Time-dependent topological systems: A study of the Bott index
TTime-dependent topological systems: a study of the Bott index
Daniele Toniolo ∗ Department Mathematik, Friedrich-Alexander-Universit¨at Erlangen-N¨urnberg, Erlangen, Germany (Dated: November 29, 2018)The Bott index is an index that discerns among pairs of unitary matrices that can or cannot be approximatedby a pair of commuting unitary matrices. It has been successfully employed to describe the approximate integerquantization of the transverse conductance of a system described by a short-range, bounded and spectrallygapped Hamiltonian on a finite two dimensional lattice on a torus and to describe the invariant of the Bernevig-Hughes-Zhang model even with disorder. This paper shows the constancy in time of the Bott index and theChern number related to the time-evolved Fermi projection of a thermodynamically large system described bya short-range and time-dependent Hamiltonian that is initially gapped. The general situation of a ramp of atime-dependent perturbation is considered, a section is dedicated to time-periodic perturbations.
I. INTRODUCTION
A recent focus of the research in condensed matter hasbeen the description of the topological properties of systemsthat are subjected to perturbations in particular to time-drivenones. Early works on this subject are the references . Theseconsider how a time-periodic drive can induce a topologicalphase in the context of the Bernevig-Hughes-Zhang modeland for a two dimensional system of fermions on a hon-eycomb lattice with spectral gap respectively. More recentstudies have realized that the Chern number is invariantunder the unitary time-evolution of the system, moreover ina two-dimensional setting when the Chern number of the ini-tial ground state and that of the ground state of the instanta-neous Hamiltonian are different then the Hall conductance isno more quantized. This has been shown both for the caseof a quench from an initial trivial state to a topological onein the Haldane model and in the case of a system of spin-less fermions of a honeycomb lattice described by an initialgapped Hamiltonian that is subjected to the linear ramp ofa periodic external electromagnetic field . The topology ofperiodically driven systems has been intensively studied, anincomplete list of works includes: the references that haveintroduced and rigorously discussed the W invariant in twodimensions and in any dimension with the use of a K-theoretic construction , the study of the chiral case and thestudy of the time-reversal invariant case , for the experimen-tal side see and references therein. The invariance of thetopological properties of the ground states, in general degen-erate, of gapped Hamiltonians with respect to local unitaritytransformations has been studied by Hastings and Wen . Re-lated ideas recently brought to an explicit formulation of theadiabatic theorem in the many body context . A differentapproach to the dynamics of topological systems is the studyof the winding of the Pancharatnam phase developed by theauthors of reference .The Bott index is an index of matrices that has beenemployed in the condensed matter realm by Loring andHastings . The index has been introduced in ref. , andits mathematical and physical foundations have been studiedin , see also and references therein. The Bott index dis-cerns among couple of unitary or almost unitary matrices thatcan or cannot be approximated by a commuting pair. In this paper we will be concerned with the index of a couple of uni-tary or quasi-unitary matrices, while the index has been usedalso for triples of Hermitian matrices and for other classes inthe case of systems with symmetries, time reversal or particlehole. Accordingly a general classification of the topologicalphases of any of the Altland-Zirnbauer symmetry classes hasbeen developed in . The Bott index of the projected posi-tion matrices on a torus, as defined by eq. (11) below, stateswhether those matrices can or cannot be approximated by acouple of commuting matrices. This encodes an informationabout the localization properties of the Fermi projection P .The existence of exponentially localized Wannier functionsimply the vanishing of the Bott index; a vanishing Bott in-dex implies a small variance of the Wannier functions withrespect to the system’s size . Recently the relation amongthe spread and the localization of Wannier functions has beeninvestigated in . The Bott index is well suited for numericalsimulations being designed for finite systems, it also handlesthe effects of disorder.The main subject of this work is to show that the Bott indexof the time-evolved Fermi projection of a two-dimensionalsystem described by a finite ranged, bounded and initiallygapped Hamiltonian is constant in time in the thermodynamiclimit when the Hamiltonian is subjected to a perturbationin general time dependent that preserves the locality of theHamiltonian. A possible change of the index in a certaintime scale is only a finite size effect, this is the content of eq.