Non-Bloch quench dynamics
NNon-Bloch quench dynamics
Tianyu Li, Jia-Zheng Sun, Yong-Sheng Zhang,
1, 2, ∗ and Wei Yi
1, 2, † CAS Key Laboratory of Quantum Information, University of Science and Technology of China, Hefei 230026, China CAS Center For Excellence in Quantum Information and Quantum Physics, Hefei 230026, China
We study the quench dynamics of non-Hermitian topological models with non-Hermitian skineffects. Adopting the non-Bloch band theory and projecting quench dynamics onto the generalizedBrillouin zone, we find that emergent topological structures, in the form of dynamic skyrmions,exist in the generalized momentum-time domain, and are correlated with the non-Bloch topolog-ical invariants of the static Hamiltonians. The skyrmion structures anchor on the fixed points ofdynamics whose existence are conditional on the coincidence of generalized Brillouin zones of thepre- and post-quench Hamiltonians. Global signatures of dynamic skyrmions, however, persist wellbeyond such a condition, thus offering a general dynamic detection scheme for non-Bloch topologyin the presence of non-Hermitian skin effects. Applying our theory to an experimentally relevant,non-unitary quantum walk, we explicitly demonstrate how the non-Bloch topological invariants canbe revealed through the non-Bloch quench dynamics.
Non-Hermitian Hamiltonians arise in open systems [1,2], and have attracted significant attention in recentyears [3–10]. As a peculiar feature of a wide class of non-Hermitian Hamiltonians, their nominal bulk eigenstatesare localized near boundaries under what is now knownas the non-Hermitian skin effects [11–22]. Remarkably,for non-Hermitian topological models with skin effects,the conventional bulk-boundary correspondence breaksdown [11–18, 23], whose restoration calls for a non-Blochband theory where topological invariants are calculatedin the generalized Brillouin zone (GBZ) under relevantboundary conditions [11–15, 21, 24, 25], rather than theconventional Brillouin zone (BZ) under a periodic bound-ary condition (PBC). Both non-Hermitian skin effectsand non-Bloch bulk-boundary correspondence have re-cently been experimentally confirmed [26–31]. However,a direct detection of non-Bloch topological invariants isyet to be demonstrated. Here we show how non-Blochtopological invariants can be determined through dy-namic topological structures in quantum quenches thatare experimentally accessible.A quantum quench describes the evolution of an eigen-state of the initial Hamiltonian H i driven by a finalHamiltonian H f . Particularly, for one-dimensional topo-logical systems, previous studies have shown that dy-namic skyrmions, a topological structure hinged uponfixed points of dynamics, should appear in the emer-gent momentum-time domain, when the initial and finalHamiltonians possess distinct topological properties [32–34]. It follows that by studying the dynamic signaturesof a quantum quench, one gains information regardingthe topological invariants of the pre- and post-quenchHamiltonians [35–53]. Similar conclusions hold for non-unitary quenches governed by non-Hermitian Hamiltoni-ans, but are predicated upon two conditions [34]: i) thedecoupling of dynamics in different momentum sectorsunder the lattice translational symmetry; and ii) realityof the eigenenergy spectra of both H i and H f , where theparity-time (PT) symmetry plays an important role [2–6]. For a non-Hermitian topological model with skin effects,however, a compromise between these two requirementsseems to be a tall order. On one hand, dynamics of dif-ferent momentum states are no longer decoupled underan open boundary condition (OBC). On the other, for asystem possessing non-Hermitian skin effects, the asso-ciated Bloch spectra in the conventional BZ necessarilyform loops in the complex plane [21, 22], such that realeigenenergy spectra only exist in the GBZ under OBCs,protected by a non-Bloch PT symmetry [31, 54, 55].We circumvent these issues by projecting the quenchdynamics onto the generalized momentum sectors of theGBZ. Such a non-Bloch analysis enables us to reveal,in the generalized momentum-time domain, dynamicskyrmions that are intimately related to the non-Blochtopological invariants of the pre- and post-quench Hamil-tonians. In particular, for a non-Hermitian Su-Schrieffer-Heeger (SSH) model with asymmetric hopping and underOBC, we prove that the existence of the underpinningfixed points for dynamic skyrmions exist, provided H i and H f are both in the non-Bloch PT unbroken phase(i.e., both having real eigen spectra) and feature thesame GBZ, but with distinct non-Bloch winding num-bers. When the GBZs of H i and H f are different, fixedpoints only exist in a perturbative sense, whereas theglobal signatures of dynamic skyrmions persist, allow-ing for the detection of non-Bloch winding numbers overa wide parameter regime. We apply our theory to therecently implemented, non-unitary topological quantumwalk, and illustrate the extraction of non-Bloch windingnumber from the non-Bloch quench dynamics. Quenching in the GBZ:—
We first consider the quenchdynamics of a non-Hermitian SSH model [11, 56] H = (cid:88) n (cid:104) ( t + γ ) | n, A (cid:105)(cid:104) n, B | + ( t − γ ) | n, B (cid:105)(cid:104) n, A | + t | n, B (cid:105)(cid:104) n + 1 , A | + t | n + 1 , A (cid:105)(cid:104) n, B | (cid:105) , (1)where each unit cell (labeled n ) has two sublattice sites a r X i v : . [ c ond - m a t . qu a n t - g a s ] J a n (labeled A and B ), with intra- and inter-cell hoppingrates given by t ± γ and t , respectively. The asym-metric intra-cell hopping here gives rise to non-Hermitianskin effects, such that topological properties of the modelunder OBC are characterized by the non-Bloch band the-ory, where a GBZ must be invoked, due to the deviationof the nominal bulk states from extended Bloch waves.Specifically, the GBZ of Hamiltonian (1) under OBCis a circle with radius r = (cid:112) | ( t − γ ) / ( t + γ ) | on thecomplex plane [11, 57], where we denote the azimuth an-gle as k ∈ ( π, π ]. In the Hermitian