Lieb-Robinson Bounds on Entanglement Gaps from Symmetry-Protected Topology
LLieb-Robinson Bounds on Entanglement Gaps from Symmetry-Protected Topology
Zongping Gong, Naoto Kura, Masatoshi Sato, and Masahito Ueda
1, 3, 4 Department of Physics, University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606-8502, Japan Institute for Physics of Intelligence, University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan RIKEN Center for Emergent Matter Science (CEMS), Wako, Saitama 351-0198, Japan (Dated: July 23, 2019)A quantum quench is the simplest protocol to investigate nonequilibrium many-body quantum dynamics.Previous studies on the entanglement properties of quenched quantum many-body systems mainly focus on thegrowth of entanglement entropy. Several rigorous results and phenomenological guiding principles have beenestablished, such as the no-faster-than-linear entanglement growth generated by generic local Hamiltonians andthe peculiar logarithmic growth for many-body localized systems. However, little is known about the dynam-ical behavior of the full entanglement spectrum, which is a refined character closely related to the topologicalnature of the wave function. Here, we establish a rigorous and universal result for the entanglement spec-tra of one-dimensional symmetry-protected topological (SPT) systems evolving out of equilibrium. Our resultis derived both for free-fermion SPT systems and interacting ones. For free-fermion systems with Altland-Zirnbauer symmetries, we prove that the single-particle entanglement gap after quenches obeys essentially thesame Lieb-Robinson bound as that on the equal-time correlation, provided that there is no dynamical symmetrybreaking. As a notable byproduct, we obtain a new type of Lieb-Robinson velocity which is related to the banddispersion with a complex wave number and reaches the minimum as the maximal (relative) group velocity.Within the framework of tensor networks, i.e., for SPT matrix-product states evolved by symmetric and trivialmatrix-product unitaries, we also identify a Lieb-Robinson bound on the many-body entanglement gap for gen-eral quenched interacting SPT systems. This result suggests high potential of tensor-network approaches forexploring rigorous results on long-time quantum dynamics. Influence of partial symmetry breaking, effects ofdisorder, and the relaxation property in the long-time limit are also discussed. Our work establishes a paradigmfor exploring rigorous results of SPT systems out of equilibrium.
I. INTRODUCTION
Recent years have witnessed remarkable experimental de-velopments in atomic, molecular and optical physics, whichhave enabled us to engineer and control artificial quantummany-body systems at the level of individual atoms, ions andphotons [1–3]. Particular attention is focused on nonequilib-rium quantum dynamics [4–7], of which the arguably simplestsituation is quantum quenches [8–10] — the system is initial-ized as a wave function | Ψ (cid:105) which then evolves unitarily bya Hamiltonian H , with respect to which | Ψ (cid:105) is typically ahighly excited superposition state. The wave function at time t is then formally given by | Ψ t (cid:105) = e − iHt | Ψ (cid:105) . To model re-alistic quantum simulators, especially ultracold atoms and su-perconducting circuits with short-range interactions, we usu-ally assume H to be local , in the sense that it can be written asa sum of short-range operators. In light of the rapid develop-ment of topological material science [11–14], there is growinginterest in topological aspects of quench dynamics [15–36].A fundamental question in this context is: given | Ψ (cid:105) as theground state of a gapped Hamiltonian H , which may be triv-ial or topological, whether the topology of | Ψ t (cid:105) will changeduring time evolution, and, if yes, in what way. To make thetopology well-defined, we may have to impose certain sym-metries. For the sake of concreteness, we assume that H and H share the same symmetries, if any. The answer to the abovequestion has recently been given and is somewhat negative:for unitary symmetries or/and anti-unitary anti-symmetries[37], | Ψ t (cid:105) stays in the same symmetry-protected topologi-cal (SPT) phase [16, 17, 33, 34]. For anti-unitary symme- tries or/and unitary anti-symmetries, the topological numberof | Ψ t (cid:105) generally becomes ill-defined (or reduces) due to dy-namical symmetry breaking [33, 34]. To understand this, weonly have to note that | Ψ t (cid:105) is the ground state of [32, 33] H ( t ) ≡ e − iHt H e iHt , (1)which shares the same spectrum as H . The conservationof topological number follows from the fact that H ( t ) isgapped and continuously deformed from H in a symmetry-preserving manner.Since topological numbers are rather abstract quantitiesand take very different forms depending on the specific sys-tems, we need a universal topological indicator to formalizethe above qualitative analysis into a general, rigorous and, inprinciple, experimentally verifiable statement. Entanglementturns out to be an ideal candidate to demonstrate the persis-tence of topology. We can trace the time evolution of the en-tanglement spectrum (ES) for a proper bipartition, which con-tains crucial information of the entanglement pattern and isarguably the most widely used universal topological indicatorthat is applicable to both noninteracting [38–41] and interact-ing systems [42–47]. The ES is expected to be accessible innear-future ultracold-atom and trapped-ion experiments [48–50]. The persistence of SPT order thus manifests itself in thatthe ES stays gapless or degenerate [33, 34].However, a vital point is missing in the above argument oftopology conservation — defining SPT phases requires local-ity in the Hamiltonian [51], while H ( t ) in Eq. (1) may becomehighly nonlocal after a long time. That is to say, the dynam-ically generated non-locality could obscure the SPT order in a r X i v : . [ qu a n t - ph ] J u l ... ...... ... ... ...... ... ... ...... ... Entanglement edge modes ES E ⇠ O (1)
Growth (c)
Degeneracy (d)
Splitting O (1)
Weyl’s perturbation theorem [84],which guarantees the spectral shift between two Hermitianoperators to be rigorously bounded by the operator norm oftheir difference. In addition to the several well-known resultssuch as the entanglement area law [85–89], the bounds on en-tanglement growth [69, 90] and the entanglement detection oftopological phases [38–47, 91, 92], our work brings about yetanother rigorous and fundamental result on the entanglementin quantum many-body systems [93–95]. As shown Fig. 2,our work lays a corner stone for exploring exact results on theentanglement properties of SPT systems out of equilibrium.Also, we hope that the ideas and methods developed in thiswork could stimulate further exploration of rigorous resultson various spectra, which appear ubiquitously in physics.This paper is structured as follows. In Sec. II, we review thebasic properties of the ES and the notion of dynamical sym-metry breaking. In Sec. III, we define the entanglement gapand present the main results, i.e., the explicit forms of Lieb-Robinson bounds. In Sec. IV, we focus on the single-particleentanglement gap in quenched free-fermion systems and de-rive the first main result (Theorem 1). In particular, we derivean almost optimal Lieb-Robinson bound for free-fermion sys-tems and justify the quasi-particle picture. In Sec. V, we fo-cus on the many-body entanglement gap in the tensor-networksetting and derive the second main result (Theorem 2). Wediscuss the impact of partial symmetry breaking, the effectsof disorder and long-time dynamics in Sec. VI. Finally, wesummarize the main results of this paper and provide someoutlook in Sec. VII. We relegate some technical details to ap-pendices to avoid digressing from the main subject.
II. PRELIMINARIES
We begin by clarifying the definitions of single-particle andmany-body ES for free-fermion and general systems. We alsobriefly review why 1D SPT order renders the many-body ESto be degenerate. Finally, we review the notion of dynamicalsymmetry breaking and point out the relevant symmetries thatcan protect SPT order in quench dynamics.
A. Definition of the ES
We consider a 1D lattice with the total number of unit cellsdenoted as L . Each unit cell contains d internal degrees offreedoms, including spins, orbitals, sublattices and so on. Forspin systems, d gives the Hilbert-space dimension of a single - - - ζ ξ spE
12 + 12 s n (1 − ξ n ) (cid:21) , (6)where s n = ± . B. SPT-enforced ES degeneracy
For free-fermion systems, it is known that { − ξ n } n is exactly the energy spectrum subject to the open bound-ary condition after band flattening [39]. To see this, weconsider a gapped quadratic Hamiltonian in the diagonalizedform H = (cid:80) n E n ψ † n ψ n , which may not correspond to atranslation-invariant system. Assuming the Fermi energy tobe zero without loss of generality, we can define the single-particle projector onto the Fermi sea as P < ≡ (cid:88) { n : E n < } | ψ n (cid:105)(cid:104) ψ n | , (7)where | ψ n (cid:105) ≡ ψ † n | vac (cid:105) is a single-particle eigenstate. With P S denoted as the projector onto the single-particle Hilbertspace of S , the single-particle ES { ξ n } n coincides with thespectrum of P S P < P S [98]. The statement made in the begin-ning of this subsection follows from the fact that the involu-tory (i.e., being its own inverse) flattened Hamiltonian is givenby H flat = I sp − P < , (8)where I sp is the identity operator in the single-particle sector.From this exact correspondence, we know that there shouldbe ξ n = modes in the single particle ES for a topologicalinsulator with boundary zero modes. According to Eq. (6), thecorresponding many-body ES is necessarily degenerate.To analyze noncritical interacting systems in 1D, we em-ploy the matrix-product-state (MPS) formalism [99–102].The validity of this formalism is rooted in the entanglementarea law [68]. For simplicity, we focus on translation-invariantMPSs, which take the form of | Ψ (cid:105) = (cid:88) { j s } Ls =1 Tr[ A j A j ...A j L ] | j j ...j L (cid:105) , (9)where j s = 1 , , .., d and A j ’s are D × D matrices with D being the bond dimension. Each MPS is associated with alinear map on C D × D : E ( · ) ≡ d (cid:88) j =1 A j ( · ) A † j . (10)Assuming the MPS to be normal [103], which rules out thepossibility of spontaneous symmetry breaking, we can alwaysperform gauge transformations of A j ’s, i.e., A j → XA j X − that leaves | Ψ (cid:105) in Eq. (9) invariant, such that E is a unitalchannel : E ( v ) = d (cid:88) j =1 A j A † j = v (11) TABLE I. Dynamical stability of unitary symmetries ( a = b = + ),anti-unitary symmetries ( a = − b = + ), unitary anti-symmetries( a = − b = − ), and anti-unitary anti-symmetries ( a = b = − ),where a and b are given in Eq. (14). a b Dynamical stability Example + + √ Parity symmetry + − × Time-reversal symmetry − + × Chiral symmetry − − √
Particle-hole symmetry where v is the identity in C D × D acting on the virtual Hilbertspace. If we further impose an on-site unitary symmetry, i.e., ρ ⊗ Lg | Ψ (cid:105) = | Ψ (cid:105) for ∀ g ∈ G with G being a group and ρ g ∈ U( d ) being a unitary representation of G , we can find a pro-jective representation V g with V g V h = ω g,h V gh , ω g,h ∈ U(1) for ∀ g, h ∈ G such that [104] d (cid:88) j (cid:48) =1 [ ρ g ] jj (cid:48) A j (cid:48) = V † g A j V g , ∀ j = 1 , , ..., d. (12)In the thermodynamic limit of subsystem S with length l ,i.e., lim l →∞ lim L →∞ , the many-body ES is exactly given by { λ α λ β } Dα,β =1 , where { λ α } Dα =1 is the spectrum of Λ ( > ),that is uniquely determined from [100] E † (Λ) ≡ d (cid:88) j =1 A † j Λ A j = Λ , Tr Λ = 1 . (13)If | Ψ (cid:105) is in an SPT phase, ω g,h must correspond to a nontrivialelement in the second cohomology group H ( G, U(1)) [70–72], leading to degeneracy in λ α ’s [43]. This is because anondegenerate λ α implies that ω g,h must be trivial, as will beproved in Sec. V B. C. Dynamical symmetry breaking
While there is a huge variety of symmetries, we can classifythem into four groups depending on whether the symmetryoperator S commutes or anti-commutes with the Hamiltonian,and whether it is unitary or anti-unitary: SHS − = aH, SiS − = bi, (14)where a, b = ± . Concretely, S is said to be symmetric (ananti-symmetric) if a = + ( a = − ), and S is said to be uni-tary (anti-unitary) if b = + ( b = − ). Note that H in Eq. (14)can be either on the single-particle level (for free-fermion sys-tems) or on the many-body level (for interacting systems).Now let us impose Eq. (14) to both H and H , i.e., theHamiltonians before and after a quench. Regarding the sym-metry action on the parent Hamiltonian (1), we have SH ( t ) S − = Se − iHt H e iHt S − = e − iabHt aH e iabHt = aH ( abt ) . (15) - π π ϵ k ( a . u . ) - π π k P < ( k )
We are now in a position to present the exact statement ofour main results — Lieb-Robinson bounds on entanglementgaps.For free-fermion systems, we examine all the Altland-Zirnbauer classes [105]. In 1D, it is known that there are fivenontrivial classes, including one complex class AIII and fourreal classes BDI, D, DIII and CII [106, 107]. According tothe previous analysis on dynamical symmetry breaking (espe-cially Table I), we know that both classes BDI and DIII reduceto class D, class CII reduces to class C and class AIII reducesto class A. Since classes C and A are both trivial, it suffices toconsider class D , which has an involutory particle-hole sym-metry and is characterized by a Z topological number in 1D[108]. This is the only Altland-Zirnbauer class in 1D that doesnot suffer from dynamical symmetry breaking so that the SPTorder persists in the thermodynamic limit [33].Since the energy spectrum of a free-fermion system in classD is paired as ( (cid:15), − (cid:15) ) , the ES should be divided as ( ξ, − ξ ) .Hence, we can define the single-particle entanglement gap as ∆ spE ≡ n (cid:12)(cid:12)(cid:12)(cid:12) ξ n − (cid:12)(cid:12)(cid:12)(cid:12) . (16)In the presence of translation invariance, we can simply dealwith the Bloch Hamiltonians and prove the following theo-rem. Theorem 1 (Free fermions)
Consider two 1D translation-invariant lattice systems in class D, whose Bloch Hamiltoni-ans are given by H ( k ) and H ( k ) and the former is gapped e iHt (1) uv (2)(3) U vu = (4) UU = u † u (5)Spec ...... p ⇤ p ⇤ ¯ AA l (6)Spec ...... (7)... ... t A U (8)... ... O j UU (9)1 e iHt (1) uv (2)(3) U vu = (4) UU = u † u (5)Spec ...... p ⇤ p ⇤ ¯ AA l (6)Spec ...... (7)... ... t A U (8)... ... O j UU (9)1 ...... U † O j U k + 1 (10)2 (a) (b) = FIG. 5. (a) Setting of Theorem 2 — stroboscopic time evolution gov-erned by an MPU (22) generated by U jj (cid:48) starting from an MPS (9)generated by A j . (b) After evolution by an MPU, an on-site oper-ator O j stays local but acts nontrivially on (at most) k + 1 sites.Here k is the smallest integer k such that the blocked tensor U k inEq. (23) is simple. and topologically nontrivial. We start from the ground state of H ( k ) and make the following two assumptions: (i) the initialBloch projector P < ( k ) ≡ (cid:73) γ < dz πi z − H ( k ) (17) is analytic on { k : | Im k | ≤ κ, | Re k | ≤ π } , where κ > and γ < is a loop that encircles all the particle bands; (ii) H ( k + iκ ) is well-defined and diagonalizable for ∀ k ∈ [ − π, π ] , i.e., H ( k + iκ ) = d (cid:88) α =1 (cid:15) k + iκ,α | u R k + iκ,α (cid:105)(cid:104) u L k + iκ,α | , (18) where | u R k + iκ,α (cid:105) ( (cid:104) u L k + iκ,α | ) is the right (left) Bloch eigenstateof the α -th band. Then the single-particle entanglement gap(16) of a length- l segment is upper bounded by ∆ spE ≤ Ce − κ ( l − vt ) (19) during the time evolution governed by H ( k ) . Here C = (cid:90) π − π dk π (cid:32) d (cid:88) α =1 (cid:107)| u R k + iκ,α (cid:105)(cid:107)(cid:107)| u L k + iκ,α (cid:105)(cid:107) (cid:33) (cid:107) P < ( k + iκ ) (cid:107) (20) with (cid:107) · (cid:107) being the operator norm and v = κ − max k ∈ [ − π,π ] ,α,β Im( (cid:15) k + iκ,α − (cid:15) k + iκ,β ) (21) depends on neither l nor t . As illustrated in Fig. 4, this theorem depends crucially onthe band picture of translation-invariant free-fermion systems.We can show that a positive κ satisfying (i) and (ii) exists un-der quite general assumptions (see Appendix A).For interacting SPT systems, we restrict ourselves to thetensor networks formalisms. That is, as illustrated in Fig. 5(a),we start from an MPS and consider the stroboscopic dynamicsgoverned by a matrix-product unitary (MPU) [109–112] U = (cid:88) { j s ,j (cid:48) s } Ls =1 Tr[ U j j (cid:48) U j j (cid:48) ... U j L j (cid:48) L ] | j j ...j L (cid:105)(cid:104) j (cid:48) j (cid:48) ...j (cid:48) L | , (22)which is a special class of matrix-product operators [113] sat-isfying U † U = ⊗ L . We assume that the MPU respects thesame symmetries of the initial MPS, i.e., [ ρ ⊗ Lg , U ] = 0 for ∀ g ∈ G , and that it belongs to the trivial cohomology class sothat the time-evolved MPS stays in the same SPT phase [111].It is known that by putting together k sites into one such thatthe building block U becomes U k : U k ≡ ... U U U k , (23)the MPU can for sufficiently large k be represented as a bi-layer unitary circuit with each unitary operator acting on twoadjacent blocked sites [114]: U k = U k = U k = U k = U k = u uv v . (24)Whenever such a representation is possible, we call thebuilding-block tensor simple [110]. In fact, given U and k such that U k is simple, U k (cid:48) is also simple for ∀ k (cid:48) ≥ k . Thesmallest k that makes U k simple, which we denote as k , has aclear physical interpretation as the Lieb-Robinson length — asshown in Fig. 5(b), any on-site operator evolved by the MPUgenerated by U acts nontrivially on at most k + 1 sites. Fur-ther details on MPUs can be found in Appendix E.Unlike free-fermion systems in class D, which are charac-terized by a Z number so that there is at most one pair ofstable topological entanglement modes near ξ = (leadingto = 4 -fold degeneracy in the many-body ES), the many-body ES of an SPT MPS can be r -fold degenerate for ∀ r =2 , , , ... in the thermodynamic limit. Here the square r arises from the fact that a subsystem has two edges. Minimalsymmetries that realize these SPT MPSs are Z r × Z r , whosesecond-order cohomology groups read H ( Z r × Z r , U(1)) = Z r . Given r and a many-body ES { ζ n } n with ζ n ≥ ζ n +1 (notethat larger ζ corresponds to lower eigenvalue of the entangle-ment Hamiltonian), we define the many-body entanglementgap to be ∆ mbE ≡ | ζ − ζ r | . (25)Our main theorem is the following. Theorem 2 (Interacting systems)
Starting from an infiniteSPT MPS with bond dimension D , the many-body entangle-ment gap (25) of a length- l subsystem after t steps of timeevolution by a trivial symmetric MPU generated by U withbond dimension D U is bounded from above by ∆ mbE ≤ C ( l − k t ) D − e − κ ( l − vt ) (26) for any l − k t ≥ µ − µ . Here k is the smallest integer block-ing number that makes U k simple and µ is the spectrum ra-dius of E − E ∞ , where E is defined in Eq. (10) for the initial MPS and E ∞ ≡ lim n →∞ E n , κ = − ln µ , v = 2 k − ln D U ln µ ,and the coefficient C = 4 e D ( D + 1) µ − D (1 + µ ) D + (1 − µ ) D − (27) depends only on the initial MPS. Let us explain why we focus on the stroboscopic dynam-ics generated by an MPU rather than a continuous dynamicsgenerated by a local Hamiltonian. First, we expect this settingto be good enough because we can efficiently approximate afinite-time evolution generated by a local Hamiltonian as abilayer unitary circuit [115], which is equivalent to an MPUor a quantum cellular automaton [110]. The efficiency is en-sured by the conventional Lieb-Robinson bound. By showingthat the spectral shift is rigorously bounded by the approx-imation error, we expect a similar Lieb-Robinson bound onthe many-body entanglement gap for continuous quench dy-namics (see Appendix F). Second, this formalism is of intrin-sic interest for its own sake — it gives exact descriptions forsome Floquet systems [116–118] and quantum circuits [119–127], which have intensively been studied in the context ofnonequilibrium phases of matter and information scrambling.Moreover, this theorem exemplifies the power of tensor net-works as analytical methods for predicting long-time dynam-ical behaviors of interacting quantum many-body systems farfrom equilibrium, which are hardly accessible in numerics.
