Floquet Topological Phases of Non-Hermitian Systems
FFloquet Topological Phases of Non-Hermitian Systems
Hong Wu and Jun-Hong An ∗ School of Physical Science and Technology, Lanzhou University, Lanzhou 730000, China
The non-Hermiticity caused breakdown of the bulk-boundary correspondence (BBC) in topologi-cal phase transition was cured by the skin effect for the systems with chiral symmetry and translationinvariance. However, periodic driving, as an active tool in engineering exotic topological phases,breaks the chiral symmetry, and the inevitable disorder destroys the translation invariance. Here,we propose a scheme to retrieve the BBC and establish a complete description of the topologicalphases of the periodically driven non-Hermitian system both with and without the translation in-variance. The demonstration of our method in the non-Hermitian Su-Schrieffer-Heeger model showsthat exotic non-Hermitian topological phases of widely tunable numbers of edge states and Floquettopological Anderson insulator are induced by the periodic driving and the disorder. Our result sup-plies a useful way to artificially synthesize exotic phases by periodic driving in the non-Hermitiansystem.
Introduction.—
Topological phases in non-Hermitiansystems have attracted much attention both theoreti-cally [1–33] and experimentally [34–41]. Many interestingcharacters have been found in different non-Hermitiansystems [16–30]. One of their unique features is thatnot only the edge states but also the nontopologicallyprotected bulk states are localized at the edges, whichis called skin effect [10–14, 42–45]. It causes that onecannot characterize the edge states by the topologi-cal properties of the bulk spectrum. This is the non-Hermiticity induced breakdown of bulk-boundary corre-spondence (BBC) [1–5], which lays the foundation for theclassification of topological phases in Hermitian systems[46–49]. To describe the topological features of the edgestates, many strategies including biorthogonal eigenstate[3], singular-value decomposition [31], gauge transforma-tion [50], and modified periodic boundary condition [51]have been proposed. A milestone among these is the non-Bloch band theory established in the generalized Bril-louin zone (BZ) for the one-dimensional (1D) chirallysymmetric and translation invariant systems [4, 8], whichis recently generalized to the system without chiral sym-metry [44, 45].Coherent control via periodic driving dubbed as Flo-quet engineering has become a versatile tool in artificiallysynthesizing exotic topological phases in systems of ultra-cold atoms [52, 53], photonics [54, 55], superconductorqubits [56], and graphene [57]. Parallel to the topolog-ical phases in static systems, the topological phases inperiodically driven systems are called Floquet topolog-ical phases (FTPs). Many intriguing FTPs absent instatic systems [58–69] have been simulated by periodicdriving in Hermitian systems. The key role played byperiodic driving is changing symmetry and inducing aneffective long-range hopping in lattice systems [70–72]. Anatural question is what controllable topological charac-ters can periodic driving bring to non-Hermitian systems.Given the fact that the chiral symmetry can be brokenby periodic driving, one cannot apply the well developednon-Bloch band theory of 1D chirally symmetric static systems [4] to the periodic ones for recovering the BBCand defining topological invariants. Without touchingthe topological characterization, the transport phenom-ena of the non-Hermitian Floquet edge states was studiedin Refs. [73, 74]. For some special cases in the absenceof the skin effect, the topological numbers were definedin the traditional BZ [75, 76]. Recent study reveals thatthe BBC is approximately recoverable only for small in-tercell coupling [77]. Further, the inevitable spatial disor-der also invalidates the non-Bloch band theory to restorethe BBC. Thus, a general theory to characterize the non-Hermitian FTPs is still lacking.In this work, we investigate the FTPs in the peri-odically driven non-Hermitian systems. A general de-scription is established to characterize the FTPs of suchnonequilibrium systems both in the momentum and thereal spaces. The main idea to characterize the FTPsin both of the spaces is to restore the chiral symme-try of the periodically driven systems by the proposedsimilarity transformations, which keep the quasienergyspectrum unchanged. Taking the non-Hermitian Su-Schrieffer-Heeger (SSH) model as an example, we findthat rich topological phases absent in the static case arecreated by the periodic driving. The studies on the real-space topological physics in the presence of disorder re-veal that the extra phases called non-Hermitian Floquettopological Anderson insulator phases are induced by thedisorder. Our results demonstrate that the periodic driv-ing and its constructive interplay with the disorder sup-ply us useful ways to engineer exotic topological phasesin the non-Hermitian systems.
