Protecting topological order by dynamical localization
Yu Zeng, Alioscia Hamma, Yu-Ran Zhang, Jun-Peng Cao, Heng Fan, Wu-Ming Liu
PProtecting topological order by dynamical localization
Yu Zeng, Alioscia Hamma, ∗ Yu-Ran Zhang, Jun-Peng Cao,
1, 4
Heng Fan,
1, 4, † and Wu-Ming Liu
1, 4, ‡ Beijing National Laboratory for Condensed Matter Physics,Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China Department of Physics, University of Massachusetts Boston, 100 Morrissey Blvd, Boston MA 02125 Theoretical Quantum Physics Laboratory, RIKEN Cluster for Pioneering Research, Wako-shi, Saitama 351-0198, Japan Collaborative Innovation Center of Quantum Matter, Beijing 100190, China
As a prototype model of topological quantum memory, two-dimensional toric code is genuinely immune togeneric local static perturbations, but fragile at finite temperature and also after non-equilibrium time evolutionat zero temperature. We show that dynamical localization induced by disorder makes the time evolution alocal unitary transformation at all times, which keeps topological order robust after a quantum quench. Weconfirm this assertion by investigating the Wilson loop expectation value and topological entanglement entropy.Our results suggest that the two dimensional topological quantum memory can be dynamically robust at zerotemperature.
Introduction.—
Novel quantum phases in many-bodysystems that feature topological order are of great impor-tance in both condensed matter physics [1] and in quan-tum information processing [2]. They possess gapped en-ergy spectrum and robust ground-state degeneracy, whichare supposed to be promising candidates of the self-correcting quantum memories [3]. These novel quan-tum phases cannot be described by the Landau paradigmof symmetry breaking and are not characterized by lo-cal order parameters. Instead, they are characterized bya long-range pattern of entanglement dubbed topologicalentropy (TE) [4–7] that serves as a nonlocal order param-eter [8].In order to obtain a self-correcting quantum memory,topological order must be robust not only under staticperturbations, but also at the dynamical level and at fi-nite temperature [9, 10]. Topologically ordered systemsor self-correcting quantum memories in two and three di-mensions, based on local Hamiltonians with commutingoperators, are not stable both at finite temperature [11–18] or when cast away from equilibrium [19–22]. Onthe other hand, both the topological phase and its self-correcting quantum memory can be robust in four orgreater spatial dimensions, which, unfortunately, is notrealistic for implementation [3, 16, 23, 24]. The deple-tion of both topological entropy and topological quan-tum memory is due to the free diffusion of point-like de-fects that ultimately destroy both features, as they are in-timately connected, although not exactly the same thing[15, 23, 25]. Several schemes have been proposed toovercome these shortcomings, from the introduction oflong-range interactions between the excitations [26, 27],to models that feature membrane condensation [28] to-gether with the absence of string-like excitations [29, 30],and the introduction of localization [31–35], the lattershowing that localization can increase the lifetime of quantum memory, also see [16] for extended references.In this Letter, we study how dynamical localization in-duced by disorder can keep the topological ordered phaserobust at all times after a quantum quench at zero tem-perature. We show that, by randomizing the toric codestabilizer coupling constants, the unitary time evolutionoperator is equivalent to a local adiabatic transformation.The unitary time evolution after a quantum quench mapsa ground state of the toric code model into a ground stateof a local Hamiltonian belonging to the same phase. Theground state degeneracy, the energy gap, and the patternof long-range entanglement are all preserved during thetime evolution. On the other hand, without disorder, thetime-evolution operator becomes highly nonlocal at longtimes, and the TE will self-thermalize after a quantumquench [22].
Toric code with external fields.—
We consider the twodimensional toric code model (TCM) [36], defined on a M × N square lattice Λ with periodic boundary condi-tions and spins / on the bonds of the lattice. The TCMHamiltonian is given by H T C = − ∑ s J s A s − ∑ p J p B p ,where A s ≡ ∏ i ∋ s σ xi and B p ≡ ∏ i ∈ ∂p σ zi are stabilizeroperators indexed by s on the lattice site (vertex) and p on the dual lattice site (face). All the coupling constants J s and J p are positive, so each stabilizer operators actstrivially as + in an arbitrary ground state. Because ofthe periodic boundary conditions, the ground-state spaceis 4-fold degenerate and can thus encode 2 qubits. Thelogical operators are two pairs of topologically nontriv-ial string operators: W αi ≡ W [ γ αi ] = ∏ l ∈ γ αi σ αl with α = x, z , and i counts the generators of the homotopygroup of the torus. The non-contractable path of γ x con-nects dual lattice sites, while γ z connects lattice sites, seeFig. 1. The ground state wave function of the TCM isgapped and possesses the property of closed-string con-densation with topological order [37]. a r X i v : . [ qu a n t - ph ] F e b γ γ γ γ s p FIG. 1. (Color online) Illustration of the square lattice Λ withphysical spins living on the bonds in odd rows (black dots) andeven rows (white dots). The Examples of star (s), plaquete (p),and the non-contractible path γ αi ( α = x, z and i = , ) areshown. The quantum quench protocol consists in preparing thesystem in one state of the ground space manifold of theTCM (without loss of generality, we choose the sector of W x = , W z = [38]), and then suddenly switching onthe external local fields. The post-quench Hamiltonianreads H ( J, h )=−∑ s J s A s −∑ p J p B p −∑ i ∈ oddrows h oi σ zi − ∑ j ∈ evenrows h ej σ xj , (1)then the initial state evolves as ∣ Ψ ( t )⟩ = U ( J, h ; t )∣ Ψ ( )⟩ , where the time-evolution opera-tor is U ( J, h ; t ) = e − itH ( J,h ) .We can map the stabilizer operators to effective spinsresiding on lattice and dual lattice sites [22, 35, 39]: A s ↦ τ zs and B p ↦ τ zp . Each external local field op-erator flips the effective spins on its two ends, so inthis so called ‘ τ -picture’, we have σ z < s,s ′ > ↦ τ xs τ xs ′ and σ x < p,p ′ > ↦ τ xp τ xp ′ , where < s, s ′ > labels the bond betweenthe two adjacent lattice sites s and s ′ , while < p, p ′ > la-bels the bond between the two adjacent dual lattice sites p and p ′ . The Hamiltonian Eq. (1) is mapped to the sumsof quantum Ising chains as H ( J, h ) = M ∑ l = N ∑ j = [− J l,j τ zl,j − h l,j τ xl,j τ xl,j + ] , (2)with period boundary conditions in the sector we choose.It is obvious that the initial state corresponds to the para-magnetic state with effective spins pointing at z directionalong each row in the τ -picture.The Hamiltonian Eq. (2) can be solved by mapping the τ spins to fermion operators via Jordan-Wigner transfor-mations: τ zj = − c † j c j and τ xj = ∏ l < j ( − c † l c l )( c j + c † j ) .Introducing a row vector ψ † = ( c † , c , c † , c , ⋯ , c † N , c N ) and its Hermitian conjugate column vector ψ , we write the Hamiltonian as the quadratic form H ( J, h ) = ψ † M ( J, h ) ψ. (3)The first quantized Hamiltonian is given as a × -block tri-diagonal Jacobi matrix (except for the boundaryterms) M ( J, h )=⎛⎜⎜⎜⎜⎜⎝ J σ z − h S h N S T − h S T J σ z ⋱ ⋱⋱ J N − σ z − h N − Sh N S − h N − S T J N σ z ⎞⎟⎟⎟⎟⎟⎠ , with S = σ z + iσ y , where σ y and σ z are the standardPauli matrices. The fermion operators in the Heisenbergpicture are c l ( t )= N ∑ j = ( e − itM ) l − , j − c j +( e − itM ) l − , j c † j . (4)Mapping back to original spin space, we can get spin op-erators in the Heisenberg picture. Dynamical localization and adiabatic connection.—