IV. FREE FERMIONS
As mentioned above, for quench dynamics within the sameAltland-Zirnbauer class [105–107], the only nontrivial classin 1D that does not suffer from (partial) “dynamical symmetrybreaking” is class D [33]. In fact, previous numerical studieson the Su-Schrieffer-Heeger (SSH) model [81] and the Kitaevchain [82] have revealed that the ξ = modes split after acharacteristic time scale t ∗ ∼ l v max , where l is the length ofthe subsystem and v max is the maximal group velocity of banddispersions. In Fig. 6(b), we reproduce the splitting dynamicsin the SSH model (Fig. 6(a)), which shows that not only thetopological entanglement mode but also the full ES splits. Inthe following, we rigorously establish the underlying Lieb-Robinson bound on the ES splitting stated in Theorem 1.
A. Zero entanglement gap for symmetric bipartition
We first point out a crucial proposition — for half-chainentanglement cut, i.e., l = L , the topological entanglementmodes will be pinned exactly at at any time (see Fig. 6(b)).Note that we can always choose the anti-unitary and involu-tory particle-hole-symmetry operator C to be the complex con-jugation K [128]. Under this basis, we have H = iR with R being a skew-symmetric real matrix. Suppose that H is flat-tened so that R = − I sp . If Pf R = − , where Pf denotesthe Pfaffian, there will be a pair of topological entanglementmodes exponentially close to ξ = . With the translation t ξ - - - - - - t EGCorrLRB J J (a)(b) (c)
30 35 400.4950.5000.505
FIG. 6. (a) Quench protocol in the SSH model. (b) Single-particleES dynamics after the quench. The parameters are quenched as ( J , J ) = (0 . , → (1 , . and l = 40 . The red solid and bluedashed curves correspond to L = ∞ and L = 2 l , respectively. Inthe former case, the single-particle ES splits after t ∗ ∼ (indicatedby the red dashed line and determined by threshold ∆ spE = 10 − as shown in (c)). Inset: Zoom in on the splitting of the topolog-ical entanglement modes. (c) Dynamics of the single-particle en-tanglement gap for L = ∞ (solid curve). The equal-time correla-tion (cid:107)(cid:104) j | P ∞ < ( t ) | j + l (cid:105)(cid:107) (purple dotted curve) and the Lieb-Robinsonbound (blue dashed curve) given by Theorem 1 with κ = 0 . aresuperimposed for the sake of comparison. invariance assumed, the system at least exhibits a half-chaintranslation symmetry , leading to R = (cid:34) R d R o R o R d (cid:35) = σ ⊗ R d + σ x ⊗ R o , (28)where R d and R o are both real and skew-symmetric and σ =[ ] , σ x = [ ] . Moreover, the half-chain ES is given by thespectrum of (1 − iR d ) . From R = − I sp we obtain R + R = − I half , { R d , R o } = half , (29)where I half and half are the half-chain identity and zero oper-ators within the single-particle sector. Provided that R < ,we can find an anti-unitary operator A ≡ i ( − R ) − R o K , (30)such that A = − I half , [ A , iR d ] = 0 . (31)Due to the interplay between the Kramers degeneracy en-forced by A and the nontrivial Z topology, two quasi-zeromodes of iR d must be pinned exactly at zero. Since the Z index stays unchanged in quench dynamics, the topologicalentanglement mode should always be pinned at , leading toa persistent zero single-particle entanglement gap.Even if R o is not invertible so that A in Eq. (30) becomesill-defined, we can still show that all the eigenvalues of iR d falling in the range ( − , are degenerate. For an arbitrarynormalized eigenvector φ with iR d φ = (cid:15)φ , (cid:15) ∈ ( − , , wecan construct ˜ φ ≡ √ − (cid:15) R o ¯ φ, (32) such that ˜ φ † ˜ φ = 1 and iR d ˜ φ = (cid:15) ˜ φ . Moreover, we have φ † ˜ φ = 1 √ − (cid:15) φ † R o ¯ φ = 1 √ − (cid:15) ( φ † R o ¯ φ ) T = − √ − (cid:15) φ † R o ¯ φ = − φ † ˜ φ, (33)which means ˜ φ is orthogonal to φ .It is worth mentioning that such an emergent symmetry inES has systematically been studied in Ref. [41]. In addition tothe above physical analysis, we also provide a rigorous proofin Appendix B. B. General idea
To highlight the finite size of a periodic lattice, we willhereafter use P ( L ) < instead of P < in Eq. (7) as the single-particle projector onto the Fermi sea, where L denotes thenumber of unit cells. As mentioned in Sec. II B, withthe single-particle projector onto a subsystem S denoted as P S , the single-particle ES coincides with the spectrum of P S P ( L ) < P S [98]. We have shown in the previous subsectionthat for L = 2 l with l being the length of S , whenever H flat in Eq. (8) is characterized by a nontrivial Z number protectedby the particle-hole symmetry, the spectrum of P S P ( L ) < P S contains two degenerate eigenstates with eigenvalue andthe entanglement gap exactly vanishes. To prove Theorem 1,it suffices to prove that the spectral shift from P S P (2 l ) < P S to P S P ( ∞ ) < P S is exponentially small until a time scale propor-tional to l . To this end, a natural idea is to utilize the followingWeyl’s perturbation theorem. Theorem 3 (Weyl’s perturbation theorem)
Consider twoHermitian operators O and O (cid:48) on a finite Hilbert space.Denoting the j th largest eigenvalue of O and O (cid:48) as λ j and λ (cid:48) j , respectively, we have | λ j − λ (cid:48) j | ≤ (cid:107) O − O (cid:48) (cid:107) . (34)For self-containedness, we provide a brief proof in Ap-pendix C. According to this theorem, denoting the n th largesteigenvalue of P S P ( L ) < P S as ξ ( L ) n , we have | ξ ( L ) n − ξ ( ∞ ) n | ≤ (cid:107) P S P ( L ) < P S − P S P ( ∞ ) < P S (cid:107) . (35)According to Eq. (16), we have the following collorary: fora topological class D system, the single-particle entanglementgap is bounded from above as ∆ spE ≤ (cid:107) P S P (2 l ) < P S − P S P ( ∞ ) < P S (cid:107) . (36)It is thus sufficient to find a Lieb-Robinson bound on the right-hand side (rhs) of Eq. (36), which measures the finite-size cor-rection to the correlation in a finite subsystem S . Re k Im k ⇡ ⇡ (1)1 v max κ v L R (a) (b) FIG. 7. (a) Contour deformation used in Eq. (42). For sufficientlysmall κ , the shaded area inside the contour ∂ { k : | Re k | ≤ π, ≤ Im k ≤ κ } contains no poles and thus (cid:82) π − π = (cid:82) π + iκ − π + iκ . Note thatthe two integrals along the vertical edges cancel out due to the π -periodicity in Re k . (b) κ dependence of the Lieb-Robinson velocity v LR = v predicted by Lemma 1 for the SSH model with ( J , J ) =(1 , . . The maximal group velocity v max = min { J , J } natu-rally appears in the limit of κ → . The red dashed lines correspondto the specific choice κ = 0 . used in Fig. 6(c). The monotonicity of v LR with respect to κ holds for general analytic Bloch Hamiltonians(see Appendix A 3). Before going into rigorous proofs, let us first give an intu-itive argument based on the
Wannier-function picture [129].Note that the projector onto the Fermi sea can be expressed as P ( L ) < = (cid:88) j ∈ Z L ,α ∈O | W ( L ) jα (cid:105)(cid:104) W ( L ) jα | , (37)where | W ( L ) jα (cid:105) is a Wannier function of the α th band centeredat the j th site, Z L ≡ { , , ..., L } consists of all the sites and O consists of all the occupied bands. Since | W ( L ) jα (cid:105) is ex-ponentially localized in real space [130–132], we expect anexponentially small difference between | W ( L ) jα (cid:105) and | W ( ∞ ) jα (cid:105) .By the same token, although P ( ∞ ) < contains infinitely moreWannier-function projectors than P ( L ) < , such a differenceshould again become exponentially small after being pro-jected by P S , provided that both l and L − l are sufficientlylarge compared with the localization length of a Wannier func-tion. When the system is driven out of equilibrium, we expectthe Wannier function to spread no faster than linearly, leadingto a light-cone behavior of (cid:107) P S P ( L ) < P S − P S P ( ∞ ) < P S (cid:107) . Wewill later make this argument rigorous for translation-invariantsystems. Note that the above argument seems to be equallyapplicable to disordered systems with exponentially localizedWannier functions [133]. C. Lieb-Robinson bound on correlations in free-fermionsystems
An important ingredient in our proof is the conventionalLieb-Robinson bound on correlation functions. While theLieb-Robinson bound for general interacting systems is cer-tainly applicable to free-fermion systems, such a bound is usu-ally very loose since it only involves a very limited amount ofinformation about the system such as the hopping range andthe maximal hopping amplitude. Here, we derive a greatly (a) (b)(c) - - ϵ k - π π - k v k FIG. 8. (a) Dynamics of spatial correlations (cid:107)(cid:104) j | P < ( t ) | (cid:105)(cid:107) ( j (cid:54) = 0 )in an SSH chain involving L = 101 unit cells. The parameters are thesame as those in Fig. 6. An additional particle-hole symmetric term H PHS = (cid:80) j,a = A,B iJ ( c † j +1 ,a c ja − H . c . ) does not change the dy-namics of correlations. The dashed lines are given by j = ± v max t ,where v max is the largest group velocity of the SSH model with(green) or without (purple) the additional term H PHS . (b) Band dis-persions and (c) the corresponding group velocities of the SSH model(purple curves) and that with the additional term (green curves) with J = 0 . . The arrows indicate where v kα reaches its maximal abso-lute value. The solid and dotted curves correspond to the particle andhole bands, respectively. improved Lieb-Robinson bound on the correlation functionsin translation-invariant free-fermion systems. Our result isalmost optimal in the sense that the Lieb-Robinson velocityreaches the maximal (relative) group velocity of band disper-sions in the semiclassical limit.A nice property of free-fermion systems is that, accordingto Wick’s theorem, all the correlation functions can be decom-posed into two-point correlation functions. Moreover, if thewave function is a particle-number eigenstate, only the corre-lators like (cid:104) c † c (cid:105) are relevant. To compute such a correlator inthe quench dynamics, we can use the formula (cid:104) Ψ t | c † j (cid:48) a (cid:48) c ja | Ψ t (cid:105) = (cid:104) j (cid:48) a (cid:48) | P < ( t ) | ja (cid:105) , (38)where | ja (cid:105) = c † ja | vac (cid:105) and P < ( t ) ≡ e − iHt P < e iHt is thetime-evolved single-particle projector onto the Fermi sea. Tomeasure the strength of correlation at the length scale | j − j (cid:48) | , we can consider the norm of (cid:104) j | P < ( t ) | j (cid:48) (cid:105) , which can beshown to obey a Lieb-Robinson bound given by the followinglemma. Lemma 1
Consider a time-evolved projector P ( L ) < ( t ) ≡ e − iHt P ( L )0 < e iHt , where P ( L )0 < is the projector onto the Fermisea of H , which is gapped, and denote the Bloch Hamilto-nians of H and H as H ( k ) and H ( k ) , respectively. Thenfor ∀ κ > such that (i) P < ( k ) ≡ (cid:80) α ∈O | u kα (cid:105)(cid:104) u kα | (seealso Eq. (17)) is analytic over { k : | Re k | ≤ π, | Im k | ≤ κ } ,where | u kα (cid:105) , α ∈ O is the normalized Bloch eigenstate of the α th occupied band, and (ii) H ( k + iκ ) is well-defined anddiagonalizable for ∀ k ∈ [ − π, π ] , we have (cid:107)(cid:104) j | P ( ∞ ) < ( t ) | j (cid:48) (cid:105)(cid:107) ≤ Ce − κ ( | j − j (cid:48) |− vt ) , ∀ j, j (cid:48) ∈ Z , (39) where | j (cid:105) is the state localized at the j th site, and C and v aregiven in Eqs. (20) and (21). Proof.—
After analytic continuation, the Hermiticity con-straints on the Bloch Hamiltonians are generalized to H ( k ) † = H (¯ k ) and H ( k ) † = H (¯ k ) (see Appendix A 1).Accordingly, the Bloch projector P < ( k ) satisfies P < ( k ) † ≡ − (cid:73) ¯ γ < d ¯ z πi z − H ( k ) † = (cid:73) γ < d ¯ z πi z − H (¯ k ) = P < (¯ k ) , (40)where γ < is a loop that encircles the energies of all the occu- pied bands. Note that (cid:104) j | P ( ∞ ) < ( t ) | j (cid:48) (cid:105) = (cid:90) π − π dk π e ik ( j − j (cid:48) ) P < ( k, t )= ( (cid:104) j (cid:48) | P ( ∞ ) < ( t ) | j (cid:105) ) † , (41)where P < ( k, t ) ≡ e − iH ( k ) t P < ( k ) e iH ( k ) t . This implies (cid:107)(cid:104) j | P ( ∞ ) < ( t ) | j (cid:48) (cid:105)(cid:107) = (cid:107)(cid:104) j (cid:48) | P ( ∞ ) < ( t ) | j (cid:105)(cid:107) , and therefore we canassume j ≥ j (cid:48) without loss of generality. By deforming thecontour of integration (see Fig. 7(a)) and applying spectral de-composition to H ( k + iκ ) , which is generally non-Hermitian,we can bound the norm of Eq. (41) from above by (cid:107)(cid:104) j | P ( ∞ ) < ( t ) | j (cid:48) (cid:105)(cid:107) = e − κ | j − j (cid:48) | (cid:13)(cid:13)(cid:13)(cid:13)(cid:90) π − π dk π e ik ( j − j (cid:48) ) e − iH ( k + iκ ) t P < ( k + iκ ) e iH ( k + iκ ) t (cid:13)(cid:13)(cid:13)(cid:13) = e − κ | j − j (cid:48) | (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:88) α,β (cid:90) π − π dk π e ik ( j − j (cid:48) ) − i ( (cid:15) k + iκ,α − (cid:15) k + iκ,β ) t | u R k + iκ,α (cid:105)(cid:104) u L k + iκ,α | P < ( k + iκ ) | u R k + iκ,β (cid:105)(cid:104) u L k + iκ,β | (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ (cid:88) α,β e − κ | j − j (cid:48) | (cid:90) π − π dk π e Im( (cid:15) k + iκ,α − (cid:15) k + iκ,β ) t (cid:13)(cid:13) | u R k + iκ,α (cid:105)(cid:104) u L k + iκ,α | P < ( k + iκ ) | u R k + iκ,β (cid:105)(cid:104) u L k + iκ,β | (cid:13)(cid:13) ≤ e − κ | j − j (cid:48) | +max k ∈ [ − π,π ] ,α,β Im( (cid:15) k + iκ,α − (cid:15) k + iκ,β ) t (cid:88) α,β (cid:90) π − π dk π (cid:107)| u R k + iκ,α (cid:105)(cid:104) u L k + iκ,α |(cid:107)(cid:107) P < ( k + iκ ) (cid:107)(cid:107)| u R k + iκ,β (cid:105)(cid:104) u L k + iκ,β |(cid:107) , (42)which completes the proof of Lemma 1. (cid:3) Let us discuss how our result is related to the conventionalgroup velocity [134] v kα = d(cid:15) kα dk . (43)In the presence of the sublattice symmetry, as is the caseof the SSH model, the eigenenergies are paired as ± (cid:15) k for ∀ k ∈ [ − π, π ] (see purple curves in Fig. 8(b)), and v in Eq. (21)becomes v = 2 κ − max k ∈ [ − π,π ] ,α | Im (cid:15) k + iκ,α | . (44)The maximal group velocity v max ≡ max k ∈ [ − π,π ] ,α (cid:12)(cid:12)(cid:12)(cid:12) d(cid:15) kα dk (cid:12)(cid:12)(cid:12)(cid:12) (45)thus naturally emerges when κ → (see Fig. 7(b)) due to therelation (cid:15) k + iκ,α = (cid:15) kα + i d(cid:15) kα dk κ + O ( κ ) . Since the boundin Eq. (39) can be made rather small by a sufficiently large l for a given small κ , we expect that the Lieb-Robinson ve-locity v LR = v [54] is essentially given by v max at largelength scales. More precisely, suppose that we scale up l and t simultaneously while keeping lt fixed; then, as long as lt < v max , we can always choose κ > such that the boundin Eq. (39) scales like e − O ( l ) and thus vanishes in the thermo-dynamic limit. This is quite reasonable since the group veloc-ity in Eq. (43) is derived in the semiclassical regime, wherethe length scale is much larger than the lattice constant [134]. In general, however, the energy dispersions are not pairedat each k . This can be the case even if there is a particle-holesymmetry, which only requires (cid:15) kα = − (cid:15) − k ¯ α with ¯ α being theparticle-hole conjugation of α (see, for example, green curvesin Fig. 8(b)). In the limit of κ → , v in Eq. (21) generallyreaches the maximal relative group velocity v mr ≡ max k ∈ [ − π,π ] ,α,β (cid:18) d(cid:15) kα dk − d(cid:15) kβ dk (cid:19) , (46)which is smaller than twice of the maximal group velocity(45) and thus gives a tighter bound on the propagation of cor-relation. For example, as shown in Fig. 8(a), the dynamics ofcorrelation in the SSH model does not change in the presenceof an additional particle-hole symmetric term, which enhancesthe maximal group velocity (see Fig. 8(c)) but leaves the rela-tive group velocity invariant over the entire Brillouin zone.One may ask whether v in Eq. (21) can be made smallerthan Eq. (46) for some κ > so that the Lieb-Robinson ve-locity can be even tighter. However, by employing a con-tinuous version of the majorization technique [135], we canprove that Eq. (21) increases monotonically with respect to κ (see Fig. 7(b) for example and Appendix A 3 for the generalproof). Therefore, our rigorous bound does not lead to anytighter bound than the maximal relative group velocity. This isphysically reasonable since a completely destructive interfer-ence between the modes with maximal relative group veloci-ties seems impossible due to the difference in wave numbers.Our result thus quantitatively justifies and refines the widelyused quasiparticle picture on the propagation of correlation in0the quench dynamics [9], which, to our knowledge, has ana-lytically been confirmed only in specific situations [136–138]. D. Proof of Theorem 1
We are now in a position to prove the first main result. Tobound the single-particle entanglement gap (19) from the ex-ponential decay in the correlation function, we need the fol-lowing lemma.