Floquet topological phases.—
A time-periodic system H ( t ) = H ( t + T ) with period T has a complete set ofbasis | u α ( t ) i determined by Floquet equation [ H ( t ) − i∂ t ] | u α ( t ) i = ε α | u α ( t ) i such that any state evolves as | Ψ( t ) i = P α c α e − iε α t | u α ( t ) i [78, 79]. Acting as station-ary states and eigenenergies of static systems, | u α ( t ) i and ε α are called quasistationary states and quasiener-gies, respectively. Being equivalent to U T | u α (0) i = e − iε α T | u α (0) i with U T the one-period evolution opera- a r X i v : . [ c ond - m a t . d i s - nn ] J u l tor, the Floquet equation defines an effective Hamilto-nian H eff = iT ln U T whose eigenvalues are the quasiener-gies. The FTPs are defined in the quasienergy spectrum.Different from the static case, they can occur at both ofthe quasienergies 0 and π/T [70].Chiral symmetry plays an important role in character-izing the non-Hermitian topological phases [2–4, 6, 8].However, it cannot be preserved if a periodic drivingis applied. Consider a non-Hermitian two-band system H with its parameters periodically driven between twospecific H and H in the respective time duration T and T . Applying the Floquet theorem, we obtain H eff from U T = e − iH T e − iH T . It can be seen that even H j ( j = 1 ,
2) have chiral symmetry SH j S − = − H j with S being the chiral operator, H eff breaks the symmetrydue to [ H , H ] = 0. The absence of the chiral symme-try in H eff makes it hard to define the FTPs in a non-Hermitian system by the non-Bloch band theory, whichis developed for the chirally symmetric static system [4].We propose the following scheme to resolve this prob-lem. Two similarity transformations G j = e i ( − j H j T j / covert U T into ˜ U T, = e − iH T / e − iH T e − iH T / and˜ U T, = e − iH T / e − iH T e − iH T / , from which the de-fined ˜ H eff ,j = iT ln ˜ U T,j share the same quasienergies with H eff while recover the chiral symmetry of H j . It canbe equivalently understood to define new chiral opera-tors G − j S − G j such that H eff obeys the chiral symme-try. The similar scheme was used in Hermitian systems[80]. As we will see later, the recovered chiral symme-try is significant to characterize the FTPs in the non-Hermitian system both for the translation-invariant andvariant cases. Translation-invariant non-Hermitian system.—
If thesystem is further translation invariant, we can developa general characterization to the FTPs in the momen-tum space. The coefficient matrices of H j are writtenin the momentum space as H j ( k ) = h j ( k ) · σσσ with σσσ being the Pauli matrices. We readily obtain H eff ( k ) = h eff ( k ) · σσσ = i ln[ e − i H ( k ) T e − i H ( k ) T ] /T with the Blochvector h eff ( k ) = − arccos( (cid:15) ) r /T and (cid:15) = cos( T E ) cos( T E ) − h · h sin( T E ) sin( T E ) , (1) r = h × h sin( T E ) sin( T E ) − h cos( T E ) × sin( T E ) − h cos( T E ) sin( T E ) , (2)where T = T + T , h j = h j /E j , and E j = p h j · h j isthe complex eigen energies of H j ( k ) [81]. The FTP tran-sition is associated with the closing of the quasienergybands, which occurs at the exceptional points for the k and driving parameters satisfying T j E j = n j π, n j ∈ Z , (3)or ( h · h = ± T E ± T E = nπ, n ∈ Z (4)at the quasienergy zero (or π/T ) if n is even (or odd)[81]. As the condition for the phase transition, Eqs. (3) and (4) supply a guideline to manipulate the exceptionalpoints via the driving parameters for engineering variousnon-Hermitian