The main result of this paper is that, in presence of dy-namical localization, a quantum quench is equivalent toa quasiadiabatic continuation ∣ Ψ ( )⟩ ↦ ∣ Ψ ( t )⟩ [40–42],and thus, the two adiabatically connected states belong tothe same phase [43]. The initial state is a ground stateof the local Hamiltonian H i : H i ∣ Ψ ( )⟩ = E ∣ Ψ ( )⟩ .The time-evolution operator is generated by a dynami-cal localized Hamiltonian H f : U ( t ) = exp (− iH f t ) , and ∣ Ψ ( t )⟩ = U ( t )∣ Ψ ( )⟩ . Define now the family of Hamil-tonians H ( t ) = U ( t ) H i U ( t ) † . (5)All the members in the family belong to the same con-nected component of iso-spectral Hamiltonians so thatadiabatic evolution is well defined [44]. Under this con-dition, H ( t )∣ Ψ ( t )⟩ = E ∣ Ψ ( t )⟩ and, following Ref. [43],states ∣ Ψ ( t )⟩ and ∣ Ψ ( )⟩ are in the same quantum phaseiff H ( t ) is a local Hamiltonian. In order to prove thisresult, we need to show that, starting with a spin Hamil-tonian H = ∑ Z I Z , where each I Z is a bounded operatorsupported on a set Z with bounded diameter, the Hamil-tonian H ( t ) = ∑ Z I Z ( t ) is also a local Hamiltonian.In a (non relativistic) quantum many-body system,locality manifests with the emergence of an effectivelight cone characterized by the Lieb-Robinson velocity v , which is the maximum velocity of signals in the model[45–48]. Signals outside the light cone are exponentiallysuppressed. We use a Lieb-Robinson bound in this form:for any two operators A X and B Y supported on subsets X and Y in Λ , ∥[ A X ( t ) , B Y ]∥⩽ c ∣ X ∣∥ A X ∥∥ B Y ∥ e − µ ( dist ( X,Y )− vt ) . (6)Here c , µ and v are nonnegative, ∥⋯∥ denotes oper-ator norm, ∣⋯∣ denotes the cardinality of the set, and dist ( X, Y ) is a well defined distance, which makes thelattice a metric space, between subsets X and Y . A quan-tum spin system is dynamical localized if v = , i.e.,the system has the zero-velocity Lieb-Robinson bound[49, 50].First, we show that each I Z ( t ) can be approximatedby a local operator with finite diameter l . Following Ref.[47], define I lZ ( t ) = ∫ dµ ( V ) V I Z ( t ) V † , (7)where the integral is over unitary operator acting on theset with a distance larger than l from set Z with Haarmeasure. Then, I lZ ( t ) is supported on the ball of radius l about set Z , denoted by B l ( Z ) . Therefore, we get ∥ I Z ( t ) − I lZ ( t )∥ ≤ ∫ dµ ( V ) ∥[ V, I Z ( t )]∥ . (8)Combining the Lie-Robinson bound (6) with dynamicallocalization, i.e. v = , we get ∥ I Z ( t ) − I lZ ( t )∥ ≤ c ∣ Z ∣ ∥ I Z ∥ e − µl , (9)where the error of the approximation is bounded by anexponential decay with l .Then, H = ∑ Z ′ H Z ′ is a local Hamiltonian if for anypoint j ∈ Λ , ∑ Z ′ ∋ j ∥ H Z ′ ∥ ∣ Z ′ ∣ exp [ ν diam ( Z ′ )] ≤ s < ∞ , (10)where ν , s are positive constants, and diam ( Z ′ ) is thediameter of set Z ′ . Here diam ( Z ′ ) can be arbitrarylarge, while ∥ H Z ′ ∥ needs to be exponentially decayingwith diam ( Z ′ ) . Equation (10) is a sufficient condi-tion for a Lieb-Robinson bound [51]. We decompose I Z ( t ) = ∑ l H lZ ( t ) by defining a sequence of operators H lZ ( t ) = I lZ ( t ) − I l − Z ( t ) , H Z = I Z ( t ) . (11) H lZ ( t ) is supported on set B l ( Z ) with diam ( B l ( Z )) ≤ diam ( Z )+ l , and its norm can be bounded using Eq. (9)and the triangle inequality ∥ H lZ ( t )∥ ≤ c ′ e µ diam ( Z ) ∣ Z ∣ ∥ I ∥ e − µ diam ( B l ( Z )) , (12)where c ′ = c ( + e µ ) is a constant. Since diam ( Z ) and ∣ Z ∣ are bounded by constants, H ( t ) = ∑ Z I Z ( t ) =∑ Z,l H lZ ( t ) , satisfying local condition Eq. (10), is a lo-cal Hamiltonian. At this point, we want to show that the topologicalphase in the model Eq. (1) and its counterpart Eq. (2) ispreserved after a quantum quench with disordered cou-plings { J } for the stabilizers. To this end, we need toshow that the model is dynamically localized. In Refs.[49, 52], it was proved that the system is dynamicallylocalized provided that the effective one-particle Hamil-tonian in Eq. (3) satisfies E ( sup t ∈ R ∥[ e − itM ] jk ∥) ≤ Ce − η ∣ j − k ∣ ζ . (13)Here, E (⋯) denotes disorder averaging, [⋯] jk is a × -matrix-valued entry, η is positive, and ζ ∈ ( , ] . Ageneral result of Ref. [53] covers the model we discussedwith conditions of large disorder as well as sufficientlysmooth distribution of { J } . The exact exponential decaywith ζ = in Eq. (13) is proved therein. For arbitrarynontrivial compactly supported distributions, Ref. [52]proved Eq. (13) with η ∈ ( , ) , where the bound decayssub-exponentially provided the gap is not closed. Noticethat we can define dist ′ ( i, j ) = ∣ i − j ∣ ζ , which is a welldefined distance as you can verify, and then the boundturns out to exponential decay. Wilson loop expectation value.—
Having shown themain result of this Letter, that is, ∣ Ψ ( t )⟩ and ∣ Ψ ( )⟩ be-long to the same phase with unchanged energy gap, wenow investigate two typical ( nonlocal ) order parametersfor topological order to confirm our conclusions. If onlyone type of external fields are turned on ( h o ≠ and h e = for clarity), the Z gauge structure is intact dur-ing the time evolution. Let us consider the closed stringconnecting the dual lattice sites and surrounding a squareregion R with side length D , and the Wilson loop oper-ator reads W R ≡ ∏ i ∈ ∂R σ xi = ∏ s ∈ R A s . In the τ picture,every A s corresponds to an effective spin τ zs , so the Wil-son loop operator is products of D rows of τ z strings.Taking advantage of the dual symmetry we transformthe τ spins to their dual µ spins: µ zl,j = τ xl,j τ xl,j + , and µ xl,j = ∏ k ≤ j τ zl,k . Then the Wilson loop operator expecta-tion value is mapped to the spin correlation function in µ picture ⟨ W R ⟩ = D ∏ l = ⟨ µ xl,r µ xl,r + D ⟩ . (14)We first consider the clean Hamiltonian. Equation (14)satisfies the perimeter law ⟨ W R ⟩ ∼ exp (− αD ) when thestate is ferromagnetic in the µ picture and deconfined(topological ordered) in the σ picture; or the area law ⟨ W R ⟩ ∼ exp (− D / ξ ) when the state is paramagnetic inthe µ picture and confined (topological trivial) in the σ picture. Nevertheless, even though the initial state is de- D -8 -6 -4 -2 C xx ( D ) =0=0.125=0.25=0.375=0.5 D S ( D ) (a) (b) FIG. 2. (Color online) (a) The spin correlation function C xx ( D ) and (b) the entanglement entropy S ( D ) are functionsof D at fixed time t = with N = . (cid:15) is