Subsystem complexity after a global quantum quench
SSubsystem complexity after a global quantum quench
Giuseppe Di Giulio and Erik Tonni
SISSA and INFN Sezione di Trieste, via Bonomea 265, 34136, Trieste, Italy
Abstract
We study the temporal evolution of the circuit complexity for a subsystem in harmoniclattices after a global quantum quench of the mass parameter, choosing the initial reduceddensity matrix as the reference state. Upper and lower bounds are derived for the temporalevolution of the complexity for the entire system. The subsystem complexity is evaluatedby employing the Fisher information geometry for the covariance matrices. We discussnumerical results for the temporal evolutions of the subsystem complexity for a block ofconsecutive sites in harmonic chains with either periodic or Dirichlet boundary conditions,comparing them with the temporal evolutions of the entanglement entropy. For infiniteharmonic chains, the asymptotic value of the subsystem complexity is studied through thegeneralised Gibbs ensemble. a r X i v : . [ h e p - t h ] F e b ontents N . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.5 Initial growth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 A.1 Covariance matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48A.2 Complexity through the matrix W TR . . . . . . . . . . . . . . . . . . . . . . . . 49A.3 GGE correlators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 B Complexity w.r.t. the unentangled product state 52C Technical details about some limiting regimes 54
C.1 Approximation for small kN at finite N . . . . . . . . . . . . . . . . . . . . . . . 54C.2 Thermodynamic limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55C.3 Continuum limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 D Further numerical results on the relaxation to the GGE 59 Introduction
The complexity of a quantum circuit is a quantity introduced in quantum information theory[1–6] which has been studied also in the context of the holographic correspondence duringthe past few years [7–16]; hence it provides an insightful way to explore a connection betweenquantum information theory and quantum gravity.A quantum circuit allows to construct a target state starting from a reference state througha sequence of gates. The circuit complexity quantifies the difficulty to obtain the target statefrom the reference state by counting the minimum number of allowed gates that is necessaryto construct the circuit in an optimal way. Besides the reference state, the target state andthe set of allowed gates, the circuit complexity can depend also on the tolerance parameterfor the target state. Many results have been obtained for the complexity of quantum circuitsmade by pure states constructed through lattice models [17–25] and in the gravitational sideof the holographic correspondence. Some proposals have been done also to study the circuitcomplexity in quantum fields theories [26–38].Quantum quenches are insightful ways to explore the dynamics of isolated quantum systemsout of equilibrium (see [39, 40] for recent reviews). Given a quantum system prepared in theground state | ψ (cid:105) of the hamiltonian (cid:98) H , at t = 0 a sudden change is performed such thatthe evolution Hamiltonian of the initial state | ψ (cid:105) becomes (cid:98) H (cid:54) = (cid:98) H . Since (cid:98) H and (cid:98) H do notcommute in general, the unitary evolution | ψ ( t ) (cid:105) = e − i (cid:98) Ht | ψ (cid:105) for t > ω in (cid:98) H to the value ω in (cid:98) H [39, 41–43]. Insightful results have been obtainedabout the asymptotic regime t → ∞ of this unitary evolution by employing the generalisedGibbs ensemble (GGE) (see the reviews [44–46]).It is worth studying the circuit complexity with the target state given by the time-evolvedpure state of certain unitary evolution and the reference state by another pure state alongthe same evolution [18, 47–50]. In particular, considering a global quench protocol, we areinterested in the optimal circuit and in the corresponding complexity where | ψ ( t ) (cid:105) and | ψ (cid:105) are respectively the target and the reference states. Within the gauge/gravity correspondence,the temporal evolution of complexity for pure states has been explored in [51–53].Entanglement of spatial bipartitions plays a crucial role both in quantum information theoryand in quantum gravity, hence it is a fundamental tool to understand the connections betweenthem (see [54–60] for reviews). The entanglement dynamics after global quantum quencheshas been largely explored by considering the temporal evolutions of various entanglementquantifiers. The entanglement entropy has been mainly investigated through various methods[40, 61–65], but also other entanglement quantifiers like the entanglement spectra [66–68], theentanglement Hamiltonians [66, 67, 69], the entanglement negativity [70] and the entanglementcontours [67, 71, 72] have been explored.In order to understand the relation between entanglement and complexity, it is useful tostudy the optimal circuits and the corresponding circuit complexity when both the referenceand the target states are mixed states [73–76]. The approach to the complexity of mixedstates based on the purification complexity [73, 76, 77] is general, but evaluating this quantity3or large systems is technically complicated. Some explicit results for large systems can befound by restricting to the simple case of bosonic Gaussian states and by employing themethods of the information geometry [78–80]. In our analysis we adopt the approach to thecomplexity of mixed states based on the Fisher information geometry [74], which allows tostudy large systems numerically. The crucial assumption underlying this approach is that allthe states involved in the construction of the circuit are Gaussian. We consider the importantspecial case given by the subsystem complexity, namely the circuit complexity correspondingto a circuit where both the reference and the target states are the reduced density matricesassociated to a subsystem.Within the gauge/gravity correspondence, the subsystem complexity has been evaluatedboth in static [12, 16, 77, 81–83] and in time dependent gravitational backgrounds [84–87].In static backgrounds, it is given by the volume identified by the minimal area hypersurfaceanchored to the boundary of the subsystem, whose area provides the holographic entanglemententropy [88] (for static black holes, this hypersurface does not cross the horizon [89–91]),while, in time dependent gravitational spacetimes, the extremal hypersurface occurring in thecovariant proposal for the holographic entanglement entropy [64] must be employed.In this manuscript we study the temporal evolution of the subsystem complexity after aglobal quantum quench in harmonic lattices where the mass parameter is suddenly changedfrom ω to ω . Considering a ground state as the initial state, the Gaussian nature of thestate is preserved during the temporal evolution. In these bosonic systems, the reduceddensity matrices are characterised by the corresponding reduced covariance matrices [92]. Byemploying the approach to the complexity of bosonic mixed Gaussian states based on theFisher information geometry [74], we evaluate numerically the subsystem complexity for one-dimensional harmonic lattices (i.e. harmonic chains) and subsystems A given by blocks ofconsecutive sites. We consider harmonic chains where either periodic boundary conditions(PBC) or Dirichlet boundary conditions (DBC) are imposed. This allows to study the roleof the zero mode. The temporal evolution of the subsystem complexity at a generic timeafter the global quench w.r.t. the initial state is compared with the temporal evolution of thecorresponding increment of the entanglement entropy.This manuscript is organised as follows. In Sec. 2 we introduce the main expressions toevaluate the circuit complexity after the global quench of the mass parameter through thecovariance matrices of the reference and the target states for harmonic lattices in a genericnumber of dimensions. In the special case where the entire system is considered, these statesare pure and bounds are obtained for the temporal evolution of the circuit complexity w.r.t.the initial state. In Sec. 3 we specify this analysis to harmonic chains with either PBC or DBC.The main results of this manuscript are discussed in Sec. 4 and Sec. 5, where the temporalevolution of the subsystem complexity for a block of consecutive sites is investigated. InSec. 4, finite harmonic chains with either PBC or DBC are studied, while in Sec. 5 we considerinfinite harmonic chains either on the line or on the semi-infinite line with DBC at the origin.In the cases of infinite chains, we employ known results about the GGE to determine theasymptotic regime of the subsystem complexity. In Sec. 6 we draw some conclusions. Sometechnical details and supplementary results are discussed in the appendices A, B, C and D.4 Complexity from the covariance matrix after the quench
In this section we discuss the expressions that allow to evaluate the temporal evolution of thecircuit complexity based on the Fisher-Rao geometry for the harmonic lattices in a genericnumber of spatial dimensions when both the reference and the target states are pure. Analyticexpressions that bound this temporal evolution are also derived.