(27). The result is model-independent within the stated hy-pothesis on the Hamiltonian. An analogous numerical resultfor a specific model limited to a time-periodic periodic per-turbation was provided in ref. . Another result of this workis to show that the Chern number of the time-evolved Fermiprojection is constant in time under the same hypothesis forthe Hamiltonian as in the Bott index case but in an infinite2-dimensional systems. This also proves that the Bott indexand the 2-dimensional Chern number are equivalent both in atime-independent setting and in a time-dependent one.A related result regarding the topological order of a set ofdegenerate ground states has been obtained in the reference .The structure of the paper is as follows: in section II the in-variance of the 2-dim. Chern number along the time-evolutionis proved. The Bott index is introduced in section III and therequirements on the physical setting for the index to be welldefined are described. The spectral flow that shows the mech- a r X i v : . [ c ond - m a t . m e s - h a ll ] N ov anism of a possible variation of the Bott index is discussed inthe unitary and in the general case. In section IV the growth intime of the norm of the commutator among the time-evolvedFermi projection and the position operator is studied and it isshown that for finite range Hamiltonians it cannot give rise toa change of the Bott index. Some technical details are in theappendix A. II. INVARIANCE OF THE 2-DIMENSIONAL CHERNNUMBER OF THE TIME-EVOLVED FERMI PROJECTION
Let us consider a two dimensional insulating system of noninteracting particles with N internal degrees of freedom onan infinite lattice, that for convenience we take equal to Z ,described by a single-particle gapped Hamiltonian H of finiterange R acting on the Hilbert space l ( Z ) ⊗ C N . H = N ∑ l , k = ∑ (cid:107) n − m (cid:107)≤ R H n , m , l , k | n , l (cid:105)(cid:104) m , k | The finite range condition reads: if (cid:107) n − m (cid:107) > R then H n , m , l , k = {| n (cid:105) , n = ( n x , n y ) ∈ Z } is the usual basis of l ( Z ) such that | n (cid:105) equals 1 at the site n and zero elsewhere.The Fermi level µ is supposed to lie in an energy gap of H .The expression of the Chern number of the Fermi projection P ≡ χ ( H ≤ µ ) , here defined by the functional calculus with χ the characteristic function, suitable for an evaluation of thistopological invariant in real space has been employed for ex-ample as eq. (19) of and more recently in the appendix C ofKitaev’s . See also eq. (7) of and .Chern ( P ) ≡ π ImTr u . a . Q [ X , P ] P [ Y , P ] (1)Note that there is no uniformity in the choice of the sign ofthe Chern number in the literature. Tr u . a . is the trace per unitarea: Tr u . a . ≡ lim A → ∞ Tr χ A A , χ A denotes multiplication by thecharacteristic function of the area A , that is 1 inside A , zerooutside. Q is the projection orthogonal to P , Q ≡ − P . Theoperators X and Y in eq. (1) are the position operators of l ( Z ) . For a discussion of the convergence of the trace thatdefines the Chern number see the so called Sobolev conditiondescribed for example in , for a reformulation of the Chernnumber using switch functions instead of position operatorsand the inclusion of a wider class of Hamiltonian with expo-nentially decreasing amplitudes see for instance . Anotherperspective with equivalent results for the quantization of theHall conductance is that of Avron et al. that considered theFredholm index of an operator associated to a couple of pro-jections.The constancy in time of the Chern number when the uni-tary time-evolution of the system is taken into account hasbeen already shown analytically in the ref. for a spatiallyperiodic system following a time-evolution where a periodicperturbation is turned on and in ref. through a numerical ev-idence for the Haldane model following a quench. In bothcases the Schroedinger picture for the unitary evolution was employed. This means that, in the present notation, the quan-tity Chern ( P ( t , t )) = (2) = π ImTr u . a . Q ( t , t ) [ X , P ( t , t )] P ( t , t ) [ Y , P ( t , t )] has been studied, where P ( t , t ) ≡ U ( t , t ) P ( t ) U † ( t , t ) andshown to be independent of t with a local Hamiltonian H ( t ) . U ( t , t ) is the unitary operator of time-evolution of the systemsatisfying: i ∂ t U ( t , t ) = H ( t ) U ( t , t ) , U ( t , t ) = (3)For a time-independent system the time-evolution is given by U ( t , t ) = e − iH ( t − t ) , then the invariance of the Chern numberis manifest, the relevant fact is the invariance in the generalcase of a system with a time dependent Hamiltonian H ( t ) , seeeq. (20). The conditions required for the invariance of theChern number under unitary evolution according to ref. arethe locality of the instantaneous Hamiltonian H ( t ) in eq. (3)and certain regularity properties of the ground state projec-tor over the Brillouin zone. In the present setting I considerthe instantaneous Hamiltonian H ( t ) finite range and in gen-eral gapped only at the initial time t , this ensures the regu-larity of the projections P ( t ) and Q ( t ) as discussed for ex-ample in . I present an alternative proof of the constancyin time of the Chern number, Chern ( P ( t , t )) = Chern ( P ( t )) .To this purpose the Chern number is expressed with the aidof switch functions Λ x and Λ y defined as follows: it ex-ists a positive integer M such that with x > M , Λ x ( x ) = x < − M , Λ x ( x ) = Λ x varying continuouslyin between. Λ y is similarly defined. Using the functionalcalculus the operators Λ x ( X ) and Λ y ( Y ) are defined, namely Λ x ( X ) | n x , n y (cid:105) = Λ x ( n x ) | n x , n y (cid:105) . With abuse of notation I willwrite in the following Λ x and Λ y for the corresponding opera-tors. According, for example, to Chern ( P ) = π ImTr Q [ Λ x , P ] P [ Λ y , P ] (4)The Tr is over the Hilbert space l ( Z ) . In what follows: t = P ≡ P ( t ) and P ( t ) ≡ P ( t , t ) . We want to show theinvariance of the Chern when P is replaced with P ( t ) . TheChern number of a projection is well defined when the tracein eq. (4) is finite, in this case the Chern number turns outto be an integer. Projectors that have well defined Chernnumber and that are homotopically equivalent have the sameChern number, for a proof of this statement in the context ofFredholm-index theory see for example . P and P ( t ) are ho-motopically equivalent therefore the task is to show that thetrace in eq. (4) is finite when replacing P with P ( t ) . In whatfollows it is convenient to consider a more general class ofHamiltonians than the finite range ones, namely the class ofHamiltonians with off diagonal elements falling exponentiallyfast: |(cid:104) n | H | m (cid:105)| ≤ Me − ν (cid:107) n − m (cid:107) , this class of operators is calledlocal. Let us consider:Chern ( P ( t )) = π Tr Q ( t ) [ Λ x , P ( t )] P ( t ) [ Λ y , P ( t )] Defining the Heisenberg picture X H ( t ) ≡ U † ( t ) XU ( t ) , withthe aid of functional calculus we have: Λ x ( X H ( t )) = U † ( t ) Λ x ( X ) U ( t ) Therefore, dropping the time index of U ( t ) , we have:Chern ( P ( t )) = π ImTr Q (cid:2) U † Λ x U , P (cid:3) P (cid:2) U † Λ y U , P (cid:3) (5)The equation of motion for X H ( t ) is: i ddt X H ( t ) = [ X , H ( t )] H . where the explicit time dependence of the Hamiltonian hasbeen put in evidence. With X = X ( t ) we have: i ( X H ( t ) − X ) = (cid:90) t ds [ X , H ( s )] H A simple manipulation of eq. (5) leads to:Chern ( P ( t )) = (6) = π ImTr Q (cid:2) ( Λ x + U † [ Λ x , U ]) , P (cid:3) P (cid:2) ( Λ y + U † [ Λ y , U ]) , P (cid:3) We replace in the equation above: U † [ Λ x , U ] = − i (cid:90) t dsU † ( s ) [ Λ x , H ( s )] U ( s ) (7)and similarly for the y -factor. We note that the operator [ Λ x , H ( s )] is confined around the y axis, this follows fromthe definition of Λ x and from the locality of H ( s ) with s ∈ [ , t ] , this is discussed for example in the references . Wewill denote this behavior saying that [ Λ x , H ( s )] is x -confined,namely that it exists a positive constant a such that the opera-tor [ Λ x , H ( s )] e a | x | is bounded, see lemma 4.4 of ref. . In thecase of a finite range H ( s ) the x -confinement of [ Λ x , H ( s )] iseasily understood.To prove that the trace in eq. (6) is finite and well defined,namely basis-independent, we need to show that the operatorthat we are tracing out is trace class, see for the definition,to do so we show that it splits up as a sum of trace class oper-ators, denoted I , II , III and IV . I ≡ Q [ Λ x , P ] P [ Λ y , P ] is trace class by hypothesis, this follows from the fact thatthe Fermi projection P of a gapped and short range, or local,Hamiltonian is local, namely it