FTPs at will. They reduce to the resultsin the Hermitian case [71, 72] as a special case when thenon-Hermitian terms in h j vanish.We see from Eq. (2) that h eff ( k ) generally has threecomponents even though the chirally symmetric h j haveonly two. It proves that the chiral symmetry is broken bythe periodic driving [81]. Thanks to the similarity trans-formation G j , we obtain ˜ H eff ,j ( k ) preserving the chiralsymmetry of H j ( k ). Then we can restore the BBC anddefine proper topological invariants in our periodicallydriven non-Hermitian system by introducing the gener-alized BZ in the similar manner as the static system [4].The topological properties of the periodic non-Hermitiansystem are fully characterized by the two winding num-bers W j defined in the generalized BZ associated with˜ H eff ,j . The number of 0- and π/T -mode edge states re-lates to W j as [41, 75] N = |W + W | / , N π/T = |W − W | / . (5)Without loss of generality, we demonstrate our methodby the 1D non-Hermitian SSH model [82–84] H = L X l =1 [( t + γ a † l b l + ( t − γ b † l a l + t ( a † l b l − + h.c.)] , (6)where a l ( b l ) are the annihilation operators on the sub-lattice A ( B ) of the l th lattice, and L is lattice length. Inmomentum space and the operator basis (˜ a k , ˜ b k ) T with˜ a k (˜ b k ) being the Fourier transform of a l ( b l ), it reads H ( k ) = d x σ x + ( d y + iγ/ σ y , (7)where d x = t + t cos k and d y = t sin k . The bandsclose at k = π (or 0) when t = t ± γ/ − t ± γ/ t = p t + γ / σ − z H ( k ) σ z = −H ( k ) [49] is cured bythe skin effect. Via replacing e ik by β = q | t − γ/ t + γ/ | e ik ,Eq. (7) is converted into H ( β ) = P n = ± R n ( β ) σ n with σ ± = ( σ x ± iσ y ) / R ± ( β ) = t ± γ + β ∓ t . Here β defines a generalized BZ. Its topological property isdescribed by the winding number W = − ( W + − W − ) / W ± = π [arg R ± ( β )] C β with [arg R ± ( β )] C β are thephase change of R ± as β counterclockwisely goes alongthe generalized BZ C β [4, 8]. When | t | < p t + γ / W = 1 and a pair of edge states is formed.Choosing the periodic driving as t ( t ) = ( f, t ∈ [ mT, mT + T ) q f, t ∈ [ mT + T , ( m + 1) T ) , m ∈ Z , (8)we now investigate the FTPs in our periodically drivennon-Hermitian SSH model. Figure 1(a) shows the FIG. 1. (a) Quasienergy spectra with the change of thedriving amplitude under the open (blue lines) and periodic(gray lines) boundary conditions. Numbers of 0-mode (b)and π/T -mode (c) edge states defined in the conventional (reddashed) and generalized (cyan solid) BZ. We use t = 2 . γ , T = T = 0 . γ − , q = 3 .
0, and L = 80. quasienergy spectrum under the open-boundary condi-tion. It indicates that even the static system when f = 0is topologically trivial, diverse topological phases at thequasienergies 0 and π/T can be created by the peri-odic driving. However, this quasienergy spectrum hasa dramatic difference from the one under the periodic-boundary condition, which takes p h eff ( k ) · h eff ( k ). Itreveals that the non-Hermiticity induced breakdown ofBBC occurs in our periodically driven system too. Tosolve this problem, we introduce the generalized BZ viareplacing e ik in H eff ( k ) by β [81]. Then the effectiveHamiltonian is converted to H eff ( β ). First, H eff ( β ) cor-rectly explains the exceptional points of the quasiener-gies under the open-boundary condition. Remembering h ( t ) = [ t + t ( t )( β + β − ) / , i [ γ + t ( t )( β − − β )] / , t > γ/ >
0, weobtain the phase-transition conditions as follows.