Lemma 2
Denoting P ( L ) < as the projector onto the Fermi seaof a length- L lattice system, we have (cid:104) j | P ( L ) < | j (cid:48) (cid:105) − (cid:104) j | P ( ∞ ) < | j (cid:48) (cid:105) = (cid:88) n ∈ Z \{ } (cid:104) j | P ( ∞ ) < | j (cid:48) + nL (cid:105) (47) for ∀ j, j (cid:48) ∈ Z L . This result arises from the combination of Eq. (37) and therelation between the Wannier functions on finite and infinitelattices (see Appendix D). Lemma 2 can be used to derive abound on the finite-size correction to the correlation matrix,which determines the ES.
Lemma 3
Consider a length- l segment embedded in a gappedtranslation invariant 1D lattice system with length L ( ≤ l
1) sinh κ , (49) where P S ≡ (cid:80) lj =1 | j (cid:105)(cid:104) j | ⊗ I is the projector onto the seg-ment.Proof.— According to Eq. (47), the norm of (cid:104) j | P ( L ) < | j (cid:48) (cid:105) −(cid:104) j | P ( ∞ ) < | j (cid:48) (cid:105) is bounded from above by (cid:107)(cid:104) j | P ( L ) < | j (cid:48) (cid:105) − (cid:104) j | P ( ∞ ) < | j (cid:48) (cid:105)(cid:107)≤ (cid:88) n ∈ Z \{ } (cid:107)(cid:104) j | P ( ∞ ) < | j (cid:48) + nL (cid:105)(cid:107)≤ C (cid:88) n ∈ Z \{ } e − κ | j − j (cid:48) − nL | = C ∞ (cid:88) n =1 [ e − κ ( nL + j (cid:48) − j ) + e − κ ( nL + j − j (cid:48) ) ]= 2 C cosh κ ( j − j (cid:48) ) e κL − , (50)where Eq. (48) has been used. Using Eq. (50) and the norminequality [139] (cid:107) O (cid:107) ≤ (cid:88) j,j (cid:48) (cid:107) O jj (cid:48) (cid:107) , (51) with O jj (cid:48) ≡ P j OP j (cid:48) , P j P j (cid:48) = δ jj (cid:48) P j and (cid:80) j P j = for anarbitrary bounded partitioned operator O = (cid:80) j,j (cid:48) O jj (cid:48) , weobtain (cid:107) P S P ( L ) < P S − P S P ( ∞ ) < P S (cid:107) ≤ l − (cid:88) j,j (cid:48) =0 (cid:107)(cid:104) j | P ( L ) < | j (cid:48) (cid:105) − (cid:104) j | P ( ∞ ) < | j (cid:48) (cid:105)(cid:107) ≤ C l (cid:88) j,j (cid:48) =1 cosh κ ( j − j (cid:48) )( e κL − = 2 C [( (cid:80) lj =1 e κj )( (cid:80) lj (cid:48) =1 e − κj (cid:48) ) + l ]( e κL − ≤ (cid:20) C sinh κl ( e κL −
1) sinh κ (cid:21) , (52)where we have used sinh κl ≥ l sinh κ for l ≥ . (cid:3) The remaining step to prove Theorem 1 is simply to com-bine Lemmas 3 and 1 with Eq. 36. By identifying C inEq. (48) with Ce κvt in Eq. (39), we find that Eq. (36) leadsto Theorem 1.Finally, let us discuss how to generalize Theorem 1 to afinite lattice with length L > l . The existence of such a Lieb-Robinson bound is already clear from the triangle inequality | ξ ( L ) n − ξ (2 l ) n | ≤ | ξ (2 l ) n − ξ ( ∞ ) n | + | ξ ( L ) n − ξ ( ∞ ) n | , the rhs of whichcan be further bounded by Lemma 3. However, this bound istoo loose since the exact degeneracy is not reproduced when L = 2 l . To obtain a tighter bound, we use the following gen-eralization of Lemma 2: (cid:104) j | P ( L ) < | j (cid:48) (cid:105) − (cid:104) j | P ( L ) < | j (cid:48) (cid:105) = (cid:88) n ∈ Z \ LCM( L ,L L Z (cid:104) j | P ( ∞ ) < | j (cid:48) + n L (cid:105)− (cid:88) n ∈ Z \ LCM( L ,L L Z (cid:104) j | P ( ∞ ) < | j (cid:48) + n L (cid:105) , (53)where LCM denotes the least common multiple. Followingthe calculations in Lemma 3, we can use Eq. (53) to derive ∆ spE ≤ C sinh κl sinh κ e κvt × (cid:18) e κL − e κl − − e κ LCM( L, l ) − (cid:19) , (54)which reproduces Eq. (19) for L = ∞ and ∆ spE = 0 for L =2 l . Moreover, the bound is O (1) when L is close to l . This isreasonable because the many-body ES of a length- l segmentshould be the same as its complement with length L − l . If L − l is comparable to or even smaller than the localization lengthof the entanglement edge modes, both many-body and single-particle entanglement gaps will be significantly nonzero. V. INTERACTING SYSTEMS
Let us move on to interacting systems. It suffices to con-sider spin (bosonic) systems since interacting fermions can1be mapped onto spin systems via the Jordan-Wigner transfor-mation, which preserves the locality in the presence of thefermion-parity superselection rule [140]. The only thing weshould be cautious about is that fermionic SPT states with Ma-jorana modes will be transformed into spontaneous symmetrybroken states, as will be discussed in details in Sec. V E. Al-though the very notion of the band is no longer applicable, wecan employ MPSs to efficiently describe the ground state of agapped local Hamiltonian [141]. While the formalism is dif-ferent, we can again upper bound the entanglement gap by aquantity closely related to the correlation of two local observ-ables at the boundaries of the subsystem, as detailed in thefollowing.
A. ES of an MPS
We consider a translation-invariant normal MPS in thecanonical form [142], as given in Eq. (9). For an arbitraryorthonormal basis {| α (cid:105)} Dα =1 on the virtual level, we can de-compose Eq. (9) into | Ψ (cid:105) = (cid:88) α,β | ψ αβ (cid:105)| Φ βα (cid:105) , (55)where | ψ αβ (cid:105) = (cid:88) { j s } ls =1 (cid:104) α | A j A j ...A j l | β (cid:105)| j j ...j l (cid:105) , | Φ βα (cid:105) = (cid:88) { j s } Ls = l +1 (cid:104) β | A j l +1 A j l +2 ...A j L | α (cid:105)| j l +1 j l +2 ...j L (cid:105) . (56)In particular, if | α (cid:105) ’s are chosen to be the eigenstates of Λ inEq. (13) with eigenvalues λ α ’s ( (cid:80) Dα =1 λ α = 1 ), we have inthe thermodynamic limit L → ∞ [100] (cid:104) Φ βα | Φ β (cid:48) α (cid:48) (cid:105) = lim L →∞ (cid:104) β (cid:48) |E L − l ( | α (cid:48) (cid:105)(cid:104) α | ) | β (cid:105) = (cid:104) β (cid:48) |E ∞ ( | α (cid:48) (cid:105)(cid:104) α | ) | β (cid:105) = (cid:104) β (cid:48) | Tr[Λ | α (cid:48) (cid:105)(cid:104) α | ] v | β (cid:105) = λ α δ β,β (cid:48) δ α,α (cid:48) . (57)where E ∞ ( · ) ≡ lim L →∞ E L ( · ) = Tr[Λ · ] v . Therefore,the reduced density matrix of subsystem [1 , l ] ⊂ Z (i.e., thesegment consisting of sites , , ..., l ) can be written as ρ [1 ,l ] = (cid:88) α,β λ α | ψ αβ (cid:105)(cid:104) ψ αβ | . (58)Unlike | Φ βα (cid:105) ’s, | ψ αβ (cid:105) ’s are not strictly orthogonal to eachother: (cid:104) ψ αβ | ψ α (cid:48) β (cid:48) (cid:105) = λ β δ α,α (cid:48) δ β,β (cid:48) + (cid:15) αβ,α (cid:48) β (cid:48) ,(cid:15) αβ,α (cid:48) β (cid:48) = (cid:104) α (cid:48) | ( E l − E ∞ )( | β (cid:48) (cid:105)(cid:104) β | ) | α (cid:105) . (59)Defining | φ α,β (cid:105) ≡ √ λ α | ψ α,β (cid:105) , we can simplify Eq. (58) as ρ [1 ,l ] = (cid:88) αβ | φ αβ (cid:105)(cid:104) φ α,β | (60) Spec ...... p ⇤ p ⇤ ¯ AA l (1)Spec ...... (2)... ... t A U (3)... ... O j UU (4)...... U † O j U k + 1 (5)1Spec ...... p ⇤ p ⇤ ¯ AA l (1)Spec ...... (2)... ... t A U (3)... ... O j UU (4)...... U † O j U k + 1 (5)1 =
Given a Hermitian operator ρ = (cid:80) Jj =1 | φ j (cid:105)(cid:104) φ j | > , where | φ j (cid:105) ’s are generally neithernormalized nor orthogonal to each other, the spectrum of ρ coincides with the nonzero part of the spectrum of M ∈ C J × J with M jj (cid:48) = (cid:104) φ j | φ j (cid:48) (cid:105) .Proof.— Denoting V = span {| φ j (cid:105)} Jj =1 on which ρ acts, wecan expand any | ψ (cid:105) ∈ V as | ψ (cid:105) = | φ (cid:105) c = (cid:80) Jj =1 c j | φ j (cid:105) ,where | φ (cid:105) ≡ [ | φ (cid:105) , | φ (cid:105) , ..., | φ J (cid:105) ] and c = [ c , c , ..., c J ] T .Defining (cid:104) φ | = [ (cid:104) φ | , (cid:104) φ | , ..., (cid:104) φ J | ] T , we have ρ = | φ (cid:105)(cid:104) φ | and M = (cid:104) φ | φ (cid:105) . Now we prove the following equivalence. There are r linearly independent eigenstates {| ψ j (cid:105)} rj =1 of ρ with the same eigenvalue λ (cid:54) = 0 . ⇔ There are r linearly independent eigenvectors { v j } rj =1 of M with eigenvalue λ (cid:54) = 0 .This statement implies Lemma 4. ⇒ : Expanding | ψ j (cid:105) as | φ (cid:105) v j , by assumption we have ρ | ψ j (cid:105) = | φ (cid:105)(cid:104) φ | φ (cid:105) v j = | φ (cid:105) M v j = λ | ψ j (cid:105) = λ | φ (cid:105) v j . (62)Multiplying (cid:104) φ | from the left of Eq. (62), we obtain M v j = λM v j , implying that M v j (cid:54) = (otherwise ρ | ψ j (cid:105) = 0 ) is aneigenvector of M with eigenvalue λ . Suppose that M v j ’s arenot linearly independent, which means r (cid:88) j =1 k j M v j = (63)for some k j ’s with (cid:80) rj =1 | k j | (cid:54) = 0 . Operating Eq. (63) on | φ (cid:105) gives (cid:80) rj =1 k j ρ | ψ j (cid:105) = λ (cid:80) rj =1 k j | ψ j (cid:105) = 0 , which con-tradicts the linear independence of {| ψ j (cid:105)} rj =1 . ⇐ : By assumption, we have M v j = λ v j for j =1 , , ..., r . Defining | ψ j (cid:105) = | φ (cid:105) v j , we again obtain Eq. (62),2implying that | ψ j (cid:105) is an eigenstate of ρ with eigenvalue λ .Suppose that | ψ j (cid:105) ’s are not linearly independent, which means r (cid:88) j =1 k j | ψ j (cid:105) = r (cid:88) j =1 k j | φ (cid:105) v j = 0 (64)for some k j ’s with (cid:80) rj =1 | k j | (cid:54) = 0 . Operating Eq. (64) on (cid:104) φ | gives (cid:80) rj =1 k j M v j = λ (cid:80) rj =1 k j v j = , which contra-dicts the linear independence of { v j } rj =1 . (cid:3) According to Lemma 4, with the unimportant zero part ne-glected, the ES is nothing but the spectrum of M αβ,α (cid:48) β (cid:48) = (cid:104) φ αβ | φ α (cid:48) β (cid:48) (cid:105) , whose entries are given in Eq. (61). B. Exact ES degeneracy in the thermodynamic limit
In the limit of l → ∞ , (cid:15) αβ,α (cid:48) β (cid:48) in Eq. (59) vanishes. Hence, M αβ,α (cid:48) β (cid:48) = (cid:104) φ αβ | φ α (cid:48) β (cid:48) (cid:105) becomes diagonalized and the ESis simply given by { λ α λ β } Dα,β =1 , which is the spectrum of Λ ⊗ . In this case, SPT order enforces the ES to be exactlydegenerate, as a result of symmetry fractionalization on thevirtual level. Lemma 5
The ES of any topologically nontrivial MPS pro-tected by unitary symmetries is at least four-fold degeneratein the thermodynamic limit of a subsystem.Proof.—
Denoting the symmetry group as G and its projectiverepresentation on the virtual level as V g , we have [104] [ V g , Λ] = 0 , ∀ g ∈ G, (65)where Λ is the unique left fixed point of the unital channel as-sociated with the MPS. Suppose that there is a non-degenerateeigenvalue λ associated with eigenstate | λ (cid:105) . From Eq. (65) wehave V g | λ (cid:105) = ν g | λ (cid:105) , ν g ∈ U(1) , ∀ g ∈ G. (66)Since V g V h = ω g,h V gh for ∀ g, h ∈ G , Eq. (66) implies ν g ν h = ω g,h ν gh ⇔ ω g,h = ν g ν h ν gh , ∀ g, h ∈ G. (67) Hence, ω g,h belongs to the trivial cohomology class, contra-dicting the assumption that | Ψ (cid:105) is topological. Therefore, theES in the thermodynamic limit of a subsystem, which is noth-ing but the spectrum of Λ ⊗ , must be at least four-fold degen-erate. (cid:3) We mention that anti-unitary symmetries can also supportnontrivial interacting SPT phases with the exactly degenerateES in the thermodynamic limit. A prototypical example is theinvolutary ( T = 1 ) time-reversal symmetry, which may befractionalized into an anti-involutary ( T = − ) anti-unitarysymmetry on the virtual level, leading to the Kramers degener-acy in Λ [43]. However, since anti-unitary symmetries sufferdynamical symmetry breaking [33], the corresponding SPTorder cannot be dynamically stable. C. Bound on the many-body entanglement gap
In general, M αβ,α (cid:48) β (cid:48) for a finite l can be decomposed into [Λ ⊗ ] αβ,α (cid:48) β (cid:48) and a perturbative term: M αβ,α (cid:48) β (cid:48) = [Λ ⊗ ] αβ,α (cid:48) β (cid:48) + P αβ,α (cid:48) β (cid:48) , (68)where P αβ,α (cid:48) β (cid:48) = (cid:112) λ α λ α (cid:48) (cid:104) α (cid:48) | ( E l − E ∞ )( | β (cid:48) (cid:105)(cid:104) β | ) | α (cid:105) . (69)It is clear from Eq. (68) that the exact eigenvalue degener-acy in Λ ⊗ is generally lifted by P . However, accordingto Weyl’s perturbation theorem, the many-body entanglementgap should be upper bounded by twice the norm of P : ∆ mbE ≤ (cid:107) P (cid:107) . (70)To proceed further, we upper bound (cid:107) P (cid:107) by (cid:107) P (cid:107) ≡ (cid:112) Tr[ P † P ] , which is the Schatten 2-norm and its square takesa rather simple form: (cid:107) P (cid:107) = D (cid:88) α,α (cid:48) ,β,β (cid:48) =1 λ α λ α (cid:48) (cid:104) α (cid:48) | ( E l − E ∞ )( | β (cid:48) (cid:105)(cid:104) β | ) | α (cid:105)(cid:104) α | ( E l − E ∞ )( | β (cid:105)(cid:104) β (cid:48) | ) | α (cid:48) (cid:105) = D (cid:88) α,α (cid:48) ,β,β (cid:48) =1 λ α λ α (cid:48) (cid:104) α (cid:48) α | ( E l − E ∞ ) ⊗ ( | β (cid:48) β (cid:105)(cid:104) ββ (cid:48) | ) | αα (cid:48) (cid:105) = D (cid:88) α,α (cid:48) ,β,β (cid:48) =1 Tr[Λ ⊗ | αα (cid:48) (cid:105)(cid:104) α (cid:48) α | ( E l − E ∞ ) ⊗ ( | β (cid:48) β (cid:105)(cid:104) ββ (cid:48) | )]= Tr[Λ ⊗ S ( E l − E ∞ ) ⊗ ( S )] , (71)where we have used [ E ( O )] † = E ( O † ) and S ≡ (cid:80) Dα,β =1 | αβ (cid:105)(cid:104) βα | is the swap operator acting on two copies of virtual Hilbert spaces. By defining the norm of a superop-3erator as (cid:107)L(cid:107) ≡ max (cid:107) O (cid:107) = (cid:107) O (cid:107) =1 | Tr[ O † L ( O )] | , we canbound (cid:107) P (cid:107) by (cid:107) P (cid:107) ≤ (cid:107) Λ ⊗ S (cid:107) (cid:107) S (cid:107) (cid:107) ( E l − E ∞ ) ⊗ (cid:107)≤ D (cid:107)E l − E ∞ (cid:107) , (72)where D arises from (cid:107) S (cid:107) , and upper bounds (cid:107) Λ ⊗ S (cid:107) =Tr[Λ ] since the spectrum of Λ is at least two-fold degenerate.Combining Eq. (72) with Eq. (70), we obtain ∆ mbE ≤ √ D (cid:107)E l − E ∞ (cid:107) . (73)Note that E l − E ∞ appears routinely in the correlation func-tions of MPSs and leads to an exponential decay [99–101].Quantitatively, (cid:107)E l −E ∞ (cid:107) can be upper bounded by c (cid:15) ( µ + (cid:15) ) l for ∀ (cid:15) > , where µ is the spectral radius of E − E ∞ and c (cid:15) does not depend on l [99], implying that ∆ mbE is also expo-nentially small just like the correlation functions mentionedabove. We emphasize that (cid:107)E l −E ∞ (cid:107) may scale like poly( l ) µ l [143], in which case a nonzero (cid:15) is necessary for giving a rig-orous bound like c (cid:15) ( µ + (cid:15) ) l . D. Proof of Theorem 2
Let us analyze how the bound given in Eq. (73) changeswhen an MPS evolves according to a symmetric and trivialMPU. Denoting the bond dimension of the MPU as D U , thatof the MPS after t steps of evolutions is no more than DD tU .Moreover, the spectrum of the associated unital channel staysinvariant during the time evolution as a result of unitarity[111] (see also Lemma 10 in Appendix E), and so does µ .Having in mind that (cid:107)E l − E ∞ (cid:107) is bounded by c (cid:15) ( µ + (cid:15) ) l ,it is natural to expect a Lieb-Robinson bound from Eq. (73).However, this expectation may fail — given (cid:15) , c (cid:15) may growfaster than exponentially in time. To rule out this possiblity,we utilize the function-algebra -based techniques developed inRef. [144] to carefully estimate the growth of (cid:107)E l − E ∞ (cid:107) .The main idea of Ref. [144] is represented by the followinglemma. Lemma 6
Given an operator M on a finite linear space gen-erating a bounded semigroup {M n } n ∈ N , i.e., (cid:107)M n (cid:107) ≤ C M for ∀ n ∈ N , and an element f in the Wiener algebra W ≡{ f ∈ Hol( D ) : f ( z ) = (cid:80) p ∈ N f p z p , (cid:107) f (cid:107) W ≡ (cid:80) p ∈ N | f p | < ∞} ( Hol( D ) : set of holomorphic functions within the unit disk D ≡ { z ∈ C : | z | < } ), we have (cid:107) f ( M ) (cid:107) ≤ C M (cid:107) f (cid:107) W/m M W , (74) where (cid:107) f (cid:107) W/m M W ≡ inf {(cid:107) g (cid:107) W : g = f + m M h, h ∈ W } and m M ∈ W is the minimal polynomial of M , i.e.,the nonzero polynomial with the lowest degree such that m M ( M ) = 0 and the leading coefficient (that of the high-est degree) is equal to .Proof.— Due to (cid:107)M n (cid:107) ≤ C M for ∀ n ∈ N , for ∀ f ∈ W , thenorm of f ( M ) can be upper bounded by (cid:107) f ( M ) (cid:107) ≤ (cid:88) p ∈ N | f p |(cid:107)M p (cid:107) ≤ C M (cid:88) p ∈ N | f p | = (cid:107) f (cid:107) W . (75)Since m M is the minimal polynomial of M , for ∀ h ∈ W , wehave f ( M ) = ( f + m M h )( M ) . (76)Applying Eq. (75) to the rhs of Eq. (76) gives (cid:107) f ( M ) (cid:107) ≤ C M (cid:107) f + m M h (cid:107) W , ∀ h ∈ W. (77)Equation (74) then follows from minimization of the rhs ofEq. (77). (cid:3) An arbitrary unital channel E satisfies the condition inLemma 6 due to the fact that E n is again a unital channel and (cid:107)E n (cid:107) ≤ (cid:113) D , where D is the Hilbert-space dimension [145].Accordingly, E − E ∞ also satisfies the condition.Regarding the minimal polynomial of E − E ∞ during timeevolution, we have the following lemma. Lemma 7