limit with γ = 0,the GBZ reduces to the conventional BZ, a unit circleon the complex plane with r = 1, with k being theBloch quasi-momentum. As the non-Hermitian, non-Bloch counterparts of the Bloch states, we then introducethe biorthogonal spatial basis in a generalized momentumspace characterized by k : | β k,R ( L ) (cid:105) = √ N (cid:80) n r ± n e ikn | n (cid:105) ( N is the total number of unit cells, α ∈ { A, B } ), whichsatisfy the biorthogonal and completeness relations [58]: (cid:104) β k,L | β k (cid:48) ,R (cid:105) = δ k,k (cid:48) and (cid:80) k | β k,R (cid:105)(cid:104) β k,L | = n , with n = (cid:80) n | n (cid:105)(cid:104) n | . Conveniently, while | β k,R/L (cid:105) become ex-tended Bloch states in the Hermitian limit, | β k,R (cid:105) ( | β k,L (cid:105) )is localized toward the right (left) boundary for r > P k = | β k,R (cid:105)(cid:104) β k,L | ⊗ α , α being the identity operator in the sublattice basis {| A (cid:105) , | B (cid:105)} , and project Hamiltonian (1) onto the GBZas H k = P k HP k . Writing H k = (cid:80) i d i ( k ) σ i ( σ i are thePauli operators with i = 1 , ν = 12 π (cid:90) dk − d ∂ k d + d ∂ k d d + d , (2)which recovers the bulk-boundary correspondence underOBC [11]. Note that while the eigen spectrum underPBC is always complex, it is completely real under OBCfor | t | > γ , due to the presence of a non-Bloch PT sym-metry [54, 55]. Following Eq. (2), the non-Bloch topo-logical phase boundaries in this PT-unbroken region aregiven by t = ± (cid:112) t − γ , which reduce to those of aHermitian SSH model when γ = 0, with Eq. (2) thenyielding the Bloch winding number.In a general quantum quench, an eigenstate of the ini-tial Hamiltonian H i evolves under the final Hamiltonian H f . For an initial state | ψ ik (cid:105) in a given k -sector, for in-stance, the time-dependent density matrix ρ ( t ) evolvesaccording to ρ ( t ) = e − iH f t | Ψ ik (cid:105)(cid:104) Ψ ik | e iH f † t , (3)where | Ψ ik (cid:105) = | β ik,R (cid:105) ⊗ | ψ ik (cid:105) , with | β ik,R (cid:105) the right spatialbasis of H i and | ψ ik (cid:105) the internal state in the sublat-tice basis. Unlike the quasi-momentum in the BZ, gen-eralized momentum k in a GBZ is not a good quantum b a FIG. 1. Schematic illustration of (a) non-Bloch and (b) Blochquench dynamics of the non-Hermitian SSH model of Eq. (1).(a) Non-Bloch quench dynamics in the GBZ is periodic in each k -sector, with either (left) identical GBZs or (right) distinctGBZs for the pre- and post-quench Hamiltonians [32, 34]. Theradius r and generalized momentum k of the GBZ, as well asthe Bloch-sphere vector n ( k, t ) are defined in the main text.Black dots in the left panel indicate fixed points. (b) Blochquench dynamics is steady-state approaching in the k -sectors. number, as (cid:104) β fk,L | H f | β ik (cid:48) ,R (cid:105) (cid:54) = 0 for γ (cid:54) = 0, such that astate initialized in a k -sector inevitably proliferates intoother k -sectors in the time evolution. However, it canbe shown that for r i = r f , i.e., the GBZs of H i and H f coincide with each other, these cross-coupling terms van-ish in the thermodynamic limit, rendering the dynamicsblock-diagonal in the GBZ, similar to the conventionalquench dynamics in the BZ.Inspired by such an observation, we first focus on thesimpler case of r i = r f , and discuss more general scenar-ios of r i (cid:54) = r f or non-circular GBZs later. To characterizequench dynamics in the GBZ, we first project ρ ( t ) ontothe GBZ of H f [57] ρ ( k, t ) = P k ρ ( t ) P † k , (4)where P † k is introduced to accommodate the time evolu-tion of (cid:104) Ψ ik | in Eq. (3). We characterize dynamics in each k -sector with the dynamic spin texture n ( k, t ) in a gen-eralized momentum-time space spanned by ( k, t ) [34, 53] n ( k, t ) = Tr[ ρ ( k, t ) η τ ]Tr[ ρ ( k, t ) η ] , (5)where τ = ( τ , τ , τ ), with τ i = (cid:80) µν = ± | ψ fk,µ (cid:105) σ µνi (cid:104) χ fk,ν | (i=1, 2, 3). Here, σ µνi are the elements of Pauli matri-ces σ i , and σ is a 2 × | ψ fk, ± (cid:105) ( (cid:104) χ fk, ± | )is the right (left) eigenstate of H fk , with H fk | ψ fk, ± (cid:105) = E fk, ± | ψ fk, ± (cid:105) ( H f † k | χ fk, ± (cid:105) = E f ∗ k, ± | χ fk, ± (cid:105) ), and ± are theband indices. The metric operator η = (cid:80) µ | χ fk,µ (cid:105)(cid:104) χ fk,µ | is introduced to normalize ρ ( k, t ), such that n ( k, t ) isa unit, real vector that can be visualized on the Blochsphere S [34, 58]. -101-1 0 1-101 -101-1 0 1-101 -1 0 10246 -101 -1 0 10246 -101 -1 0 10246 -101 -101-1 0 1-101 -101-1 0 1-101 (f) (g) (h) (i) (j) -1 0 10246 -101 -2 0 200.511.5 (a) (b) (c) (d) (e) -2 0 200.511.5 FIG. 2. Quench dynamics projected in (top row) the GBZ and (lower row) BZ, respectively, where t is taken as the unit ofenergy. (a) Non-Bloch topological phase diagram under OBC. All quenches are performed in the region with γ <
1, where theeigen spectra are completely real. (b) Dynamic spin texture n ( k, t ) in the generalized momentum-time space, for a quantumquench from point A ( H i with non-Bloch winding number ν i = 0) to F ( H f with non-Bloch winding number ν f = 1). (c) n ( k, t ) across the GBZ at different times t for the quench in (b). (d) n ( k, t ) for a quench from point B ( H i with non-Blochwinding number ν i = 1) to F. No skyrmion structures are present, in contrast to (b). (e) n ( k, t ) across the GBZ at differenttimes t for the quench in (d). (f) Bloch topological phase diagram under PBC. Eigen spectra are complex throughout thephase diagram. (g) n ( k, t ) in the momentum-time space, for a quantum quench from point A ( H i with Bloch winding number ν i = 1 /
2) to F ( H f with Bloch winding number ν f = 1 / n ( k, t ) across the BZ at different times t for the quench in (g).(i) n ( k, t ) in the momentum-time space for a quench from point B to F. (j) n ( k, t ) across the BZ at different times t for thequench in (i). Dashed vertical lines in (b)(d) denote the locations of fixed points at k m = 0 , ± π . The parameters for pointsA, B, and F are t = 0 . , , .