The Hamiltonian of the harmonic lattice made by N sites with nearest neighbour spring-likeinteraction reads (cid:98) H = N (cid:88) i =1 (cid:18) m ˆ p i + mω q i (cid:19) + (cid:88) (cid:104) i,j (cid:105) κ q i − ˆ q j ) = 12 ˆ r t H phys ˆ r (2.1)where the position and the momentum operators ˆ q i and ˆ p i are hermitean operators satisfyingthe canonical commutation relations [ˆ q i , ˆ q j ] = [ˆ p i , ˆ p j ] = 0 and [ˆ q i , ˆ p j ] = i δ i,j . The matrix H phys in (2.1) has been defined by collecting the position and the momentum operators into thevector ˆ r ≡ (ˆ q , . . . , ˆ q N , ˆ p , . . . , ˆ p N ) t .In the Heisenberg picture, the unitary temporal evolution of the position and the momentumoperators ˆ q j ( t ) and ˆ p j ( t ) through the evolution Hamiltonian (cid:98) H readsˆ q j ( t ) = e i (cid:98) Ht ˆ q j (0) e − i (cid:98) Ht ˆ p j ( t ) = e i (cid:98) Ht ˆ p j (0) e − i (cid:98) Ht . (2.2)In order to study the temporal evolution of the harmonic lattices after the global quantumquench of the mass parameter that we are considering, we need to introduce the N × N correlation matrices for operators (2.2) whose elements read Q i,j ( t ) ≡ (cid:104) ψ | ˆ q i ( t ) ˆ q j ( t ) | ψ (cid:105) P i,j ( t ) ≡ (cid:104) ψ | ˆ p i ( t ) ˆ p j ( t ) | ψ (cid:105) M i,j ( t ) ≡ Re (cid:2) (cid:104) ψ | ˆ q i ( t ) ˆ p j ( t ) | ψ (cid:105) (cid:3) (2.3)where | ψ (cid:105) is the ground state of the Hamiltonian (cid:98) H , defined by (2.1) with ω replaced by ω .At any time t > γ ( t ), which is the following 2 N × N real, symmetric and positive definite matrix γ ( t ) = (cid:18) Q ( t ) M ( t ) M ( t ) t P ( t ) (cid:19) (2.4)where the elements of the N × N block matrices are given by (2.3). This covariance matrixhas been already used to study the entanglement dynamics e.g. in [61, 62, 70].In the appendix A.1 we discuss the fact that, for the global quench we are exploring, theblocks of the covariance matrix (2.4) can be decomposed as Q ( t ) = (cid:101) V Q ( t ) (cid:101) V t P ( t ) = (cid:101) V P ( t ) (cid:101) V t M ( t ) = (cid:101) V M ( t ) (cid:101) V t (2.5)5here (cid:101) V is a real orthogonal N × N matrix, while Q ( t ), P ( t ) and M ( t ) are N × N diagonalmatrices whose k -th element along the diagonal is [43] Q k ( t ) ≡ Q k,k ( t ) = 12 m Ω k (cid:18) Ω k Ω ,k [cos(Ω k t )] + Ω ,k Ω k [sin(Ω k t )] (cid:19) P k ( t ) ≡ P k,k ( t ) = m Ω k (cid:18) Ω k Ω ,k [sin(Ω k t )] + Ω ,k Ω k [cos(Ω k t )] (cid:19) M k ( t ) ≡ M k,k ( t ) = 12 (cid:18) Ω ,k Ω k − Ω k Ω ,k (cid:19) sin(Ω k t ) cos(Ω k t ) (2.6)in terms of the dispersion relations Ω ,k and Ω k of the Hamiltonians (cid:98) H and (cid:98) H respectively,which depend both on the dimensionality of the lattice and on the boundary conditions.At t = 0, the expressions in (2.6) simplify respectively to Q k (0) = 12 m Ω ,k P k (0) = m Ω ,k M k (0) = 0 . (2.7)From the above discussion, one realises that γ ( t ) is a function of t determined by the setof parameters given by { m, κ, ω, ω } .When the dispersion relation vanishes for certain value of k , e.g. k = N , the correspondingmode is a zero mode. The relations (2.6) and (2.7) are well defined when Ω ,k does not vanish;hence Ω k can have a zero mode, while Ω ,k cannot. This highlights the asymmetric role ofΩ ,k and Ω k . The circuit complexity is proportional to the length of the optimal quantum circuit thatcreates a target state from a reference state. In this manuscript we evaluate the complexitythrough the Fisher-Rao distance between two bosonic Gaussian states with vanishing firstmoments [92–94]. This approach allows to study also the circuits made by mixed states [74].Denoting by γ R and γ T the covariance matrices with vanishing first moments of the referenceand of the target state respectively, the Fisher-Rao distance between them [79, 95] providesthe following definition of complexity C ≡ √ (cid:114) Tr (cid:110)(cid:2) log (cid:0) γ T γ − R (cid:1)(cid:3) (cid:111) . (2.8)When both γ R and γ T characterise pure states, this complexity corresponds to the one definedthrough the F cost function [18].The analysis of the circuits made by bosonic Gaussian states based on the Fisher-Raometric provides also the optimal circuit between γ R and γ T . It reads [95] G s ( γ R , γ T ) ≡ γ / R (cid:16) γ − / R γ T γ − / R (cid:17) s γ / R (cid:54) s (cid:54) γ R when s = 0 and γ T when s = 1. The length of the optimal circuit (2.9)evaluated through the Fisher-Rao distance is proportional to the circuit complexity (2.8),which has been explored both for pure states [18] and for mixed states [74].6n this manuscript we are interested in the temporal evolution of the circuit complexityafter a global quench. In the following discussion and in Sec. 3 we consider first the casewhere both the reference and the target states are pure states, while in Sec. 4 and Sec. 5 westudy the case where both the reference and the target states are mixed states.Denoting by t R and t T the values of t corresponding to the reference state and to the targetstate respectively, let us adopt the following notation γ R = γ ( t R ) γ T = γ ( t T ) . (2.10)In the most general setup, γ R is a function of t R characterised by the set of parameters { m R , κ R , ω R , ω , R } , while γ T is a function of t T parameterised by { m T , κ T , ω T , ω , T } . Thismeans that the reference and target states are obtained as the time-evolved states at t = t R (cid:62) t = t T (cid:62) t R respectively, through two different global quenches determined by { κ R , m R , ω R , ω , R } and { κ T , m T , ω T , ω , T } respectively.The covariance matrix (2.4) at a generic value of t can be written as follows γ ( t ) = V t Γ( t ) V V = (cid:101) V ⊕ (cid:101) V (2.12)where V is an orthogonal and symplectic matrix because (cid:101) V is orthogonal and the blockdecomposition of Γ( t ) reads Γ( t ) = (cid:18) Q ( t ) M ( t ) M ( t ) P ( t ) (cid:19) (2.13)in terms of the diagonal matrices whose elements have been defined in (2.6).Hereafter we enlighten the expressions by avoiding to indicate explicitly the dependence on t , wherever this is possible. The inverse of (2.13) is Γ − = (cid:0) Q P − M (cid:1) − (cid:18) P − M− M Q (cid:19) . (2.15)Since γ in (2.4) describes a pure state, the condition (i J γ ) = holds; hence the blocks Q , P and M are not independent. More explicitly, this constraint reads(i J γ ) = (cid:18) P Q − ( M t ) P M − M t PQM t − M Q QP − M (cid:19) = V t (cid:18) PQ − M PM − M t PQM t − MQ Q P − M (cid:19) V = 14 (2.16) We used that (cid:18)
A BC D (cid:19) = (cid:18) S A S † S B T † T C S † T D T † (cid:19) = (cid:18) S T (cid:19) (cid:18) A BC D (cid:19) (cid:18) S † T † (cid:19) (2.11)where A , B , C and D are diagonal matrices. The expression (2.15) is a special case of the following general formula γ ≡ (cid:18) A BB t C (cid:19) γ − ≡ (cid:18) A BB t C (cid:19) A ≡ (cid:0) A − B C − B t (cid:1) − C ≡ (cid:0) C − B t A − B (cid:1) − B ≡ − A − B (cid:0) C − B t A − B (cid:1) − . (2.14) Q P − M = 14 ⇐⇒ Q k P k − M k = 14 1 (cid:54) k (cid:54) N . (2.17)This result allows to further simplify (2.15), which becomesΓ − = 4 (cid:18) P −M−M Q (cid:19) . (2.18)In this manuscript we restrict to cases where a symplectic matrix V exists such that γ R = V t Γ R V γ T = V t Γ T V (2.19)where both Γ R and Γ T have the form (2.13), in terms of the corresponding diagonal matrices.When (2.19) holds, the matrix occurring in the argument of the logarithm in (2.8) becomes γ T γ − R = V t Γ T Γ − R V − t (2.20)which tells us that the complexity (2.8) is provided by the eigenvalues of Γ T Γ − R . Thus, thematrix V does not influence the temporal evolution of the complexity after the global quenchwhen both the reference and the target states are pure states. Instead, they play a crucialrole for the temporal evolution of the subsystem complexity discussed in Sec. 4 and Sec. 5.By using (2.13) for Γ T and (2.18) for Γ − R , we obtain the following block matrixΓ T Γ − R = 4 (cid:18) P R Q T − M R M T Q R M T − M R Q T P R M T − M R P T Q R P T − M R M T (cid:19) (2.21)whose blocks are diagonal matrices. By