Suppose that a single step of evolution by an MPUgenerated by U changes the associated unital channel of anMPS from E into E (cid:48) . Denoting the minimal polynomials of E − E ∞ and E (cid:48) − E (cid:48)∞ as m and m (cid:48) , respectively, we have m (cid:48) ( z ) | z k m ( z ) , (78) namely, m (cid:48) ( z ) is a divisor of z k m ( z ) , where k is the small-est integer such that U k is simple.Proof.— By assumption, for ∀ k ≥ k , the open boundarytensor U k U k can be decomposed into... UU UU UU k ≡ U k U k = U k U k ρ U k U k ⊗ k − k .(79)We express m explicitly as (cid:80) k c k z k , which satisfies c = 0 (due to that the spectrum of E − E ∞ contains zero) and m ( E − E ∞ ) = (cid:88) k c k ( E k − E ∞ ) = 0 ⇔ (cid:88) k c k E k = (cid:88) k c k E ∞ . (80)Then, using Eq. (79), we can evaluate z k m ( z ) | z = E (cid:48) as4 (cid:80) k c k ... ¯ A UU A ¯ A UU A ¯ A UU Ak + 2 k = (cid:80) k c k ¯ A k U k ρ U k A k ¯ A k A k ¯ A k U k U k A k = lim l →∞ (cid:80) k c k ¯ A k U k ρ U k A k ¯ A l A l ¯ A k U k U k A k (81) = ( (cid:80) k c k ) lim l →∞ U k U k U l U l U k U k ¯ A k A k ¯ A l A l ¯ A k A k ρ ρ = ( (cid:80) k c k ) lim l →∞ ¯ A l U l U l A l ,where we have used Eq. (80) (enclosed in dashed rectangles)and Eq. (79). Lemma 7 follows immediately from Eq. (81),which is the tensor-network representation of E (cid:48) k m ( E (cid:48) ) = (cid:80) k c k E (cid:48)∞ ⇔ z k m ( z ) | z = E (cid:48) −E (cid:48)∞ = 0 . (cid:3) A direct corollary of Lemma 7 (applying t times) is m t ( z ) | z k t m ( z ) , where m t ( z ) ( m ( z ) ≡ m ( z ) ) is the mini-mal polynomial of E t −E ∞ t , with E t determined from the MPSat time t . Using this fact, as long as l > k t , we have (cid:107) z l (cid:107) W/m t W ≤ inf {(cid:107) g (cid:107) W : g = z l + z k t mh, h ∈ W } = (cid:107) z l − k t (cid:107) W/mW , (82)where we have used (cid:107) g (cid:107) W = (cid:107) z n g (cid:107) W for ∀ n ∈ N . Accord-ing to the main result of Ref. [144], which is summarized asTheorem 6 in Appendix G, when l − k t > µ − µ , we canbound (cid:107)E lt − E ∞ t (cid:107) from above by µ l − k t +1 C t e (cid:112) | m E | ( | m E | + 1)( l − k t )[1 − (1 + l − k t ) µ ] × sup | z | = µ (1+ l − k t ) | B ( z ) | , (83)where C t ≡ sup n ∈ N (cid:107)E nt (cid:107) and B ( z ) is the Blaschke product(G1) with respect to the spectrum of E ≡ E . If we furtherrequire l − k t ≥ µ − µ , even in the worst case, (cid:107)E lt − E ∞ t (cid:107) isbounded by an exponentially small quantity up to polynomialcorrections (see Eq. (20) in Ref. [144]): (cid:107)E lt − E ∞ t (cid:107) ≤ e C t (cid:112) | m | ( | m | + 1) (cid:18) µ − µ (cid:19) × (cid:20) − µ µ ( l − k t ) (cid:21) | m |− µ l − k t . (84)Denoting the bond dimension of the initial MPS | Ψ (cid:105) and thatof | Ψ t (cid:105) = U t | Ψ (cid:105) after t steps of time evolutions as D and D t , respectively, we have D t ≤ DD tU = De t ln D U , (85)and hence [145] C t ≤ (cid:114) D t ≤ (cid:114) D e ln DU t . (86)Combing Eq. (73) and Eqs. (84)-(86) with | m | ≤ D ( D ≥ ,otherwise | Ψ (cid:105) is a product state), we obtain Theorem 2.Our theorem rigorously establishes the dynamical stabilityfor general SPT systems in 1D. Yet another important im-plication of Theorem 2 is that the topological discrete time-crystalline oscillation , which is a toggle between differentSPT phases [146] generated by an MPU with nontrivial co-homology [111], persists up to a time scale which increases atleast linearly with respect to the system size. To see this, wehave only to apply the theorem to U p with trivial cohomology,where p is the order of the cohomology group. E. Applicability to interacting fermions
As mentioned in the beginning of this section, a fermionicsystem in 1D can always be mapped into a spin chain throughthe Jordan-Wigner transformation: c † ja = σ +( j − d + a (cid:89) l< ( j − d + a σ zl , (87)where σ z = (cid:2) − (cid:3) and σ + ≡ ( σ x + iσ y ) = [ ] . Herewe follow the setup in Sec. II A, i.e., we consider d internalstates in each site of the original fermion system, so that thelength of the corresponding spin chain is Ld . The fermion-parity operator reads P f = ( − ) (cid:80) j,a c † ja c ja = (cid:89) j,a σ zjd + a , (88)5which sets the superselection rule and is thus a Z symmetryof the Hamiltonian. Rigorously speaking, to keep the localityof the obtained spin Hamiltonian, we have to assume the openboundary condition. However, as long as the chain is suffi-ciently long, its subsystem property should not depend on theboundary condition. While a fermionic system never breaksthe Z symmetry, the corresponding spin system may sponta-neously break the Z symmetry, whenever the fermionic sys-tem exhibits a Majorana mode at the open boundaries [46, 71].If the Z symmetry is not broken, the proof for 1D bosonicSPT systems applies directly to the fermionic SPT systems,which have even parities. Remarkably, after a few modifi-cations, essentially the same proof applies even to fermionicSPT systems with odd parities.While it is possible to directly describe fermionic SPTphases using fermionic MPSs [147, 148], where the anti-commutativity of fermionic operators is encoded in an addi-tional Z -graded algebraic structure of the tensors, we wouldrather like to follow the approach in Refs. [46, 71] and workin the spin picture after the Jordan-Wigner transformation. Inthe spin picture, denoting | Ψ SB (cid:105) as a “physical” symmetry-broken ground state, the corresponding exact (cat) groundstates read [46] | Ψ (cid:105) = 1 √ | Ψ SB (cid:105) + ( − ) η P f | Ψ SB (cid:105) )= 1 √ (cid:88) { j s } Ls =1 Tr[ Z η A j ...A j L ] | j ...j L (cid:105) , (89)where η = 0 , , | Ψ SB (cid:105) is generated by a normal tensor B j without P f symmetry and Z = σ z ⊗ v , A j = ( σ z ) | j | ⊗ B j , j = 1 , , ..., d , (90)with | j | being the parity of state | j (cid:105) , i.e., P f | j (cid:105) = ( − ) | j | | j (cid:105) .Let us consider the spectral property of the associated quan-tum channel: E ( ρ ) ≡ (cid:80) d j =1 A j ρA † j = λρ , where λ isan eigenvalue and ρ can generally be expressed as ρ = (cid:80) w =0 ,x,y,z σ w ⊗ ρ w . Defining | w | ∈ Z from σ w σ z =( − ) | w | σ z σ w , we obtain E ( ρ ) = (cid:88) w =0 ,x,y,z σ w ⊗ d (cid:88) j =1 ( − ) | w || j | B j ρ w B † j = (cid:88) w =0 ,x,y,z σ w ⊗ λρ w , (91)implying that the spectrum of E is the union of those of E ± defined by E ± ( · ) = (cid:80) d j =1 ( ± ) | j | B j ( · ) B † j , with each eigen-value doubled. The doubling arises from the fact that, given ρ + ( ρ − ) as an eigenstate of E + ( E − ) with eigenvalue λ + ( λ − ), σ ⊗ ρ + and σ z ⊗ ρ + ( σ x ⊗ ρ + and σ y ⊗ ρ + ) are two de-generate eigenstates of E with the same eigenvalue λ + ( λ − ).Recalling that B j is a normal tensor, we know that, after nor-malization, the spectral radius of E + is one and that it onlyhas a single eigenvalue equal to one. Let µ − be the spectralradius of E − . Since B j is not P f -symmetric, we have µ − < [104]. By gauge transforming B j ’s such that E + ( v ) = v and E + (Λ + ) = Λ + , we have E ∞ ( · ) ≡ lim l →∞ E l ( · ) = 12 σ ⊗ v Tr[ σ ⊗ Λ + ( · )]+ 12 σ z ⊗ v Tr[ σ z ⊗ Λ + ( · )] , (92)which satisfies Z E ∞ ( Z · Z ) Z = E ∞ ( · ) . According to Fig. 9,the ES in the thermodynamic limit of the subsystem is givenby the spectrum of ( σ ⊗ σ + σ z ⊗ σ z ) ⊗ Λ ⊗ , which doesnot depend on η in Eq. (89) and is at least two-fold degen-erate. If there are additional symmetries, then following theanalysis for symmetric normal MPSs we know that a nontriv-ial projective representation on the virtual level may enforcean r -fold degeneracy in the spectrum of Λ + , leading to a totalof r -fold degeneracy in the ES.When the parity symmetry-broken state is evolved by aparity symmetric MPU U , we find that the cat-state struc-ture (89) stays valid since P f U = U P f by assumption. Foran individual | Ψ SB (cid:105) , we know that the degeneracy in thetime-evolved fixed point Λ + ( t ) determined from U t | Ψ SB (cid:105) should stay unchanged. Since the convergence bound given inRef. [144] applies equally to the quantum channels with mul-tiple steady states, the Lieb-Robinson bound on the entangle-ment gap given in Theorem 2 is also applicable to fermionicSPT states with Majorana modes. Moreover, the coeffecient C in Eq. (27) can be tightened by a factor of due to the normal-ization prefactor √ in Eq. (89). Also, denoting the spectralradius of E + − E ∞ + as µ + , we have µ = min { µ + , µ − } . VI. DISCUSSIONS
While the Lieb-Robinson bound places a rigorous upperbound on the maximal splitting between degenerate ES val-ues, it is usually a highly nontrivial problem to determine thedegree of degeneracy, especially when the system undergoespartial symmetry breaking. In addition, the derivation of therigorous Lieb-Robinson bounds on entanglement gaps makesfull use of the translation invariance and the results are mean-ingful only up to t ∗ ∼ lv . It is natural to ask what happenswhen the translation invariance breaks down or/and in longertime scales. Here we give some heuristic arguments to addressthese issues. A. Partial symmetry breaking
1. Free fermions
For free-fermion systems with Altland-Zirnbauer symme-tries, we have explained the effect of dynamical symmetrybreaking and the reduction of symmetry classes [33]. Theonly two classes whose reduced class is nontrivial are classesBDI and DIII, both of which reduce to class D. In starkcontrast, the former gives a surjective group homomorphism Z → Z , while the latter gives a trivial one Z → ( ⊂ Z ). If6 … … … … S p ec ...... p ⇤ p ⇤ ¯ A A l ( ) S p ec ...... ( ) ...... t A U ( ) ...... O j U U ( ) ...... U † O j U k + ( ) N
2. Interacting systems
The Haldane phase [149], which is a prototypical SPTphase corresponding to the nontrivial element in H ( Z × Z , U(1)) = Z , always becomes trivial upon any symmetrybreaking quench since H ( Z , U(1)) = H (0 , U(1)) = 0 .See Ref. [33] for a numerical demonstration for an immediateopening of the entanglement gap. Here, we consider a moregeneral situation where the initial SPT is protected by someunitary symmetries that form a group G , and the postquenchHamiltonian H only respects the symmetries in a subgroup ˜ G < G .It is, in general, very difficult to determine the reduced ˜ G -symmetric SPT phase and the corresponding ES degeneracyfrom a given G -symmetric SPT phase. See Appendix I 2 foran exact mathematical formulation of this problem in terms ofcategory-theoretic languages. Here, let us consider a class ofminimal but yet nontrivial examples — G = Z N × Z N → ˜ G = Z n × Z n , where N = np with p ∈ Z + . Suppose that theinitial state correspond to ν ∈ Z N = H ( Z N × Z N , U(1)) .By choosing the coboundary gauge properly, the projectiverepresentation V on the virtual level satisfies V ( a,b ) V ( a (cid:48) ,b (cid:48) ) = ω νa (cid:48) bN V ( a + a (cid:48) ,b + b (cid:48) ) , (95)where ( a, b ) , ( a (cid:48) , b (cid:48) ) ∈ Z N × Z N . Regarding the elements inthe subgroup Z n × Z n , we find ˜ V a,b ˜ V a (cid:48) ,b (cid:48) = V ap,bp V a (cid:48) p,b (cid:48) p = ω νa (cid:48) bp N V ( a + a (cid:48) ) p, ( b + b (cid:48) ) p = ω pνa (cid:48) bn ˜ V a + a (cid:48) ,b + b (cid:48) , (96)where we have used the fact that ( a, b ) ∈ Z n × Z n cor-responds to ( ap, bp ) ∈ Z N × Z N . This implies that thegroup homomorphism from Z N = H ( Z N × Z N , U(1)) to Z n = H ( Z n × Z n , U(1)) is given by ν → ˜ ν = pν mod n. (97)7Note that such a group homomorphism can be trivial, as is thecase for N = 4 and n = 2 due to ˜ ν = 2 ν mod 2 = 0 .The minimal realizations of nontrivial group homomorphismsturn out to be N = 6 and n = 3 , which gives rise to Z → Z : ν → ˜ ν = − ν mod 3 , and N = 6 and n = 2 ,which gives rise to Z → Z : ν → ˜ ν = ν mod 2 . Using thefact that the ES of a Z N × Z N -symmetric SPT state charac-terized by ν is (at least) N GCD( ν,N ) -fold degenerate under theopen boundary condition [111], where GCD is the greatestcommon divisor, we can figure out how many branches theoriginal ES should split into. The results are summarized inTable II and are numerically verified by some minimal models(see Appendix I 3).
B. Effects of disorder
As schematically illustrated in the right top panel in Fig. 2,disorder can dramatically alter the universal dynamical behav-ior of the entanglement growth. In this subsection, we quali-tatively address the impact of disorder on the ES dynamics in1D SPT systems.Since the entanglement-gap opening is ultimately relatedto operator spreading and propagation of correlation, we ex-pect disorder in the postquench Hamiltonian H to stabilize theSPT order during quench dynamics, provided that the disor-der does respect the symmetry. For free fermions in 1D, theAnderson localization occurs at an arbitrarily small disorderstrength and introduces a new length scale ξ [150], which isa typical localization length of the eigenfunctions of H andmeasures how far a wave packet can diffuse. As illustrated inFig. 11(a), for l (cid:29) ξ , we expect that ∆ spE is saturated at e − O ( l ) and the SPT order survives even in the limit of t → ∞ (seenumerical signatures in Appendix J 1). On the other hand, forvery weak disorder, l may be comparable to or even smallerthan ξ , and we can still observe the opening of the entangle-ment gap. This occurs even for a half-chain bipartition, unlessthe disorder configuration happens to respect the half-chaintranslation symmetry.In the presence of interactions, we may expect qualitativelydifferent behavior because the entanglement-entropy growthis unbounded in the many-body localized phase [151], al-though it is logarithmical and hence extremely slow [152–156]. In this case, we conjecture that, after a possible tran-sient similar to the noninteracting case, ∆ mbE grows no fasterthan a power law , which is similar to the behavior of out-of-time-order correlators [157–159]. This is supported by thenumerics on a phenomenological model (see Appendix J 2). C. Longer time scales
While our rigorous results ensure the persistence of (ap-proximate) ES degeneracy until a time scale that grows lin-early long with the subsystem size, these results are not usefulfor describing the dynamical behavior at longer time scales.In fact, previous numerical studies on some two-band free-fermion models have revealed the possibility for a single- t ES t ES (a) (b) ⇠ & l
The Lieb-Schultz-Mattis-Oshikawa-Hastings theorem[165–167], the Lieb-Robinson bound [53] and the entangle-ment area law [68] are of critical importance in quantummany-body systems. These rigorous results are of greatcurrent interest in light of theoretical insights from topolog-ical material science [168–170] and quantum information[54, 73–75], and of rapidly growing experimental rele-vance [55–57, 76–78]. In addition to these fundamentalachievements, we have established a general principle forthe dynamics of entanglement gaps in SPT systems that aredriven out of equilibrium by a local Hamiltonian or MPU.For free fermions, we have extensively used both the bandand the Wannier-function pictures to derive a Lieb-Robinsonbound on the single-particle entanglement gap (Theorem 1).As a byproduct, we have clarified the relation between theLieb-Robinson velocity and the group velocity of banddispersions. For interacting SPT systems, we have employedthe tensor-network approaches and the techniques of functionalgebra to derive the Lieb-Robinson bound on the many-bodyentanglement gap (Theorem 2). This result illustrates thatthe shortcoming of tensor-network approaches as numericaltools does not hinder its powerfulness as analytical tools fordealing with long-time quantum dynamics. We have alsoconsidered the impact of partial symmetry breaking, in whichcase the ES degeneracy may immediately be lifted partiallyor completely. Possible effects of disorder and relaxationbehaviors at longer time scales have also been discussed.As future studies, it would be of fundamental importanceto go beyond the tensor-network formalism and prove a Lieb-Robinson bound on the many-body entanglement gap for con-tinuous quench dynamics starting from exact ground statesof local Hamiltonians. As discussed in Appendix F, whileimproving MPUs to continuously generated unitaries aloneseems plausible, it is far from clear how we can improvethe assumption of MPS initial states to exact ground states.It is also natural to consider the generalization to higher di-mensions [47], where we may have to use other indicators tomeasure the sharpness of SPT order. Moreover, some tech-niques used here may break down. For example, we cannotconstruct an exponentially localized Wannier function for atwo-dimensional Chern insulator [132]. Finally, we note thatconsiderable efforts have recently been made for generalizingthe notions of Lieb-Robinson bounds and topological phasesto lattice systems with long-range hoppings and interactions[171–174]. It would also be interesting to relax the localityassumption and consider long-range systems, which can nat-urally be implemented with trapped ions [6, 7, 56, 57], Ryd-berg atoms [5], polar molecules [175] and nitrogen-vacancycenters [4].
