C C C < − C · C C C > − C · C C C < − 1, it is convenient to rewrite Eq.(B1) asfollows Tr( M · M ) =2 cos (cid:16) π C T l + π C T l (cid:17) + 2 (cid:16) C · C C C + 1 (cid:17) · sin (cid:16) π C T l (cid:17) · sin (cid:16) π C T l (cid:17) . (B2) One can find that for C T l ’ + and C T l ’ + , one has0 < Tr( M · M ) < 2, and therefore the system is in anon-heating phase.Similarly, for C · C C C > 1, one can rewrite Eq.(B1) as Tr( M · M ) =2 cos (cid:16) π C T l − π C T l (cid:17) + 2 (cid:16) C · C C C − (cid:17) · sin (cid:16) π C T l (cid:17) · sin (cid:16) π C T l (cid:17) . (B3) For C T l ’ + and C T l ’ − + , or C T l ’ − + and C T l ’ + , we have − < Tr( M · M ) < 0, and thereforethe system is in a non-heating phase.For the two regions corresponding to non-heatingphases as discussed above, one can see Fig. 8 for example.In short, when the two non-commuting driving Hamil-tonians are both elliptic, there are both heating and non-heating phases in the phase diagram. (ii) Elliptic-parabolic Without loss of generality, we consider the case that H is elliptic, and H is parabolic. Then we have Tr( M · M ) =2 cos (cid:16) π C T l (cid:17) + 2 C · C C · πT l · sin (cid:16) π C T l (cid:17) , (B4) C · C = 0. For C · C < 0, one can find thatfor π C T l ’ + and T /l ’ + , we always have 0 < Tr( M · M ) < 2, and therefore the system is in a non-heating phase. On the other hand, for finite π C T l , as T goes to infinity, we always have a heating phase. For C · C > 0, one can find that for π C T l ’ − + and π C T l ’ + , we always have 0 < Tr( M · M ) < 2, andtherefore the system is in a non-heating phase. (iii) Elliptic-hyperbolic Without loss of generality, we consider the case that H is elliptic, and H is hyperbolic. Then we have Tr( M · M ) =2 cos (cid:16) π C T l (cid:17) · cosh (cid:16) π C T l (cid:17) + 2 C · C C · C · sin (cid:16) π C T l (cid:17) · sinh (cid:16) π C T l (cid:17) . (B5) For C · C = 0, it is straightforward to check that boththe heating and non-heating phases can exist in the phasediagram. For example, the driven CFT is always in thenon-heating phase along the lines C T l = + n , where n ∈ Z (See Fig. 3). If C T l = + n , the driven CFTwill be in the heating phase for large enough T l . For C · C = 0, one can always write Eq. (B5) in the formof N cos (cid:0) π C T l + φ (cid:1) , where | N | increases exponentiallywith C T /l for C T /l (cid:29) 1. Then the system is in thenon-heating phase along the lines π C T l + φ = (1 / n ) π where n ∈ Z , and in the heating phase when π C T l + φ = (1 / n ) π and C T l is large enough. Therefore,for arbitrary C · C , we always have both heating andnon-heating phases in the phase diagram. (iv) Parabolic-parabolic If both the driving Hamiltonians are parabolic, we have Tr( M · M ) =2 + 2 C · C · πT l · πT l , (B6) where C · C = 0. Since T , T > 0, one can find thatthere are both heating and non-heating phases if C · C < 0. On the other hand, for C · C > 0, one always hasTr( M · M ) > (v) Parabolic-hyperbolic Without loss of generality, we consider the case that H is parabolic, and H is hyperbolic. Then we have Tr( M · M ) =2 cosh (cid:16) π C T l (cid:17) + 2 C · C C · πT l · sinh (cid:16) π C T l (cid:17) . (B7) Recall that T , T > 0, it is straightforward to check thatthere are both heating and heating phases if C · C < . There is only a heating phase if C · C ≥ . (vi) Hyperbolic-hyperbolic If both the driving Hamiltonians are hyperbolic, thenwe have Tr( M · M ) =2 cosh (cid:16) π C T l (cid:17) · cosh (cid:16) π C T l (cid:17) + 2 C · C C · C · sinh (cid:16) π C T l (cid:17) · sinh (cid:16) π