Case I: h · h = 1. We can check that Eqs. (4) induce T | κ + e iα f | + T | κ + e iα qf | = n α π, ( n α ∈ Z ) (9)for k in β being α = 0 or π , where κ = p t − γ /
4. Heresgn[( κ − f )( κ − qf )] = 1 is further needed for α = π . Case II: h · h = − k = π when sgn[( κ − f )( κ − qf )] = −
1. Then Eqs. (4) give T | κ − f | − T | κ − qf | = n π π. (10) Case III:
According to Eq. (3), any k in β satisfying T E = n π, T E = n π, ( n , n ∈ Z ) (11)contributes to the band closing.Taking care of the skin effect via introducing β ,Eqs. (9)-(11) perfectly describe the band closing of thequasienergy spectrum under the open-boundary condi-tion. The π/T -mode band-closing points at f ’ . γ FIG. 2. Trajectories of R ± in ˜ H eff,1 ( β ) with k in β runningfrom 0 to 2 π when f crosses the phase boundaries. The wind-ing number W changes from 0 when f = 0 . γ (red dashed)to 1 when 0 . γ (blue solid) in (a) and (b); and from 1 when f = 0 . γ (red dashed) to 0 when 1 . γ (blue solid) in (c) and(d). Others parameters are the same as Fig. 1. and 2 . γ in Fig. 1(a) are obtainable from Eqs. (9) with n = n π = 1. The 0-mode ones at f ’ . γ and 1 . γ are obtainable from Eqs. (10) with n π = 0 and (9) with n = 2, respectively. Thus the BBC has been successfullyretrieved in our periodically driven system.Second, the FTPs of the quasienergy spectrum underthe open boundary condition are well characterized bythe two winding numbers W j defined in ˜ H eff ,j . Accord-ing to Eq. (5), we plot in Figs. 1(b) and 1(c) the numbersof 0-mode and π/T -mode edge states calculated fromthe conventional and generalized BZs. Although qualita-tively capturing the exceptional points of the quasienergyunder the periodic-boundary condition, the ill-definedtopological numbers from the conventional BZ nonphys-ically take half integers. However, the ones from thegeneralized BZ correctly count the number of the edgestates. It is called the non-Bloch BBC [4, 8]. Note that,absent in the static system, such correspondence for the π/T -mode edge states is unique in our periodic system.Third, the topological change of the quasienergy spec-trum is reflected by H eff ( β ). We plot in Fig. 2 the tra-jectories of R ± in ˜ H eff,1 ( β ) when f increases across thephase borders. Figures 2(a) and 2(b) show that R ± haveno wrapping to the origin and thus W = 0 before the π/T -mode phase transition. When f increases across thecritical point, R ± at the neighbourhood of k = 0 changessuch that ε = p R + R − crosses π/T . Due to its peri-odicity, ε abruptly jumps to − π/T keeping the directionof R ± unchanged. Then an anticlockwise and a clock-wise wrappings to the origin are formed by R + and R − ,respectively, and thus W = 1. Figures 2(c) and 2(d)show that W changes from 1 to 0, where R ± at theneighbourhood of k = π changes such that ε crosses thequasienergy 0. This gives a geometric picture to the FTP FIG. 3. Phase diagram characterized by W (a) and W (b).The red solid, the black dashed, and the blue dot-dashed linesare the phase boundaries from Eqs. (9), (10), and (11), re-spectively. We use t = 1 . γ , f = 2 γ , and q = 0. transition in Fig. 1.As a useful tool in controlling the exceptional points,the periodic driving enables us to realize not only thetopological phases inaccessible in the same static-systemcondition, but also rich phases completely absent in itsoriginal static system. Figure 3 shows the phase dia-gram in the T - T plane. A widely tunable number of W j and edge states are induced by changing the drivingparameters. The presence of such rich phases originatesfrom the distinguished role of periodic driving in sim-ulating an effective long-range hopping in different lat-tices [70–72]. The phase boundaries in red solid lines(black dashed lines) are perfectly described by Eq. (9)with α = 0 [by Eq. (10)]. T E = π in Eqs. (11) issatisfied by T ’ . /γ . T E = n π is satisfied by T ’ n π/ ( γ √ .