ACKNOWLEDGMENTS
We acknowledge Y. Ashida, M. A. Cazalilla, M.-C. Chung,I. Danshita, K. Fujimoto, R. Hamazaki, K. Kawabata, N. Mat-sumoto, and M. McGinley for valuable discussions. In partic- ular, Z. G. appreciates Z. Wang for providing a simple pictureabout the monotonicity of Eq. (21). This work was supportedby KAKENHI Grant No. JP18H01145, No. JP17H02922and a Grant-in-Aid for Scientific Research on Innovative Ar-eas “Topological Materials Science (KAKENHI Grant No.JP15H05855) from the Japan Society for the Promotion ofScience. Z. G. was supported by MEXT. N. K. was supportedby the Leading Graduate Schools “ALPS. The authors thankthe Yukawa Institute for Theoretical Physics at Kyoto Uni-versity, where this work was initiated during the InternationalMolecule Program YITP-T-18-01 on “Floquet Theory: Fun-damentals and Applications”.
Appendix A: Band theory with a complex wave number
In this appendix, we discuss how to define 1D Bloch Hamil-tonians with complex wave numbers through analytic continu-ation. This is always possible for quasi-local hoppings, whichdecay exponentially with respect to the hopping range. We ar-gue that the analytically continued Hamiltonian should gener-ally be diagonalizable, even if the original one respects certainanti-unitary symmetries. In addition, we prove and explainwhy the Lieb-Robinson velocity in Eq. (21) is monotonic withrespect to the imaginary wave number and thus upper boundsthe maximal relative group velocity. Finally, we provide anexample of the SSH model.
1. Analytic continuation
A Bloch Hamiltonian H ( k ) can generally be expanded as H ( k ) = (cid:88) n ∈ Z e ikn H n , (A1)where each Fourier component can be obtained as H n = (cid:90) π − π dk π H ( k ) e − ikn = H †− n . (A2)A sufficient condition for the rhs of Eq. (A1) to converge is (cid:88) n ∈ Z (cid:107) H n (cid:107) < ∞ , (A3)which is valid even if H ( k ) is non-Hermitian and is obviouslysatisfied by any model with a finite hopping range R , i.e., H n = H − n = , ∀ n > R. (A4)For such a class of models, the analytic continuation to thecomplex wave number H ( k + iκ ) ≡ (cid:88) n ∈ Z e ikn − κn H n (A5)is well-defined for ∀ κ ∈ R and satisfies H ( k + iκ + 2 π ) = H ( k + iκ ) and H ( k + iκ ) † = H ( k − iκ ) for ∀ k ∈ [ − π, π ] .9The analytic continuation can be applied to a wider class ofBloch Hamiltonians whose Fourier components satisfy (cid:107) H n (cid:107) ≤ C e − κ | n | , ∀ n ∈ Z , (A6)where C , κ ∈ R + do not depend on n . One can easily checkthe validity of Eq. (A3). In this case, for ∀ κ ∈ ( − κ , κ ) , wehave (cid:107) e − κn H n (cid:107) ≤ C e − ( κ −| κ | ) | n | , ∀ n ∈ Z , (A7)so that H ( k + iκ ) in Eq. (A5) converges. Now let us showthat if H ( k ) is gapped and satisfies Eq. (A6), then the flattenedHamiltonian or the Bloch projector (17) also satisfies Eq. (A6)but with C and κ modified. To see this, we note that the n thFourier component of the Bloch projector can be written as P n = (cid:90) π − π dk π (cid:73) γ < dz πi e − ikn − κn z − H ( k − iκ ) (A8)by deforming the integral contour. With length of γ < denotedas l γ < ≡ (cid:72) γ < | dz | , such an integral (A8) can be bounded fromabove by (cid:107) P n (cid:107) ≤ (cid:90) π − π dk π (cid:73) γ < | dz | π (cid:13)(cid:13)(cid:13)(cid:13) z − H ( k − iκ ) (cid:13)(cid:13)(cid:13)(cid:13) e − κn ≤ l γ < π max k ∈ [ − π,π ] ,z ∈ γ < (cid:13)(cid:13)(cid:13)(cid:13) z − H ( k − iκ ) (cid:13)(cid:13)(cid:13)(cid:13) e − κn , (A9)provided that det[ z − H ( k − iκ )] (cid:54) = 0 for ∀ z ∈ γ < , which canalways be satisfied by sufficiently small κ due to a finite gap.This result ensures the existence of κ > such that P < ( k ) can be analytically continued to { k : | Re k | ≤ π, | Im k | ≤ κ } , implying that condition (i) in Theorem 1 can always besatisfied.On the other hand, the analytic continuation cannot be ap-plied to long-range hopping, i.e., (cid:107) H n (cid:107) ∼ O ( n − γ ) with γ ≥ — while H ( k ) stays well-defined for γ > , Eq. (A5)diverges whenever κ (cid:54) = 0 . Therefore, Theorem 1 cannot beapplied to long-range systems.
2. Diagonalizability
We argue that condition (ii) in Theorem 1 is also satisfied ingeneral — that is, there always exists a nonzero κ such that H ( k + iκ ) is diagonalizable for ∀| κ | < κ . To show this, it issufficient to show that H ( k + iκ ) becomes non-diagonalizableonly at a discrete set of complex wave numbers. This is ex-pected to be true if H ( k + iκ ) does not respect any sym-metries, since an obvious mechanism for being nondiagonal-izable is the emergence of a second-order exceptional point[176], which requires the fine-tuning of two parameters. Here,these two parameters are k and κ . However, if H ( k + iκ ) re-spects an anti-unitary symmetry or anti-symmetry S for given k and κ , i.e., S H ( k + iκ ) S − = ± H ( k + iκ ) , (A10) we only need to fine tune a single parameter to create an ex-ceptional point. This type of Hamiltonians have been activelystudied in the context of non-Hermitian topological systems[177–180], where S is typically the parity-time symmetryand H ( k + iκ ) may not be analytic and thus κ is no morethan a control parameter. In the following, we will show that H ( k + iκ ) , which is an analytic continuation of H ( k ) with ananti-unitary symmetry S , satisfies S H ( k + iκ ) S − = ± H ( k − iκ ) (A11)instead of Eq. (A10), so we still need to fine-tune both k and κ to make H ( k + iκ ) non-diagonalizable.By imposing Eq. (A10) for κ = 0 , we can use Eq. (A2) toobtain S H n S − = (cid:90) π − π dk π S H ( k ) S − e ikn = ± (cid:90) π − π dk π H ( k ) e ikn = ± H − n . (A12)Therefore, the action of S on H ( k + iκ ) gives S H ( k + iκ ) S − = (cid:88) n ∈ Z e − ikn − κn S H n S − = ± (cid:88) n ∈ Z e i ( k − iκ )( − n ) H − n = ± H ( k − iκ ) . (A13)Similarly, we can show that S H ( k ) S − = ± H ( − k ) , whichis the case of the particle-hole symmetry, gives S H ( k + iκ ) S − = ± H ( − k + iκ ) by analytic continuation and thus H ( k + iκ ) is expected to stay diagonalizable for sufficientlysmall κ .Finally, we note that even if H ( k + iκ ) happens to be non-diagonalizable, Eq. (19) should still be valid with C replacedby a polynomial of t . This is because H ( k + iκ ) can always betransformed into the Jordan normal form and a nontrivial Jor-dan block with size s ≥ contributes a polynomial prefactorwith degree s − in e − iH ( k + iκ ) t .
3. Monotonicity of Eq. (21)
We prove that Eq. (21) is monotonic in terms of κ and thusreaches its minimum at κ = 0 . To simplify the notation, wefirst introduce ∆ αβ ( k ) ≡ (cid:15) kα − (cid:15) kβ , (A14)which satisfies ∆ αβ ( k + iκ ) = ∆ αβ ( k − iκ ) for k, κ ∈ R . Forfurther simplification, we omit the subscript “ αβ ” and define v ( k, κ ) ≡ Re∆ (cid:48) ( k + iκ ) . (A15)We then have v ( k, κ ) = v ( k, − κ ) and V ( k, κ ) ≡ κ (cid:90) κ − κ dκ (cid:48) v ( k, κ (cid:48) ) = Im∆( k + iκ ) κ . (A16)0To show the monotonicity of max k ∈ [ − π,π ] | V ( k, κ ) | , it is suf-ficient to show that V ( k, κ ) as a function of k is majorized by V ( k, κ (cid:48) ) for ∀ κ (cid:48) > κ . By the statement that an integrablereal function f : I ≡ [ a, b ] → R majorizes f : I → R , wemean that there exists a kernel K ( x ; x (cid:48) ) : I × I → R + (cid:83) { } satisfying (cid:90) I dxK ( x ; x (cid:48) ) = (cid:90) I dx (cid:48) K ( x ; x (cid:48) ) = 1 (A17)and f ( x ) = (cid:90) I dx (cid:48) K ( x ; x (cid:48) ) f ( x (cid:48) ) . (A18)Such a definition is a straightforward generalization of themajorization for real vectors, which can also be regarded asfunctions with I being a discrete set. That is, a real vector a is majorized by b if there exists a doubly stochastic matrix M ds , whose entries are all non-negative and the sum of eachrow/column equals to one, such that a = M ds b [135]. Notethat K ( x ; x (cid:48) ) is a continuous version of M ds . If f majorizes f , for ∀ x ∈ I , we have f ( x ) ≤ (cid:90) I dx (cid:48) K ( x ; x (cid:48) ) max x ∈ I f ( x ) = max x ∈ I f ( x ) (A19)due to K ( x ; x (cid:48) ) ≥ , f ( x (cid:48) ) ≤ max x ∈ I f ( x ) for ∀ x (cid:48) ∈ I and Eq. (A17). Since Eq. (A19) is true for ∀ x ∈ I , wehave max x ∈ I f ( x ) ≤ max x ∈ I f ( x ) . Similarly, we have min x ∈ I f ( x ) ≤ min x ∈ I f ( x ) and thus max x ∈ I | f ( x ) | ≥ max x ∈ I | f ( x ) | .Now let us consider the kernel that transforms V ( k, κ ) into V ( k, κ ) with κ ≥ κ . Since v ( k, κ ) is a harmonic function which is periodic in k and even in κ , it can generally be ex-panded as v ( k, κ ) = ∞ (cid:88) n =1 [ a n cos( nk ) + b n sin( nk )] cosh( nκ ) , (A20)where a n , b n ∈ R . Accordingly, we can obtain a general formof V ( k, κ ) in Eq. (A16): V ( k, κ ) = ∞ (cid:88) n =1 [ A n cos( nk ) + B n sin( nk )] sinh( nκ ) κ , (A21)where we have redefined the coefficients as A n = a n n and B n = b n n . This general form (A21) implies the followingkernel that transforms V ( k, κ ) into V ( k, κ ) : K ( k, κ ; k (cid:48) , κ ) = 12 π + 1 π ∞ (cid:88) n =1 κ sinh( nκ ) κ sinh( nκ ) cos[ n ( k − k (cid:48) )] , (A22)where the constant term is determined by Eq. (A17). To seewhy the kernel takes this form, we have only to note that cos[ n ( k − k (cid:48) )] = cos( nk ) cos( nk (cid:48) ) + sin( nk ) sin( nk (cid:48) ) and (cid:82) π − π dk (cid:48) π cos( nk (cid:48) ) V ( k (cid:48) , κ ) ( (cid:82) π − π dk (cid:48) π cos( nk (cid:48) ) V ( k (cid:48) , κ ) ) givesthe Fourier coefficient before cos( nk ) ( sin( nk ) ), whichshould be modified in an n -dependent manner following the N = = = = - - - k - k' P a r t i a l s u m s N = = = - - - k - k' (a) (b) Re k Im k R R [(2 m + 1) ] i [(2 m + 1) + ] i (1)1 (c) FIG. 12. Partial sums (cid:80) ( N − / n = − ( N − / of the kernel function K ( k, κ ; k (cid:48) , κ ) with κ = 2 and κ = 0 expanded as in (a)Eq. (A23) and (b) Eq. (A26). The curve labeled by “Fourier” in (b) isnothing but the curve in (a) for N = 9 . Note that the partial sum ofthe Fourier series may be not positive for certain k − k (cid:48) (see arrowsin (a)), but K ( k, κ ; k (cid:48) , κ ) is clearly positive from Eq. (A26). (c)Contour integral and poles of Y ( k ; κ , κ ) . In the limit of R → ∞ ,as in the case in Eq. (A29), we should sum up all the residues withrespect to the poles on the lower-half imaginary axis. change of κ . We can rewrite Eq. (A22) into a more compactform K ( k, κ ; k (cid:48) , κ ) = 12 π (cid:88) n ∈ Z κ sinh( nκ ) κ sinh( nκ ) e in ( k − k (cid:48) ) . (A23)We can use Eq. (A23) to check that K ( k, κ ; k (cid:48) , κ ) = 12 π (cid:88) n ∈ Z e in ( k − k (cid:48) ) = δ ( k − k (cid:48) ) . (A24)Another special case that transforms V ( k, κ ) into V ( k,
0) = v ( k, reads K ( k, k (cid:48) , κ ) = 12 π (cid:88) n ∈ Z nκ sinh( nκ ) e in ( k − k (cid:48) ) . (A25)In general, whenever κ > κ , the Fourier coefficients inEq. (A23) decays as e − ( κ − κ ) | n | for large | n | , implying theconvergence.The remaining problem is to confirm whether Eq. (A23) isnon-negative for ∀ κ > κ = 0 . This is not clear at firstglance since the partial sum in Eq. (A23) generally containsnegative parts (see Fig. 12(a)), especially when κ is close to κ . Nevertheless, the positivity becomes clear in a differentexpansion: K ( k, κ ; k (cid:48) , κ ) = (cid:88) n ∈ Z Y ( k − k (cid:48) + 2 nπ ; κ , κ ) , (A26)where Y ( k ; κ , κ ) = sin( κ κ π )2 κ [cosh( πκ k ) + cos( κ κ π )] (A27)1 f m j- ,j -f m j- ,j-1 f m j ,j -f m j- ,j-1 f m j ,j -f m j ,j-1 j - - (a)(b) (c) ( m j , j )( m j-1 , j - )( m j+1 , j + ) FIG. 13. (a) 2D array with the maximum of each row marked as ( m j , j ) . The solid arrows denote the differences between the startingpoint to the end point, which are all positive. The dashed arrowshighlight the order relations (descent along the arrow) between thesedifferences. (b) A randomly generated array with L = 500 and (c)the differences in (a) for the j th rows with j = 1 , , ..., . is positive over R . In perticular, when κ = 0 , Eq. (A27)becomes Y ( k ; 0 , κ ) = π κ [cosh( πκ k ) + 1] . (A28)To prove Eq. (A26), we have only to calculate the n th Fouriercoefficient of the rhs: (cid:90) π − π dk π (cid:88) m ∈ Z Y ( k + 2 mπ ; κ , κ ) e − ink = (cid:90) ∞−∞ dk π Y ( k ; κ , κ ) e − ink = (cid:90) ∞−∞ dk πiκ (cid:34) e − ink e − π ( k + iκ κ + 1 − e − ink e − π ( k − iκ κ + 1 (cid:35) = κ πκ ∞ (cid:88) m =0 ( e − n [(2 m +1) κ − κ ] − e − n [(2 m +1) κ + κ ] )= κ πκ sinh( nκ )sinh( nκ ) , (A29)where we have used the residue theorem in deriving the fourthequality (see Fig. 12(c)). The final result in Eq. (A29) indeedcoincides with that in Eq. (A23).In fact, a simple picture is available from a discrete versionof the problem. We consider a 2D array f : Z × Z → R , whichsatisfies that for ∀ j, j (cid:48) ∈ Z , f j,j (cid:48) = − f j, − j (cid:48) (so that f j, = 0 ), f j + L,j (cid:48) = f j,j (cid:48) ( L ∈ Z + ) and the discrete Laplace equation f j,j (cid:48) = f j +1 ,j (cid:48) + f j − ,j (cid:48) + f j,j (cid:48) +1 + f j,j (cid:48) − . (A30)By imposing the boundary condition f j, = − f j, − = h ∆ (cid:48) (cid:18) πjL (cid:19) , (A31) we have f ( k, κ ) ≡ Im∆( k + iκ ) = lim L →∞ ,h → f (cid:98) kL π (cid:99) , (cid:98) κh (cid:99) since f ( k, κ ) is a harmonic function satisfying f ( k, κ ) = f ( k + 2 π, κ ) = − f ( k, − κ ) ,∂ κ f ( k, κ ) | κ =0 = ∆ (cid:48) ( k ) . (A32)The discrete counterpart of the monotonicity of κ − max k ∈ [ − π,π ] f ( k, κ ) reads j (cid:48) + 1 max j ∈ Z L f j,j (cid:48) +1 ≥ j (cid:48) max j ∈ Z L f j,j (cid:48) (A33)for ∀ j (cid:48) ≥ . In fact, we can prove a stronger result max j ∈ Z L f j,j (cid:48) +1 − max j ∈ Z L f j,j (cid:48) ≥ max j ∈ Z L f j,j (cid:48) − max j ∈ Z L f j,j (cid:48) − , (A34)which implies Eq. (A33). Denoting m j (cid:48) as the horizontal labelwhere f m j (cid:48) ,j (cid:48) reaches the maximum for a given j (cid:48) , we have max j ∈ Z L f j,j (cid:48) +1 − max j ∈ Z L f j,j (cid:48) ≥ f m j (cid:48) ,j (cid:48) +1 − f m j (cid:48) ,j (cid:48) =3 f m j (cid:48) ,j (cid:48) − f m j (cid:48) − ,j (cid:48) − f m j (cid:48) +1 ,j (cid:48) − f m j (cid:48) ,j (cid:48) − ≥ f m j (cid:48) ,j (cid:48) − f m j (cid:48) ,j (cid:48) − ≥ max j ∈ Z L f j,j (cid:48) − max j ∈ Z L f j,j (cid:48) − , (A35)which completes the proof. We provide a schematic illustra-tion and a numerical verification in Fig. 13. Similarly, wecan prove the monotonicity of j (cid:48) min j ∈ Z L f j,j (cid:48) and thus themonotonicity of j (cid:48) max j ∈ Z L | f j,j (cid:48) | .The above idea can also be implemented directly in a con-tinuous manner, provided that k ∗ , which makes f ( k ∗ , κ ) max-imal or minimal for a given κ , forms a smooth curve of κ . Letus focus on the case of maximum since the minimum counter-part is quite similar. By definition, along this curve we have ∂ k f dk + ∂ k ∂ κ f dκ = 0 , ∂ k f = − ∂ κ f ≤ , (A36)so the second-order differential along this curve satisfies d f = ∂ k f dk + ∂ κ f dκ + 2 ∂ k ∂ κ f dkdκ = ∂ κ f dκ − ∂ k f dk + 2( ∂ k f dk + ∂ k ∂ κ f dκ ) dk = ∂ κ f (cid:34) (cid:18) dkdκ (cid:19) (cid:35)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k = k ∗ dκ . (A37)Introducing F ( κ ) ≡ f ( k ∗ ( κ ) , κ ) , we find from Eqs. (A36)and (A37) that F (cid:48)(cid:48) ( κ ) ≥ . (A38)This result is sufficient to show κ − F ( κ ) ≥ κ − F ( κ ) , ∀ κ > κ ≥ . (A39)To see this, we consider the equivalent inequality F ( κ ) − F ( κ ) κ − κ ≥ F ( κ ) − F (0) κ . (A40)2According to the mean-value theorem, there exists ξ ∈ [ κ , κ ] and ξ ∈ [0 , κ ] such that F (cid:48) ( ξ ) = F ( κ ) − F ( κ ) κ − κ and F (cid:48) ( ξ ) = F ( κ ) − F (0) κ , so the above inequality becomes F (cid:48) ( ξ ) ≥ F (cid:48) ( ξ ) , which is ensured by Eq. (A38).