has off diagonal componentsfalling off exponentially fast: |(cid:104) n | P | m (cid:105)| ≤ Ke − µ (cid:107) n − m (cid:107) . II ≡ − i (cid:90) t dsQ [ Λ x , P ] PU † ( s ) [ Λ y , H ( s )] U ( s ) (8)is also trace class in fact [ Λ x , P ] and [ Λ y , H ( s )] are respectively x and y -confined therefore their product is trace class, sincethe product of a trace class operator and a bounded opera-tor is trace class it follows that PU † [ Λ y , H ( s )][ Λ x , P ] is traceclass. Using the cyclic property of the trace it follows that [ Λ x , P ] PU † [ Λ y , H ( s )] is trace class. Using again the fact thatthe trace class operators are an ideal of the bounded operatorswe get that (8) is trace class. III ≡ − i (cid:90) t ds (cid:48) Q [ Λ y , P ] PU † ( s (cid:48) ) (cid:2) Λ x , H ( s (cid:48) ) (cid:3) U ( s (cid:48) ) IV ≡ (cid:90) t dsds (cid:48) QU † ( s ) [ Λ x , H ( s )] U ( s ) PU † ( s (cid:48) ) (cid:2) Λ y , H ( s (cid:48) ) (cid:3) U ( s (cid:48) ) Applying a similar reasoning as for II to III and IV we obtainthat both are trace class.The above trace class discussion together with thehomotopy-equivalence of P and P ( t ) shows that the value ofthe Chern number of P ( t ) is constant in time under the hy-pothesis that the Hamiltonian of the system H ( s ) is spatiallylocal with s ∈ [ , t ] . This also means that the sum of the con-tributions to the trace of the operators II , III , and IV is zero.We note that in the reasoning above we have exchangedthe time integration and the trace, this can be justified simply.We suppose that the time-dependence of the Hamiltonian is atleast strongly-continuous, this ensures the existence of a dy-namics for the system, namely the propagator U ( t , t ) of eq.(3) is well defined, see paragraph X.12 of for a discussion.We can approximate the time-integration by a finite sum plusa small remainder. We can safely exchange the trace and thefinite sum, moreover since we conclude that the contributionof the finite sum is independent from t that means the contri-bution of the remainder is vanishing. III. RECOLLECTING THE DEFINITION OF THE BOTTINDEX AND ITS RELATION WITH THE 2-DIMENSIONALCHERN NUMBER
In the references it has been shown that for a finitemany-body system of non-interacting particles described bya short-ranged, bounded and gapped Hamiltonian living ona lattice on a two-torus the invariant of matrices called Bottindex coincides in the thermodynamic limit with the Chernnumber. In finite systems the correction is of order L − , being L the linear size of the system. This in particular implies thequantization of the Hall conductance on a torus of finite sizewith an error of order L − . This has been shown in the ref. exploiting the definition of the Hall conductance as the longtime transverse current response of the system to an electricfield adiabatically turned. The proof of ref. relies instead ona direct comparison of the invariants as they appear in eq. (1)and eq. (18).Following ref. we consider a representation of the posi-tion operators X and Y such that the positions of all the par-ticles of the system ( x i , y i ) are disposed on the diagonal of X and Y respectively: X i , j = x i δ i , j . Being L the linear size of thesystem and assuming a lattice spacing equal to 1, X and Y arematrices of size of the order of NL × NL . Denoting as insection II the Fermi projection with PP = W (cid:18) n (cid:19) W † (9)with n = dim P and W the unitary matrix of basis change fromenergy to position, so we also have: Q ≡ − P = W (cid:18) m
00 0 (cid:19) W † (10)In a system with periodic boundary conditions we consider theunitary matrices e ( i π XL ) and e ( i π YL ) , then for the projected po-sition operators on a torus Pe ( i π XL ) P and Pe ( i π YL ) P , we have: Pe ( i π XL ) P = W (cid:18) U (cid:19) W † (11) Pe ( i π YL ) P = W (cid:18) U (cid:19) W † (12)where U and U are non singular. With R (cid:28) L the range ofthe Hamiltonian, J a bound for the norm of H and ∆ E thespectral gap of the Hamiltonian it turns out that: (cid:107) [ X , H ] (cid:107) ≤ O ( RJ ) (13) (cid:107) (cid:104) e ( i π XL ) , P (cid:105) (cid:107) ≤ O (cid:18) RJL ∆ E (cid:19) (14)A couple of words on these relations: eq. (13) follows fromthe fact that X is a diagonal matrix then [ X , H ] looks like theoff diagonal part of H with elements multiplied by factors ofmodulus at most equal to R because H connects lattice pointsthat are at most R far apart. A similar bound occurs in thecase of local Hamiltonian on l ( Z ) where R ∝ ν . Eq. (14)follows from (13) writing P as a contour integral of the re-solvent R ( z ) ≡ ( z − H ) − , see and for more details. Thisimplies that the matrices U and U almost commute and arequasi unitarity: (cid:107) [ U , U ] (cid:107) ≤ O (cid:18) RJL ∆ E (cid:19) (15) (cid:107) U a U † a − n (cid:107) ≤ O (cid:18) RJL ∆ E (cid:19) , a ∈ { , } (16)For a discussion of these results see . The Bott index of U and U is defined as :Bott ( U , U ) ≡ π ImTr log (cid:16) U U U †1 U †2 (cid:17) (17)The branch cut of the log is assumed on the real negativeaxis then the definition is well posed when U U U †1 U †2 hasno real negative eigenvalue. Bott ( U , U ) = U and U are arbitrarily close to a couple of commut-ing quasi unitaries . This has been shown to be in relationwith the existence of exponentially localized Wannier func-tions. More precisely the existence of exponentially localizedWannier functions implies the vanishing of the Bott index,while the vanishing of the Bott implies a spread of the Wan-nier functions, quantified by their variance, that is small com-pared with the linear size of the system, . An equivalentdefinition of the Bott index is given employing the matrices V ≡ Q + Pe ( i π XL ) P and V ≡ Q + Pe ( i π YL ) P , then:Bott ( U , U ) = π ImTr log (cid:16) V V V †1 V †2 (cid:17) (18)The proof of the equivalence is immediate using the represen-tation of the projections P and Q , eqs. (9), (10).With the use of conditions (15) and (16) it is possible toshow that:Bott ( U , U ) = π ImTr (cid:16) Pe i θ x Pe i θ y Pe − i θ x Pe − i θ y P (cid:17) + O ( L − ) (19) with θ x ≡ π XL and θ y ≡ π YL . This expression is particularlywell suited for numerical investigations.Let us consider how the Bott index varies starting with thesimpler case of unitary matrices namely when U and U ineq. (17) are replaced by unitaries. In this case the Bott indexis well defined when (cid:107) [ U , U ] (cid:107) <
2, because with U and U unitary U U U †1 U †2 is unitary as well, that means its spectrumlies on the unit circle of the complex plane. It is easy to seethat (cid:107) [ U , U ] (cid:107) = (cid:107) U U U †1 U †2 − (cid:107) then recalling that the op-erator norm of a matrix A is (cid:107) A (cid:107) ≡ max {| λ | , λ ∈ σ ( A ) } wesee in the left panel of Fig. 1 that (cid:107) U U U †1 U †2 − (cid:107) = − U U U †1 U †2 . Then whenan eigenvalue crosses the real negative axis, that correspondsto cross the branch cut of the log, its phases changes of 2 π then the Bott index (17) changes. In the ref. it is discussedhow a deformation (a homotopy) of a couple of unitary matri-ces that preserves their unitarity can lead to a change of theirBott index only if at an intermediate point (cid:107) [ U , U ] (cid:107) = t → ( U t , V t ) with ( U t , V t ) unitarymatrices ∀ t ∈ [ , ] then Bott ( U , V ) (cid:54) = Bott ( U , V ) only ifit exists ¯ t ∈ [ , ] such that (cid:107) [ U ¯ t , V ¯ t ] (cid:107) =
2. See and refer-ences therein for more theorems about it. In the case of thedefinition eq. (17) where U and U are not unitary we mustconsider where the spectrum of U U U †1 U †2 is located. It ispossible to see, appendix (A) for the details, that in for a time-independent system the spectrum is located close by the point ( , ) of the complex plane, this follows directly from equa-tions (15) and (16). The time evolution generated by H ( t ) such that the operator U ( t , t ) commutes with P ( t ) leaves thespectrum in the same region, when instead U ( t , t ) does notcommute with P ( t ) , as explained in the next section, thetime evolution causes the eigenvalues to move within the discof unit radius, see the right panel of Fig. 1. . .. . Re z Im z Re z Im z FIG. 1. (Color on line) Left panel: with U and U unitary the spec-trum of U U U †1 U †2 − ( U U U †1 U †2 ) =
1. The only point at a distance equalto 2 from the origin is on the real negative axis. The thick red lineindicates the branch cut of the log. Right panel: the spectrum of V V V †1 V †2 as given in eq. (18) is confined in a small region close to ( , ) . The unitary time-evolution of the system is such that P and Q are replaced by U ( t , t ) P ( t ) U † ( t , t ) and U ( t , t ) Q ( t ) U † ( t , t ) asa consequence the eigenvalues of V ( t ) V ( t ) V †1 ( t ) V †2 ( t ) can move intime but nevertheless they stay confined within the unit disc in anarea close by ( , ) schematically drawn as an ellipse. The thick redline indicates the branch cut of the log. IV. INVARIANCE OF THE BOTT INDEX OF THETIME-EVOLVED FERMI PROJECTION