66 cos k ). When k runs from 0 to π for given n , a series line segments with a common T ’ . /γ (see the blue dot-dashed line in Fig. 3) areformed, which all give the phase boundaries. We see thatour analytical method successfully describes the FTPs inthe periodically driven non-Hermitian system. The re-sult reveals that, without changing the intrinsic parame-ters in the static system, the periodic driving supplies usanother control dimension to adjust the numbers of thenon-Hermitian topological edge states. This is useful inthe application of non-Hermitian topological physics. Translation-variant non-Hermitian system.—
Whenthe translation invariance of Eq. (6) is broken by thedisorder dξ l in the non-Hermitian term γ , where ξ l ∈ [ − . , .
5] is the disorder in the l th cell with strength d , we cannot work in the momentum space anymore.The non-Bloch band theory is inapplicable too. How-ever, we still may characterize the FTPs by the chirallysymmetric ˜ H eff ,j in the real space. Regarding l ∈ [1 , ‘ ]and [ L − ‘ +1 , L ] of the chain as the boundaries, we define FIG. 4. Real-space winding numbers W (blue solid) and W (red dashed) in (a), (b) and the corresponding quasienergiesin (c), (d) with the change of the disorder strength. T = T =0 . t − in (a), (c) and 0 . t − in (b), (d). Other parametersare γ = 0 . t , f = 0 . t , q = 3 . L = 160, and ‘ = 40. (a),(b) is obtained after 250 times average to the disorder. the real-space winding numbers [85] W j = 12 L Tr ( SQ j [ Q j , X ]) . (12)Here S ls,l s = δ ll ( σ z ) ss is the chiral operator and X ls,l s = lδ ll δ ss with s, s = A, B being the sublattices, Q j = P n ( | n R j ih n L j | − S | n R j ih n L j | S † ) with ˜ H eff ,j | n R j i = ε j,n | n R j i and ˜ H † eff ,j | n L j i = ε ∗ j,n | n L j i , and Tr denotes thetrace over the middle interval with length L = L − ‘ and P n denotes the summation to the bulk states. Wecan check that W j return to W when d = 0. Thus, moregeneral than W j , the real-space W j can give a unified de-scription to the FTPs of the non-Hermitian system bothfor the translation-invariant and variant cases.Figure 4 shows the winding numbers W j and thequasienergies with the change of the disorder strength.We can see from Figs. 4(a) and 4(c) that the topologicaltrivial character of the disorder-free case is robust whenthe disorder is weak for d . t . With the increase of d , it is remarkable to find that a 0-mode edge state istriggered in a wide range d ∈ (2 , t . The disorder-induced edge state has been found in static Hermitiansystems [86–89]. Analogous to that, we call the similarstate occurred in our periodically driven non-Hermitiansystem as Floquet topological Anderson insulator phase.Its presence can be further confirmed by Figs. 4(b) and4(d), where a π/T -mode edge state exists in the disorder-free case. Here, it is interesting to observe a coexistedregime of the π/T -mode edge state and 0-mode Floquettopological Anderson insulator state. Both of the statesare absent in the static system. However, in the strongdisorder regime, the bands close and all the edge statesdisappear, which is compatible to the result in the Her-mitian case [90]. Discussions and conclusion.—