using also (2.17), for the eigenvalues of (2.21) we find g ( ± ) TR ,k ≡ (cid:18) P R ,k Q T ,k + Q R ,k P T ,k − M R ,k M T ,k (2.24) ± (cid:113)(cid:0) P R ,k Q T ,k − Q R ,k P T ,k (cid:1) + 4 (cid:0) Q R ,k M T ,k − M R ,k Q T ,k (cid:1)(cid:0) P R ,k M T ,k − M R ,k P T ,k (cid:1) (cid:19) labelled by 1 (cid:54) k (cid:54) N , which can be written as g ( ± ) TR ,k = C TR ,k ± (cid:113) C TR ,k − Considering a 2 N × N matrix M partitioned into four N × N blocks A , B , C and D which are diagonalmatrices, its eigenvalues equation can be written through the formula for the determinant of a block matrix,finding M = (cid:18) A BC D (cid:19) det( M − λ ) = det (cid:2) D − λ (cid:3) det (cid:2) A − λ − B C ( D − λ ) − (cid:3) = 0 (2.22)where is the identity matrix. Since the matrices in (2.22) are diagonal, this equation becomes (cid:81) Nk =1 [( d k − λ )( a k − λ ) − b k c k ] = 0; hence the 2 N eigenvalues of M in (2.22) are λ ( ± ) k = a k + d k ± (cid:112) ( a k − d k ) + 4 b k c k (cid:54) k (cid:54) N . (2.23) C TR ,k ≡ (cid:0) Q T ,k P R ,k + P T ,k Q R ,k − M T ,k M R ,k (cid:1) (2.26)in terms of the expressions in (2.6) specialised to the reference and the target states.From (2.25) and (2.26), one observes that g (+) TR ,k g ( − ) TR ,k = 16 (cid:0) Q R ,k P R ,k − M R ,k (cid:1)(cid:0) Q T ,k P T ,k − M T ,k (cid:1) . (2.27)By employing (2.17) in this result, we find g (+) TR ,k = 1 /g ( − ) TR ,k for pure states, for any 1 (cid:54) k (cid:54) N .From (2.21), (2.25) and (2.27), for the complexity (2.8) one obtains C = 12 (cid:118)(cid:117)(cid:117)(cid:116) N (cid:88) k =1 (cid:2) log( g (+) TR ,k ) (cid:3) = 12 (cid:118)(cid:117)(cid:117)(cid:116) N (cid:88) k =1 (cid:2) log( g ( − ) TR ,k ) (cid:3) = 12 (cid:118)(cid:117)(cid:117)(cid:116) N (cid:88) k =1 (cid:2) arccosh( C TR ,k ) (cid:3) . (2.28)In the most general setup described below (2.10), the complexity can be found by writing(2.6) for the reference and the target states first and then and plugging the results into (2.26)and (2.28). The final result is a complicated expressions which can be seen as a function of t R and t T parameterised by { κ R , m R , ω R , ω , R } and { κ T , m T , ω T , ω , T } . We remark that (2.28)can be employed when (2.19) holds. Furthermore, we consider only cases where the matrix V in (2.19) depends on the geometric parameters of the system and of the subsystem but it isindependent of the physical parameters occurring in the Hamiltonians (see Sec. 3.1).In the appendix A.2, the expression (2.28) is obtained through the Williamson’s decompo-sition [96] of the covariance matrices (2.10).A remarkable simplification occurs when the reference and the target states are pure statesalong the time evolution of a given quench. In this case, the parameters to fix in (2.6) are m R = m T = m , κ R = κ T = κ , ω R = ω T = ω , and ω , R = ω , T = ω ; hence (2.26) simplifies to C TR ,k = 1 + 12 (cid:32) Ω k − Ω ,k Ω k Ω ,k sin[Ω k ( t R − t T )] (cid:33) (2.29)which must be plugged into (2.28) to get the complexity of pure states after the global quench.Notice that (2.29) is not invariant under the exchange Ω k ↔ Ω ,k for a given k . We remarkthat (2.29) and the corresponding complexity depend on | t R − t T | . This is not the case for themost generic choice of the parameters. A very natural choice for the reference state is the initial state | ψ (cid:105) , which is a crucial ingredientof the quench protocol. This corresponds to choose t R = 0 in (2.10). In this case, from (2.7)and (2.17) we have that M R = and Q R P R = , which allow to write (2.26) as C TR ,k ≡ (cid:18) Q T ,k Q R ,k + P T ,k P R ,k (cid:19) . (2.30) The last step expression in (2.28) is obtained through the identity log( x + √ x − x ) for x (cid:62) m R = m T = m for simplicity and t T = t and t R = 0 in the most general setupdescribed below (2.10) and then using (2.6) and (2.7), this expression becomes C TR ,k = (cid:0) Ω , T ,k + Ω , R ,k (cid:1) Ω T ,k [cos(Ω T ,k t )] + (cid:0) Ω T ,k + Ω , T ,k Ω , R ,k (cid:1) [sin(Ω T ,k t )] T ,k Ω , R ,k Ω , T ,k (2.31)in terms of the dispersion relations Ω , S ,k (with S ∈ { R , T } ) before the quenches providingthe reference and the target states and of the dispersion relations Ω T ,k after the quench (Ω R ,k does not occur because t R = 0, hence (2.7) must be employed).The expression (2.30) is consistent with the result reported in [48], where the temporalevolution of the complexity of this free bosonic system has been also studied through a differentquench profile that does not include the quench protocol that we are considering. In manystudies the reference state is the unentangled product state [17–19, 47]. In appendix B webriefly discuss the temporal evolution of the complexity given by (2.28) and (2.31) in the casewhere the initial state is the unentangled product state.When the same quench is employed to construct the reference and the target states Ω , R ,k =Ω , T ,k = Ω ,k for any k and (2.31) simplifies. This choice corresponds to evaluate the com-plexity between the initial state and the state at time t after the quench. Specialising (2.31)to this case and renaming Ω T ,k ≡ Ω k , we obtain C TR ,k = 1 + 12 (cid:32) Ω k − Ω ,k Ω k Ω ,k sin(Ω k t ) (cid:33) (2.32)which coincides with (2.29) for t R = 0 and t T = t , as expected. Plugging (2.32) into (2.28)and using the identity | arccosh(1 + x / | = 2 | arcsinh( x/ | , one finds C = (cid:118)(cid:117)(cid:117)(cid:116) N (cid:88) k =1 (cid:34) arcsinh (cid:32) Ω k − Ω ,k k Ω ,k sin(Ω k t ) (cid:33)(cid:35) . (2.33)In this expression the dispersion relations Ω k and Ω ,k (which depend on the number of spatialdimensions and on the boundary conditions of the lattice) do not occur in a symmetric way.We find it worth highlighting the contribution of the N -th mode by denoting c ≡ (cid:20) arcsinh (cid:18) Ω N − Ω ,N N Ω ,N sin(Ω N t ) (cid:19)(cid:21) C ≡ N − (cid:88) k =1 (cid:34) arcsinh (cid:32) Ω k − Ω ,k k Ω ,k sin(Ω k t ) (cid:33)(cid:35) (2.34)which lead to write (2.33) as C = η c + C (2.35)where either η = 1 or η = 0, depending on whether the N -th mode plays a particular role,as one can read from the dispersion relation. This is the case e.g. for the zero mode inthe harmonic lattices that are invariant under spatial translations, which is briefly discussedalso at the end of Sec. 2.1; hence hereafter we refer to c as the zero mode contribution. Forinstance, η = 1 in the harmonic chains with PBC, while η = 0 when DBC are imposed, asdiscussed later in Sec. 3.1. The result (2.33), which can be applied for harmonic lattices in10eneric number of dimensions and for diverse boundary conditions, has been already reportedin [49] for harmonic chains with PBC.It is interesting to determine the initial growth of the complexity by considering the seriesexpansion of (2.33) as t →
0. The function C obtained from (2.33) is an even function of t ,hence its expansion for t → t . Since C| t =0 = 0, we have C = b t + b t + b t + O ( t ) = ⇒ C = (cid:112) b t (cid:18) b b t + 4 b b − b b t + O ( t ) (cid:19) (2.36)where the coefficients b , b and b are b = 14 N (cid:88) k =1 (cid:32) Ω k − Ω ,k Ω ,k (cid:33) b = − N (cid:88) k =1 (cid:32) Ω k − Ω ,k Ω ,k (cid:33) (2.37)and b = 1360 N (cid:88) k =1 (Ω k − Ω ,k ) (Ω k + Ω ,k − Ω k Ω ,k )Ω ,k . (2.38)Since b >