The Chern number is invariant under unitary evolution ofa generic local and time dependent Hamiltonian. We ask thesame question about the Bott index. A way to realize the timedriving of an initial Hamiltonian H i towards a final Hamilto-nian H f ( t ) is the ramp of a perturbation V ( t ) in general time-dependent. H f ( t ) = H i + r ( t ) V ( t ) , r ( t ) = , t < t v ( t ) , t ≤ t ≤ t , t > t (20)with v ( t ) a monotonic regular function interpolating betweenzero and one. When the slope of v ( t ) increases significantlythe driving becomes a so called quantum quench. In this casethe operator of unitary evolution from the initial Hamiltonian H i to the final Hamiltonian H f = H i + V is given in the caseof V time-independent by U ( t , t ) = U ( t , t ) U ( t , t ) = e − i ( t − t ) H f U ( t , t ) (21)With t − t (cid:28) (cid:0) ∆ ψ ( t ) V (cid:1) − (22)the operator U ( t , t ) (cid:39) . In eq. (22) ∆ ψ ( t ) V ≡ (cid:113) (cid:104) ψ ( t ) | V | ψ ( t ) (cid:105) − ( (cid:104) ψ ( t ) | V | ψ ( t ) (cid:105) ) (23)is the variance of the perturbing potential over the initial state | ψ ( t ) (cid:105) . This is discussed for example in .Let us study the time evolution of the Bott index, thiscan be done in the Schroedinger picture replacing P ( t ) with U ( t , t ) P ( t ) U † ( t , t ) and Q ( t ) with U ( t , t ) Q ( t ) U † ( t , t ) .The invariance for a time-independent system is manifest, infact P ( t ) and U ( t , t ) = e − i ( t − t ) H commute. I stress that thisis different from considering the instantaneous Bott index thatwe would get replacing P with P ( t ) = ∑ i | ψ i ( t ) (cid:105)(cid:104) ψ i ( t ) | be-ing | ψ i ( t ) (cid:105) the instantaneous eigenvector of the Hamiltonian H ( t ) | ψ i ( t ) (cid:105) = E i ( t ) | ψ i ( t ) (cid:105) , E i ( t ) ≤ µ . Is the invariance of theBott index of the time-evolved Fermi projection also grantedfor a general time-dependent system with Hamiltonian (20)?A variation of the Bott index has been numerically shown inthe ref. , in the rest of this section I show that this can happenonly as a finite size effect, the Bott index does not change dueto the unitary evolution generated by a local Hamiltonian inthe thermodynamic limit.The analysis of the unitary case showed that the variation ofthe Bott index is due to the growth of (cid:107) [ U , U ] (cid:107) . In our phys-ical context the matrices U and U are not unitary and theincrease of their commutator with time is due to the growth of (cid:107) [ e i θ x , U ( t , t ) H ( t ) U † ( t , t )] (cid:107) = (cid:107) [ e i θ x , H ( t ) , H ] (cid:107) with θ x , H ( t ) ≡ U † ( t , t ) θ x U ( t , t ) the Heisenberg picture of θ x . This deter-mines in principle the growth of (cid:107) [ e i θ x , U ( t , t ) PU † ( t , t )] (cid:107) = (cid:107) [ e i θ x , H ( t ) , P ] (cid:107) in time. Let us examine this explicitly.A time-independent, short-ranged, bounded and gappedHamiltonian implies in a system large compared to the range R the relations (13), (14), (15) and (16) above. In the gen-eral setting of a time-dependent Hamiltonian, that might beassociated to the ramp of a time-dependent perturbation, theunitary operator of time evolution, also called the propaga-tor, is: U ( t , t ) = T exp (cid:16) − i (cid:82) tt dsH ( s ) (cid:17) , T denotes the op-erator of time ordering. U ( t , t ) does not commute in gen-eral with the Hamiltonian H ( t ) . A possible way to ex-amine (cid:107) [ U † ( t , t ) XU ( t , t ) , H ] (cid:107) is to use the Lieb-Robinsonbounds but these are better suited for operators that havesupports that do not overlap, therefore I employ a differentstrategy.The equation of motion for X H ( t ) ≡ U † ( t , t ) XU ( t , t ) is: i ddt X H ( t ) = [ X , H ( t )] H . The explicit time dependence of theHamiltonian has been put in evidence. With X ( t ) = X wehave: i ( X H ( t ) − X ) = (cid:90) tt ds [ X , H ( s )] H Being (cid:107) [ X , U ( t , t )] (cid:107) = (cid:107) U † ( t , t ) XU ( t , t ) − X (cid:107) it followsthat: (cid:107) [ X , U ( t , t )] (cid:107) ≤ | t − t | sup s ∈ [ t , t ] (cid:107) [ X , H ( s )] (cid:107) using eq. (13) and denoting R ( t ) the range of H ( t ) and J ( t ) itsnorm, it follows: (cid:107) [ X , U ( t , t )] (cid:107) ≤ | t − t | sup s ∈ [ t , t ] R ( s ) J ( s ) (24)Our interest is in the growth with time of (cid:107) [ U † ( t , t ) XU ( t , t ) , H ( t )] (cid:107) . It follows from eq. (13)and eq. (24), we drop the time-indexes of U ( t , t ) and set J ≡ J ( t ) , that: (cid:107) [ U † XU , H ( t )] (cid:107) = (cid:107) [ U † [ X , U ] + X , H ( t )] (cid:107) = (cid:107) [ U † [ X , U ] , H ( t )] + [ X , H ( t )] (cid:107)≤ (cid:107) [ U † [ X , U ] , H ( t )] (cid:107) + RJ ≤ J (cid:107) [ X , U ] (cid:107) + RJ ≤ J | t − t | sup s ∈ [ t , t ] ( R ( s ) J ( s )) + RJ (25)In analogy to the static case, where eq. (13) implied eq. (14),in the time-dependent case we have, being P ≡ P ( t ) the Fermiprojection: (cid:107) [ U † XU , P ] (cid:107) L ≤ | t − t | sup s ∈ [ t , t ] ( R ( s ) J ( s )) L + RJL ∆ E (26)A necessary condition for the change of the Bott index is: (cid:107) [ U † XU , P ] (cid:107) L (cid:39) t − t that would give rise to a change of the index such that:¯ t − t (cid:39) L − RJ ∆ E s ∈ [ t , ¯ t ] ( R ( s ) J ( s )) (27)For a finite range Hamiltonian in the thermodynamic limit LR → ∞ therefore if the instantaneous Hamiltonian H ( t ) hasalways a finite range R ( t ) the time scale for the variation ofthe index diverges. In a finite size setting the interplay of theratios LR (cid:29) J ∆ E (cid:29) . A. Periodically driven systems
Let us consider the case in eq. (20) of the ramp of a timeperiodic perturbation V ( t ) = V ( t + T ) . This implies that theHamiltonian H ( t ) when t ≥ t is time-periodic. In this casethe estimate given by eq. (24) can be made sharper. In gen-eral the propagator U of a time-periodic Hamiltonian is notperiodic, but when U has a spectral gap then there is a ho-motopy that maps U to a time-periodic propagator preservingthe given gap as described in the reference . The Hamil-tonian that generates this periodic propagator is called therelative Hamiltonian, its construction is described for exam-ple in . The propagator U and the homotopically equiv-alent time-periodic propagator share the same topological in-dex W ε of , see for a different naming, that characterizeseach spectral gap of the propagator placed at e − iT ε and there-fore the Chern number of the spectral projection in betweenthe various gaps. In fact denoting P ε , ε (cid:48) the spectral projectionfor the spectrum of U in between the gaps e − iT ε and e − iT ε (cid:48) itholds: W ε − W ε (cid:48) = Chern ( P ε (cid:48) , ε ) . This is eq. 14 of or eq.3.22 of .The propagator U of a time periodic Hamiltonian of pe-riod T is such that ∀ n ∈ Z : U ( t + nT , t + nT ) = U ( t , t ) .This property together with the supposed periodicity U ( t + T , t ) = U ( t , t ) = imply that U ( t + nT , t ) = . There-fore given t and t , since it exists a positive integer N suchthat t − ( t + NT ) < T , we have that: U ( t , t ) = U ( t , t + NT ) U ( t + NT , t ) = U ( t , t + NT ) . We suppose that accord-ing to eq. (20) the time needed to fully turn on the time-periodic perturbation V ( t ) = V ( t + T ) is equal to t − t thenthe propagator of the Hamiltonian of eq. (20) for a suitable N ,denoting U per the periodic propagator, obeys the decomposi-tion: U ( t , t ) = U per ( t , t ) U ramp ( t , t )= U per ( t , t + NT ) U ramp ( t , t )= U per ( t − NT , t ) U ramp ( t , t ) In this way we conclude that the time interval | t − t | of eq.(27) is less than t + T − t . Therefore for a periodic drivingthe change of the Bott index is not only forbidden at any givenfixed | t − t | in the thermodynamic limit but also disfavoredwith respect to the general case for a finite size setting.I stress that the relative Hamiltonian of a space-local Hamil-tonian under the hypothesis of existence of a spectral gap forthe propagator is also space-local. This has been show in proposition 5.6 of reference , see also the specific notion oflocality employed in that reference.This discussion ignores possible heating effects that mighttakes place in the context of periodic driving, nevertheless re-cent works indicate the stability of such a phases over almostexponentially long times . V. CONCLUDING DISCUSSION
The Bott index introduced in the physics’ realm in thereferences has been investigated in a general (20) time-dependent setting and the constancy of the index of the time-evolved Fermi projection has been established in the thermo-dynamic limit over. The time scale of a possible change of theindex is identified in eq. (27), this is a mere finite size effect,in particular it looks disfavored in the time-periodic case.A fundamental issue unexplored in this work is the meaningas a physical quantity of the Bott index for a general time-dependent Hamiltonian. In fact if for a static system it isequivalent to the Chern number so it measures the Hall con-ductance, what about instead a general time-dependent sys-tem? We should recall that the Hall conductance is not themean value of an operator over a state but a transport coef-ficient computed with the aid of the Kubo formula. The Hallconductance has been found to be not quantized after a quenchaccording