Note the chiral symme-try is not recoverable in some driving cases [91], where thenon-Hermitian FTPs can be described by a Z topologi-cal invariant [81]. Our result is realizable in the presentexperimental state of art of photonics, where the non-Hermitian topological phases of the SSH model [34, 92]and the Hermitian FTPs [54, 55] have been observed.We have investigated the topological phases in pe-riodically driven non-Hermitian systems. A scheme isproposed to retrieve the BBC, based on which a com-plete description to the FTPs is established for such non-Hermitian systems both with and without the transla-tion invariance. Taking the SSH model as an example,we have found that diverse exotic FTPs can be createdfrom the topologically trivial static system by the pe-riodic driving. Further study reveals that the Floquettopological Anderson topological insulator phases can betriggered by the moderate-strength disorder. Exhibitinga wide perspective of controllability, our results hopefullypromote further studies of both fundamental physics andpotential applications of rich non-Hermitian FTPs. Acknowledgments.—
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B , 245135 (2019). upplemental material for “Floquet Topological Phases of Non-Hermitian Systems” Hong Wu and Jun-Hong An ∗ School of Physical Science and Technology, Lanzhou University, Lanzhou 730000, China
DERIVATION OF EQS. (1)-(4)
According to the Euler’s formula of the Pauli matrices e − i h j · σσσT j = cos( E j T j ) − i sin( E j T j ) h j · σσσ (S1)with E j = p h j · h j and h j = h j /E j , we have U ( T ) = e − i h · σσσT e − i h · σσσT = (cid:15)I × + i r · σσσ ≡ e − iH eff T , (S2) (cid:15) = cos( E T ) cos( E T ) − h · h sin( E T ) × sin( E T ) , (S3) r = h × h sin( T E ) sin( T E ) − h cos( T E ) × sin( T E ) − h cos( T E ) sin( T E ) , (S4)where and T = T + T . Using the Euler’s formula again,we can infer H eff from Eq. (S2) as H eff = − arccos( (cid:15) ) r · σσσ/ [sin(arccos( (cid:15) )) T ] (S5)Due to (cid:15) + r · r = 1, H eff can be simplified into H eff = arccos( (cid:15) ) r · σσσ/ [ p − (cid:15) T ] = arccos( (cid:15) ) r · σσσ/T. (S6)The quasienergy bands touch at 0 and ± π/T , which oc-curs when (cid:15) = +1 and −
1, respectively. Therefore, weobtain that the bands close from Eq. (S3) when T j E j = n j π, n j ∈ Z , (S7)or ( h · h = ± T E ± T E = nπ, n ∈ Z (S8)at the quasienergy zero (or π/T ) if n is even (or odd). CHIRAL SYMMETRY IN MOMENTUM SPACE
For a two-band non-Hermitian system, the coefficientmatrix of its Hamiltonian in operator basis can be writ-ten in the momentum space as H ( k ) = h ( k ) · σσσ , where σσσ are the Pauli matrices. If the parameters in h are period-ically driven between two specific h and h within therespective time duration T and T , then we, accordingto the Floquet theorem, can obtain the effective Hamli-tonian as H eff ( k ) ≡ h eff · σσσ = i ln[ e − i H ( k ) T e − i H ( k ) T ]with the Bloch vector h eff ( k ) = − arccos( (cid:15) ) r /T and (cid:15) = cos( T E ) cos( T E ) − h · h sin( T E ) sin( T E ) , r = h × h sin( T E ) sin( T E ) − h cos( T E ) × sin( T E ) − h cos( T E ) sin( T E ) , (S9) where T = T + T , h j = h j /E j , and E j = p h j · h j isthe complex eigen energies of H j ( k ). We can see that H eff ( k ) does not inherit the chiral symmetry of H j ( k )due to the presence of the first term of Eq. (S9).We propose the following scheme to resolve this prob-lem. A similarity transformation G = e − i H ( k ) T / converts the evolution operator U T to ˜ U T, = U U with U = e − i H ( k ) T / e − i H ( k ) T / and U = e − i H ( k ) T / e − i H ( k ) T / . According to Eq. (S9),we have U j = (cid:15) j I × + i r j · σσσ with (cid:15) = (cid:15) and r j = ( − j a h × h − b h − c h , where a =sin( T E /