5, respectively, with γ = 0 .
8. Points C and D are used in Fig. 3, with parameters t = 0 . , . γ = 0 . r i = r f = 0 .
33; whereas for the Bloch quenches(lower), r i = r f = 1. Without loss of generality, we consider the initial statein each k -sector to lie within the lower band | ψ ik, − (cid:105) , with H ik | ψ ik, − (cid:105) = E ik, − | ψ ik, − (cid:105) . Importantly, when E fk,µ is real, n ( k, t ) rotates around the north pole of the Bloch spherewith a period T k = π/E fk [see Fig. 1(a) for schemat-ics] [34, 57]. Consequently, dynamic fixed points k m inthe GBZ can occur when c ± = (cid:104) χ fk m , ± | ψ ik m , − (cid:105) = 0, i.e.,when the initial n ( k m , t = 0) vector is aligned with thenorth ( c − = 0) or south ( c + = 0) pole of the Blochsphere, leading to stationary evolutions at k m .For the quench dynamics of a non-Hermitian SSHmodel without skin effects, it has been shown that thepresence of fixed points are related to the Bloch wind-ing numbers of the pre- and post-quench Hamiltonians,provided both Hamiltonians belong to the PT-unbrokenregime with completely real eigen spectra [34, 53]. Here,it is straightforward to show that, when r i = r f , thesame conclusions still hold, albeit now the fixed pointsare only visible in the GBZ, and the protecting symmetryfor real eigen spectra is instead the non-Bloch PT sym-metry. By contrast, if the quench dynamics is projected onto the BZ, i.e., replacing spatial basis states | β fk,R/L (cid:105) in the projector P k with Bloch states, the dynamics isalways steady-state approaching, as Bloch spectra E fk, ± are necessarily complex for k ∈ BZ with r f = 1. Asillustrated in Fig. 1(b), n ( k, t ) then asymptotically ap-proaches the north (Im E fk >
0) or south (Im E fk < n ( k, t ), in the generalizedmomentum-time domain, for a quench between Hamilto-nians with different non-Bloch winding numbers. Whilethe fixed points ( k m = 0 , ± π ) and periodic dynamics(blue curves indicating multiples of T k ) divide the gener-alized momentum-time space into various submanifolds,arrays of momentum-time skyrmions can be identified,which are characterized by dynamic Chern numbers inthe corresponding submanifolds [32, 33, 57]. Such a dy-namic topological structure, however, is absent when oneperforms the quench in the BZ under the same param-eters [see Fig. 2(g)(h)], where the spin dynamics is es-sentially steady-state approaching. For comparison, weshow in Fig. 2(d)(e) and Fig. 2(i)(j), quench dynamics (a) -1 0 10246 -101 (c) (d)(b) -1 0 10246 FIG. 3. Non-Bloch quench for r i (cid:54) = r f . (a) Spin texture n ( k, t ) in the generalized momentum-time space, for a quenchfrom point C in Fig. 2(a) ( ν i = 0, r i = 0 .
5) to point F( ν f = 1, r f = 0 . n ( k, t ) in the gen-eralized momentum-time space, for a quench from point D inFig. 2(a) ( ν i = 1, r i = 1) to point F. (c) Time-averaged spincomponents ¯ n ( k = 0) and ¯ n ( k = 0) as functions of vary-ing r i /r f , for quench processes ending at F and starting frompoints close to C, with ν i = 0 and a fixed t = 0 .
5. (d) ¯ n (0)and ¯ n ( π ) as functions of varying r i /r f , for quench processesending at F and starting from points close to D, with ν i = 1and a fixed t = 1 .