4. Example: SSH model
We consider a general quench in the SSH model: H ( k ) = − ( J + J cos k ) σ x − J sin kσ y → H ( k ) = − ( J (cid:48) + J (cid:48) cos k ) σ x − J (cid:48) sin kσ y , (A41) where σ x and σ y are Pauli matrices. To evaluate v LR , we haveonly to numerically maximize | Im (cid:15) k + iκ, ± | = (cid:115) (cid:20)(cid:113) ( J (cid:48) + J (cid:48) + 2 J (cid:48) J (cid:48) cos k cosh κ ) + (2 J (cid:48) J (cid:48) sin k sinh κ ) − ( J (cid:48) + J (cid:48) + 2 J (cid:48) J (cid:48) cos k cosh κ ) (cid:21) (A42)over k ∈ [ − π, π ] . Note that κ can be chosen freely, as longas | κ | < | ln J J | (otherwise P < ( k + iκ ) may become non-analytic). We plot v LR = κ − max k ∈ [ − π,π ] | Im (cid:15) k + iκ, ± | in Fig. 7(b) for the quench protocol used in the main text: J = J (cid:48) = 0 . and J = J (cid:48) = 1 . The κ dependence of v LR turns out to be rather weak, at least for κ ∈ [0 , ln 2) .To evaluate C , we first write down the initial Bloch projec- tor P < ( k ) = 12 J + J e − ik √ J + J +2 J J cos kJ + J e ik √ J + J +2 J J cos k . (A43)By introducing F ( k, κ, J , J ) = J + J e κ + 2 J J e κ cos k (cid:112) ( J + J ) + 4 J J ( J + J ) cosh κ cos k + 2 J J (cos 2 k + cosh 2 κ ) , (A44)we can express the norm of Eq. (A43) with respect to a com-plex wave number as (cid:107) P < ( k + iκ ) (cid:107) = 12 [ F ( k, κ, J , J ) + F ( k, − κ, J , J )] . (A45)Moreover, the left and right eigenvectors are found to be (cid:104) k = 0 , a | u R k + iκ, ± (cid:105) = 1 √ (cid:34) ∓ (cid:113) J (cid:48) + J (cid:48) e ik − κ J (cid:48) + J (cid:48) e − ik + κ (cid:35) , (cid:104) k = 0 , a | u L k + iκ, ± (cid:105) = 1 √ (cid:34) ∓ (cid:113) J (cid:48) + J (cid:48) e ik + κ J (cid:48) + J (cid:48) e − ik − κ (cid:35) , (A46)where a labels the sublattice degrees of freedom, leading to (cid:107)| u R k + iκ, ± (cid:105)(cid:107) = (cid:114)
12 [1 + F ( k, − κ, J (cid:48) , J (cid:48) )] , (cid:107)| u L k + iκ, ± (cid:105)(cid:107) = (cid:114)
12 [1 + F ( k, κ, J (cid:48) , J (cid:48) )] . (A47) Substituting Eqs. (A45) and (A47) into Eq. (20) yields C = (cid:90) π − π dk π [1 + F ( k, − κ, J (cid:48) , J (cid:48) )][1 + F ( k, κ, J (cid:48) , J (cid:48) )] × [ F ( k, κ, J , J ) + F ( k, − κ, J , J )] . (A48)In Fig. 6(c), we choose κ = 0 . , leading to C (cid:39) . . Appendix B: Exact zero modes
In Sec. IV A, we have argued that the emergent Kramersdegeneracy and the nontrivial Z topology necessarily supportzero modes in R d in Eq. (28). Here, we prove this statementby showing that the invertibility of R d implies a trivial Z topological index. Theorem 4
Given two skew-symmetric real matrices R d and R o such that R ≡ σ ⊗ R d + σ x ⊗ R o is unitary and Pf R = − , we must have det R d = 0 .Proof.— If det R d (cid:54) = 0 , we can apply the formula Pf R = Pf R d Pf( R d − R o R − R o ) . (B1)3This formula can be derived from the identity Pf(
BAB T ) = det B Pf A, (B2)which is valid for arbitrary A T = − A and B , and (cid:32) R d R d − R o R − R o (cid:33) = (cid:32) − R o R − (cid:33) (cid:32) R d R o R o R d (cid:33) (cid:32) − R − R o (cid:33) . (B3)Since R is unitary, we have { R d , R o } = and R + R = − , leading to R d − R o R − R o = R d + R − R = R d + R − ( − − R ) = − R − . (B4)Combining Eq. (B4) with Eq. (B1), we have Pf R = Pf R d Pf( − R − ) = (Pf R d ) det R d = 1 , (B5)where we have used Eq. (B2) with A = B = R − and det A = (Pf A ) for ∀ A T = − A . This result (B5) contra-dicts the assumption Pf R = − , implying det R d = 0 . (cid:3) Appendix C: Proof of Weyl’s perturbation theorem
In this appendix, we follow Ref. [84] to prove Weyl’s per-turbation theorem. To this end, we first introduce the min-maxprinciple . Lemma 8 (Min-max principle)
For any Hermitian operator O on an n -dimensional Hilbert space, its j th largest eigen-value λ j ( j = 1 , , ..., n ) is given by λ j = max dim V = j min | ψ (cid:105)∈ V (cid:104) ψ | O | ψ (cid:105) = min dim W = n +1 − j max | ψ (cid:105)∈ W (cid:104) ψ | O | ψ (cid:105) , (C1) where V and W are Hilbert subspaces and | ψ (cid:105) is a normal-ized state vector.Proof.— We denote the eigenstate with eigenvalue λ j as | ψ j (cid:105) ( j = 1 , , ..., n ) and define the following two special classesof Hilbert spaces with dimensions j and n +1 − j , respectively: V j ≡ span {| ψ (cid:105) , | ψ (cid:105) , ..., | ψ j (cid:105)} ,W j ≡ span {| ψ j (cid:105) , | ψ j +1 (cid:105) , ..., | ψ n (cid:105)} . (C2)With V chosen to be V j , the rhs of the first line in Eq. (C1)gives λ j , implying λ j ≤ max dim V = j min | ψ (cid:105)∈ V (cid:104) ψ | O | ψ (cid:105) . (C3)Similarly, with a choice of W = W j , the second line inEq. (C1) gives λ j , implying λ j ≥ min dim W = n +1 − j max | ψ (cid:105)∈ W (cid:104) ψ | O | ψ (cid:105) . (C4) On the other hand, for an arbitrary Hilbert subspace V withdimension j , we must have dim( V (cid:84) W j ) ≥ . Otherwise, if V (cid:84) W j = ∅ , the full Hilbert space containing V (cid:83) W j willbe at least n + 1 dimensional, contradicting the assumption.This implies that for ∀ V with dim V = j , we have min | ψ (cid:105)∈ V (cid:104) ψ | O | ψ (cid:105) ≤ min | ψ (cid:105)∈ V (cid:84) W j (cid:104) ψ | O | ψ (cid:105)≤ max | ψ (cid:105)∈ W j (cid:104) ψ | O | ψ (cid:105) = λ j , (C5)leading to λ j ≥ max dim V = j min | ψ (cid:105)∈ V (cid:104) ψ | O | ψ (cid:105) . (C6)Combining Eqs. (C3) and (C6), we obtain the first line inEq. (C1). By analogy, from dim( V j (cid:84) W ) ≥ for ∀ W with dim W = n − j + 1 , we can derive λ j ≤ min dim W = n +1 − j max | ψ (cid:105)∈ W (cid:104) ψ | O | ψ (cid:105) . (C7)Combining Eqs. (C4) and (C7), we obtain the second line inEq. (C1). (cid:3) Now let us turn to the proof of Theorem 3. For ∀ V with dim V = j , we can decompose O as O (cid:48) + ( O − O (cid:48) ) to obtain min | ψ (cid:105)∈ V (cid:104) ψ | O | ψ (cid:105) = min | ψ (cid:105)∈ V ( (cid:104) ψ | O (cid:48) | ψ (cid:105) + (cid:104) ψ | O − O (cid:48) | ψ (cid:105) ) ≥ min | ψ (cid:105)∈ V (cid:104) ψ | O (cid:48) | ψ (cid:105) + min | ψ (cid:105)∈ V (cid:104) ψ | O − O (cid:48) | ψ (cid:105)≥ min | ψ (cid:105)∈ V (cid:104) ψ | O (cid:48) | ψ (cid:105) − (cid:107) O − O (cid:48) (cid:107) . (C8)After maximizing the rhs of Eq. (C8) with respect to V andusing the first line in Eq. (C1), we obtain λ j ≥ λ (cid:48) j − (cid:107) O − O (cid:48) (cid:107) . (C9)Following a similar procedure, we can derive max | ψ (cid:105)∈ W (cid:104) ψ | O | ψ (cid:105) ≤ max | ψ (cid:105)∈ W (cid:104) ψ | O (cid:48) | ψ (cid:105) + (cid:107) O − O (cid:48) (cid:107) , (C10)which gives rise to (due to the second line in Eq. (C1)) λ j ≤ λ (cid:48) j + (cid:107) O − O (cid:48) (cid:107) . (C11)Theorem 3 follows from the combination of Eqs. (C9) and(C11). Appendix D: Proof of Lemma 2
We first show the following lemma which provides the re-lation between | W ( L ) jα (cid:105) and | W ( ∞ ) jα (cid:105) . Lemma 9
Given | W ( ∞ ) jα (cid:105) as a Wannier function of an infinite1D lattice system, the Wannier function | W ( L ) jα (cid:105) of the corre-sponding finite system with length L reads | W ( L ) jα (cid:105) = (cid:88) n ∈ Z P Z L | W ( ∞ ) j + nL,α (cid:105) , (D1)4 W ( L )0
We briefly review Ref. [110] and introduce the several basicproperties of MPUs, which are crucial for proving the mainresult. By definition, for ∀ L ∈ Z + , U in Eq. (22) in the maintext obeys U † U = I ≡ ⊗ L ⇒ d − L Tr[ U † U ] = 1 , (E1)5implying that the spectrum of the quantum channel E U ( · ) = d − (cid:80) dj,j (cid:48) =1 U jj (cid:48) · U † jj (cid:48) consists of a single and all the othersbeing zero. This property enforces d − U jj (cid:48) to be a normaltensor [103], in the sense that E U has a unique fixed point,which is Hermitian and positivie-definite. By choosing thegauge properly, i.e., performing a similarity transformation U jj (cid:48) → X − U jj (cid:48) X on the virtual level, U jj (cid:48) can always bemade to satisfy d UU ρ = ρ , d UU = , (E2)where ρ † = ρ , ρ > and Tr ρ = 1 . It can be proved [110]that for ∀ k ∈ Z + U k U k ρ = ⊗ k , (E3)where U k is related to U by putting together k sites into one,as given in Eq. (23).An MPU is locality-preserving, in the sense that a local op-erator acting nontrivially on a finite segment stays local afterbeing evolved by an MPU. This property can be understoodfrom the following theorem [110]. Theorem 5
It is always possible to make an MPU generatedby U simple by putting together k ≤ D U ( D U : bond dimen-sion of the MPU) sites into one (see Eq. (23)). By simple, wemean that the building block U satisfies UU UU = UU UU ρ , (E4) where ρ is given in Eq. (E2). Denoting k as the smallest integer such that U k is simple, theabove theorem (5) implies that a local operator spreads by nomore than k sites upon being evolved by the MPU. To seethis, consider a general local operator O L acting nontriviallyon a segment with length l . Applying Eqs. (E3) and (E4) to O (cid:48) L = U O L U † yields ......... ... U U U U U U UU U U U U U U O L l = U k U l U k U k U l U k O L ρ (E5) = ...... O (cid:48) L l + 2 k ,where local identities are omitted. Note that Fig. 2(c) in themain text is the special case of l = 1 , which is neverthe-less sufficient for obtaining Eq. (E5) since a local operatorcan generally be expressed as O L = (cid:80) { j s } l s =1 c j j ...j l O j ⊗ O j ⊗ ... ⊗ O j l with O j s being an on-site operator [111].Besides the locality-preserving property, an MPU is by def-inition unitary, implying the following lemma. Lemma 10
Given an MPS | Ψ (cid:105) generated by { A j } dj =1 and anMPU U generated by {U jj (cid:48) } dj,j (cid:48) =1 , with the associated unitalchannel of the evolved MPS | Ψ (cid:48) (cid:105) = U | Ψ (cid:105) denoted as E (cid:48) ( · ) ≡ (cid:80) dj =1 A (cid:48) j · A (cid:48)† j , the spectrum of E (cid:48) is the same as that of E ( · ) ≡ (cid:80) dj =1 A j · A † j , the unital channel associated with | Ψ (cid:105) .Proof.— The main idea is already mentioned in Ref. [111]— we only have to combine Lemma 9 in Ref. [181] with
Tr[( (cid:80) j A (cid:48) j ⊗ ¯ A (cid:48) j ) L ] = Tr[( (cid:80) j A j ⊗ ¯ A j ) L ] , ∀ L ∈ Z + , whichresults from the unitary nature of the time evolution:... ¯ A (cid:48) A (cid:48) ¯ A (cid:48) A (cid:48) ¯ A (cid:48) A (cid:48) L = ... ¯ A UU A ¯ A UU A ¯ A UU AL = ... ¯ AA ¯ AA ¯ AAL . (cid:3) (E6)6A direct corollary of Lemma 10 is that the spectrum of E , i.e.,the transfer matrix of an MPS, stays invariant during the stro-boscopic time evolution governed by an MPU. In particular, µ as the spectral radius of E − E ∞ is conserved. Appendix F: Interacting systems undergoing continuousevolution
We argue that the Lieb-Robinson bound on the many-bodyentanglement gap for MPUs implies qualitatively the same re-sult for continuous time evolutions generated by local Hamil-tonians. To this end, we first prove that the difference in time-evolution operators, which can be highly nonlocal, rigorouslybounds the difference in any reduced density operator after thetime evolution.