a positiveparmeter to control the disorder strength. Black line: (cid:15) = ;cyan line: (cid:15) = . ; blue line: (cid:15) = . ; green line: (cid:15) = . ;red line: (cid:15) = . . confined, the post-quench Hamiltonian with nonzero ex-ternal fields, whatever it is confined or deconfined, willevolve the expectation value of Wilson loop operator tosatisfy area law [54–56]. The above scenario of Z gaugetheory analysis is compatible with the calculation of thetopological R´enyi entropy, where the initial topologicalentropy collapses to the half after the quench [22]. Thehalf residual topological entropy is believed to originatefrom the gauge structure [57].Let us now see what happens when dynamical local-ization is induced by disorder in the couplings J j . Weset J j = + (cid:15)η j where η j ∈ [− , ] are i.i.d random vari-ables, and (cid:15) is a positive parameter to control the disorderstrength. Setting h = . and t = fixed, the numericalresults for different (cid:15) , each with 1000 realizations, shownin Fig. 2 (a), indicate that, as disorder increase (withoutclosing the gap), the spin correlation function tends to re-silience with the distance, which results in the perimeterlaw of Wilson loop expectation value and thus deconfine-ment of the phase. The numerical data for a long timeevolution of correlation function (also entanglement en-tropy in the later discussion) lead to the same conclusion[54]. Topological entanglement entropy.—
To capture thelong-range entanglement of the system after a quantumquench, we consider the topological entanglement en-tropy. To this end we calculate the Von Neumann en-tropy of an extended cylindrical subregion R , which is S ( ρ R ) = − tr ρ R log ρ R . The subregion boundary con-tains only left and right sides at a distance of D , and thelength of each side is M , which equal to the vertical sizeof the lattice. For the model Hamiltonian Eq. (1), weconsider the state ρ in sector W x = , W z = . The VonNeumann entropy of the reduced density operator in the σ picture, ρ σR , equals the sum of entropy of each row in the µ picture [54]: S ( ρ σR ) = M ∑ k = S ( ρ µR k ) . (15)For the ground state ρ of the TCM, the l.h.s of aboveequation can be directly obtained: S ( ρ σ R ) = M [5],and the r.h.s equals the sums of the bipartite entanglemententropy of the GHZ state, S ( ρ µ R k ) = , so the equationis satisfied. Notice that the topological entropy term ismissing, this paradox being caused by the subregion andthe sector we choose. The ground state in the sector isan equal weighted superposition of all topological trivialclosed strings and a topological non-trivial string alongpath γ x . Unlike local subregions, path γ x always goesacross the boundary of R and cannot bypass it by contin-uous deformation. Nevertheless, the ground state ρ ′ insector W z = , W z = contains only topological trivialclosed strings, in which case the r.h.s of Eq. (15) turns outto be S ( ρ ′ σ R ) = M − , where the topological entropyappears as log = .After a quantum quench, each S ( ρ µ R k ) grows lin-early in time for clean systems [48, 58, 59]. Thoughthe entanglement boundary law is satisfied and the topo-logical order cannot be completely destroyed at a shorttime [47], the entanglement entropy reachs a value pro-portional to the area of subsystem over a long enoughtime [59], and the TE vanishes [22]. However, In theregime of dynamical localization, S ( ρ µR k ) grows loga-rithmically in a short time, and then reaches to a satu-ration value, which is convergent as the size of the sub-system increases [54, 60] (but diverges logarithmically inthe critical regime [61, 62]). Actually, as we have shownbefore, ρ µ after the time evolution is a ground state ofa gapped local Hamiltonian. Therefore, the correlationsin the state is always exponentially decayed [51, 63–65],and the entanglement entropy is bounded by a constant atall times [66].The numerical results for distinct disorder strengths (cid:15) , as shown in Fig. 2(b), at a fixed time, also suggestthat S ( ρ µR k ( (cid:15) )) ≤ α ( (cid:15) ) , where α ( (cid:15) ) is a positive num-ber independent of the size of subsystem. Therefore, S ( ρ σ R ) = ∑ Mk = S ( ρ µ R k ) ≤ α ( (cid:15) ) M , which implies theentanglement boundary law. For the same reason as thestatic ground states, topological entanglement entropy is log = in the thermodynamic limit of both system andsubsystem. Conclusion and remarks.—
In this Letter, we investi-gate the fate of topological order after a quantum quenchat zero temperature in the two dimensional toric code inpresence of disorder. We show that disorder induces dy-namical localization, and in turn this makes the time evo-lution equivalent to a local quasiadiabatic transformation,which keeps the state within the same topological phase.Thus, dynamical localization makes topological order ro-bust after a quantum quench. We have verified this resultby a mapping to free fermions and numerically comput-ing both the Wilson loop expectation values and the en-tanglement entropy. Some of this paper’s authors alsocalculated the topological R´enyi entropy directly by thescheme of Levin and Wen [7] for small subsystems inRef. [35]. The results therein show the TE is resilientas disorder increases, which complements the contentsof this letter. Our conclusion is also compatible with theresult in Ref. [31], where the storage time of the mem-ory, after which the storage fidelity drops below a giventhreshold, grows exponentially with the system size.Some remarks are in order. The time evolution in thedynamical-localization regime is analogous to the quasi-adabatic continuation introduced in the scenario of time-independent local perturbation [40, 41], where weak lo-cal perturbations lift the ground state degeneracy expo-nentially small with an open spectrum gap [42], such thatthe code space is preserved. Quasiadiabatic continuationallows to define dressed operators which are equivalent totheir time evolution in the Heisenberg picture, so the fam-ily of Hamiltonians Eq. (5) is local and isospectral, thuspreserving both the code space and the quantum mem-ory. Moreover, the dressed Wilson loop operators and de-formed local Z gauge transformations can be derived byanalogy with Ref.[40], where the “zero law” of dressedWilson loop indicating deconfinement can be obtained.Dressed anyons, as argued in Ref. [42], can be realizedin the similar fashion.The entanglement entropies of all the energy eigen-states of a dynamical localized Hamiltonian are alsobounded by a constant [67, 68]. As a consequence,the entanglement of a thermal state, e.g., the entangle-ment of formation [69], satisfies an area law [50]. Thismay hint at the possibility of self-correcting low dimen-sional quantum memory at finite temperature. Whilewe tackle the perturbed TCM in a regime where thetwo-dimensional system can be decoupled in many spinchains, results for general two dimensional models arestill lacking. We leave these important issues to the fu-ture exploration.This work was supported by the National Key R&DProgram of China (grants No. 2016YFA0301500),NSFC (grant No. 61835013), Strategic Priority Re-search Program of the Chinese Academy of Sciences(grants Nos. XDB01020300, XDB21030300)(W. -M.L.); the National Key R&D Program of China (grantNo. 2017YFA0304300), Strategic Priority Research Pro-gram of the Chinese Academy of Sciences (grant No.XDB28000000)(H. F.); NSFC (Grant Nos. 12074410, 12047502, 11934015)(J. -P. C); the JSPS PostdoctoralFellowship (Grant No. P19326), the JSPS KAKENHI(Grant No. JP19F19326)(Y. -R. Z); NSF (award No.2014000) (A. H.). ∗ [email protected] † [email protected] ‡ [email protected][1] X. -G. Wen, Quantum field theory of many body systems (Oxford university press, 2004).[2] C. Nayak, S. H. Simon, A. Stern, M. Freedman, andS. Das Sarma,