5. We take t = 1 as the unit of energy. in the GBZ and BZ, respectively, between Hamiltonianswith the same winding numbers. While fixed points stillexist in the GBZ, the dynamic Chern numbers vanish onall submanifolds in Fig. 2(d). Finally, it is worth notingthat, in our analysis, projected quench dynamics in dif-ferent k -sectors are calculated independently. For a half-filled system initialized in the lower band, however, ad-ditional cross-coupling terms from other k -sectors wouldappear, but which are typically negligibly small under allparameters considered here [57]. Beyond the simple case:—
While the GBZ of Hamil-tonian (1) under OBC is always circular in the complexplane, it is easy to choose parameters such that r i (cid:54) = r f .In this case, dynamics in different k -sectors of the GBZare coupled. Nevertheless, one can still perform the pro-jection in Eq. (4), and focus solely on dynamics in differ-ent k -sectors of H f . While fixed points no longer rigor-ously exist in this case [see Fig. 1(a) for schematics] [57],remnants of the dynamic skyrmions for r i /r f = 1 (for ν i (cid:54) = ν f ), or the lack thereof (for ν i = ν f ), would carryover and leave identifiable signatures in the dynamic spintexture, over a considerable range of r i /r f .We numerically confirm such a picture in Fig. 3(a)(b),where r i /r f = 1 .
5. For both cases, the system is ini- -2-10
FAI
FAI AF (a) (b) FIG. 4. (a) Non-Bloch topological phase diagram underOBC for the effective Hamiltonian H of the Floquet oper-ator U in Eq. (6). The horizontal green dashed lines arelocations where quasi-local eigenstates exist. (b) ∆¯ n (seemain text for definition) evaluated for quench processes start-ing from I , and ending at different points along the line I–F in (a). The vertical dashed and dash-dotted lines indi-cate, respectively, the Bloch and non-Bloch topological phaseboundaries. Inset: variation of r i /r f as H f changes from Ito F. The parameters for points I, A and F are ( θ , θ ) =(0 . π, . π ) , (0 . π, . π ) , (0 . π, . π ), respectively. tialized in the lower-band with occupied | ψ ik, − (cid:105) of each k -sector, and under the same parameters with ν i = 0.We then quench the system under final Hamiltoniansfeaturing ν f = 1 and ν f = 0, respectively. As ex-pected, skyrmion-like structures only emerge in Fig. 3(a)where ν i (cid:54) = ν f . To capture key features of these struc-tures away from r i /r f = 1, we define the time-averagedspin component ¯ n ( k ) = T k (cid:82) T k n ( k, t ) dt , which shouldgive ¯ n ( π ) = − ¯ n (0) ≈ n (0) = ¯ n ( π ) ≈ n ( k ) at k = 0 and k = π over arange of different H i (hence varying r i /r f ), where thepresence [see Fig. 3(c)] and absence [see Fig. 3(d)] ofskyrmion-like structures can be clearly identified. Cru-cially, knowing ¯ n ( k m ) and one of the non-Bloch windingnumbers ν i/f , one can immediately infer the other. Thus,while dynamic Chern numbers cannot be rigorously de-fined for r i /r f (cid:54) = 1, the remnant of dynamic skyrmions,due to their robustness as a global topological structurein the generalized momentum-time space, can still beused to determine non-Bloch topological invariants over aconsiderable range of r i /r f . Remarkably, similar conclu-sions can be drawn for quench dynamics between Hamil-tonians with distinct, non-circular GBZs [57]. Non-unitary quantum walk:—
Quantum-walk dynam-ics is an ideal platform for demonstrating non-Blochquenches discussed here, particularly in light of the recentexperimental observation of dynamic skyrmion structurestherein [53]. For such a purpose, we consider a non-unitary quantum walk along a one-dimensional lattice,governed by the Floquet operator [27] U = R ( θ ) S R ( θ ) M R ( θ ) S R ( θ ) , (6)where the coin operator R ( θ i ) = w ⊗ e − iθ i σ y ( i = 1 , M = w ⊗ ( e γ | A (cid:105)(cid:104) A | + e − γ | B (cid:105)(cid:104) B | ). Here w = (cid:80) x | x (cid:105)(cid:104) x | , x labels the lat-tice sites, {| A (cid:105) , | B (cid:105)} are the internal coin states, and γ is the gain-loss parameter. The shift operator S = (cid:80) x | x (cid:105)(cid:104) x | ⊗ | A (cid:105)(cid:104) A | + | x + 1 (cid:105)(cid:104) x | ⊗ | B (cid:105)(cid:104) B | , and S = (cid:80) x | x − (cid:105)(cid:104) x |⊗| A (cid:105)(cid:104) A | + | x (cid:105)(cid:104) x |⊗| B (cid:105)(cid:104) B | . In the presence ofboundaries, quantum walks governed by U exhibit non-Hermitian skin effects, as state evolutions are found tobe localized near boundaries, due to the localization ofall eigenstates.In a discrete-time quantum walk, U is repeatedly en-forced upon an initial state, such that the time evolutionconstitutes a stroboscopic simulation of the dynamicsgoverned by an effective Hamiltonian H , with U = e − iH .Such a time evolution can be further regarded as a quenchprocess if the initial state is an eigenstate of a certain U i ,or equivalently, an eigenstate of H i with U i = e − iH i . InFig. 