Lemma 11
Given an arbitrary wave function | Ψ (cid:105) defined ona bipartite system S (cid:83) ¯ S and two unitaries U and U (cid:48) sat-isfying (cid:107) U − U (cid:48) (cid:107) ≤ (cid:15) , denoting ρ S ≡ Tr ¯ S | Ψ (cid:105)(cid:104) Ψ | and ρ (cid:48) S ≡ Tr ¯ S | Ψ (cid:48) (cid:105)(cid:104) Ψ (cid:48) | as the density operators of the evolvedwave functions | Ψ (cid:105) ≡ U | Ψ (cid:105) and | Ψ (cid:48) (cid:105) ≡ U (cid:48) | Ψ (cid:105) , we have (cid:107) ρ S − ρ (cid:48) S (cid:107) ≤ (cid:15) .Proof.— We first point out a useful norm inequality for thecommutator of a positive-semidefinite Hermitian operator A ( A † = A and A ≥ ) and an arbitrary operator B [182]: (cid:107) [ A, B ] (cid:107) ≤ (cid:107) A (cid:107)(cid:107) B (cid:107) . (F1)Note that there is an improvement of factor compared with (cid:107) [ A, B ] (cid:107) ≤ (cid:107) A (cid:107)(cid:107) B (cid:107) , which holds for arbitrary A and B .Let us move on to prove Lemma 11. Noting that ρ S − ρ (cid:48) S is Hermitian, according to the definition of the operator norm,we have (cid:107) ρ S − ρ (cid:48) S (cid:107) = max (cid:107)| ψ (cid:105)(cid:107) =1 (cid:104) ψ | ρ S − ρ (cid:48) S | ψ (cid:105) = max P ψ Tr[ P ψ ⊗ ¯ S ( | Ψ (cid:105)(cid:104) Ψ | − | Ψ (cid:48) (cid:105)(cid:104) Ψ (cid:48) | )]= max P ψ (cid:104) Ψ | U ( P ψ ⊗ ¯ S ) U † − U (cid:48) ( P ψ ⊗ ¯ S ) U (cid:48)† | Ψ (cid:105)≤ max P ψ (cid:107) U ( P ψ ⊗ ¯ S ) U † − U (cid:48) ( P ψ ⊗ ¯ S ) U (cid:48)† (cid:107)≤ max P ψ (cid:107) [ P ψ ⊗ ¯ S , U † U (cid:48) ] (cid:107) , (F2)where P ψ ≡ | ψ (cid:105)(cid:104) ψ | is a rank-one projector, i.e., P ψ = P ψ and Tr P ψ = 1 . Since P ψ ⊗ ¯ S ≥ and (cid:107) P ψ ⊗ ¯ S (cid:107) = 1 , wefind from Eq. (F1) that for ∀ P ψ (cid:107) [ P ψ ⊗ ¯ S , U † U (cid:48) ] (cid:107) = (cid:107) [ P ψ ⊗ ¯ S , U † U (cid:48) − ] (cid:107)≤(cid:107) U † U (cid:48) − (cid:107) = (cid:107) U − U (cid:48) (cid:107) . (F3)Combining Eq. (F3) with Eq. (F2), we obtain (cid:107) ρ S − ρ (cid:48) S (cid:107) ≤ (cid:107) U − U (cid:48) (cid:107) ≤ (cid:15). (cid:3) (F4)Now let us discuss how to combine Lemma 11 with Theo-rem 2 and the main result of Ref. [115] to bound the many-body entanglement gap in a continuous quench dynamics e iHt (1) uv (2)(3) U vu = (4) UU = u † u (5)Spec ...... p ⇤ p ⇤ ¯ AA l (6)Spec ...... (7)... ... t A U (8)... ... O j UU (9)1 e iHt (1) uv (2)(3) U vu = (4) UU = u † u (5)Spec ...... p ⇤ p ⇤ ¯ AA l (6)Spec ...... (7)... ... t A U (8)... ... O j UU (9)1 e iHt (1) uv (2)(3) U vu = (4) UU = u † u (5)Spec ...... p ⇤ p ⇤ ¯ AA l (6)Spec ...... (7)... ... t A U (8)... ... O j UU (9)1 e iHt (1) uv (2)(3) U vu = (4) UU = u † u (5)Spec ...... p ⇤ p ⇤ ¯ AA l (6)Spec ...... (7)... ... t A U (8)... ... O j UU (9)1 e iHt (1) uv (2)(3) U vu = (4) UU = u † u (5)Spec ...... p ⇤ p ⇤ ¯ AA l (6)Spec ...... (7)... ... t A U (8)... ... O j UU (9)1 (a)(b)(c)(d) (e) FIG. 15. (a) Continuous time evolution e − iHt generated by a localHamiltonian H and its (b) partial and (c) complete approximationsby local unitaries, which are denoted as U (cid:48) c and U c . If we focuson a subsystem marked in the red rectangle, (b) and (c) make nodifference — starting from the same initial state, the reduced den-sity operators of the states evolved by (b) and (c) coincide with eachother. (d) Building block of (c), which can be regarded as an MPU.(e) Quantum channel associated with (d) takes the form ρ Tr[ ... ] . starting from an SPT MPS. According to Ref. [115], given alocal Hamiltonian H = (cid:80) j h j , we can approximate it with abilayer unitary circuit or quantum cellular automaton U c suchthat (cid:107) e − iHt − U c (cid:107) ≤ O (cid:18) L | Ω | (cid:19) e − κ c ( | Ω |− v c t ) , (F5)where L is the total system size, | Ω | is the number of sites thata single unitary acts on, and κ c and v c are constants indepen-dent of L . While the bound diverges in the thermodynamiclimit, we can make a cut off of the circuit approximation atthe length scale of the subsystem without changing the re-duced density operator. For such a truncated circuit U (cid:48) c , wehave (cid:107) e − iHt − U (cid:48) c (cid:107) ≤ O (cid:18) l | Ω | (cid:19) e − κ c ( | Ω |− v c t ) , (F6)where the rhs is finite in the thermodynamic limit. SeeFigs. (15)(a), (b) and (c) for the relations and distinctions in e − iHt , U (cid:48) c and U c .On the other hand, to apply Theorem 2, we should still re-gard the approximated reduced density operator as resultingfrom the time evolution by U c . After putting | Ω | sites into one,we can regard U c as an MPU generated by the block given inFig. 15(d), which is simple (see Fig. 15(e)). Under such arescaling, the parameters in Theorem 2 reads k = t = 1 , D U = d | Ω | and µ → µ | Ω | , l → l | Ω | , (F7)7while D stays invariant. Therefore, the many-body entan-glement gap of the approximated density operator ρ (cid:48) [1 ,l ] isbounded by ∆ (cid:48) mbE ≤ poly (cid:18) l | Ω | (cid:19) e − κl + κ (cid:48) | Ω | , (F8)where κ = − ln µ and κ (cid:48) = ln d + ( D + 1) κ, (F9)provided that l | Ω | − ≥ coth (cid:16) κ | Ω | (cid:17) . (F10)As for the many-body entanglement gap of the exact den-sity operator ρ [1 ,l ] , we have ∆ mbE ≡ | ζ r − ζ |≤ | ζ r − ζ (cid:48) r | + | ζ r − ζ (cid:48) | + | ζ (cid:48) r − ζ (cid:48) |≤ (cid:107) ρ [1 ,l ] − ρ (cid:48) [1 ,l ] (cid:107) + ∆ (cid:48) mbE ≤ (cid:107) e − iHt − U (cid:48) c (cid:107) + ∆ (cid:48) mbE , (F11)where we have used Lemma 11 in deriving the last inequality.Combining Eqs. (F6) and (F8) with Eq. (F11) and taking (weuse (cid:39) since | Ω | should be an even integer dividing l ) | Ω | (cid:39) κ c v c t + κ (cid:48) lκ c + κ (cid:48) , (F12)we obtain ∆ mbE ≤ O (1) e − κ c κκ c+ κ (cid:48) ( l − κ (cid:48) κ v c t ) , (F13)which indeed takes the form of Lieb-Robinson bound. Wenote that, at large length scales such that coth( κ | Ω | ) = 1 + o (1) , Eq. (F10) can be satisfied by t < κ c + κ (cid:48) κ c (cid:18) − κκ c + κ (cid:48) − o (1) (cid:19) lv c , (F14)where the rhs is positive due to κ (cid:48) > κ , which arises fromEq. (F9) and D ≥ for an arbitrary SPT MPS.On the other hand, it seems that we cannot derive a Lieb-Robinson bound by simply combining Theorem 2 with theerrors in approximating ground states with MPSs [141]. Evenif we only require the MPS approximation to be locally good[183, 184], the needed bond dimension scales like a polyno-mial of the inverse of error, implying an exponentially largebond dimension and a doubly exponentially large (due to theprefactor) bound predicted by Theorem 2 for an exponentiallysmall error. We leave this problem for future work, whichprobably requires new ideas to directly estimate the entangle-ment gap in the exact ground state. Appendix G: Convergence bounds for unital channels
We briefly review Ref. [144] and discuss how to bound (cid:107)E l − E ∞ (cid:107) by using function algebra. Applying Lemma 6to M = E − E ∞ leads to the following theorem. Theorem 6 (Main result of Ref. [144] for unital channels)
Let E be a unital channel acting on a D × D -dimensionaloperator space such that (cid:107)E n (cid:107) ≤ C for ∀ n ∈ N . Let µ ≡ lim n →∞ (cid:107)E n − E ∞ (cid:107) n be the spectral radius of E − E ∞ . We denote the minimal polynomial of E − E ∞ as m ( z ) = m E−E ∞ ( z ) = (cid:80) Jj =1 ( z − µ j ) s j , where µ j ’sare different eigenvalues of E − E ∞ and s j is the size ofthe largest Jordan block with eigenvalue µ j , and define thecorresponding Blaschke product as B ( z ) ≡ J (cid:89) j =1 (cid:18) z − µ j − ¯ µ j z (cid:19) s j . (G1) Then, for l > µ − µ , we have (cid:107)E l − E ∞ (cid:107) ≤ C (cid:107) z l (cid:107) W/mW ≤ µ l +1 e C (cid:112) | m | ( | m | + 1) l [1 − (1 + l − ) µ ] × sup | z | =(1+ l − ) µ (cid:12)(cid:12)(cid:12)(cid:12) B ( z ) (cid:12)(cid:12)(cid:12)(cid:12) , (G2) where | m | = (cid:80) Jj =1 s j is the degree of m and C is upperbounded by (cid:113) D [145]. Since the detailed proof in Ref. [144] is rather technical, itis worthwhile to sketch the outline here. First, we note that“ sup ” in Eq. (G2) arises from the following Cauchy-Schwarzinequality: (cid:107) f r (cid:107) W = (cid:88) p ∈ N r p | f p |≤ (cid:118)(cid:117)(cid:117)(cid:117)(cid:116)(cid:88) p ∈ N r p (cid:88) p ∈ N | f p | = (cid:115) − r sup ≤ ρ< (cid:90) π dφ π | f ( ρe iφ ) | ≤ (cid:107) f (cid:107) H ∞ √ − r , ∀ f ∈ W, (G3)where r ∈ (0 , can arbitrarily be chosen, f r ( z ) ≡ f ( rz ) and (cid:107) f (cid:107) H ∞ ≡ sup z ∈ D | f ( z ) | . In the typical case with s j = 1 ,i.e., if M = E − E ∞ is diagonalizable [185], (cid:107) z l (cid:107) W/mW is,by definition, upper bounded by (cid:107) g r (cid:107) W as long as g r ( µ j ) = g ( rµ j ) = µ lj for ∀ j = 1 , , ..., J . An example is g ( z ) = J (cid:88) j =1 µ lj ˜ B j ( z )˜ B j ( rµ j ) , (G4)where ˜ B j ( z ) ≡ ˜ B ( z ) z − rµ j and ˜ B ( z ) ≡ (cid:81) Jj =1 z − rµ j − r ¯ µ j z is the mod-ified Blaschke product. Combining Eqs. (G3) and (G4), we8obtain (cid:107) z l (cid:107) W/mW ≤ (cid:107) g r (cid:107) W ≤ (cid:107) g (cid:107) H ∞ √ − r = 1 √ − r sup | z | =1 | g ( z ) | = 1 √ − r sup | z | =1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) J (cid:88) j =1 µ lj ( z − rµ j ) ˜ B j ( rµ j ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (G5)where | ˜ B ( z ) | = 1 for ∀| z | = 1 is used. We will eventuallyarrive at Eq. (G2) by further bounding the rightmost expres-sion in Eq. (G5), which can be rewritten in terms of a contourintegral as (cid:73) | w | =(1+ l − ) µ dw πi w l ˜ B r ( w )( z − rw ) (G6)for l > µ − µ , and set r = (cid:115) − − (1 + l − ) µ | m | . (G7) Appendix H: Generalization to finite interacting systems
We generalize Theorem 2 to the case of finite L . While westill have the decomposition given in Eq. (55), | Φ αβ (cid:105) ’s are nolonger orthogonal to each other. To compute the ES, we usethe following generalization of Lemma 4: Lemma 12
For a bipartite state | Ψ (cid:105) = (cid:80) Jj =1 | φ j (cid:105)| ψ j (cid:105) ,where | φ j (cid:105) ’s and | ψ j (cid:105) ’s are generally neither normalized nororthogonal to each other, the entanglement spectrum undersuch a bipartition coincides with the spectrum of ¯ M ψ M φ ¯ M ψ ( ¯ M ψ : complex conjugation of M ψ ) or ¯ M φ M ψ ¯ M φ (except forzeros), where [ M φ ] jj (cid:48) = (cid:104) φ j | φ j (cid:48) (cid:105) and [ M ψ ] jj (cid:48) = (cid:104) ψ j | ψ j (cid:48) (cid:105) .Proof.— It is equivalent to consider the spectrum of ρ φ =Tr ψ | Ψ (cid:105)(cid:104) Ψ | and that of ρ ψ = Tr φ | Ψ (cid:105)(cid:104) Ψ | . To be specific, wefocus on the former, which can be explicitly written as ρ φ = (cid:88) j,j (cid:48) | φ j (cid:105)(cid:104) φ j (cid:48) | Tr[ | ψ j (cid:105)(cid:104) ψ j (cid:48) | ]= (cid:88) j,j (cid:48) [ M ψ ] jj (cid:48) | φ j (cid:105)(cid:104) φ j (cid:48) | . (H1)Since M ψ is Hermitian, it can be expressed as U † Λ U with Λ = diag { Λ k } Jk =1 and U being unitary. To be concrete,we have [ M ψ ] jj (cid:48) = (cid:80) j (cid:48)(cid:48) ¯ U j (cid:48)(cid:48) j (cid:48) Λ j (cid:48)(cid:48) U j (cid:48)(cid:48) j . Introducing | ˜ φ j (cid:105) = (cid:80) j (cid:48) (cid:112) Λ j U jj (cid:48) | φ j (cid:48) (cid:105) , we can rewrite ρ φ as ρ φ = (cid:88) j | ˜ φ j (cid:105)(cid:104) ˜ φ j | (H2) to which we can apply Lemma 4 – the spectrum of ρ φ coin-cides with that of (cid:104) ˜ φ j | ˜ φ j (cid:48) (cid:105) = (cid:88) j (cid:48)(cid:48) ,j (cid:48)(cid:48)(cid:48) (cid:112) Λ j Λ j (cid:48) U j (cid:48) j (cid:48)(cid:48)(cid:48) ¯ U jj (cid:48)(cid:48) (cid:104) φ j (cid:48)(cid:48) | φ j (cid:48)(cid:48)(cid:48) (cid:105) = (cid:88) j (cid:48)(cid:48) ,j (cid:48)(cid:48)(cid:48) (cid:112) Λ j ¯ U jj (cid:48)(cid:48) [ M φ ] j (cid:48)(cid:48) j (cid:48)(cid:48)(cid:48) U T j (cid:48)(cid:48)(cid:48) j (cid:48) (cid:112) Λ j (cid:48) = [ √ Λ ¯
U M φ U T √ Λ] jj (cid:48) . (H3)From the fact that unitary conjugation preserves the spectrum,we know that the spectrum of ρ φ should be given by that of U T √ Λ ¯
U AU T √ Λ ¯ U = ¯ M ψ M φ ¯ M ψ . (H4)Note that the spectrum of ¯ M ψ M φ ¯ M ψ is nothing but thesquared absolute values of the singular values of M φ ¯ M ψ ,which are the same as those of ( M φ ¯ M ψ ) T = M ψ ¯ M φ andthus give the same spectrum as that of ¯ M φ M ψ ¯ M φ (this candirectly be obtained by considering ρ ψ following a similaranalysis as above). (cid:3) In fact, this result has already been obtained in Ref. [47],where it is used to calculate the ES of a projected entangled-pair state.To bound the many-body entanglement gap, we need thefollowing lemma.