0, the expansion (2.36) tells us that the initial growth of the complexity (2.33) islinear in t .The temporal evolution of the circuit complexity for a bosonic system after a global quenchhas been studied also in [47], by employing a smooth quench and the unentangled productstate as the reference state. This smooth quench becomes the one that we are consideringin the limit of sudden quench but it is different from the quench considered in [48]. Inappendix B, where the unentangled product state is considered as the initial state, we find adifferent result with respect to [47] because of the different sets of allowed gates. We find it worth studying some bounds for the complexity with respect to the initial state.From (2.33), it is straightforward to observe that η c (cid:54) C (cid:54) (cid:101) C , where c is the timedependent expression defined in (2.34) and (cid:101) C ≡ η c + N − (cid:88) k =1 (cid:34) arcsinh (cid:32) Ω k − Ω ,k k Ω ,k (cid:33)(cid:35) (2.39)hence for the complexity (2.33) we find √ η c (cid:54) C (cid:54) (cid:101) C . (2.40)The zero mode contribution determines the behaviour of these bounds for large t .The occurrence of a zero mode in the dispersion relation Ω k e.g. for k = N means thatΩ N = 0. In the absence of a zero mode, Ω k is non vanishing for any value of k ; hence c and (cid:101) C are finite for any t and (2.40) tells us that the complexity (2.33) is always finite afterthe quench. Instead, when a zero mode for k = N occurs, the time dependent zero modecontribution c in (2.34) becomes c = (cid:2) arcsinh(Ω ,N t/ (cid:3) (2.41)11hich diverges at large t because arcsinh( x ) ∼ log(2 x ) as x → + ∞ . The terms labelled by1 (cid:54) k (cid:54) N − t because Ω k is non vanishing.Thus, in the presence of a zero mode, the bounds (2.40) tell us that the complexity for purestates in (2.33) diverges logarithmically when t → ∞ .The bounds (2.40) can be significantly improved by employing the decomposition (2.35).The following integral representationarcsinh( x ) = (cid:90) x √ x s ds (2.42)leads to rewrite C in (2.34) as C = N − (cid:88) k =1 (cid:34) (cid:90) (cid:113) x k sin (Ω k t ) s ds (cid:35) ˜ x k (cid:2) sin(Ω k t ) (cid:3) ˜ x k ≡ Ω k − Ω ,k k Ω ,k . (2.43)Then, by using (2.42), one observes thatarcsinh(˜ x k )˜ x k (cid:54) (cid:90) (cid:113) x k sin (Ω k t ) s ds (cid:54) N − (cid:88) k =1 (cid:2) arcsinh(˜ x k ) sin(Ω k t ) (cid:3) (cid:54) C (cid:54) N − (cid:88) k =1 ˜ x k (cid:2) sin(Ω k t ) (cid:3) . (2.45)This result, combined with (2.34), provides the following bounds for the complexity (2.33) C L (cid:54) C (cid:54) C U (2.46)where we have introduced C B ≡ η c + N − (cid:88) k =1 f B (˜ x k ) (cid:2) sin(Ω k t ) (cid:3) = (cid:32) η c + 12 N − (cid:88) k =1 f B (˜ x k ) (cid:33) − N − (cid:88) k =1 f B (˜ x k ) cos(2Ω k t ) (2.47)with B ∈ { L , U } and f L ( x ) = (cid:2) arcsinh( x ) (cid:3) f U ( x ) = x (2.48)in terms of ˜ x k defined in (2.43), of the time dependent zero mode contribution c introducedin (2.34) and of the parameter η , which is either η = 1 or η = 0, depending on whether thezero mode contribution occurs or not respectively.The bounds (2.46) can be employed to improve the bounds reported in (2.40). Indeed, inthe presence of a zero mode, C L (cid:62) c and therefore C L provides a better lower bound than(2.34). Instead, the relation between C U in (2.47) and (cid:101) C in (2.39) depends on the parameters;hence the optimal upper bound is given by min (cid:2) C U ( t ) , (cid:101) C ( t ) (cid:3) .12 Complexity for harmonic chains
In this section we apply the results discussed in Sec. 2 to the harmonic chains where eitherPBC or DBC are imposed. The numerical data reported in all the figures of the manuscripthave been obtained by setting κ = 1 and m = 1. The Hamiltonian of the harmonic chain made by N oscillators with the same frequency ω ,the same mass m and coupled through the elastic constant κ is (2.1) specialised to one spatialdimension, i.e. (cid:98) H = N (cid:88) i =1 (cid:18) m ˆ p i + mω q i + κ q i − ˆ q i − ) (cid:19) = 12 ˆ r t H phys ˆ r (3.1)where the vector ˆ r ≡ (ˆ q , . . . , ˆ q N , ˆ p , . . . , ˆ p N ) t collects the position and momentum operators.Imposing PBC means that ˆ q = ˆ q N , while DBC are satisfied when ˆ q = ˆ q N = 0 and ˆ p N = 0.When PBC hold, the orthogonal matrix (cid:101) V defined in (2.5), when N is even, is [94] (cid:101) V i,k ≡ (cid:112) /N cos(2 π i k/N ) 1 (cid:54) k < N/ − i / √ N k = N/ (cid:112) /N sin(2 π i k/N ) N/ (cid:54) k < N − / √ N k = N (3.2)while, when N is odd, it reads (cid:101) V i,k ≡ (cid:112) /N cos(2 π i k/N ) 1 (cid:54) k < ( N − / (cid:112) /N sin(2 π i k/N ) ( N − / (cid:54) k < N − / √ N k = N . (3.3)The dispersion relations of ˆ H and ˆ H for PBC are respectivelyΩ ,k = (cid:114) ω + 4 κm [sin( πk/N )] Ω k = (cid:114) ω + 4 κm [sin( πk/N )] (cid:54) k (cid:54) N . (3.4)When DBC hold, only N − i = 0and i = N are fixed by the boundary conditions; hence the vector ˆ r contains 2( N −
1) operatorsand, correspondingly, the covariance matrix γ ( t ) is the (2 N − × (2 N −
2) symmetric matrixgiven by (2.4), where Q , P and R are ( N − × ( N −
1) matrices. For DBC and independentlyof the parity of N , the matrix (cid:101) V defined in (2.5) becomes (cid:101) V i,k = (cid:114) N sin( i k π/N ) 1 (cid:54) i, k (cid:54) N − . (3.5)The dispersion relations of ˆ H and ˆ H for DBC read respectivelyΩ ,k = (cid:114) ω + 4 κm [sin( πk/ (2 N ))] Ω k = (cid:114) ω + 4 κm [sin( πk/ (2 N ))] (cid:54) k (cid:54) N − . (3.6)13e remark that, both for PBC and DBC, the matrix V = (cid:101) V ⊕ (cid:101) V defined in (2.12) dependsonly on N ; hence the corresponding harmonic chains can be studied as special cases of theharmonic lattices considered in Sec. 2.2 because the condition (2.19) is satisfied. Since η = 1for PBC and η = 0 for DBC, the complexity (2.33) for these harmonic chains becomes C = (cid:118)(cid:117)(cid:117)(cid:116) N − η (cid:88) k =1 (cid:20) arcsinh (cid:18) ω − ω k Ω ,k sin(Ω k t ) (cid:19)(cid:21) (3.7)= (cid:118)(cid:117)(cid:117)(cid:116) η (cid:20) arcsinh (cid:18) ω − ω ω ω sin( ωt ) (cid:19)(cid:21) + N − (cid:88) k =1 (cid:20) arcsinh (cid:18) ω − ω k Ω ,k sin(Ω k t ) (cid:19)(cid:21) where the dispersion relations Ω ,k and Ω k are given by (3.4) for PBC and by (3.6) for DBC.When PBC are imposed, the first term under the square root in the last expression of (3.7)comes from the zero mode k = N and it does not occur for DBC. This crucial differencebetween the two models leads to different qualitative behaviours for the complexity.The dispersion relations of the harmonic chain with PBC given in (3.4) are invariant underthe exchange k ↔ N − k . This symmetry leads to an expression for the complexity which issimpler to evaluate numerically. Indeed, by introducing c ≡ (cid:20) arcsinh (cid:18) ω − ω ω ω sin( ωt ) (cid:19)(cid:21) (3.8)and c N/ ≡ (cid:20) arcsinh (cid:18) ω − ω N/ Ω ,N/ sin(Ω N/ t ) (cid:19)(cid:21) even N N (3.9)one observes that (3.7) for PBC can be written as C = (cid:118)(cid:117)(cid:117)(cid:117)(cid:116) c + 2 (cid:98) N − (cid:99) (cid:88) k =1 (cid:20) arcsinh (cid:18) ω − ω k Ω ,k sin(Ω k t ) (cid:19)(cid:21) + c N/ (3.10)where (cid:98) x (cid:99) denotes the integer part of x . Notice that c N/ in (3.9), as function of t , is boundedby a constant.We find it worth considering the small quench regime, defined by setting ω = ω + δω andtaking | δω | (cid:28) δω →
0, the leading term of the expansion reads C = ω δω (cid:118)(cid:117)(cid:117)(cid:116) η [sin( ωt )] ω + N − (cid:88) k =1 [sin(Ω k t )] Ω k + O (cid:0) δω (cid:1) (3.11)This result simplifies to C = η δω t + O ( δω ) when ω →
0; which tells us that the O ( δω ) termdoes not occur in this limit when DBC hold. 14 .2 Critical evolution An important case that we find worth emphasising is the global quench where the evolutionHamiltonian is gapless, i.e. when ω = 0.When PBC are imposed, by specialising (2.32) and (3.4) to ω = 0, we obtain C TR ,k = 1 + ω (cid:2) sin (cid:0) (cid:112) κ/m t sin( πk/N ) (cid:1)(cid:3) κ/m ) [sin( πk/N )] (cid:0) ω + 4( κ/m )[sin( πk/N )] (cid:1) (3.12)which satisfies the following bounds1 < C TR ,k < ω κ/m ) [sin( πk/N )] (cid:0) ω + 4( κ/m )[sin( πk/N )] (cid:1) (cid:54) k (cid:54) N − . (3.13)For k = N , the expression (3.12) simplifies to C TR ,N = 1 + ω t , which diverges as t → ∞ .Instead, when DBC hold and therefore the zero mode does not occur, by using (3.6) and(2.32) with ω = 0, we obtain C TR ,k = 1 + ω (cid:2) sin (cid:0) (cid:112) κ/m t sin( πk/ (2 N )) (cid:1)(cid:3) κ/m )[sin( πk/ (2 N ))] (cid:0) ω + 4 κ/m [sin( πk/ (2 N ))] (cid:1) (3.14)which is finite when t → ∞ , for any allowed value of k .Plugging the expressions discussed above for C TR ,k into (2.28), we find that, when theevolution Hamiltonian is critical, the complexity of the pure state at time t with respect tothe initial state