4(a), we show the topological phase diagram of theeffective Hamiltonian under OBC. The phase diagram isthe same as that of U in the time frame given by Eq. (6)and under OBC. Note that the GBZ of U is always cir-cular, which simplifies the problem.To facilitate experimental implementation, a quasi-local initial state is preferred, which, to simultaneouslyfulfil the conditions of our scheme outlined above, shouldcorrespond to an eigenstate of U i . As a concrete exam-ple, we choose a U i with θ i = 0 . π and θ i = 0 . π [point I in Fig. 4(a)], and with a quasi-local eigenstate | Ψ i (cid:105) ∝ | (cid:105) ⊗ ( | A (cid:105) + | B (cid:105) ) + 2 i/r i | − (cid:105) ⊗ | A (cid:105) + 1 /r i | − (cid:105) ⊗ ( −| A (cid:105) + | B (cid:105) ). Writing the initial state in the GBZas | Ψ i (cid:105) = (cid:80) k | β ik,R (cid:105) ⊗ | ψ ik (cid:105) and evolve it repeatedly with U f , we follow the prior practice of projecting the time-evolved density matrix into different k -sectors of the GBZof H f (with U f = e − iH f ), from which n ( k, t ) can beextracted [57]. We then vary parameters of H f alongthe line I → F in Fig. 4(a), and show the change of∆¯ n = ¯ n ( k = 0) − ¯ n ( k = π ) in Fig. 4(b). Despite thedeviation of r i /r f from unity [see inset of Fig. 4(b)], ∆¯ n features an abrupt jump at the non-Bloch topologicalphase transition, thus providing a sensitive and robustsignal for the detection of non-Bloch winding number ν f of the effective Hamiltonian, once ν i is known. Discussion:—
We show that projecting quench dynam-ics onto the GBZ reveals dynamic topological structuresthat can be used as signals for detecting non-Bloch topo-logical invariants. We explicitly demonstrate how sucha non-Bloch quench dynamics can be simulated usingquantum walks, which are experimentally accessible. Inthe spirit of the non-Bloch quench dynamic discussedhere, it is hopeful that various previously observed dy-namic topological structures in unitary quench processes,such as vortices [49], links [50], and rings [51], can alsobe explored in systems with non-Hermitian skin effects,which would then serve to detect non-Bloch topological invariants for more general models in higher dimensions.We acknowledge helpful discussions with Tian-ShuDeng. This work is supported by the National Natu-ral Science Foundation of China (Grant Nos. 11974331,11674306, 92065113) and the National Key R&D Pro-gram (Grant Nos. 2016YFA0301700, 2017YFA0304100). ∗ [email protected] † [email protected][1] H. J. 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Here we provide more details on the non-Hermitian Su-Schieffer-Heeger (SSH) model, the projection of quenchdynamics in the generalized Brillouin zone (GBZ), derivation for the conditional existence of fixed points and dynamicChern numbers, analysis and numerical calculations demonstrating the persistence of skyrmions-like structures in moregeneral settings, as well as theoretical and numerical analysis of quantum-walk dynamics.
Non-Hermitian SSH model with skin effects
For the non-Hermitian SSH model [Eq. (1) in the main text] under the open boundary condition (OBC), all of itsnominal bulk eigenstates are localized near the boundaries (for γ (cid:54) = 0) [11], and Bloch quasi-momenta of the Brillouinzone (BZ) are no longer a good bulk quantum numbers even in the thermodynamic limit. Central to our analysishere is the projection of dynamics onto GBZs characterized by the generalized momentum k associated with thebiorthogonal spatial basis | β k,R ( L ) (cid:105) = 1 √ N (cid:88) n r ± n e ikn | n (cid:105) , (S1)where r is the radius the GBZ, k is identified as the generalized momentum of the GBZ, and the basis states satisfy (cid:104) β k,L | β k (cid:48) ,R (cid:105) = δ k,k (cid:48) and (cid:80) k | β k,R (cid:105)(cid:104) β k,L | = 1. While any single-particle state of the system can be generically writtenas | Ψ (cid:105) = (cid:80) k | β k,R (cid:105) ⊗ | ψ k (cid:105) , with | ψ k (cid:105) = (cid:104) β k,L | Ψ (cid:105) its internal state in the sublattice space {| A (cid:105) , | B (cid:105)} , we introducethe projection operators P k = | β k,R (cid:105)(cid:104) β k,L | ⊗ α and Q k = − P k , where α and are the identity operators in thesublattice space and the full Hilbert space of the system, respectively. This enables us to write, in the thermodynamiclimit, H = P k HP k + Q k HQ k = H k + Q k HQ k , where H is given in Eq. (1) of the main text, H k = P k HP k , and wehave used the fact that P k HQ k and Q k HP k vanish in the thermodynamic limit, though neither of them are exactlyzero for a finite-size system.Under the sublattice basis defined above, H k can be written in the matrix form H k = (cid:18) t + γ + t r − e − ik t − γ + t re ik (cid:19) . (S2)The right and left eigenstates of H k are defined through H k | ψ k, ± (cid:105) = ± E k | ψ k, ± (cid:105) and H † k | χ k, ± (cid:105) = ± E ∗ k | χ k, ± (cid:105) , respec-tively. They further satisfy the biorthogonal and completeness relations: (cid:104) χ k,µ | ψ k,ν (cid:105) = δ µν and (cid:80) µ | ψ k,µ (cid:105)(cid:104) χ k,µ | = α .Throughout the work, we mainly focus on the regime with t > γ , where, under an exact non-Bloch parity-time (PT)symmetry, the eigen spectrum E k is completely real. Here E k = (cid:113) t + t + 2 t t cos k , with t = (cid:112) ( t − γ )( t + γ ),and ± are the band indices. The eigenstates are given as | ψ k, ± (cid:105) = (cid:32) ± / (cid:113) t + t + 2 t t cos kr/ ( t + t e − ik ) (cid:33) , (S3) | χ k, ± (cid:105) = (cid:32) ± / (cid:113) t + t + 2 t t cos k /r ( t + t e − ik ) (cid:33) . (S4)While the biorthogonal spatial basis of the GBZ can be seen as the non-Bloch analog of Bloch waves, the complicationof non-Bloch quench dynamics, i.e., the analysis of quench dynamics in the GBZ, comes from the general non-orthogonality of spatial basis states of the pre- and post-quench Hamiltonians, even if they are of the same k -sector. Quench dynamics and fixed points in the GBZ