Non-Abelian anyons and topological quan-tum computation , Rev. Mod. Phys. , 1083 (2008).[3] E. Dennis, A. Kitaev, A. Landahl, and J. Preskill, Topo-logical quantum memory , J. Math. Phys. , 4452 (2002).[4] A. Hamma, R. Ionicioiu, and P. Zanardi, Bipartite entan-glement and entropic boundary law in lattice spin systems ,Phys. Rev. A , 022315 (2005).[5] A. Hamma, R. Ionicioiu, P. Zanardi, Ground state entan-glement and geometric entropy in the Kitaev model , Phys.Lett. A , 22 (2005).[6] A. Kitaev and J. Preskill,
Topological entanglement en-tropy , Phys. Rev. Lett. , 110404 (2006).[7] M. Levin and X. -G. Wen, Detecting topological order ina ground state wave function , Phys. Rev. Lett. , 110405(2006).[8] A. Hamma, W. Zhang, S. Haas, D. Lidar, Entanglement, fi-delity, and topological entropy in a quantum phase transi-tion to topological order , Phys. Rev. B , 155111 (2008).[9] S. Chesi, D. Loss, S. Bravyi, and B. M. Terhal, Thermo-dynamic stability criteria for a quantum memory based onstabilizer and subsystem codes , New J. Phys. , 025013,(2010).[10] S. Iblisdir, D. P´erez-Garcia, M. Aguado, and J. Pachos, Thermal states of anyonic systems , Nucl. Phys. B , 401(2010).[11] S. Bravyi, and B. Terhal,
A no-go theorem for a two-dimensional self-correcting quantum memory based onstabilizer codes , New J. Phys. , 043029 (2009).[12] O. Landon-Cardinal and D. Poulin, Local topological or-der inhibits thermal stability in 2D , Phys. Rev. Lett. ,090502 (2013).[13] B. Yoshida,
Feasibility of self-correcting quantum mem-ory and thermal stability of topological order , Ann. Phys.(Amsterdam) 326, 2566 (2011).[14] C. Castelnovo and C. Chamon,
Entanglement and topo-logical entropy of the toric code at finite temperature ,Phys. Rev. B , 184442 (2007).[15] C. Castelnovo and C. Chamon, Topological order in athree-dimensional toric code at finite temperature , Phys.Rev. B , 155120, (2008).[16] B. J. Brown, D. Loss, J. K. Pachos, C. N. Self, andJ. R. Wootton, Quantum memories at finite temperature ,Rev. Mod. Phys. , 045005 (2016).[17] M. B. Hastings, Topological order at nonzero temperature , Phys. Rev. Lett. , 210501 (2011).[18] R. Mohseninia,
Thermal stability of the two-dimensionaltopological color code , Phys. Rev. A , 022306 (2016).[19] A. Kay, Nonequilibrium reliability of quantum memories ,Phys. Rev. Lett , 070503 (2009).[20] A. Kay, capabilities of a perturbed toric code as a quan-tum memory , Phys. Rev. Lett , 270502 (2011).[21] F. Pastawski, A. Kay, N. Schuch, and I. Cirac,
Limitationsof passive protection of quantum information , QuantumInf. Comput. , 0580 (2010).[22] Y. Zeng, A. Hamma, and H. Fan, Thermalization of topo-logical entropy after a quantum quench , Phys. Rev. B ,125104 (2016).[23] D. Maz´aˇc and A. Hamma, Topological order, entangle-ment, and quantum memory at finite temperature , Ann.Phys. (N.Y.) , 2096 (2012).[24] R. Alicki, M. Horodecki, P. Horodecki, and R. Horodecki,
On thermal stability of topological qubit in Kitaev’s 4Dmodel , Open Syst. Inf. Dyn. , 1 (2010).[25] Z. Nussinov and G. Ortiz, Autocorrelations and thermalfragility of anyonic loops in topologically quantum or-dered systems , Phys. Rev. B , 064302 (2008).[26] A. Hamma, C. Castelnovo, and C. Chamon, Toric-bosonmodel: Toward a topological quantum memory at finitetemperature , Phys. Rev. B , 245122 (2009).[27] S. Chesi, B. R¨othlisberger, and D. Loss, Self-correctingquantum memory in a thermal environment , Phys. Rev. A , 022305, (2010).[28] A. Hamma, P. Zanardi, X.-G. Wen, String and Membranecondensation on 3D lattices , Phys. Rev. B , 035307(2005).[29] J. Haah, Local stabilizer codes in three dimensions withoutstring logical operators , Phys. Rev. A , 042330 (2005).[30] S. Bravyi and J. Haah, Quantum self-correction in the 3Dcubic code model , Phys. Rev. Lett. , 200501 (2013).[31] S. Bravyi and R. K¨onig,
Disorder-assisted error correc-tion in majorana chains , Comm. Math. Phys. , 641(2012).[32] J. R. Wootton and J. K. Pachos,
Bringing order throughdisorder: localization of errors in topological quantummemories , Phys. Rev. Lett. , 030503 (2011).[33] C. Stark, L. Pollet, A. Imamo˘glu, and R. Renner,
Local-ization of toric code defects , Phys. Rev. Lett. , 030504(2011).[34] H. Yarloo, A. Langari, and A. Vaezi,
Anyonic self-induceddisorder in a stabilizer code: Quasi many-body localiza-tion in a translational invariant model
Phys. Rev. B ,054304 (2018)[35] Y. Zeng, A. Hamma, and H. Fan, Disorder-protected topological entropy after a quantum quench ,arXiv:1704.08819[36] A. Y. Kitaev,
Fault-tolerant quantum computation byanyons , Ann. Phys. (N.Y.) , 2 (2003).[37] M. A. Levin and X. -G. Wen
String-net condensation: Aphysical mechanism for topological phases , Phys. Rev. B , 045110 (2005).[38] These constraints dose not change the general result. Weillustrate in [22] that the density matrices for a local sub-system in different topological sectors are identical, thus the entanglement entropy and the topological entropy areunchanged.[39] J. Yu, S.-P. Kou, and X.-G. Wen, Topological quantumphase transition in the transverse Wen-plaquette model ,Europhys. Lett. , 17 004 (2008).[40] M. B. Hastings and X. -G. Wen, Quasiadiabatic con-tinuation of quantum states: The stability of topologicalground-state degeneracy and emergent gauge invariance ,Phys. Rev. B , 045141 (2005).[41] T. J. Osborne, Simulating adiabatic evolution of gappedspin systems , Phys. Rev. A , 032321 (2007).[42] S. Bravyi, M. Hastings, and S. Michalakis, Topologicalquantum order: Stability under local perturbations , J.Math. Phys. , 093512 (2010).[43] X. Chen, Z. -C. Gu, and X. -G. Wen, Local unitary trans-formation, long-range quantum entanglement, wave func-tion renormalization, and topological order , Phys. Rev. B , 155138 (2010).[44] A. Hamma, P. Zanardi, Quantum entangling power ofadiabatically connected Hamiltonians , Phys. Rev. A ,062319 (2004).[45] E. H. Lieb and D. W. Robinson, The finite group veloc-ity of quantum spin systems , Comm. Math. Phys. , 251(1972).[46] B. Nachtergaele, Y. Ogata, and R. Sims, Propagation ofcorrelations in quantum lattice systems , J. Stat. Phys. ,1 (2006).[47] S. Bravyi, M. B. Hastings, and F. Verstraete,