can be written by highlighting the zero mode contribution as follows C = η (cid:20) log (cid:18) ω t ) ω t (cid:112) ( ω t ) + 4 (cid:19)(cid:21) + 14 N − (cid:88) k =1 (cid:2) arccosh (cid:0) C TR ,k (cid:1)(cid:3) (3.15)where either η = 1 for PBC or η = 0 for DBC (see the text above (3.7)) and C TR ,k is given by(3.12) for PBC and by (3.14) for DBC. In particular, (3.15) tells us that, for PBC and finite N , the complexity diverges logarithmically as t → ∞ because of the zero mode contribution.Instead, for DBC (i.e. η = 0) and finite N , all the terms in (3.15) are finite as t → ∞ .In Fig. 1 we show the temporal evolution of the complexity (3.15) for various ω ’s, wheneither PBC (left panels) or DBC (right panels) are imposed. Since N is finite, the revivalsalready studied in the temporal evolutions of other quantities [97] are observed also in thetemporal evolution of the complexity, with a period given by N/ N forDBC. The most important qualitative difference between PBC and DBC is the overall growthobserved for PBC, which does not occur for DBC. This growth is due to the zero modecontribution occurring in the complexity (3.15) for PBC. Indeed, when the correspondingterm is subtracted, as done in the bottom left panel of Fig. 1, the resulting curve is similar tothe temporal evolution of the complexity when DBC hold.Finally, let us remark that the effect of the decoherence as t increases is more evident forhigher values of ω . For PBC this is observed once the zero mode contribution has beensubtracted (see the bottom left panel of Fig. 1).15
200 400 600 800 1000 1200 1400051015202530
Figure 1: Temporal evolution of the complexity after the global quench w.r.t. the initial stateat t = 0 for harmonic chains with either PBC (left panels) or DBC (right panels) made by N = 100 sites. The solid lines correspond to the complexity (3.15). In the top left panel,the dashed lines show the zero mode term c (i.e. the expression multiplyed by η in (3.15),which has been subtracted to obtain the bottom left panel), with the same colour code forthe corresponding value of ω .In [18] the temporal evolution of the complexity of a thermofield double state is consideredby taking the unentangled product state as the reference state (in this setup, the choice ω = 0is not allowed). Despite this temporal evolution is different from the one investigated in thismanuscript, it also exhibits an overall logarithmic growth due to the zero mode contribution. It is instructive to discuss further the bounds for the complexity introduced in Sec. 2.3.1 inthe special cases of the harmonic chains with either PBC or DBC.In Fig. 2 we show the complexity (3.7) and the corresponding bounds (2.40) for harmonicchains with PBC. In this case the zero mode term influences the bounds in a crucial way. InFig. 2, the bounds (2.40) correspond to the red and blue dashed lines, while in the top left16
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Figure 2: Temporal evolution of the complexity (3.7) (solid lines) and of the correspondingbounds in (2.40) for PBC. The blue and red dashed lines show the lower and the upper bounds,from (2.34) and (2.39) respectively.panel of Fig. 1, where ω = 0, the lower bound in (2.40) is shown through the dashed curves.In the temporal evolutions of the complexity for PBC displayed in the top panel of Fig. 2,we can identify two periods approximatively given by π/ω and N/
2. Considering also thebottom panels of Fig. 2, the revivals observed for the critical evolution in Fig. 1 for PBC and ω = 0 occur also when ω > πω (cid:29) N . The bottom panels in Fig. 2 highlight thatthe revivals are not observed when ω is large enough with respect to 1 /N .For PBC, by comparing the top panel with the bottom ones in Fig. 2, which differ for thesize N of the chain, we notice that the bounds (2.40) are very efficient when πω (cid:29) N , whilethey become not useful away from this regime. In our numerical investigations we have alsoobserved that the bounds (2.40) are not useful when ω > ω .When DBC hold, the lower bound in (2.40) is trivial and the upper bound is a constant.The bounds (2.46) can be written explicitly for the harmonic chains that we are consideringby setting either η = 1 or η = 0 and employing either (3.4) or (3.6) for the dispersion relationswhen either PBC or DBC respectively are imposed. The resulting expressions for these boundsrequire to sum either N or N −
50 100 150 200 250 300024681012
Figure 3: Temporal evolution of the complexity (3.7) (solid lines) and of the correspondingbounds in (3.16) for harmonic chains with either PBC (left panels) or DBC (right panels) and N = 100. The blue (red) dashed lines correspond to the lower (upper) bound (see (3.18) and(3.17) for the left panels and (3.19) for the right panels). In all the panels ω = 0 . C L ,k L (cid:54) C (cid:54) C U ,k U (3.16)where k L and k U are independent parameters related to the number of terms in the sum keptto define the corresponding bound. Since the explicit expressions of the dispersion relationsare important to write explicitly the bounds in (3.16), the cases of PBC and DBC must bestudied separately.Considering PBC first, one observes that the corresponding f L (˜ x k ) as function of k (thatcan be constructed from (2.48), (2.43) and (3.4)) is large when k (cid:39) k (cid:39) N −
1, whileit becomes negligible in the middle of the interval [1 , N − k = 1 , . . . , k L and k = N − k L , . . . , N −
1, for some k L . Thus, by employing also the symmetry k ↔ N − k of the dispersion relations (3.4), the lower bound in (3.16) for PBC reads C L ,k L = c + 2 k L (cid:88) k =1 f L (˜ x k ) [sin(Ω k t )] . (3.17)18he upper bound C U ,k U can be found through similar considerations applied to the function f U (˜ x k ) introduced in (2.48). This leads to sum the terms whose k is close to the boundaryof [1 , N −
1] keeping their dependence on t and to set sin (Ω k t ) = 1 in the remaining ones,which must not be discarded. The resulting bound is C U ,k U = c + 2 k U (cid:88) k =1 f U (˜ x k ) [sin(Ω k t )] + (cid:98) N − (cid:99) (cid:88) k = k U +1 f U (˜ x k ) + f U (˜ x N/ ) (cid:12)(cid:12) cos( πN/ (cid:12)(cid:12) . (3.18)The bounds (2.46) are recovered when k L = k U = (cid:98) N − (cid:99) , by setting to zero the second sum inthe r.h.s. of (3.18) and by restoring the time dependence in the term having k = N/ N is even, both in (3.17) and (3.18).When DBC are imposed, a similar analysis can be carried out, with the crucial differencethat the symmetry k ↔ N − k in the dispersion relations (3.6) does not occur in this case.Setting η = 0 and employing the dispersion relations (3.6), one obtains (3.16) with C L ,k L ≡ k L (cid:88) k =1 f L (˜ x k ) [sin(Ω k t )] C U ,k U ≡ k U (cid:88) k =1 f U (˜ x k ) [sin(Ω k t )] + N − (cid:88) k = k U +1 f U (˜ x k ) (3.19)where 1 (cid:54) k L , k U (cid:54) N −
1. In order to recover (2.46) from (3.16), we have to choose k L = k U = N − C L ,k L (cid:54) C L and C U ,k U (cid:62) C U , but C L ,k L and C U ,k U contain less termsthan C L and C U respectively, hence they are easier to evaluate and to study analytically. Forboth PBC and DBC, considering either the lower or the upper bound in (3.16), it improvesas either k L or k U respectively increases.In Fig. 3 we show the bounds (3.16) when either PBC (left panels) or DBC (right panels)are imposed and small values of k L and k U are considered. For given values of k L and k U , theagreement between the bounds and the exact curve improves as | ω − ω | decreases. Noticethat higher values of k L and k U are needed for DBC to reach an agreement with the exactcurve comparable with the one obtained for PBC. N It is important to study approximate expressions for the temporal evolution of the complexitywhen large values of N are considered.In our numerical analysis, we noticed that, for finite but large enough values of N (cid:38) ωN , ω N and t/N . This function,which depends on whether PBC or DBC are imposed, can be written by introducing theapproximation sin( x ) (cid:39) x into the dispersion relations and keeping only the