Consider a quench process initialized in the lower band of the initial Hamiltonian H i , i.e., with all eigenstates | ψ ik, − (cid:105) occupied. The initial state in a given k -sector is then | Ψ ik (cid:105) = | β ik,R (cid:105) ⊗ | ψ ik, − (cid:105) , where | β ik,R (cid:105) is the right spatial basischaracterized, according to Eq. (S1), by the GBZ radius r i of H i . Evolving the initial state with the Hamiltonian H f , the time-dependent density matrix is given by ρ ( t ) = e − iH f t | Ψ ik (cid:105)(cid:104) Ψ ik | e iH f † t . (S5)Importantly, when γ (cid:54) = 0, (cid:104) β fk,L | β ik (cid:48) ,R (cid:105) (cid:54) = 0 (and (cid:104) β fk,R | β ik (cid:48) ,L (cid:105) (cid:54) = 0), and dynamics in different k -sectors are coupled.Here | β fk,R/L (cid:105) are the right/left spatial bases associated with the GBZ of H f according to Eq. (S1).To overcome this difficulty, we project the dynamics onto the GBZ of H f , such that the density matrix in each k -sector becomes ρ ( k, t ) = P k ρ ( t ) P † k = N e − iH fk t | ψ ik, − (cid:105)(cid:104) ψ ik, − | e iH f † k t (S6)where P k = | β fk,R (cid:105)(cid:104) β fk,L | ⊗ α , Q k = − P k , N = (cid:104) β fk,L | β ik,R (cid:105) = N (cid:80) n ( r i r f ) n , and we have used H fk = P k H f P k , P k H f Q k = 0.As discussed in the main text, the projected dynamics can be visualized on a Bloch sphere through n ( k, t ) = Tr[ ρ ( k, t ) η τ ]Tr[ ρ ( k, t ) η ] , (S7)where η = (cid:80) µ | χ fk,µ (cid:105)(cid:104) χ fk,µ | is the metric operator, τ = ( τ , τ , τ ), with τ i = (cid:80) µν = ± | ψ fk,µ (cid:105) σ µνi (cid:104) χ fk,ν | (i=0, 1, 2, 3).These definitions ensure that n ( k, t ) = ( n , n , n ) is always a real, unit vector.When the eigen spectra E fk of H f is real, we have [34, 53] n =( c ∗ + c − e iE fk t + c.c. ) /n , (S8) n = i ( c ∗ + c − e iE fk t − c.c. ) /n , (S9) n =( c ∗ + c + − c ∗− c − ) /n , (S10)where c µ ( k ) = (cid:104) χ fk,µ | ψ ik, − (cid:105) , and n = c ∗ + c + + c ∗− c − . In this case, n is time independent, and n ( k, t ) in a given k -sectorrotates around the north pole of the Bloch sphere with a period T k = π/E fk [see Fig. 1(a) of the main text], wherethe north and south poles are given by the fixed-point conditions c − ( k m ) = 0 and c + ( k m ) = 0, respectively. Here k m is the location of fixed point, where local density matrix does not evolve in time.When E fk is purely imaginary, n =( c ∗ + c − + c.c. ) /n , (S11) n = i ( c ∗ + c − − c.c. ) /n , (S12) n =( c ∗ + c + e − i E fk t − c ∗− c − e i E fk t ) /n , (S13)where n = c ∗ + c + e − i E fk t + c ∗− c − e i E fk t . As t tends to infinity, n ( k, t ) asymptotically approaches either the north (forIm E k >
0) or south (for Im E k <
0) pole [see Fig. 1(b) of the main text].Further, from the analytical expression of c µ c ± ( k ) = ∓ (cid:113) t i + t i + 2 t i t i cos k (cid:113) t f + t f + 2 t f t f cos k + r i r f t i + t i e − ik )( t f + t f e ik ) , (S14)we see that when r i = r f , there are always two fixed points at k m = 0 and k m = π , respectively.The existence of fixed points and periodic dynamics in the k -sectors (under a completely real E fk ) divide thegeneralized momentum-time space into a series of S submanifolds, on each of which a quantized, dynamic Chernnumber can be defined C mn = 14 π (cid:90) k n k m dk (cid:90) T k dt [ n ( k, t ) × ∂ t n ( k, t )] · ∂ k n ( k, t ) , (S15)where k m , k n ∈ { , ± π } are two adjacent fixed points. It is straightforward to show that: when c + ( k m ) = 0 and c − ( k n ) = 0, C mn = 1; when c − ( k m ) = 0 and c + ( k n ) = 0, C mn = −
1; otherwise, C mn = 0. Hence, if we categorizefixed points into two different kinds, with either c + ( k m ) = 0 or c − ( k m ) = 0, the dynamic Chern number can onlytake non-zero values when the corresponding submanifold is pinned by fixed points of different kinds. While thesedynamic Chern numbers serve as skyrmion numbers for the dynamic skyrmions in the generalized momentum-timespace shown in the main text, they are linked with the non-Bloch winding numbers of the pre- and post-quenchHamiltonians. Specifically, from Eq. (S14) and in combination with the expression for non-Bloch winding numbers[Eq. (2) of the main text], we find that, for ν i (cid:54) = ν f , two fixed points of different kinds emerge at 0 , π , ensuring theappearance of dynamic skyrmion structure. However, for ν i = ν f , the two fixed points at 0 and π are of the samekind, and dynamic skyrmions are absent. We note that Eq. (S14) as well sa the conclusions above are