Lieb-Robinson bounds and the generation of correlations andtopological quantum order , Phys. Rev. Lett. , 050401(2006).[48] J. Eisert and T. J. Osborne, General entanglement scal-ing Laws from time evolution , Phys. Rev. Lett. , 150404(2006).[49] E. Hamza, R. Sims, and G. Stolz, Dynamical localizationin disordered quantum spin systems , Comm. Math. Phys.315, 215 (2012).[50] H. Abdul-Rahman, B. Nachtergaele, R. Sims, and G.Stolz,
Localization properties of the disordered XY spinchain , Ann. Phys. (Berlin) , 1600280 (2017).[51] M. B. Hastings, T. Koma,
Spectral Gap and ExponentialDecay of Correlations , Commun. Math. Phys. , 781(2006).[52] J. Chapman and G. Stolz,
Localization for random blockoperators related to the XY spin chain , Ann. HenriPoincar´e 16, 405 (2015).[53] A. Elgart, M. Shamis, and S. Sodin,
Localisation for non-monotone Schr¨odinger operators , J. Eur. Math. Soc. ,909 (2014).[54] See Supplemental Material.[55] K. Sengupta, S. Powell, and S. Sachdev, Quench dynamicsacross quantum critical points , Phys. Rev. A , 053616(2004).[56] P. Calabrese, F. H. L. Essler, and M. Fagotti, Quantumquench in the transverse-field Ising chain , Phys. Rev. Lett. , 227203 (2011)[57] C. Castelnovo and C. Chamon,
Topological order andtopological entropy in classical systems , Phys. Rev. B ,174416 (2007). [58] P. Calabrese and J. Cardy, Evolution of entanglement en-tropy in onedimensional systems , J. Stat. Mech. (2005)P04010.[59] M. Fagotti and P. Calabrese,
Evolution of entanglemententropy following a quantum quench: Analytic results forthe XY chain in a transverse magnetic field , Phys. Rev. A , 010306(R) (2008).[60] C. K. Burrell and T. J. Osborne, Bounds on the speed of in-formation propagation in disordered quantum spin chains ,Phys. Rev. Lett. , 167201 (2007).[61] F. Igl´oi, Z. Szatm´ari, and Y. -C. Lin, Entanglement entropydynamics of disordered quantum spin chains , Phys. Rev. B , 094417 (2012).[62] Y. Zhao, F. Andraschko, and J. Sirker, Entanglement en-tropy of disordered quantum chains following a globalquench , Phys. Rev. B , 205146 (2016).[63] M. B. Hastings, Lieb-Schultz-Mattis in higher dimensions
Phys. Rev B , 104431 (2004).[64] M. B. Hastings, Locality in quantum and Markov dynam- ics on lattices and networks , Phys. Rev. Lett. , 140402(2004).[65] B. Nachtergaele, and R. Sims, Lieb-Robinson bounds andthe exponential clustering theorem , Commun. Math. Phys.bf 265, 119 (2006).[66] F. G. S. L. Brand ˜ ao and M. Horodecki, An area lawfor entanglement from exponential decay of correlations ,Nat. Phys. , 721 (2013).[67] H. Abdul-Rahman, B. Nachtergaele, R. Sims, andG. Stolz, Entanglement dynamics of disordered quantumXY chains , Lett. Math. Phys. , 649 (2016).[68] H. Abdul-Rahman and G. Stolz,
A uniform Area Lawfor the entanglement of eigenstates in the disordered XY-chain , J. Math. Phys. , 121901 (2015).[69] C. H. Bennett, D. P. DiVincenzo, J. A. Smolin, andW. K. Wootters, Mixed-state entanglement and quantumerror correction , Phys. Rev. A , 3824 (1996). upplemental Material for “Protecting topological order by dynamical localization” Yu Zeng, Alioscia Hamma, ∗ Yu-Ran Zhang, Jun-Peng Cao,
1, 4
Heng Fan,
1, 4, † and Wu-Ming Liu
1, 4, ‡ Beijing National Laboratory for Condensed Matter Physics,Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China Department of Physics, University of Massachusetts Boston, 100 Morrissey Blvd, Boston MA 02125 Theoretical Quantum Physics Laboratory, RIKEN Cluster for Pioneering Research, Wako-shi, Saitama 351-0198, Japan Collaborative Innovation Center of Quantum Matter, Beijing 100190, China
CONTENTS
A. Wilson loop expectation value in clean system 1B. Von Neumann entanglement entropy for a cylindrical subsystem 2C. Calculation of correlation function and entanglement entropy 41. Formalism 42. Analytical and numerical results 7References 11
Appendix A: Wilson loop expectation value in clean system
The perturbed toric code Hamiltonian in the original σ picture, Eq. (1) in the main text, can be mapped to the sumof uncorrelated quantum Ising chains in the τ picture. Applying the dual transformation, we get the Hamiltonian in the µ picture H ( J, h ) = M ∑ l = N ∑ j = [− J l,j µ xl,j µ xl,j + − h l,j µ zl,j ] (A.1) D D R FIG. A1. (Color online) Illustration of Wilson loop surrounding a square (blue) region R with side length D = . The Wilson loopoperator is the product of the σ x operators crossed by the dashed line, which equals the product of A s operators inside the blueregion R . ∗ [email protected] † [email protected] ‡ [email protected] a r X i v : . [ qu a n t - ph ] F e b with periodic boundary condition in the sector of W x = , W z = . The expectation value of Wilson loop, surroundinga square region R with side length D , see Fig. A1, equals to a product of equal-time spin correlation functions in the µ picture: ⟨ W R ⟩ = D ∏ l = ⟨ µ xl,r µ xl,r + D ⟩ . (A.2)We first consider the static case. In the ferromagnetic phase with J / h > , the correlation function tends to a constant,which reads lim D →∞ ⟨ µ xr µ xr + D ⟩ F = ( − ( hJ ) ) . (A.3)So we get the perimeter law ⟨ W R ⟩ ∼ exp (− αD ) , where the coefficient α = − ln [ − ( hJ ) ] , which implies the groundstate is in the Z deconfined phase. On the other hand, the paramagnetic phase with J / h < goes with the exponentialdecay of the correlation function: lim D →∞ ⟨ µ xr µ xr + D ⟩ P F ∼ exp (− D / ξ ) , (A.4)where the correlation length is ξ = ( − Jh ) − [1]. So the area law ⟨ W R ⟩ ∼ exp (− D / ξ ) is followed, which indicatesthat the ground state is in the Z confined phase. It is clear that, in the original σ picture, the Hamiltonian parameterscontrol the topological quantum phase transition in the toric code model, which corresponds to the quantum phasetransition in the transverse field Ising model in the µ picture.The second related example is the quantum quench with clean Hamiltonian. The initial state is prepared to be theferromagnetic ground state in the µ picture and then evolves with the suddenly changed Hamiltonian. The analyticalresults indicates that the correlation function decays exponentially with the distance between the spins after long time[4]. explicitly, when < h / J < , lim t →∞ ,D →∞ ⟨ Ψ ( t ) ∣ µ xr µ xr + D ∣ Ψ ( t )⟩ ∼ ⎛⎜⎝ + √ − ( hJ ) ⎞⎟⎠ D + ; (A.5)when h / J > , lim t →∞ ⟨ Ψ ( t ) ∣ µ xr µ xr + D ∣ Ψ ( t )⟩ = D . (A.6)So the area law is always appeared as ⟨ W R ⟩ ∼ exp (− α ′ D ) after a quantum quench, where α ′ = − ln [ ( +√ − ( h / J ) )] if < h / J < ; or α ′ = ln 2 if h / J > . This result indicates that, in the clean spin system, theinitial deconfined phase collapses to the confined phase after a quantum quench with added external fields, no matterhow small the field strength is. Appendix B: Von Neumann entanglement entropy for a cylindrical subsystem