leading term (seeappendix C.1 for a more detailed discussion). For PBC we find C approx = (cid:118)(cid:117)(cid:117)(cid:117)(cid:116) c ( t ) + 2 (cid:98) N − (cid:99) (cid:88) k =1 (cid:34) arcsinh (cid:32) ( ωN ) − ( ω N ) (cid:101) Ω (P) k (cid:101) Ω (P) ,k sin (cid:0)(cid:101) Ω (P) k t/N (cid:1)(cid:33)(cid:35) (3.20)19here c ( t ) is (3.8); while for DBC we get C approx = (cid:118)(cid:117)(cid:117)(cid:116) N − (cid:88) k =1 (cid:34) arcsinh (cid:32) ( ωN ) − ( ω N ) (cid:101) Ω (D) k (cid:101) Ω (D) ,k sin (cid:0)(cid:101) Ω (D) k t/N (cid:1)(cid:33)(cid:35) (3.21)where (cid:101) Ω (P) k = (cid:114) ( ωN ) + 4 π κm k (cid:101) Ω (D) k = (cid:114) ( ωN ) + π κm k (3.22)while (cid:101) Ω (P) ,k and (cid:101) Ω (D) ,k are obtained by replacing ω with ω in these expressions. Notice thatboth (3.20) and (3.21) depend on ωN , ω N and t/N . These approximate expressions havebeen used to plot the dashed light grey curves in the top panels of Fig. 4, which nicely agreewith the corresponding solid coloured curves.The thermodynamic limit N → ∞ of the complexity can be studied through the standardprocedure. Introducing θ ≡ πk/N and substituting (cid:80) k → Nπ (cid:82) π dθ in (3.7), at the leadingorder we find C TD = (cid:114) Nπ (cid:115)(cid:90) π (cid:20) arcsinh (cid:18) ω − ω θ Ω ,θ sin(Ω θ t ) (cid:19)(cid:21) dθ (3.23)where the dispersion relations for PBC and DBC become respectivelyΩ ,θ = (cid:114) ω + 4 κm (sin θ ) Ω θ = (cid:114) ω + 4 κm (sin θ ) (3.24)and Ω ,θ = (cid:114) ω + 4 κm [sin( θ/ Ω θ = (cid:114) ω + 4 κm [sin( θ/ . (3.25)Notice that, for PBC, the zero mode does not contribute because c /N → N → ∞ .When DBC hold, by using the dispersion relations (3.25), changing of variable ˜ θ = θ/ x ) in the interval [0 , π ], one findsthat (3.23) with (3.24) holds for both PBC and DBC. Thus, the leading order of this limit isindependent of the boundary conditions. This means that the complexity does not distinguishthe boundary conditions in this regime. Indeed, in the left and right panels of Fig. 4, the samefunction (just described) has been used to plot the dashed black curves.The boundary conditions become crucial in the subleading term of the expansion of (3.7)as N → ∞ , which can be studied through the Euler-Maclaurin formula [98]. The details ofthis analysis are discussed in appendix C.2 and the final result is C − C TD = R (B) , ∞ B ∈ (cid:8) P , D (cid:9) R (B) , ∞ = R (P) , ∞ PBC R (D) , ∞ = R (P) , ∞ ζ DBC (3.26)where R (P) , ∞ and ζ are the time-dependent functions given in (C.17) and in (C.19) respectively.Numerical checks for these results are shown in Fig. 4. In the top panels of this figure we havedisplayed also C approx /N from (3.20) (left panel) and ( C approx − ζ ) /N from (3.21) (right panel)through dashed light grey lines. 20
50 100 150 200 2500.000.020.040.060.080.10
Figure 4: Temporal evolutions of the complexity for harmonic chains with either PBC (leftpanels) or DBC (right panels). The solid lines show C /N for PBC and ( C − ζ ) /N for DBC,with C given by (3.7) and ζ by (C.19). The dashed black lines represent C TD /N , from (3.23).21n the continuum limit, N → ∞ and the lattice spacing a ≡ (cid:112) m/κ → N a ≡ (cid:96) is kept fixed. In this limit, the expression (3.7) for the complexity (which holds forboth PBC and DBC) becomes C cont = (cid:114) (cid:96) π (cid:115)(cid:90) ∞−∞ (cid:20) arcsinh (cid:18) ω − ω p Ω ,p sin(Ω p t ) (cid:19)(cid:21) dp (3.27)(see appendix C.3 for a detailed discussion) whereΩ ,p = (cid:113) ω + p Ω p = (cid:112) ω + p . (3.28)Since Ω p (cid:39) p when p (cid:29) ω , the vanishing of the integrand in (3.27) as p → ±∞ is such that thecomplexity is UV finite. We remark that, instead, when the reference state is the unentangledproduct state, the continuum limit of the complexity is UV divergent, as discussed in appendixC.3; hence a UV cutoff in the integration domain over p must be introduced. It is worth discussing the initial growth of the complexity for the harmonic chains that weare considering. Since the complexity (3.7) is a special case of (2.33), its expansion as t → b in (2.37)), which provides the lineargrowth, but a similar analysis can be applied straightforwardly to the coefficients of the higherorder terms in the t → C = | ω − ω | (cid:32) N − η (cid:88) k =1 Ω − ,k (cid:33) / t + O ( t ) (3.29)where η = 1 and (3.4) must be used for PBC, while η = 0 and (3.6) must be employed forDBC. We remark that the slope of the initial linear growth in (3.29) is proportional to | ω − ω | .In Fig. 5, we consider the initial growth of the complexity (3.7) when PBC are imposed,comparing the exact curve against its expansion (2.36). The corresponding analysis for DBCprovides curves that are very similar to the ones displayed in Fig. 5; hence it has not beenreported in this manuscript.Let us conclude our discussion about the temporal evolution of the complexity of pure stateswith a brief qualitative comparison between the results discussed above and the correspondingones for the temporal evolution of the holographic complexity [8–10, 13, 14, 51–53].The Vaidya spacetimes are the typical backgrounds employed as the gravitational dualsof global quantum quenches in the conformal field theory on their boundary. They describethe formation of a black hole through the collapse of a matter shell. In Vaidya spacetimes,the temporal evolution of the holographic entanglement entropy has been largely studied[64, 65, 99–104] and the temporal evolutions of the holographic complexity for the entire spatial22 .0 0.5 1.0 1.50.00.51.01.52.02.5 Figure 5: Initial growth of the complexity for harmonic chains with PBC and N = 100. Theevolution Hamiltonian is either massive (left panel) or massless (right panel), for three values ω . The solid lines show the complexity (3.7) (with (3.4)) and the dashed lines represent itsexpansion (2.36) up to the O ( t ) (grey), O ( t ) (yellow) and O ( t ) (green) term included.section of the conformal field theory on the boundary has been investigated in [51–53, 105].Considering the temporal evolution of the rate d C dt allows to avoid the problem of choosingthe reference state, which deserves further clarifications for the holographic complexity, evenfor static gravitational backgrounds. The analysis of d C dt in Vaidya spacetimes, both for theCV and for the CA prescriptions, shows that these temporal evolutions are linear in timeboth at very early and at late time [51, 52]. While also the initial growth of the complexitythat we have explored is linear (see (3.29)), the late time growth is at most logarithmic. Thisdisagreement, which deserves further analysis, has been discussed in [18].We find it worth observing also that the coefficient of the initial growth (3.29) is propor-tional to | ω − ω | and that the corresponding coefficient for the holographic complexity isproportional to the mass of the final black hole [51, 52]. In this section we study the temporal evolution of the subsystem complexity after a globalquench. The reference and the target states are the reduced density matrices associated to agiven subsystem. We focus on the simple cases where the subsystem A is a block made byconsecutive sites in harmonic chains with either PBC or DBC. In the harmonic lattices that we are considering, the reduced density matrix associated to A characterises a Gaussian state which can be described equivalently through its reducedcovariance matrix γ A . This matrix is constructed by considering the reduced correlation ma-trices Q A , P A and M A , whose elements are respectively given by ( Q A ) i,j = (cid:104) ψ | ˆ q i ( t ) ˆ q j ( t ) | ψ (cid:105) ,23 P A ) i,j = (cid:104) ψ | ˆ p i ( t ) ˆ p j ( t ) | ψ (cid:105) and ( M A ) i,j = Re (cid:2) (cid:104) ψ | ˆ q i ( t ) ˆ p j ( t ) | ψ (cid:105) (cid:3) with i, j ∈ A , which de-pend also on the time after the global quench. These matrices provide the following blockdecomposition of the reduced covariance matrix γ A ( t ) = (cid:18) Q A ( t ) M A ( t ) M A ( t ) t P A ( t ) (cid:19) . (4.1)For the harmonic chains with either PBC or DBC introduced in Sec. 