Lemma 13
Let M , M (cid:48) , M , M (cid:48) be non-negative definiteHermitian matrices and let the j th largest eigenvalue of ¯ M (cid:48) M ¯ M (cid:48) and that of ¯ M (cid:48) M ¯ M (cid:48) be denoted as λ j and λ j , respectively. Then for ∀ j , we have | λ j − λ j | ≤ min {(cid:107) ¯ M (cid:48) δ (cid:107) + (cid:107) ¯ M δ (cid:48) ¯ M (cid:107) , (cid:107) ¯ M δ (cid:48) (cid:107) + (cid:107) ¯ M (cid:48) δ ¯ M (cid:48) (cid:107)} + (cid:107) ¯ δ (cid:48) δ (cid:107) , (H5) where δ ≡ M − M and δ (cid:48) ≡ M (cid:48) − M (cid:48) .Proof.— We first note the following useful norm inequality:for any two Hermitian matrices A and B with B ≥ (so that B is well-defined), we have (cid:107) B AB (cid:107) ≤ (cid:107) AB (cid:107) . (H6)This is because the spectrum of B AB coincides with thatof AB due to Tr[( B AB ) n ] = Tr[( AB ) n ] for ∀ n ∈ N [181]. Moreover, B AB is Hermitian so its norm is nothingbut the spectral radius, which is no more than the norm of AB .This result is a special case of Proposition IX.1.1 in Ref. [84].Let us turn to the proof of the lemma. Denoting the j thlargest eigenvalue of ¯ M (cid:48) M ¯ M (cid:48) as λ (cid:48) j , which is also the j thlargest eigenvalue of ¯ M M (cid:48) ¯ M , we use Weyl’s perturbation9theorem to obtain | λ j − λ j | ≤ | λ j − λ (cid:48) j | + | λ (cid:48) j − λ j |≤ (cid:107) ¯ M (cid:48) M ¯ M (cid:48) − ¯ M (cid:48) M ¯ M (cid:48) (cid:107) + (cid:107) ¯ M M (cid:48) ¯ M − ¯ M M (cid:48) ¯ M (cid:107) = (cid:107) ¯ M (cid:48) δ ¯ M (cid:48) (cid:107) + (cid:107) ¯ M δ (cid:48) ¯ M (cid:107)≤ (cid:107) ¯ M (cid:48) δ (cid:107) + (cid:107) ¯ M δ (cid:48) ¯ M (cid:107)≤ (cid:107) ¯ M (cid:48) δ (cid:107) + (cid:107) ¯ M δ (cid:48) ¯ M (cid:107) + (cid:107) ¯ δ (cid:48) δ (cid:107) , (H7)where we have used (cid:107) A + B (cid:107) ≤ (cid:107) A (cid:107) + (cid:107) B (cid:107) in the last step.Replacing M and M with M (cid:48) and M (cid:48) , respectively, and fol-lowing a similar procedure, we obtain | λ j − λ j | ≤ (cid:107) ¯ M δ (cid:48) (cid:107) + (cid:107) ¯ M (cid:48) δ ¯ M (cid:48) (cid:107) + (cid:107) ¯ δδ (cid:48) (cid:107) . (H8)Combining Eqs. (H7) and (H8), we obtain Eq. (H5). (cid:3) To apply Lemma 13 to the many-body entanglement gap,we have only to choose M αβ,α (cid:48) β (cid:48) = a L (cid:104) α (cid:48) |E l ( | β (cid:48) (cid:105)(cid:104) β | ) | α (cid:105) ,M (cid:48) αβ,α (cid:48) β (cid:48) = a L (cid:104) β (cid:48) |E L − l ( | α (cid:48) (cid:105)(cid:104) α | ) | β (cid:105) ,M = a L v ⊗ Λ , M (cid:48) = a L Λ ⊗ v , (H9)where a L = (Tr E L ) − is a finite-size normalization factor.Following the derivation of Eq. (73) in the main text, we canupper bound each term on the rhs of Eq. (H5) as (cid:107) ¯ M (cid:48) δ (cid:107) ≤ a L (cid:18) D (cid:19) (cid:107)E l − E ∞ (cid:107) , (cid:107) ¯ M δ (cid:48) (cid:107) ≤ a L (cid:18) D (cid:19) (cid:107)E L − l − E ∞ (cid:107) , (cid:107) ¯ M (cid:48) δ ¯ M (cid:48) (cid:107) ≤ a L (cid:114) D (cid:107)E l − E ∞ (cid:107) , (cid:107) ¯ M δ (cid:48) ¯ M (cid:107) ≤ a L (cid:114) D (cid:107)E L − l − E ∞ (cid:107) , (cid:107) ¯ δ (cid:48) δ (cid:107) ≤ a L D (cid:107)E l − E ∞ (cid:107)(cid:107)E L − l − E ∞ (cid:107) . (H10)Using the techniques in Sec. V D to bound (cid:107)E l − E ∞ (cid:107) as-sociated with a time evolved MPS, we obtain the followingtheorem. Theorem 7 (Finite interacting systems)
Starting from anSPT MPS with length L and bond dimension D subject to theperiodic boundary condition, the many-body entanglementgap of a length- l subsystem after t time-evolution steps by a trivial symmetric MPU with bond dimension D U is boundedfrom above by ∆ mbE ≤ (Tr E L ) − [min { b / ( l, t ) + b / ( L − l, t ) ,b / ( L − l, t ) + b / ( l, t ) } + 4 b ( l, t ) b ( L − l, t )] (H11) for any min { l, L − l } − k t ≥ µ − µ , where b α ( l, t ) = C α ( l − k t ) D − e − l − vαtξ , (H12) with k , µ and ξ being the same as those in Theorem 2, v α =2 k − ( α + ) ln D U ln µ , and the coefficient C α = e − α D + α ( D +1) µ − D (1+ µ ) D + (1 − µ ) D − (H13) depends only on the initial state. Two remarks are in order here. First, we note that in thethermodynamic limit of the entire system, we have Tr E L = 1 and b α ( ∞ , t ) = 0 so that Theorem 2 is reproduced. Even if L is finite, we still find that ∆ mbE is exponentially small up to t ∼ min (cid:26) min { l, L − l } v / , max { l, L − l } v / , Lv (cid:27) , (H14)provided that min { l, L − l } > v / (1+ µ )( v / − k )(1 − µ ) . Second, in thespecial case of the zero correlation length, i.e., µ = 0 , whichcorresponds to fixed points of entanglement renormalization[186], we can infer from Theorem 7 that the degeneracy is exact up to t ∼ l k . This is intuitively rather clear since theentanglement edge modes are absolutely localized (without anexponential tail) for fixed-point states and it takes a finite timefor a nonzero overlap to develop between the edge modes bya locality-preserving MPU. Appendix I: Details on ES dynamics upon partial symmetrybreaking1. Flat-band model for class BDI → class D Thanks to the flat-band nature of the Hamiltonians given inEqs. (93) and (94), it suffices to consider the N sites clos-est to the entanglement cut (purple dashed line in Fig. 10(a)).Restricted to the single-particle Hilbert subspace of thesesites, the projector onto the Fermi sea of H reads P = (cid:76) Nj =1 ( σ + σ x ) . The ES dynamics is then determinedby the spectrum of E S ( t ) = P S P ( t ) P S , where P S = (cid:76) Nj =1 ( σ + σ z ) is the projector onto the left half N sitesand P ( t ) = e − iH sp t P e iH sp t , H sp = 1 ⊕ N − (cid:77) j =1 e itσ x ⊕ , (I1)where we have already set J = 1 . After straightforward cal-culations, we obtain the following matrix form of E S ( t ) :0 E S ( t ) = 12 − i sin t · · · i sin t − i sin t cos t · · · i sin t cos t − i sin t cos t · · · i sin t cos t · · · ... ... ... ... . . . ... ... · · · − i sin t cos t · · · i sin t cos t N × N , (I2)whose characteristic polynomial reads f N ( ξ ; t ) ≡ det[ ξ I N × N − E S ( t )]= (cid:18) ξ − (cid:19) F N − (cid:18) ξ −
12 ; sin 2 t (cid:19) − sin t F N − (cid:18) ξ −
12 ; sin 2 t (cid:19) . (I3)Here, F N ( x ; a ) is defined recursively as F N ( x ; a ) = xF N − ( x ; a ) − a F N − ( x ; a ) , (I4)with initial conditions F ( x ; a ) = x and F ( x ; a ) = x − a (or F − ( x ; a ) = 0 and F ( x ; a ) = 1 ). In fact, we have an an-alytic expression F N ( x ; a ) ≡ (cid:81) Nj =1 ( x − a cos jπN +1 ) , whichenjoys the properties F N ( bx ; ab ) = b N F ( x ; a ) , F N ( x ; a ) = F N ( x ; − a ) and, in particular, F N ( − x ; a ) = ( − ) N F N ( x ; a ) .Using the recursive relation (I4), we can rewrite f N ( ξ ; t ) into f N ( ξ ; t ) = F N (cid:18) ξ −
12 ; sin 2 t (cid:19) − sin t F N − (cid:18) ξ −
12 ; sin 2 t (cid:19) . (I5)The roots of f N ( ξ ; t ) give the ES dynamics. Since F N ( x ; a ) isan odd function of x for odd N , we have F n +1 (0; a ) = 0 sothat ξ = is always a root of f N ( ξ ; t ) if N is odd. Moreover,defining g n +1 ( ξ ; t ) ≡ f n +1 ( ξ ; t ) / ( ξ − ) , we find g n +1 (cid:18)
12 ; t (cid:19) = (cid:18) − sin t (cid:19) n (cid:16) n cos t (cid:17) ,f n (cid:18)
12 ; t (cid:19) = (cid:18) − sin t (cid:19) n t , (I6)which are generally nonzero except for some special timepoints. Therefore, for an even N , all the initial topologicalentanglement modes at ξ = split, while one and only one ξ = mode survives for an odd N .Let us calculate the full ES dynamics for N = 1 , , , , .When N = 1 , we have f ( ξ ; t ) = ξ − , implying the persis-tence of the topological entanglement mode at ξ = . This isa trivial result since H = 0 and there is no dynamics. When N = 2 , we find f ( ξ ; t ) = (cid:0) ξ − (cid:1) − sin t , so the twoentanglement modes oscillate as ξ = 12 (1 ± sin t ) . (I7) When N = 3 , we find f ( ξ ; t ) = ξ − )[( ξ − ) − sin t (1+cos t )] , implying a persistent topological mode at ξ = andtwo oscillating modes ξ = 12 (1 ± sin t (cid:112) t ) . (I8)When N = 4 , we find f ( ξ ; t ) = ( ξ − ) − sin t (1 +2 cos t )( ξ − ) + sin t cos t , so all of the four modesoscillate in time as ξ = 12 ± sin t (cid:113) t ± (cid:112) t. (I9)Finally, we find the characteristic polynomial for N = 5 to be f ( ξ ; t ) = ( ξ − )[( ξ − ) − sin t (1 + 3 cos t )( ξ − ) + sin t cos t (2 + cos t )] . Except for a constant solution ξ = , the other four modes oscillate as ξ = 12 ± sin t (cid:113) t ± (cid:112) t − t + 1 . (I10)These exact ES dynamics are plotted in Fig. 10(b).
2. Mathematical formulation
Having the above simple example in mind, we are ready tointroduce a general mathematical formalism for dealing withpartial symmetry breaking. We first recall that the classifica-tion of gapped free-fermion systems at equilibrium is given bythe homotopy group π d s ( S ) , where d s is the spatial dimensionand S is the classifying space satisfying symmetry constraints.If we are interested in the stable topology, we can take thelimit of infinite bands and obtain the well-known K -theoryclassification [107]. For example, S = R ≡ lim n →∞ O( n ) for class BDI and S = R ≡ lim n →∞ O(2 n ) / U( n ) forclass D. However, we emphasize that the general formalismapplies equally to the finite-band case, although the practi-cal calculations could be intractable. Denoting S and ˜ S asthe classifying spaces subject to symmetries G and ˜ G with ˜ G < G , we have
S ⊂ ˜ S since a G -symmetric system isalways ˜ G -symmetric but the converse is generally not true.Therefore, we have a natural inclusion ι : S → ˜ S , which isa morphism (continuous map) in Top ∗ (category of pointedtopological spaces). Such an inclusion induces a group ho-momorphism ι ∗ : π d s ( S ) → π d s ( ˜ S ) through [ f ] → [ ι ◦ f ] ,where f : S d s → S is a continuous map from d s D sphere1to S and [ f ] is its homotopy equivalence class. In fact, π d s can be regarded as a functor from Top ∗ to Grp (categoryof groups) or Ab (category of Abelian groups) if d s ≥ ,which maps not only pointed topological spaces into homo-topy groups but also continuous maps between topologicalspaces into group homomorphisms between the correspond-ing homotopy groups [187]. In particular, ι ∗ is the image of ι that makes the following diagram commute: S ˜ S π d s ( S ) π d s ( ˜ S ) ιι ∗ π d s π d s (I11)For class BDI → class D in 1D, we have d s = 1 , π ( R ) = Z , π ( R ) = Z and ι ∗ ( N ) = N mod 2 for ∀ N ∈ Z , asillustrated in the previous subsection.Let us move on to discuss interacting SPT systems clas-sified by group cohomology [188]. We note that the group-cohomology classification is actually complete in 1D [70–72], although not in higher dimensions [189]. Instead ofthe classifying spaces, we focus directly on the symmetrygroups. The inclusion ι of ˜ G into G , which is a natu-ral group homomorphism, induces another group homomor-phism from H d st ( G, U(1)) to H d st ( ˜ G, U(1)) , where d st ≡ d s + 1 ∈ Z + is the spacetime dimension. To see this, weonly have to note that an d st -cocycle ω : G × d st → U(1) can naturally be restricted to ˜ ω : ˜ G × d st → U(1) through ˜ ω (˜ g , ˜ g , ..., ˜ g d st ) = ω (˜ g , ˜ g , ..., ˜ g d st ) , which obviously sat-isfies the cocycle property d ˜ ω = 1 . In fact, any group co-homology h : G → H d st ( G, U(1)) ( U(1) can actually bereplaced by other Abelian groups) is a contravariant functorfrom
Grp to Ab , which map not only groups into Abelian co-homology groups but also group homomorphisms into thosebetween cohomology groups [190]. By contravariant, wemean that the directions of morphisms are reversed by thefunctor. In particular, the reduction of SPT phases is deter-mined by the induced map ι ∗ that makes the following dia-gram commute: ˜ G G H d st ( ˜ G, U(1)) H d st ( G, U(1)) ιι ∗ h h (I12)In the main text, we have given the simplest nontrivial exam-ple with d st = 2 , H ( G, U(1)) = Z N , H ( ˜ G, U(1)) = Z n and ι ∗ ( ν ) = Nn ν mod n .Finally, let us comment on the impact of SPT-order reduc-tion on the ES dynamics. For free-fermion systems, the bulk-edge correspondence usually has a very simple form — thenumber of edge states, or the degeneracy in ES, is simplygiven by the bulk topological number [52, 191, 192]. As aprototypical example (e.g., class BDI → class D in 1D), a sin-gle (no) ξ = mode survives a surjective Z → Z reduction ifthe original topological number is odd (even). For interacting SPT systems in 1D, the open boundary ES degeneracy r is de-termined by the minimal dimension of the irreducible projec-tive representations of the symmetry group. Such a minimaldimension is if the projective representation can be lifted to alinear representation, but is otherwise no less than accordingto Lemma 5. In fact, there is a character theory for projectiverepresentations [193], which shares many similarities with theconventional character theory for linear representations. Forexample, denoting the dimension of the α th irreducible pro-jective representation with respect to a -cocycle ω as d α , wehave d α || G | and (cid:80) R ω α =1 d α = | G | , where R ω is the numberof ω -regular conjugacy classes , i.e., those conjugacy classeswith a representative element g satisfying ω ( g, h ) = ω ( h, g ) for ∀ h ∈ N g ≡ { h ∈ G : gh = hg } < G . An immediatecollorary is that, for G = Z N × Z N with N being a prime, wehave r = N for any G -symmetric SPT state. By simple anal-ysis, we can also infer that possible r > for G = Z × Z is , and . These conclusions are consistent with the resultsin Table II.
3. Minimal models for quenched interacting SPT systems
Let us introduce the interacting counterparts of flat-bandfree-fermion models, in the sense that these minimal mod-els have zero correlation lengths. In (1 + 1)
D spacetime,a G -symmetric SPT state with zero correlation length canbe built from a -cocyle (which satisfies ω ( gh, k ) ω ( g, h ) = ω ( g, hk ) ω ( h, k ) for ∀ g, h, k ∈ G ) as [188] | Ψ (cid:105) = 1 | G | L (cid:88) { g j } Lj =1 L (cid:89) j =1 ω ( g − j g j +1 , g − j +1 ) − | g g ...g L (cid:105) . (I13)Here the local Hilbert space C | G | is spanned by {| g (cid:105) : g ∈ G } and the on-site symmetry representation is regular: ρ g = (cid:80) h ∈ G | gh (cid:105)(cid:104) h | for ∀ g ∈ G . Note that such a construction(I13) applies equally to continuous symmetries if we replace (cid:80) g by (cid:82) dg [188]. To demonstrate the impact of partial sym-metry breaking on the ES dynamics, we consider the simplestcase in which the Hamiltonian is a sum of commutative two-site operators: H = L (cid:88) j =1 h j , h j = (cid:88) g j ,g j +1 h ( g j , g j +1 ) | g j g j +1 (cid:105)(cid:104) g j g j +1 | , (I14)whose eigenstates are simply Fock states. To partially breakthe symmetry from G to ˜ G , we require h ( g, g (cid:48) ) = h (˜ gg, ˜ gg (cid:48) ) ∈ R , ∀ g, g (cid:48) ∈ G and ˜ g ∈ ˜ G, (I15)and otherwise h ( g, g (cid:48) ) (cid:54) = h ( g (cid:48)(cid:48) g, g (cid:48)(cid:48) g (cid:48) ) , ∀ g (cid:48)(cid:48) ∈ G \ ˜ G. (I16)As illustrated in Fig. 16(a), due to the fact that h j ’s commutewith each other, the open boundary ES at time t are simply thesquared singular values of [ M ( t )] gg (cid:48) = e − ih ( g,g (cid:48) ) t ω ( g − g (cid:48) , g (cid:48)− ) − . (I17)2 - - - - t ζ - - - - t ζ - - - - t ζ - - - - t ζ - - - - t ζ - - - - t ζ !
We can then numerically determine the degeneracies r and ˜ r in the initial ES and that after the quench.In Figs. 16(b) and (c), we plot the ES dynamics for twodifferent partial symmetry breaking quenches G = Z × Z → ˜ G = Z × Z and G = Z × Z → ˜ G = Z × Z . Werandomly sample h ( g, g (cid:48) ) among [ − , while keeping the ˜ G -symmetry requirement (I15). The behaviors of ES splittingagree perfectly with the results in Table II. While not shownhere, we have also checked that the ES always becomes non-degenerate upon the partial symmetry breaking quench G = Z × Z → ˜ G = Z × Z . This is fully consistent with thefact that the induced group homomorphism from Z to Z istrivial, as mentioned in Sec. VI A 2. - - - - - - - - - - t Δ E l = = = = = t S E (a)(b) FIG. 17. Dynamics of (a) the entanglement entropy and (b) thesingle-particle entanglement gap after a quench to the disorderedSSH model (J1) from a clean state. Inset in (b): the same as the mainpanel but in the log-linear scale. We choose L = 2 l +1 so that the en-tanglement bipartition is asymmetric and the initial entanglement gapis finite. The numbers of disorder realizations for l = 10 , , , and are , × , × , and × , respectively. Theparameters are quenched as ( ¯ J, ¯ J (cid:48) , f ) = (0 . , , → (1 , . , . .Note that the early-time data of ∆ spE for l = 50 (only a part of whichis visible) are not reliable due to a finite numerical resolution. Appendix J: Numerical simulations for disordered systems
In this appendix, we provide some numerical pieces of evi-dence to support the qualitative discussions in Sec. VI B.
1. Disordered SSH model
We consider the entanglement dynamics in a disorderedSSH model described by the Hamiltonian H = − (cid:88) j ( J j b † j a j + J (cid:48) j a † j +1 b j + H . c . ) , (J1)where a j and b j denote the sublattice fermionic modes in the j th unit cell, and the hopping amplitudes J j ∈ [(1 − f ) ¯ J, (1+ f ) ¯ J ] , J (cid:48) j ∈ [(1 − f ) ¯ J (cid:48) , (1+ f ) ¯ J (cid:48) ] (J2)are uniformly and independently sampled. We start from atopological state with ( ¯ J, ¯ J (cid:48) , f ) = (0 . , , (no disorder)and quench the parameters to ( ¯ J, ¯ J (cid:48) , f ) = (1 , . , . . Thelength of the subsystem is chosen to be l = ( L − so thatthe entanglement gap in the initial state becomes nonzero dueto the asymmetric entanglement bipartition.3 L = = = = = t S E - - - - t Δ E (a)(b) FIG. 18. Dynamics of (a) the half-chain entanglement entropy and(b) the many-body entanglement gap starting from the Z × Z SPTMPS (J3) and evolving in time according to the disordered Hamil-tonian given in Eq. (J7). The numbers of disorder realizations for L = 4 , , , and are × , , × , × and × , respectively. The parameters in Eqs. (J4) and (J7) are set tobe p = q = 0 . , J = 3 and κ = 3 . As shown in Fig. 17(a), we find that the entanglement en-tropy grows logarithmically — a feature usually associatedwith many-body localized systems. Since there is no inter-action in our model, one may na¨ıvely expect that the entan-glement grows extremely slowly like ln ln t [194], as mightbe inferred from a dynamical version of the strong-disorderrenormalization group [154]. However, there is a crucial dif-ference in our setup — the intial state | Ψ (cid:105) is not a productstate (in real space) and has a nonzero correlation length. Wedenote U as a local unitary such that U | Ψ (cid:105) becomes a prod-uct state; then the quench dynamics in this new frame is gov-erned by U HU † ( H is given by Eq. (J1) with the postquenchparameters), which now involves small but finite long-rangehoppings. We expect this effect to dramatically change thecommon paradigm of entanglement growth in Anderson insu-lators. On the other hand, the entanglement entropy eventuallybecomes saturated at a constant independent of (sufficientlylarge) l , as a manifestation of a finite localization length.In stark contrast to the slow growth of entanglement en-tropy, the numerical results (see Fig. 17(b)) suggests an expo-nentially fast growth of the single-particle entanglement gapbefore saturation, which is similar to the clean case. Never-theless, due again to the finiteness of the localization length,the saturation value decreases exponentially with respect to l , implying that the topological entanglement edge modes arestable even in the long-time limit.
2. Phenomenological model for many-body localization
The minimal group that supports an interacting SPT phaseprotected by unitary symmetries is G = Z × Z = { ( m, n ) : m, n ∈ Z } , whose second cohomology group reads H ( Z × Z , U(1)) = Z . By specifying the on-site symmetry as theregular representation ρ ( m,n ) = Z m ⊗ Z n with Z = | (cid:105)(cid:104) | −| (cid:105)(cid:104) | acting on a local qubit, we can construct a nontrivialMPS as | Ψ (cid:105) = (cid:88) { j s =00 , , , } Tr[ A j A j ...A j L ] | j j ... j L (cid:105) , (J3)where a local state | j (cid:105) consists of two qubits and the injectivetensor A j is given by A = (cid:112) (1 − p )(1 − q ) σ , A = (cid:112) q (1 − p ) σ x ,A = i (cid:112) p (1 − q ) σ y , A = √ pqσ z , (J4)with p, q ∈ (0 , . We can check that the projective represen-tation on the virtual level is nontrivial: V (0 , = σ , V (1 , = σ x ,V (0 , = σ y , V (1 , = σ z . (J5)To slightly lift the exact four-fold degeneracy in the ES of afinite segment, we choose p = q = 0 . in Eq. 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