For the M × N square lattice on the torus, we consider the entanglement entropy between a cylindrical subsystem R ′ and its complement. R ′ contains two separate boundary at a distance of D , and the length of each boundary equals M , which is the vertical size of the lattice, see Fig. B2. For arbitrary density matrix ρ , the reduced density operatorcan be expressed as [2, 3] ρ R ′ = − M ( D + ) ∑ α j ∈{ ,x,y,z } j ∈ R ′ ∏ j ∈ R ′ σ α j j tr [( ∏ j ∈ R ′ σ α j j ) † ρ ] . (B.1)Notice that we apply the trace inner product. The vector space of local Hermitian operators acting on the spin spaceis 4 dimensional, so the summation counts the four orthogonal local bases. The normalization coefficient − M ( D + ) D R ’ FIG. B2. (Color online) Illustration of a cylindrical (blue) region R ′ . The distance between the two boundaries is D = . the bond,indexed by j , belongs to R ′ , if it is inside the blue region or crossed by the dashed line. results from 2-dimensional local spin space and the number of spins in the R ′ . For an arbitrary pure state ρ = ∣ Ψ ⟩⟨ Ψ ∣ ,Eq. (B.1) is ρ R ′ = − M ( D + ) ∑ α j ∈{ ,x,y,z } j ∈ R ′ ∏ j ∈ R ′ σ α j j ⟨ Ψ ∣( ∏ j ∈ R ′ σ α j j ) † ∣ Ψ ⟩ . (B.2)As mentioned in the main text, W x = ∏ l ∈ γ x σ xl and W z = ∏ l ∈ γ z σ zl commute with the Hamiltonian, and ∣ Ψ ⟩ is in thesector of W x = , W z = . It is worth noting that the non-contractible path γ x ( γ z ) can be arbitrary even (odd) closedhorizontal line. So it is obvious that, for any operator O satisfying ⟨ Ψ ∣ O ∣ Ψ ⟩ ≠ , O must commutes with arbitrary W x and W z . For this reason, the reduced density operator can be written as ρ R ′ = − M ( D + ) ∑ g ∈ G R ′ ,h ∈ H R ′ x ∈ X R ′ ,z ∈ Z R ′ ⟨ Ψ ∣ xhzg ∣ Ψ ⟩ g R ′ z R ′ h R ′ x R ′ , (B.3)where the notation follows Ref. [5]. We first define 4 groups denoted by G , H , X and Z . G is generated by allindependent A s = ∏ j ∋ s σ xj ; H is generated by all independent B p = ∏ j ∈ p σ zj ; X is generated by all σ x on the bondsbelonging to even rows; and Z is generated by all σ z on the bonds belonging to odd rows. Then the subgroup G R ′ canbe defined as G R ′ = { g ∈ G ∣ g = g R ′ ⊗ ¯ R ′ } , where R ′ denotes the complement of R . It means that the elements of G R ′ are supported on R ′ . The subgroups of H R ′ , X R ′ and Z R ′ can be defined in a similar way. The elements of all thesegroups can be mapped to the µ picture unambiguously.The Hamiltonian in the µ picture, Eq. (A.1), is a sum of uncorrelated quantum Ising chains, so the reduced densityoperator has the tensor product form ρ µR ′ = M ⊗ l = ρ µR ′ l , (B.4)where R ′ l denotes the l th row. Explicitly, ρ µR ′ l = D + ∑ g ∈ G R ′ l ,h ∈ H R ′ l x ∈ X R ′ l ,z ∈ Z R ′ l ⟨ Ψ µl ∣ xhzg ∣ Ψ µl ⟩ g R ′ l z R ′ l h R ′ l x R ′ l (B.5)when l is odd; while ρ µR ′ l = D ∑ g ∈ G R ′ l ,h ∈ H R ′ l x ∈ X R ′ l ,z ∈ Z R ′ l ⟨ Ψ µl ∣ xhzg ∣ Ψ µl ⟩ g R ′ l z R ′ l h R ′ l x R ′ l (B.6)when l is even. As a consequence, the Von Neumann entropy of ρ σR ′ in the σ picture equals the sum of the entropies of M uncorrelated Ising chains in the µ piture, S ( ρ σR ′ ) = M ∑ l = S ( ρ µR ′ l ) . (B.7) Appendix C: Calculation of correlation function and entanglement entropy1. Formalism
As mentioned previously, the Wilson loop expectation value and the entanglement entropy for the two dimensionalmodel, Eq. (1) in the main text, can be evaluated by uncorrelated one dimensional quantum Ising chains lying on therows of the lattice. The Hamiltonian of each Ising chain in the µ picture has the common form of ˆ H ( J, h ) = N ∑ j = − J j µ xj µ xj + − h j µ zj (C.1)with the periodic boundary condition, where the hat is adopted to distinguish from the two dimensional Hamiltonian.We concern the sector of ∏ j µ zj = . The scenario of quantum quench is as follows: the initial state ∣ Ψ i ⟩ is a groundof the pre-quench Hamiltonian ˆ H i , and at t = the Hamiltonian is changed to the post-quench Hamiltonian ˆ H f ,then the initial state will evolve as ∣ Ψ ( t )⟩ = e − it ˆ H f ∣ Ψ i ⟩ . The mission we met is to calculate the correlation function C xx ( j, l, t ) = ⟨ Ψ ( t )∣ µ xj µ xl ∣ Ψ ( t )⟩ and the Von Neumann entropy S ( ρ A ( t )) = − tr [ ρ A ( t ) log ρ A ( t )] . We apply thestandard method of Jordan-Wigner transformation to map the spin model to the free fermion model, which is ana-lytically solvable in the clean case. However, one has to resort to numeric if the parameters in the Hamiltonian arerandom. Fortunately, the complexity of diagonalizing a Hermitian matrix is polynomial with the matrix size, so we candeal with sufficiently large finite system.We apply the Jordan-Wigner transformation to define the Dirac fermions c l = ⎛⎝ l − ∏ j = µ zj ⎞⎠ µ xl + iµ yl ,c † l = ⎛⎝ l − ∏ j = µ zj ⎞⎠ µ xl − iµ yl . (C.2)Then the spin Hamiltonian, Eq. (C.1), turns out to be a quadratic fermion Hamiltonian ˆ H ( J, h ) = N ∑ j = − J j ( c † j c j + − c j c † j + + c † j c † j + − c j c j + ) + µ j ( c † j c j − c j c † j ) . (C.3)The general form of a quadratic fermion Hamiltonian with real parameters is H = ∑ mn c † m A mn c n − c m A mn c † n + c † m B mn c † n − c m B mn c n , (C.4)where A mn = A nm , and B mn = − B nm . The Hamiltonian can be diagonalized as H = ∑ k ω k ( η † k η k − η k η † k ) = ∑ k ω k η † k η k − ∑ k ω k . (C.5)The quasi-particle operators of η k and η † k are fermion operators because of the linear transformation η k = ∑ l g kl c l + h kl c † l ,η † k = ∑ l h kl c l + g kl c † l , (C.6)with conditions of ∑ l g kl g k ′ l + h kl h k ′ l = δ kk ′ , ∑ l g kl h k ′ l + h kl g k ′ l = . (C.7)Therefore, we can write the diagonalization process as a form of block matrix: ( g hh g ) ( A B − B − A ) ( g T h T h T g T ) = ( ω − ω ) (C.8)and ( cc † ) = ( g T h T h T g T ) ( ηη † ) . (C.9)Here η , η † , c and c † are the shorthand notations of columns of fermion operators.The correlation function C xx ( j, l, t ) , we assume j < l without loss of generality, is ⟨ Ψ ( t )∣ µ xj µ xl ∣ Ψ ( t )⟩ = ⟨ Ψ ( t )∣ B j A j + B j + ⋯ A l − B l − A l ∣ Ψ ( t )⟩ , (C.10)where A j = c † j + c j ,B j = c † j − c j . (C.11)Applying the Wick’s theorem, the correlation function above can be expressed as a Pfaffian [6], ∣⟨ Ψ i ∣ µ xj ( t ) µ xl ( t )∣ Ψ i ⟩∣ =∣ pf Γ ( j, l, t )∣ , where the antisymmetric matrix Γ ( j, l, t ) = ( S ( j, l, t ) G ( j, l, t )− G ( j, l, t ) T Q ( j, l, t ) ) . (C.12)The dimension of each block is l − j + , and the blocks of S ( j, l, t ) and Q ( j, l, t ) are purely imaginary and anti-symmetric, while the block of G ( j, l, t ) is purely real. Explicitly, the elements of the matrix are two-point correlationfunctions: S ( j, l, t ) mn = δ mn + ⟨ Ψ i ∣ B j + m − ( t ) B j + n − ( t )∣ Ψ i ⟩ ,Q ( j, l, t ) mn = − δ mn + ⟨ Ψ i ∣ A j + m ( t ) A j + n ( t )∣ Ψ i ⟩ ,G ( j, l, t ) mn = ⟨ Ψ i ∣ B j + m − ( t ) A j + n ( t )∣ Ψ i ⟩ . (C.13)Here we use the properties of { A j , A l } = δ jl , { B j , B l } = − δ jl , and { A j , B l } = . Finally, applying the relationbetween the Pfaffian and the determinant, we have ∣⟨ Ψ i ∣ µ xj ( t ) µ xl ( t )∣ Ψ i ⟩∣ = ∣ pf Γ ( j, l, t )∣ = √ det Γ ( j, l, t ) . (C.14)The initial state ∣ Ψ i ⟩ is the vacuum state of H i , namely, η ik ∣ Ψ i ⟩ = for every k , while the time evolution is generatedby H f . To calculate the two-point correlation function in Eq. (C.13), we need to expand A l ( t ) and B l ( t ) , which is inthe Heisenberg picture, by η i and η i † in the sch¨ordinger picture: A l ( t ) = ∑ k ˜ φ ∗ lk ( t ) η i † k + ˜ φ lk ( t ) η ik ,B l ( t ) = ∑ k ˜ ψ ∗ lk ( t ) η i † k − ˜ ψ lk ( t ) η ik . (C.15)So we have ⟨ Ψ i ∣ A m ( t ) A n ( t )∣ Ψ i ⟩ = ∑ k ˜ φ mk ( t ) ˜ φ ∗ nk ( t ) = ( ˜ φ ( t ) ˜ φ † ( t )) mn , ⟨ Ψ i ∣ B m ( t ) B n ( t )∣ Ψ i ⟩ = − ∑ k ˜ ψ mk ( t ) ˜ ψ ∗ nk ( t ) = −( ˜ ψ ( t ) ˜ ψ † ( t )) mn , ⟨ Ψ i ∣ A m ( t ) B n ( t )∣ Ψ i ⟩ = ∑ k ˜ φ mk ( t ) ˜ ψ ∗ nk ( t ) = ( ˜ φ ( t ) ˜ ψ † ( t )) mn , ⟨ Ψ i ∣ B m ( t ) A n ( t )∣ Ψ i ⟩ = − ∑ k ˜ ψ mk ( t ) ˜ φ ∗ nk ( t ) = −( ˜ ψ ( t ) ˜ φ † ( t )) mn . (C.16)The linear transformation matrices ˜ φ ( t ) and ˜ ψ ( t ) can be expressed in a closed form. To this end, we first considerthe Heisenberg equation of the quasi-particle operator: ddt η fk ( t ) = i [ H f , η fk ( t )] = − iω f η fk ( t ) . (C.17)The solution is ( η f ( t ) η f † ( t ) ) = ( e − itω f e itω f ) ( η f η f † ) . (C.18)Then we have ( c ( t ) c † ( t ) ) = ( g Tf h Tf h Tf g Tf ) ( η f ( t ) η f † ( t ) )= ( g Tf h Tf h Tf g Tf ) ( e − itω f e itω f ) ( g f h f h f g f ) ( cc † )= ( g Tf h Tf h Tf g Tf ) ( e − itω f e itω f ) ( g f h f h f g f ) ( g Ti h Ti h Ti g Ti ) ( η i η i † ) . (C.19)Combining Eq. (C.11) in the Heisenberg picture, we get the linear transformation matrices in Eq. (C.15): ˜ φ ( t ) = φ Tf cos ( ω f t ) φ f φ Ti − iφ Tf sin ( ω f t ) ψ f ψ Ti , ˜ ψ ( t ) = ψ Tf cos ( ω f t ) ψ f ψ Ti − iψ Tf sin ( ω f t ) φ f φ Ti , (C.20)where φ and ψ are the combinations of g and h in Eq. (C.6) φ = g + h,ψ = g − h ; (C.21)and their subscript i and f correspond to the Hamiltonian H i and H f . As a result, as long as H i and H f are numericallydiagonalized, we can compute the correlation function C xx ( j, l, t ) = ⟨ Ψ ( t )∣ µ xj µ xl ∣ Ψ ( t )⟩ within the numerical precision.The entanglement entropy is defined as S A ( t ) = − tr [ ρ A ( t ) log ρ A ( t )] , where the subsystem consists of spins onthe contiguous lattice cites A = [ , , ⋯ , L ] . We introduce the Majorana fermions d l − = ⎛⎝ l − ∏ j = µ zj ⎞⎠ µ xl ,d l = ⎛⎝ l − ∏ j = µ zj ⎞⎠ µ yl . (C.22)Combining Eq. (C.2) and Eq. (C.11), we have d l − = c l + c † l = A l ,d l = c l − c † l i = iB l . (C.23)The reduced density matrix can be expanded as ρ A ( t ) = − L ∑ α ,α , ⋯ ,α L ∈{ , } ⟨ Ψ ( t )∣ d α d α ⋯ d α L L ∣ Ψ ( t )⟩ ( d α d α ⋯ d α L L ) † . (C.24)Notice that the fermionic parity is conserved, so if ∑ Lj = α j = ( ) , then ⟨ Ψ ( t )∣ d α d α ⋯ d α L L ∣ Ψ ( t )⟩ = . Thenone zero components can be evaluated by the Wick’s theorem. It is clear that { d j , j = , , ⋯ , L } is an orthogonalbasis which span the space of the linear operators supported on L . We can also find another orthogonal basis e m = L ∑ l = V ml d l , V ∈ O ( L ) , (C.25)to expand the reduced density matrix ρ L ( t ) , such that it has a simple direct product form.To this end, we construct the correlation matrix ⟨ Ψ ( t )∣ d m d n ∣ Ψ ( t )⟩ = δ mn + i Γ ( t ) mn , m, n = , , ⋯ , L, (C.26)and Γ ( t ) l − , s − = − i ⟨ Ψ i ∣ A l ( t ) A s ( t )∣ Ψ i ⟩ = − i ( ˜ φ ( t ) ˜ φ † ( t )) ls , Γ ( t ) l, s = i ⟨ Ψ i ∣ B m ( t ) B n ( t )∣ Ψ i ⟩ = − i ( ˜ ψ ( t ) ˜ ψ † ( t )) mn , Γ ( t ) l − , s = ⟨ Ψ i ∣ A l ( t ) B s ( t )∣ Ψ i ⟩ = ( ˜ φ ( t ) ˜ ψ † ( t )) ls , Γ ( t ) l, s − = ⟨ Ψ i ∣ B l ( t ) A s ( t )∣ Ψ i ⟩ = −( ˜ ψ ( t ) ˜ φ † ( t )) ls . (C.27)So Γ ( t ) is a real antisymmetric matrix, thus can be block diagonalized by an orthogonal matrix as V Γ ( t ) V T = L ⊕ m = ν m ( t ) ( − ) , (C.28)where V has appeared in Eq. (C.25). In the new basis, the reduced density matrix is ρ A ( t ) = L ∏ m = (⟨ Ψ ( t )∣ e m − e m ∣ Ψ ( t )⟩ e m e m − + )= L ∏ m = ( iν m e m e m − + )= L ∏ m = ( − ν m b † m b m + + ν m b m b † m ) , (C.29)where the Dirac fermion operators b m = ( e m − + ie m ) and b † m = ( e m − − ie m ) are introduced. In the end, wederive that the Von Neumann entropy is the sum of binary entropies of L uncorrelated modes [2, 7], S ( ρ A ( t )) = L ∑ m = H b ( − ν m ) , (C.30)where H b ( x ) ≡ − x log x − ( − x ) log ( − x ) , (C.31)with ≤ x ≤ , is the binary entropy.
2. Analytical and numerical results
The clean quantum Ising chain ˆ H ( h ) = N ∑ j = − µ xj µ xj + − hµ zj (C.32)can be diagonalized in term of quasi-particle operators as ˆ H ( h ) = ∑ p ω h ( p ) ( η † p η p − ) (C.33)with quasi-particle energy ω p = √ − h cos p + p . (C.34)The quasi-momentum p ’s are good quantum numbers, and the quench dynamics can be picturized as the spin precessionin each two dimensional subspace indexed by (− p, p ) pair. Consider that the initial state ∣ Ψ ⟩ is the ground state of thepre-quench Hamiltonian ˆ H ( h ) , and the time evolution is generated by the post-quench Hamiltonian ˆ H ( h ) . Then thequench dynamic is characterized by the differences of Bogoliubov angles ∆ p = arccos 4 ( + hh − ( h + h ) cos p ) ωω . (C.35)The expectation value of the quasi-particle number operator ˆ n p = η † p η p has the simple form of ⟨ Ψ ∣ ˆ n p ∣ Ψ ⟩ = − cos ∆ p , (C.36)which is conserved during the time evolution.For ∣ h ∣ , ∣ h ∣ ≤ , the equal-time spin correlation function C xx ( D, t ) = ⟨ Ψ ∣ µ xj ( t ) µ xj + D ( t )∣ Ψ ⟩ and the entanglemententropy S ( D, t ) = − tr [ ρ D ( t ) ln ρ D ( t )] have closed forms in the thermodynamic limit N → ∞ and in the limit of alarge subsystem D ≫ [8, 9], which are C xx ( D, t ) ∝ exp [ t ∫ ∣ ω ′ p ∣ t < D d p π ∣ ω ′ p ∣ ln ( cos ∆ p ) + D ∫ ∣ ω ′ p ∣ t > D d p π ln ( cos ∆ p )] , (C.37)and S ( D, t ) = t ∫ ∣ ω ′ p ∣ t < D d p π ∣ ω ′ p ∣ H b ( − cos ∆ p ) + D ∫ ∣ ω ′ p ∣ t > D d p π H b ( − cos ∆ p ) . (C.38)In the above formulas, ω ′ p = d ω p / d p is the group velocity of the mode p . The maximum group velocity for thetransverse field Ising model Eq. (C.32) is v M = max p ∣ ω ′ p ∣ = h , which can be viewed as the Lieb-Robinson velocityin the system.Since the model is analytical, the state after time evolution in the limit of t → ∞ is not thermal, but characterized bya generalized Gibbs ensemble (GGE) ρ GGE = e − ∑ p ωpT eff ( p ) ˆ n p Z , (C.39)where the partition function Z = tr [ e − ∑ p ωpT eff ( p ) ˆ n p ] . Combining Eq. (C.36), the quasi-momentum-dependent effectivetemperature T eff is determined by ⟨ Ψ ∣ ˆ n p ∣ Ψ ⟩ = ⟨ ˆ n p ⟩ GEE = e ωpT eff ( p ) + , (C.40)where ⟨ ˆ n p ⟩ GEE = tr [ ˆ n p ρ GEE ] . The expectation value of fermion occupation operator is analogous to the form of Fermi-Dirac distribution. In the limit of t → +∞ , the correlation function Eq. (C.37) decays exponentially with the distancebetween the spins as C xx ( D, +∞) ∝ exp [− D / ξ eff ] , (C.41)where the effective correlation length is ξ eff = − ∫ π − π d p π ln ( tanh ω p T eff ( p ) ) . (C.42)This is reminiscent of the correlation length in a Gibbs state at temperature T ξ T = − ∫ π − π d p π ln ( tanh ω p T ) . (C.43)Therefore, the correlation function at long time is not thermal.The entanglement entropy in a long time limit has a simple relation with the thermodynamic entropy of the GEE S ( D, +∞) = DN S ( ρ GEE ) , (C.44) t -8 -7 -6 -5 -4 -3 -2 -1 C xx ( D ,t ) t -8 -7 -6 -5 -4 -3 -2 -1 t -8 -7 -6 -5 -4 -3 -2 -1 C xx ( D ,t ) t -8 -7 -6 -5 -4 -3 -2 -1 (a1) (b1)(b2)(a2) N=512N=512 N=1024N=1024 FIG. C3. (Color online) Equal-time correlation function C xx ( D, t ) = ⟨ Ψ ∣ µ xj ( t ) µ xj + D ( t )∣ Ψ ⟩ for clean system with system size(a) N = and (b) N = . The pre-quench parameter is h = and the post-quench parameter is h = . . The curvesfrom top to bottom correspond to D = , , ⋯ , . The maximum group velocity is v M = h = . The equal-time correlationfunctions exhibit partial revival with quasi-period of T q = N / v M . After a long time evolution, deviation from volume law tends tobe significant as D / N increases, which is caused by the non-linear dispersion. where S ( ρ GEE ) = − tr [ ρ GEE ln ρ GEE ] = ∑ p H b (⟨ ˆ n p ⟩ GEE ) . (C.45)The analytical result of Eqs. (C.37) and (C.38) have a simple physical interpretation of semiclassical theory. Theinitial state ∣ Ψ ⟩ has high energy relative to the ground state of the post-quench Hamiltonian, and therefore the timeevolution can be characterized by the ballistic movement of each pair of quasiparticles with velocities of (− ω ′ p , ω ′ p ) .The details of the semiclassical theory are in Ref. [11–13]An outstanding advantage of the semiclassical theory is that it can apply not only to the thermodynamic limit but alsothe finite system [12, 13]. Consider the finite N with periodic boundary condition, and a model with linear dispersion.Thus the velocity, v , of the particle is independent on the momentum. Both ln [ C xx ( D, t )] and S ( D, t ) show perfectrevival with period T = N / v . Explicitly, for D ≤ N , S ( D, t ) grows linearly as S ( D, t ) ∝ vt if vt mod ( N ) < D ; stay on the plateau S ( D, t ) ∝ D if D ≤ vt mod ( N ) ≤ N − D ; decreases linearly as S ( D, t ) ∝ D − vt if vt mod ( N ) > N − D [13]. The behavior of ln [ C xx ( D, t )] is similar.For the model with non-linear dispersion, ln [ C xx ( D, t )] and S ( D, t ) exhibit quasi-periodic behavior with partialrevival in contrast to the case with linear dispersion. The reason is that the trajectory of each mode has differentgroup velocity ω ′ p and thus momentum-dependent period T p = N / ω ′ p . We show the numerical data of C xx ( D, t ) and S ( D, t ) with different system size and time scale in Figs. C3 and C4. As we can see, the quasi-period is determinedby the maximum group velocity T q = N / v M . Since the non-linear dispersion, S ( D, t ) grows linearly with time up to D / v M , then slowly approachs the value in the limit N → ∞ , and the plateaus are not exactly flat. The long behaviorof the entropy (and also the correlation function) deviates the volume law since the cancelation caused by differentperiods of trajectory for each mode, and the deviation tends to be significant as D / N increases.In the end, we discuss the time evolution in the regime of dynamical localization, where the post-quench Hamiltonianis Eq. (C.1). Adding disorder breaks the lattice translation symmetry, thus the picture of quasi-particle excitation cannot be applied. The asymptotic behavior of physical quantities after a long time can not be described in term of GEE0 t S ( D ,t ) t (a1) (b1) t S ( D ,t ) t (a2) (b2)N=512N=512 N=1024N=1024 FIG. C4. (Color online) Entanglement entropy S ( D, t ) = − tr [ ρ D ( t ) ln ρ D ( t )] for clean system with system size (a) N = and (b) N = . The pre-quench parameter is h = and the post-quench parameter is h = . . The curves from bottom totop correspond to D = , , ⋯ , . The maximum group velocity is v M = h = . The entanglement entropy exhibit partialrevival with quasi-period of T q = N / v M . After a long time evolution, deviation from volume law tends to be significant as D / N increases, which is caused by the non-linear dispersion. t -1 C xx ( D ,t ) t S ( D ,t ) (a) (b) =0.25=0.375=0.5 FIG. C5. (Color online) (a) Equal-time correlation function and (b) entanglement entropy in the dynamical-localization regime with N = , h = . and J j = + (cid:15)η j where η j ∈ [− , ] are i.i.d random variables, and (cid:15) is a parameter to control the disorder strength.blue lines: (cid:15) = . ; green lines: (cid:15) = . ; red lines: (cid:15) = . . For each disorder strength, the curves from (a) top to bottom, or (b)bottom to top, correspond to D = , , ⋯ , . average [14]. Setting h j = . , J j = + (cid:15)η j where η j ∈ [− , ] are i.i.d random variables and (cid:15) is a positive parameter tocontrol the disorder strength, we show the numerical data of C xx ( D, t ) and S ( D, t ) with different disorder strength andsubsystem sizes, but fixed system size N = , in Fig. C5. We have used 2000 realizations for each disorder strength, (cid:15) = . , . , . , to obtain the disorder average. In contrast to the linear growth in the clean case, S ( D, t ) growslogarithmically in a short time, then reaches to a saturation value, which is dependent on D at large t . However, thissaturation value tends to converge for large D . The logarithmical growth of entanglement entropy is predicted in Ref.[15] , where the radius of the effective light cone grows logarithmically with time in the dynamical-localization regimeas proved by the Lieb-Robinson bound. The convergence of the entropy at a long time is expected by the zero-velocity1Lieb-Robinson bound. [1] S. Suzuki, J. Inoue, and B. K. Chakrabarti, Quantum Ising Phases and Transitions in Transverse Ising Models , (Springer,2013).[2] J. I. Latorre, E. Rico, and G. Vidal,
Ground state entanglement in quantum spin chains , Quant. Inf. Comput. , 48 (2004).[3] B. -Q. Jin and V. E. Korepin, Quantum spin chain, Toeplitz determinants and the Fisher-Hartwig conjecture , J. Stat. Phys. ,79 (2004).[4] K. Sengupta, S. Powell, and S. Sachdev,
Quench dynamics across quantum critical points , Phys. Rev. A , 053616 (2004).[5] Y. Zeng, A. Hamma, and H. Fan, Thermalization of topological entropy after a quantum quench , Phys. Rev. B , 125104(2016).[6] E. Barouch and B. M. McCoy, Statistical Mechanics of the XY Model. II. Spin-Correlation Functions , Phys. Rev. A , 786(1971).[7] G. Vidal, J. I. Latorre, E. Rico, and A. Kitaev, Entanglement in Quantum Critical Phenomena , Phys. Rev. Lett. , 227902(2003).[8] M. Fagotti and P. Calabrese, Evolution of entanglement entropy following a quantum quench: Analytic results for the XY chainin a transverse magnetic field , Phys. Rev. A , 010306(R) (2008).[9] P. alabrese, F. H. L. Essler, and M. Fagotti, Quantum Quench in the Transverse-Field Ising Chain , Phys. Rev. Lett. , 227203(2011).[10] S. Sachdev and A. P. Young,
Low Temperature Relaxational Dynamics of the Ising Chain in a Transverse Field , Phys. Rev.Lett. , 2220 (1997).[11] P. Calabrese and J. Cardy, Evolution of entanglement entropy in onedimensional systems , J. Stat. Mech. (2005) P04010.[12] H. Rieger and F. Igli,
Semiclassical theory for quantum quenches in finite transverse Ising chains , Phys. Rev. B , 165117(2011).[13] R. Modak, V. Alba, and P. Calabrese, Entanglement revivals as a probe of scrambling in finite quantum systems , J. Stat. Mech.(2020) 083110.[14] T. Caneva, E. Canovi, D. Rossini, G. E. Santoro, and A. Silva,
Applicability of the generalized Gibbs ensemble after a quenchin the quantum Ising chain , J. Stat. Mech. (2011) P07015.[15] C. K. Burrell and T. J. Osborne,
Bounds on the Speed of Information Propagation in Disordered Quantum Spin Chains , Phys.Rev. Lett.99