3 and A made by L consecutive sites, Q A and P A are L × L symmetric matrices and γ A is a real, symmetric andpositive definite 2 L × L matrix.Adapting the analysis made in Sec. 3 for pure states to the mixed states described bythe reduced covariance matrices γ A ( t ), we have that the reference state is given by the re-duced density matrix for the interval A at time t R (cid:62) (cid:0) κ R , m R , ω R , ω , R (cid:1) and the target state by the reduced density matrix forthe same interval at time t T (cid:62) t R constructed through the quench protocol described by (cid:0) κ T , m T , ω T , ω , T (cid:1) . The corresponding reduced covariance matrices are denoted by γ R ,A ( t R )and γ T ,A ( t T ) respectively. These reduced covariance matrices are decomposed in terms of thecorrelation matrices of the subsystem like in (4.1).The approach to the circuit complexity of mixed states based on the Fisher informationgeometry [74] allows to construct the optimal circuit between γ R ,A ( t R ) and γ T ,A ( t T ). Thecovariance matrices along this optimal circuit are G s ( γ R ,A ( t R ) , γ T ,A ( t T )) ≡ γ R ,A ( t R ) / (cid:16) γ R ,A ( t R ) − / γ T ,A ( t T ) γ R ,A ( t R ) − / (cid:17) s γ R ,A ( t R ) / (4.2)where 0 (cid:54) s (cid:54) C A = 12 √ (cid:114) Tr (cid:110)(cid:2) log (cid:0) γ T ,A ( t T ) γ R ,A ( t R ) − (cid:1)(cid:3) (cid:111) . (4.3)Considering harmonic chains made by N sites where PBC are imposed, by using (2.5), (2.6)and either (3.2) or (3.3), one obtains the elements of the correlation matrices whose reductionto A provides (4.1). They read Q i,j ( t ) = 1 N N (cid:88) k =1 Q k ( t ) cos (cid:2) ( i − j ) 2 πk/N (cid:3) P i,j ( t ) = 1 N N (cid:88) k =1 P k ( t ) cos (cid:2) ( i − j ) 2 πk/N (cid:3) (4.4) M i,j ( t ) = 1 N N (cid:88) k =1 M k ( t ) cos (cid:2) ( i − j ) 2 πk/N (cid:3) (cid:54) i, j (cid:54) N ; while for DBC, by using (3.5), one obtains the following correlators Q i,j ( t ) = 2 N N (cid:88) k =1 Q k ( t ) sin (cid:0) i πk/N (cid:1) sin (cid:0) j πk/N (cid:1) P i,j ( t ) = 2 N N (cid:88) k =1 P k ( t ) sin (cid:0) i πk/N (cid:1) sin (cid:0) j πk/N (cid:1) (4.5) M i,j ( t ) = 2 N N (cid:88) k =1 M k ( t ) sin (cid:0) i πk/N (cid:1) sin (cid:0) j πk/N (cid:1) where 1 (cid:54) i, j (cid:54) N −
1. In these correlators, the functions Q k ( t ), P k ( t ) and M k ( t ) are givenby (2.6), with either (3.4) for PBC or (3.6) for DBC.The reduced covariance matrices γ R ,A ( t R ) and γ T ,A ( t T ) for the block A providing the op-timal circuit (4.2) and its complexity (4.3) are constructed as in (4.1), through the reducedcorrelation matrices Q A , P A and M A , obtained by restricting to i, j ∈ A the indices of thecorrelation matrices whose elements are given in (4.4) and (4.5).We remark that the matrix (cid:101) V in (2.5) (given in (3.2) or (3.3) for PBC and in (3.5) for DBC)is crucial to write (4.4) and (4.5); hence it enters in a highly non-trivial way in the evaluationof the subsystem complexity. Instead, it does not affect the complexity for the entire system,where both the reference and the target states are pure states, as remarked below (2.20). Considering the global quench that we are exploring, in the following we discuss some numer-ical results for the temporal evolution of the subsystem complexity of a block A made by L consecutive sites in harmonic chains made by N sites, where either PBC or DBC are imposed.We focus on the simplest setup where the reference state is the initial state (hence t R = 0) andthe target state corresponds to a generic value of t T = t (cid:62) ω , R = ω , T ≡ ω , ω R = ω T ≡ ω , κ R = κ T = 1 and m R = m T = 1. Inthe case of DBC, we consider both A adjacent to the boundary and separated from it.In this setup, the subsystem complexity (4.3) can be written as C A = 12 √ (cid:114) Tr (cid:110)(cid:2) log (cid:0) γ A ( t ) γ A (0) − (cid:1)(cid:3) (cid:111) . (4.6)It is natural to introduce also the entanglement entropy S A ( t ) and its initial value S A (0),which lead to define the increment of the entanglement entropy w.r.t. its initial value, i.e.∆ S A ≡ S A ( t ) − S A (0) (4.7)where S A ( t ) and S A (0) can be evaluated from the symplectic spectrum of γ A ( t ) and of γ A (0)respectively in the standard way [54, 57, 62, 92, 106–111].In all the figures discussed in this section we show the temporal evolutions of the subsystemcomplexity C A in (4.6) or of the increment ∆ S A of the entanglement entropy in (4.7) after the25 ○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○ × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × ○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○ × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × ○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○ × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × ○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○ × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × ×× ○■■■ ■ ○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○ × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × ○○○○○○○○○○○○○○○○○○○○○ ○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○ × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × ○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○ × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × ○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○ × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × ×× ○■■■ ■ Figure 6: Temporal evolution of C A in (4.6) after the global quench with gapless evolutionHamiltonian and ω N = 20, for a block A made by L consecutive sites in harmonic chainswith either PBC (left panels) or DBC (right panels) made by N sites (in the latter case A is adjacent to a boundary). When L = N , the complexity (3.7) is shown for N = 100 (solidblack lines) and N = 200 (dashed green lines).global quench. In particular, we show numerical results corresponding to N = 100 and N =200, finding nice collapses of the data when L/N , ω N and ωN are kept fixed, independentlyof the boundary conditions. The data reported in all the left panels have been obtained inharmonic chains with PBC, whose dispersion relations are (3.4), while the ones in all theright panels correspond to a block adjacent to a boundary of harmonic chains where DBCare imposed, whose dispersion relations are (3.6), if not otherwise indicated (like in Fig.12).The evolution Hamiltonian is gapless in Fig. 6, Fig. 7, Fig. 8 and Fig. 12, while it is gapped inFig. 9 and Fig. 10, with ωN = 5. In Fig. 11, where the initial growth is explored, both gaplessand gapped evolution Hamiltonians have been employed. When L = N , the complexity (3.7)for pure states has been evaluated with either N = 100 (black solid lines) or N = 200 (dashedgreen lines).In Fig. 6, Fig. 7 and Fig. 8 all the data have been obtained with ωN = 0 and either ω N = 20(Fig. 6 and Fig. 8) or ω N = 100 (Fig. 7). Revivals are observed and the different cyclescorrespond to p < t/N < p + 1 for PBC and to p < t/N < p + 1 for DBC, where p is anon-negative integer.The qualitative behaviour of the temporal evolution of the subsystem complexity cruciallydepends on the boundary conditions of the harmonic chain. For DBC, considering the datahaving L/N < / t/N < /
2, we can identify three regimes: an initial growth until alocal maximum is reached, a decrease and then a thermalisation regime after certain value of t/N , where the subsystem complexity remains constant. For PBC and
L/N < /