2) sin( T E / b = cos( T E /
2) sin( T E / c = cos( T E /
2) sin( T E / U T, = (cid:15) I × + i r · σσσ with (cid:15) = ( (cid:15) ) − r · r and r = 2 h ( − (cid:15) c + ac h · h + ab ) − h ( (cid:15) b + ac + ab h · h ) . (S10)Equation (S10) implies that if H ( k ) and H ( k ) havethe chiral symmetry with a common symmetry operator,then ˜ U T, would inherit this symmetry. The similar re-sult can be obtained by G = e i H ( k ) T / , which converts U T into ˜ U T, = U U . Leaving the quasienergy spectrumunchanged, the similarity transformations G j have suc-ceeded in making ˜ H eff ,j ( k ) ≡ iT ln ˜ U T,j preserve the chiralsymmetry in H j ( k ). GENERALIZED BRILLOUIN ZONE
According to Ref. [1], the skin effect in the static non-Hermitian SSH model can be curled by a similarity trans-formation, which reads S = diag(1 , r, r, r , · · · , r L − , r L )in the real-space sublattice basis. It converts Eq. (6) into¯ H = S − H S with¯ H = L X l =1 [ r ( t + γ a † l b l + r − ( t − γ b † l a l + t ( a † l b l − +h.c.)] . (S11)One can readily check that, as long as r = p | ( t − γ/ / ( t + γ/ | , all the bulk states of ¯ H do notreside in the edges anymore and thus the skin effect iscurled. Comparing the forms of H and ¯ H in the momen-tum space, we see that the usual BZ e ik in H is replacedby re ik ≡ β in ¯ H . Thus β defines a generalized BZ.An important observation is that the similarity trans-formation S has nothing effect on the t term. When theperiodic driving t ( t ) = ( f, t ∈ [ mT, mT + T ) q f, t ∈ [ mT + T , ( m + 1) T ) , m ∈ Z , (S12) a r X i v : . [ c ond - m a t . d i s - nn ] J u l is applied on t , we readily have¯ U T = S − U T S = e − i ¯ H T e − i ¯ H T , (S13)where ¯ H j are Eq. (S11) with t = f and qf , respectively.Since no skin effect is present in ¯ H j , neither to H eff = iT ln U T . We have successfully remove the skin effect bythe same generalized BZ defined by S as the static case.On the other hand, if the periodic driving is applied on t , the generalized BZ would be changed. One can resortto the method in Refs. [2, 3] to calculate the generalizedBZ in the general case. Z TOPOLOGICAL INVARIANT
In the widely used application of Floquet engineering,there exist some periodic driving cases in which even theproposed similarity transformations cannot restore thechiral symmetry. A typical example is the three-step pe-riodic driving to create the discrete time crystal [4]. Ifthe chiral symmetry cannot be recovered, the periodi-cally driven system intrinsically belongs to the D topo-logical class, where the winding number is ill-defined [5].We can define a new Z topological invariant in the realspace to characterize this kind of non-Hermitian Flo-quet topological phases. Inspired by the definition ofelectronic polarization of the two-dimensional Hermitiansystem in the static case [6], we can construct an open-boundary bulk-band Z topological invariant ν = 2 P forour one-dimensional periodically driven non-Hermitiansystem from the electronic polarization P = h π Im ln det