C T l (cid:17) , (B8) which may be rewritten as Tr( M · M ) = 2 cosh (cid:16) π C T l − π C T l (cid:17) + 2 (cid:16) C · C C · C (cid:17) · sinh (cid:16) π C T l (cid:17) · sinh (cid:16) π C T l (cid:17) (B9) For C · C C ·C = − 1, the driven CFT will stay at the phasetransition (or critical phase) along the line C T l = C T l . (B10)Away from this critical line, the driven CFT will alwaysbe in the heating phase.For C · C C ·C > − 1, one always has Tr( M · M ) > 2, andtherefore the system is in the heating phase.The non-heating phase can appear if and only if C · C C · C < − . (B11)It is interesting to check how the non-heating phases dis-appear as C · C C ·C approaches − C · C C ·C < − 1. For convenience, we denote C · C C ·C = − − (cid:15) , where (cid:15) = 0 + . In this limit, Eq. (B9) can be rewritten as Tr( M · M ) = 2 cosh (cid:16) π C T l − π C T l (cid:17) − (cid:15) · sinh (cid:16) π C T l (cid:17) · sinh (cid:16) π C T l (cid:17) (B12) One can find that the non-heating phases are composedof three lines connected by a tri-junction (or island), asshown in Fig. 5 for example. This can be understood asfollows. First, let us consider the line along that definedin Eq. (B10). By requiring Tr( M · M ) = 0, one canobtain C T l = C T l = 1 π arcsinh r (cid:15) . (B13) For all C T l , C T l < π arcsinh q (cid:15) along the line inEq. (B10), one has 0 < Tr( M · M ) < (cid:15) → + , Eq. (B13) can be simplified as follows C T l = C T l ’ π log 2 (cid:15) . (B14) The upper boundary of the non-heating phase along theline in (B10) can be obtained by considering Tr( M · M ) = − 2, based on which one can obtain C T l = C T l = π arcsinh q (cid:15) .3There are several interesting features: (i) The non-heating phases are composed of three lines connected byan island. (ii) As we approach C C · C C = − C C · C C < − 1, the island will move to the infinity.As a summary of this appendix, for N = 2 non-commuting driving Hamiltonians, the phase diagram inthe parameter space spanned by T /l and T /l dependson the Hamiltonian types of both H and H . If at leastone of the two Hamiltonians is elliptic, then there mustexist non-heating phases in the phase diagram. 2. Heating line in the elliptic-elliptic driving In this appendix, we show that for two arbitrary non-commuting driving Hamiltonians that are elliptic, therealways exist heating lines (lines along which the systemis in heating phases) in the phase diagram.For later use, let us first define the reflection ma-trix: M ∈ SU(1 , 1) is called reflection if M = − I andTr( M ) = 0 .From the definition, a reflection matrix is always el-liptic. In addition, it can be proved that the product oftwo non-commuting reflection matrices is always hyper-bolic . In other words, if both M and M are reflectionmatrices, and they do not commute with each other, thenwe always have | Tr( M · M ) | > H , T ) and ( H , T ), the corresponding Möbius trans-formations are expressed in Eq. (2.8). One can find thatby choosing T j = l C j , (B15)where j = 1 , 2, one has α j = − i σ j C j , β j = − i σ + j + iσ − j C j , andTr( M · M ) = Tr( M · M ) = 2 C · C C C . (B16)For non-commuting elliptic Hamiltonians H and H , onecan find that | Tr( M · M ) | > 2. This can be understoodas follows. One can always find a SU(1 , 1) matrix U ,such that Tr( U M U − · U M U − ) = 2 C · C C C , where thenormalized vector C C is rotated to ( i, , 0) or ( − i, , M = ± M , we have C C = ( iσ , σ +2 , σ − ), with − ( σ ) + ( σ +2 ) + ( σ − ) = − 1. Since at least one of σ +2 or σ − is nonzero, we always have ( σ ) > | Tr( M · M ) | = | σ | > 2, or equivalently (cid:12)(cid:12)(cid:12) C · C C C (cid:12)(cid:12)(cid:12) > . (B17) C C C C FIG. 7. Two vectors C C and C C corresponding to two non-commuting reflection matrices M and M in (B16). In short, by choosing two non-commuting Hamtilto-nians which are both elliptic, we always have a heatingphase at the point ( T , T ) = ( l C , l C ).Now we will show that there always exits a ‘heatingline’ in the phase diagram if both driving Hamiltoniansare elliptic.Now we only focus on a ‘unit cell’ with 0 < T j ≤ l/ C j ( j = 1 , 2) in the phase diagram. The locations of heatingline depends on the sign of C · C as follows:1. If C · C < 0, the heating line is determined by T l , eff + T l , eff = 1 , < T < l , eff , (B18)where we have defined l i, eff = l/ C i in the ellipticcase.2. If C · C > 0, the heating line is determined by T l , eff − T l , eff = 0 , < T < l , eff . (B19)Examples corresponding to these two cases can be foundin Fig. 8. Now we give the proofs of these two claims asfollows.Let us consider C · C < Tr( M · M ) = − (cid:18) πT l , eff (cid:19) + 2 C · C C C · sin (cid:18) πT l , eff (cid:19) (B20) Since C · C < 0, we have C · C C C < − Tr( M · M ) = − (cid:16) C · C C C + 1 (cid:17) · sin (cid:18) πT l , eff (cid:19) < − . Therefore, we always have a heating phase along the linedefined in Eq. (B18).Second, let us consider C · C > 0. Based on Eqs. (B19)and (B1) one can find that Tr( M · M ) = 2 + 2 (cid:16) C · C C C − (cid:17) · sin (cid:18) πT l , eff (cid:19) > , where we have considered C · C > C · C C C > C · C < C · C > FIG. 8. Heating line (in green) determined by Eqs.(B18)and (B19) in the unit cell with 0 < T j ≤ l/ C j ( j = 1 , C = (1 , , 0) and C = (1 , . , 0) (left) and C = (1 , , 0) and C = ( − , . , 0) (right). Here we choose l = 1. The blue dots represent the heating point as defined inEq.(B15). The regions in red (blue) corresponds to a heating(non-heating) phase. Appendix C: N = 3 As discussed in Sec. III C, when all the three drivingHamiltonians are non-elliptic, the intrinsic- N ( N = 3)condition is η N < – 1 parabolic and 2 hyperbolic Hamiltonians Now let us consider the case there are one parabolicand two hyperbolic driving Hamiltonians. Without lossof generality, let us choose H is parabolic, and H , H are hyperbolic. Based on Eqs. (2.9) and (2.10), one has Tr( M · M · M ) = 2 cosh (cid:16) π C T l (cid:17) cosh (cid:16) π C T l (cid:17) + 2 C · C C · πT l · cosh (cid:16) π C T l (cid:17) sinh (cid:16) π C T l (cid:17) + 2 C · C C · πT l · cosh (cid:16) π C T l (cid:17) sinh (cid:16) π C T l (cid:17) + 2 C · C C C · sinh (cid:16) π C T l (cid:17) sinh (cid:16) π C T l (cid:17) + 2 C ∗ C ∗ C C C · πT l · sinh (cid:16) π C T l (cid:17) sinh (cid:16) π C T l (cid:17) .. (C1) The intrinsic- N condition in (3.14) can be understood asfollows. For T i l → 0, one has Tr( M · M · M ) ’ 2. On theother hand, by taking the limit T i l → ∞ , one has Tr( M · M · M ) ’ η · πT l cosh (cid:0) π C T l (cid:1) cosh (cid:0) π C T l (cid:1) → −∞ for η < 0, where we have considered cosh π C i T i l ’ sinh π C i T i l and η = 2 C · C C +2 C · C C +2 C ∗ C ∗ C C C . Here one can checkexplicitly that η = Tr( P · P · P ), with P i expressedin (3.5) and (3.6). Then as we tune the parameters T i l continuously, there must exist non-heating phases with | Tr( M · M · M ) | < – 2 parabolic and 1 hyperbolic Hamiltonians Now let us consider the case with two parabolic andone hyperbolic driving Hamiltonians. Without loss of generality, let us choose H and H to be parabolic, and H to be hyperbolic. Based on Eqs. 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