conditional onthe reality of the eigen spectra E i,fk of H i,fk , which is the regime of study here.For the more general case of r i (cid:54) = r f , i.e., the GBZs of the pre- and post-quench Hamiltonians do not match, fixedpoints do not exist in general. Firstly, this can be concluded by observing that c ± ( k ) (cid:54) = 0 in Eq. (S14). Furthermore,as a state initialized in a given k -sector would proliferate into other k -sectors during the time evolution, Eq. (S14)cannot be used to characterize dynamics for the quench dynamics of a topological system at half-filling, where all k -sectors of the lower band are occupied initially at t = 0. However, as long as r i /r f does not significantly deviate fromunity, the impact of the these cross coupling terms are typically small [see for instance, a related analysis followingEq. (S19)]; and in each k -sector, n (0 , t ) and n ( π, t ) (for completely real E fk ) precess closely around the poles of theBloch sphere, and fixed points exist in a perturbative sense. Although dynamic Chern numbers cannot be rigorouslydefined in this case, global signatures of dynamic skyrmions (for ν i (cid:54) = ν f ), or the lack there of (for ν i = ν f ), wouldstill be discernable beyond r i = r f . In fact, in the following section, we show that dynamic skyrmions even persistwhen GBZs of neither Hamiltonians are circular. Such an understanding underlies our proposal for a general detectionscheme of non-Bloch topological invariants. Non-Hermitian SSH model with next-nearest neighbor hopping
To further illustrate the applicability of our scheme on models with more general GBZs, we consider the quenchdynamics of a non-Hermitian SSH model with next-nearest neighbor hopping, i.e., with an additional term H nn = (cid:88) n ( t | n, A (cid:105)(cid:104) n + 1 , B | + H.c. ) , (S16)where t is the hopping rate. For a non-Hermitian SSH model with a finite t , the GBZ is no longer a circle on thecomplex plane. The spatial basis involves a k -dependent radius | β k,R ( L ) (cid:105) = 1 √ N (cid:88) n r ± n ( k ) e ikn | n (cid:105) , k ∈ ( − π, π ] , (S17)which does not satisfy the biothorgonal relations in the main text, i.e., (cid:104) β k (cid:48) ,L | β k,R (cid:105) (cid:54) = 0. Further, non-Bloch PTsymmetry is broken with a finite t , such that the eigen spectrum under OBC is always complex, leading to steady-state approaching quench dynamics in general.However, when t is small, the GBZ does not deviate too much from a circle, remnants of the dynamic skymionstructures still exist for ν i (cid:54) = ν f , and can still be used to determine the post-quench non-Bloch winding number oncethat of the initial Hamiltonian is known.More concretely, the projected Hamiltonian in a given k -sector is given by H k = (cid:18) t + γ + t r − e − ik + t re ik t − γ + t re ik + t r − e − ik (cid:19) , (S18)and one can perform the same analysis outlined above to study projected quench dynamics in the GBZ of H f .In Fig. S1, we show the numerical simulation of such a quench process, with the state in each k -sector of the GBZinitialized in the lower-band of H ik , and projecting the dynamics onto the GBZ of H f . In Fig. S1(a), ν i = 0 and ν f = 1,so that skyrmion-like structures are visible. By contrast, in Fig. S1(b), ν i = 1 and ν f = 1, and no visible skyrmionstructures are present. Furthermore, we characterize in Fig. S1(c) the variation of GBZs with increasing t , and showthe resulting ¯ n as a function of t in Fig. S1(d). Similar to the main text, ¯ n = (1 /T (cid:48) k ) (cid:82) T (cid:48) k [ n (0 , t ) − n ( π, t )] dt ,where T (cid:48) k = Re E fk , is used as the signature for the skyrmion-like structure. Apparently, ¯ n ∼ n ∼
0) for ν i (cid:54) = ν f ( ν i = ν f ), for over a considerable range of t . Quench dynamics in a quantum walk
As discussed in the main text, we consider quantum walk dynamics governed by the Floquet operator U given byEq. (6). Under an OBC, the GBZ of U is a circle on the complex plane, its radius r = (cid:113) | cosh γ cos 2 θ − sinh γ cosh γ cos 2 θ +sinh γ | [31].0 -1 0 10246 -101 -1 0 10246 -101 (a) (b) (c) -0.8 -0.4 0 0.4 0.8-0.8-0.400.40.8 (d) FIG. S1. (a)(b) Spin texture n ( k, t ) on the generalized momentum-time space for non-Bloch quench dynamics under non-Hermitian SSH model with next-nearest-neighbor hopping, with γ = 0 .
5. (a) Quench from H i with t i = 0 . t i = 0(non-Bloch winding number ν i = 0) to H f with t f = 1 . t f = 0 .
05 (non-Bloch winding number ν f = 1). (b) Quenchfrom t i = 1 and t i = 0 (non-Bloch winding number ν i = 1) to t f = 1 . t f = 0 .
05. (c) ∆¯ n with increasing t f and a fixed t f = 1 .
5. The initial parameters are t i = 0 . t i = 1 (blue), respectively. (d) GBZs of H f in (c), with t f = 0 (blue)and t f = 0 .
05 (red), respectively. GBZs of H i for both cases coincide with the blue circle. We take t = 1 as the unit of energy. -1 0 102468 -101 -1 0 102468 -101 -1 0 102468 -101 -1 0 102468 -101 -101-1 0 1-101 -101-1 0 1-101 -101-1 0 1-101 -101-1 0 1-101 (f) (g) (h) (i) (j)(a) (b) (c) (d) (e) FIG. S2. Stroboscopic simulation of quench dynamics using a quantum walk governed by U in Eq. (6), with γ = 0 .