U − X l,l ,s,s X ls,l s L i mod 1 , (S14)where the elements of U read U mn ≡ h m L | e i πX/L | n R i with H eff | n R i = ε n | n R i , H † eff | n L i = ε ∗ n | n L i , and X isthe coordinate operator, namely X ls,l s = lδ ll δ ss with s, s = A, B being the sublattices. The Z topologicalinvariant ν can only distinguishes between even or oddnumber of pairs of edge states.The one-period evolution operator for a three-step pe-riodic driving is given by [4] U T = e − i H ( k ) T e − i H ( k ) T e − i H ( k ) T . (S15)In the similar manner as the two-step periodic driving inthe main text, we can derive that the phase transitionoccurs for the k and driving parameters satisfying T j E j = n j π, n j ∈ Z , (S16)or ( h · h = ± , h · h = ± ,T E ± T E ± T E = nπ, n ∈ Z (S17)at the quasienergy zero (or π/T ) if n is even (or odd). Taking the non-Hermitian Su-Schrieffer-Heeger modelas an example, we illustrate the performance of the Z topological invariant ν in characterizing the Floquettopological phases. After introducing the generalizedBrillouin zone via replacing e ik by β = q t − γ/ t + γ/ e ik , thenon-Bloch Hamiltonian with h = [ t + t ( β + β − ) / , i [ γ + t ( β − − β )] / ,
0] is obtained [1]. We consider an exper-imentally accessible periodic-driving protocol t ( t ) = q f, t ∈ [ mT, mT + T ) q f, t ∈ [ mT + T , mT + T + T ) q f, t ∈ [ mT + T + T , ( m + 1) T ) . (S18)Next, we derive the condition for the band closing. Tosatisfy the first line of Eq. (S17), k in β can only be α = 0 or π , at which the Hamiltonians read H j | k = α = e iα q j f + κκ ( t σ x + iγ σ y ) . (S19)with κ = p t − γ /
4. Due to [ H i , H j ] = 0, we obtainthe effective Hamiltonian at k = α H eff | k = α = ( H T + H T + H T ) /T. (S20)Then the quasienergy bands close at the quasienergies 0and π/T when X j =1 ( κ + e iα q j f ) T j = n α π (S21)for n α being even and odd numbers, respectively. Thisdetermines the critical points for topological phase tran-sition. Here we have assumed that the parameters satisfy t > γ/ > π/T -mode edge states is induced by theperiodic driving. The π/T -mode band-closing points at f = 0 . γ , 1 . γ , and 1 . γ can be obtained from Eq.(S21) with n α = 1 , − π , and 3 , respectively. The 0-mode ones at f = 0 . γ , 1 . γ , and 2 . γ is explainablefrom Eq. (S21) with n α = 0 π , 2 , and − π , respectively.The Z topological invariant ν depicted in Fig. S1(b)perfectly describes the Floquet topological phases in ourperiodically driven non-Hermitian system. If ν = 1, thenthe system holds an odd number of pairs of edge states.If ν = 0, then the system has an even number of pairs ofedge states. ∗ [email protected][1] S. Yao and Z. Wang, Phys. Rev. Lett. , 086803 (2018).[2] K. Yokomizo and S. Murakami, Phys. Rev. Lett. ,066404 (2019). FIG. S1. (a) Quasienergy spectra with the change of thedriving amplitude under the open-boundary condition. (b)the Z topological invariant ν . We use t = 1 . γ , T = T = T = 0 . γ − , q = 1 . q = 2, q = 3, and L = 80. [3] Z. Yang, K. Zhang, C. Fang, and J. Hu, “Auxiliary gen-eralized brillouin zone method in non-hermitian band the-ory,” (2019), arXiv:1912.05499 [cond-mat.mes-hall].[4] J. Zhang, P. W. Hess, A. Kyprianidis, P. Becker, A. Lee,J. Smith, G. Pagano, I.-D. Potirniche, A. C. Potter,A. Vishwanath, N. Y. Yao, and C. Monroe, Nature ,217 (2017).[5] C.-K. Chiu, J. C. Y. Teo, A. P. Schnyder, and S. Ryu,Rev. Mod. Phys. , 035005 (2016).[6] W. A. Wheeler, L. K. Wagner, and T. L. Hughes, Phys.Rev. B100