to the references . The physical meaning of thequantized Bott index for a general time-dependent setting re-mains to be investigated.I conclude with some final comments about the literature.In ref. the Bott index has been claimed to be the right suitedinvariant to study finite systems that are disordered and pe-riodically driven being the counterpart of the winding num-ber invariant W introduce in the ref. for the study of cleanand thermodynamically large periodically driven systems. Wcounts the number of edge states supported by a periodicallydriven two dimensional system. Let me comment briefly onthe relation among the references and : one of the most in-teresting results of is to show that after ramping up a cir-cularly polarized electric field on a graphene sheet with astaggering sublattice potential the initial ground state evolveskeeping a vanishing Chern number despite the fact that theground state of the final periodic Hamiltonian has a non triv-ial Chern number. This was shown in section II using theinvariance of the Chern number of homotopically equivalentprojections and discussing the trace class properties of the op-erator on the RHS of eq. (6). The ref. on the other hand con-siders a system that is already in a periodically driven regimedisregarding the effects of the ignition of the drive. The is-sues related to the preparation of a periodically driven systemsare also discussed e.g. in the reference . Finally it needs tobe mentioned that all the effects associated with phonons andtheir coupling with electrons have been neglected, for a studythat takes these phenomena into account in the context of aquench of topological phases see . VI. ACKNOWLEDGEMENTS
It is a pleasure to thank Yang Ge and Marcos Rigol forexchange of correspondence, Hermann Schulz-Baldes, YosiAvron, Jacob Shapiro and Florian Dorsch for discussions.
Appendix A
We want to estimate the eigenvalues with maximum modu-lus (that is the norm) and the minimum modulus of the matrix Q + Pe i θ x , H Pe i θ y , H Pe − i θ x , H Pe − i θ y , H P that is argument of the logthat defines the Bott index (18). The subscript H indicatesthe Heisenberg picture. For simplicity we start considering Q + Pe i θ x , H P . This matrix is not normal, but it admits a sin-gular value decomposition: Q + Pe i θ x , H P = T † DS , T and S are unitary, D is the diagonal matrix of eigenvalues. Thenthe modulus square of the eigenvalues of Q + Pe i θ x , H P are theeigenvalues of G ≡ Q + Pe − i θ x , H Pe i θ x , H P . In fact: G = ( Q + Pe i θ x , H P ) † ( Q + Pe i θ x , H P ) (A1) = S † D † T T † DS = S † D † DS (A2)In the energy basis: P = (cid:18) n (cid:19) , Q = (cid:18) m
00 0 (cid:19) (A3)Then it is immediate to see that: G = (cid:18) m W †4 W (cid:19) (A4)With W the lower diagonal block of the unitary matrix W † e i θ x , H W W † e i θ x , H W = (cid:18) W W W W (cid:19) (A5)From equation (A4) we get: (cid:107) G (cid:107) = max { , (cid:107) W †4 W (cid:107)} = W satisfies, for example, W †2 W + W †4 W = , thenbeing W †2 W semipositive definite we have (cid:107) W (cid:107) ≤ W can be vanishing only when W commutes with e − i θ x , H , and more-over when U ( t , t ) commutes with H ( t ) that is not the casewe are interested in here. Then it follows that: (cid:107) Q + Pe i θ x , H P (cid:107) = (cid:107) Q + Pe i θ x , H Pe i θ y , H Pe − i θ x , H Pe − i θ y , H P (cid:107) = Q + Pe i θ x , H Pe i θ y , H Pe − i θ x , H Pe − i θ y , H P . We again start considering thematrix Q + Pe i θ x , H P for simplicity. Its eigenvalue of minimummodulus is the square root of the smallest eigenvalue of G thatI indicate with λ S . It is easy to see using the positivity of G and (cid:107) G (cid:107) = − λ S = (cid:107) − G (cid:107) = (cid:107) − Q − Pe − i θ x , H Pe i θ x , H P (cid:107) (A9) = (cid:107) P ( − e − i θ x , H Pe i θ x , H ) P (cid:107) (A10) = (cid:107) Pe − i θ x , H Qe i θ x , H P (cid:107) (A11) = (cid:107) [ P , e − i θ x , H ] Q [ e i θ x , H , P ] (cid:107) (A12)The eq. (A12) is a clear indication that when [ e i θ x , H , P ] is small then λ S is close to 1. This follows directly fromthe equation (14) that in turn follows from the hypothe-sis on the Hamiltonian’s properties: short ranged, bounded,gapped. We have seen that a result of the time-evolution isto make the norm (cid:107) [ e i θ x , H , P ] (cid:107) growing, this determines thedecrease of λ S . We guess that when (cid:107) [ e i θ x , H , P ] (cid:107) (cid:39) λ S is small. We have seen that this is forbidden in the ther-modynamic limit L → ∞ . 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