4. Quenchdynamics is projected in (top) the GBZ and (lower) BZ, respectively. (a) Non-Bloch topological phase diagram of the effectiveHamiltonian H under OBC, where U = e − iH . The horizontal green lines represent locations of localized eigenstates. Thevertical dashed lines are the non-Bloch exceptional lines separating the exact non-Bloch PT symmetric and PT-broken regimes,where we only focus on the PT-unbroken regime. Regions filled in yellow, cyan and red respectively feature non-Bloch windingnumbers ν = 1 , , −
1. (b) Dynamic spin texture n ( k, t ) in the generalized momentum-time space, for a quantum quench frompoint I (with non-Bloch winding number ν i = 1) to A (with non-Bloch winding number ν f = 0). (c) n ( k, t ) in the GBZ atdifferent times t for the quench in (b). (d) Spin texture n ( k, t ) for a quench from point I to B (with non-Bloch winding number ν f = 1). No skyrmion structures are present, in contrast to (b). (e) Components of n ( k, t ) in the GBZ at different times t forthe quench in (d). (f) Bloch topological phase diagram under PBC. Eigen spectra are complex throughout the phase diagram.Bloch winding numbers are marked in the phase diagram, where regions with different winding numbers are colored differently.(g) Dynamic spin texture n ( k, t ) in the momentum-time space (BZ), for a quantum quench from point I (with Bloch windingnumber ν i = 1) to A (with Bloch winding number ν f = 1 / n ( k, t ) in the BZ at different times t forthe quench in (g). (i) Spin texture n ( k, t ) in the momentum-time space for a quench from point I to B (with Bloch windingnumber ν f = 1 / n ( k, t ) in the BZ at different times t for the quench in (i). The parameters for points A , B , and I are ( θ , θ ) = (0 . π, . π ) , (0 . π, . π ) , (0 . π, . π ), respectively. | Ψ i (cid:105) ∝ | , A (cid:105) + | , B (cid:105) + 2 i/r i | − , A (cid:105) + 1 /r i ( −| − , A (cid:105) + | − , B (cid:105) ), whichis an eigenstate of an initial Floquet operator U i with θ i = 3 π/ θ i = 0 . π . Formally writing the initial stateas | Ψ i (cid:105) = 1 / √ N (cid:80) k | Ψ ik (cid:105) , where | Ψ ik (cid:105) = | β ik,R (cid:105) ⊗ | ψ ik (cid:105) , we evolve it under the Floquet operator U = e − iH f t . Thetime-evolved state is then | Ψ( t ) (cid:105) = (cid:80) k e − iH f t | Ψ ik (cid:105) . Projecting the dynamics onto the GBZ of the final effectiveHamiltonian H f , we have | ψ ( k, t ) (cid:105) = P k | Ψ( t ) (cid:105) = P k (cid:88) k e − iH f t | Ψ ik (cid:105) = N (0)( e − iH fk t | ψ ik (cid:105) + (cid:88) k (cid:48) (cid:54) = k N ( k − k (cid:48) ) N (0) e − iH fk t | ψ ik (cid:48) (cid:105) ) , (S19)where N ( k − k (cid:48) ) = (cid:104) β fk,L | β ik (cid:48) ,R (cid:105) = N (cid:80) n ( r i r f ) n e − i ( k − k (cid:48) ) n . Since |N ( k − k (cid:48) ) / N (0) | ∼ | ( r i − r f ) / ( r i + r f ) | , the secondterm on the right-hand side of Eq. (S19) vanishes for r i = r f , and is negligibly small for | r i − r f | (cid:28) r i + r f . Wenote that a similar situation should occur for the quench dynamics of lattice models at half filling, as we discussedpreviously.To see the relation between non-Bloch winding numbers of the effective Hamiltonian and non-Bloch quench dy-namics, we first project the Floquet operator U onto the GBZ U ( k ) = P k U P k , (S20)which, under the parameters considered in the main text | cos 2 θ | > | tanh γ | , can be decomposed as U ( k ) = d σ − id σ − id σ − id σ , (S21)where d = λ cos 2 θ cos k − sin 2 θ sin 2 θ cosh γ, (S22) d = 0 , (S23) d = λ sin 2 θ cos k − cos 2 θ sin 2 θ cosh γ, (S24) d = − λ sin k., (S25)with λ = [(cosh γ cos 2 θ − sinh γ )(cosh γ cos 2 θ + sinh γ )] / .The non-Bloch winding number of the effective Hamiltonian H, with U = e − iH , is calculated through [27] ν = 12 π (cid:90) dk − d ∂ k d + d ∂ k d d + d , (S26)which reduces to the Bloch winding number for γ = 0.Notably, U ( k ) is unitary and chiral symmetric with σ x U ( k ) σ x = U − ( k ), where σ x is the chiral symmetry operator.It follows that H fk in Eq. (S19) is Hermitian and chiral symmetric, and the proof regarding the existence of topologicalfixed points and dynamic Chern number (hence dynamic skyrmions) is essentially the same as that in Ref. [32] .Importantly, these conclusions are exact for r i = r f , when the second term in Eq. (S19) is zero, and approximatelytrue when the term is negligibly small.In Fig. S2, we perform a numerical simulation of quantum walk dynamics, starting from a localized state in thebulk. The quench dynamics is projected onto the GBZ in the first row of Fig. S2, and onto the BZ in the second row.We focus here on the case with r i = r f , as more general cases are already shown in the main text. Here dynamicskyrmions emerge in Fig. S2(b), where ν i = 0 and ν f = 1. In Fig. S2(d), on the other hand, no dynamic skyrmions arepresent, as ν i = 1 and ν f = 1. Therefore, if either the non-Bloch winding number of the initial or the final effectiveHamiltonian is known, one can infer the value of the other one by the observation of spin textures. Consistent withthese conclusions, we find that fixed points exist at k m = 0 and k m = π for both cases [see Fig. S2(c)(e)], but theyare of the same kind in Fig. S2(d)(e), hence the dynamic Chern numbers are zero therein.By contrast, when projected on the BZ, as shown in Fig. S2(f-j), the spin textures feature no skyrmion structures,and the dynamics is steady-state approaching in all kk