2, the latterregime is not observed and C A keeps growing. Comparing the right panel in Fig. 6 with thetop right panel in Fig. 7, one realises that, for DBC, the height of the plateaux increases aseither L/N or ω N increases, as expected. The absence of thermalisation regimes for PBC26 ○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○ × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × ○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○ × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × ○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○ × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × ○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○ × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × ×× ○■■■ ■ ○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○ × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × ○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○ × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × ○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○ × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × ○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○ × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × ×× ○ ■■■ ■ ○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○ × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × ○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○ × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × ○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○ ××××××××××××××××××××××××××××××××××××××××××××××××××× ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × ○ ■■■ ■ ○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○ × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × ○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○ × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × ○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○ ××××××××××××××××××××××××××××××××××××××××××××××××××× ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × ○■■■■ ○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○ × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × ○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○ × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × ○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○ × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × ×× ○■■■ ○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○ × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × ○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○ × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × ○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○ × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × ×× ○■■■ Figure 7: Temporal evolution of C A in (4.6) (top panels), of ∆ S A in (4.7) (middle panels) andof (cid:112) L/N ∆ S A / C A (bottom panels) after the global quench with gapless evolution Hamiltonianand ω N = 100, for a block A made by L consecutive sites in a harmonic chains with eitherPBC (left panels) or DBC (right panels) made by N sites (in the latter case A is adjacent toa boundary). When L = N the complexity (3.7) is shown for N = 100 (solid black lines) and N = 200 (dashed green lines). 27 ○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○ ×××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××× ■■■■○ × 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○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○ ×××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××× Figure 8: Temporal evolution of C A in (4.6) (top panels), of ∆ S A in (4.7) (middle panels)and of (cid:112) L/N ∆ S A / C A (bottom panels) after the global quench with a gapless evolutionHamiltonian and ω N = 20, for harmonic chains with either PBC (left panels) or DBC (rightpanels), in the same setups of Fig. 6 are considered.28ould be related to the occurrence of the zero mode, as suggested by the fact that, for purestates, the zero mode contribution provides the logarithmic growth of the complexity (3.15).However, we are not able to identify explicitly the zero mode contribution in the subsystemcomplexity, hence we cannot subtract it as done in the bottom left panel of Fig. 1 for thetemporal evolution of the complexity of pure states.For DBC, the plateau in the thermalisation regime is not observed when L/N (cid:62) / t/N ∈ [ ν, ν +1] with ν = { , } , it approximately begins at t − νN (cid:39) L and ends at t − νN (cid:39) N − L + 1. The straight dashed grey lines approximatively indicate thebeginning of the plateaux for different L/N < / t/N = 0 . C A in infinite chains is made by the three regimesmentioned above (see Fig. 14, Fig. 15 and Fig. 16), as largely discussed in Sec. 5.Comparing the temporal evolutions of C A and ∆ S A for the same quench protocol and thesame subsystem in Fig. 7, we observe that the initial growth of C A in the first revival is fasterthan the linear initial growth of ∆ S A , as highlighted by the straight dashed black lines inFig. 7. Within the first revival, we do not observe a long range of t/N where the evolution of C A is linear. Nonetheless, the straight line characterising the initial growth of ∆ S A intersectsthe first local maximum corresponding to the end of the initial growth of C A when L/N < / L/N < / t/N corresponding tohalf of the first revival, we notice that, while the temporal evolution of ∆ S A displays a lineargrowth followed by a saturation regime, the temporal evolution of C A is characterised by thethree regimes described above. The saturation regimes of C A and ∆ S A are qualitatively verysimilar and begin approximatively at the same value of t/N . Notice that the amplitude of thedecrease of C A at the end of the first revival is smaller than the one of ∆ S A .The temporal evolutions of C A and ∆ S A can be compared for L/N (cid:54) /
2. Indeed, fora bipartite system in a pure state the entanglement entropy of a subsystem is equal to theentanglement entropy of the complementary subsystem. This property, which does not holdfor C A , implies the overlap between the data for ∆ S A corresponding to L/N = 3 /
10 and to
L/N = 7 /
10. Furthermore, ∆ S A = 0 identically when L = N .In the bottom panels of Fig. 7 we have reported the temporal evolutions of the ratio ∆ S A / C A for the data reported in the other panels of the figure. The curves of ∆ S A / C A correspondingto PBC (left panel) and DBC (right panel) are very similar. For instance, the curves for (cid:112) L/N ∆ S A / C A have the same initial growth for different values of L/N . However, we remarkthat a mild logarithmic decrease occurs in the thermalisation regime for PBC.In Fig. 8 the range 0 (cid:54) t/N (cid:54)
10 is considered, which is made by 20 revivals for PBCand by 10 cycles for DBC. The temporal evolutions of C A in the top panels show that, up tooscillations due to the revivals, after the initial growth C A keeps growing logarithmically forPBC (the solid coloured lines in the top left panel are two-parameter fits through the function a + b log( t/N ) of the corresponding data), while it remains constant for DBC. This featureis observed also in the corresponding temporal evolutions of ∆ S A (middle panels of Fig. 8).These two logarithmic growths for PBC are very similar, as shown by the temporal evolution29 ○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○ × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × ○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○ × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × ○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○ × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × ○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○ × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × ×× ○■■■ ■ ○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○ × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × ○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○ × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × ○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○ × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × ○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○ × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × ×× ○■■ ■■ Figure 9: Temporal evolution of C A in (4.6) after the global quench with a gapped evolutionHamiltonian for a block A made by L consecutive sites in harmonic chains with either PBC(left panels) or DBC (right panels) made by N sites (in the latter case A is adjacent to aboundary of the segment). When L = N the complexity (3.7) is shown for N = 100 (solidblack lines) and N = 200 (dashed green lines).of ∆ S A / C A displayed in the bottom left panel of Fig. 8.In Fig. 9 and Fig. 10 we show some temporal evolutions of C A when the evolution Hamilto-nian is massive ( ω < ω in Fig. 9 and ω > ω in Fig. 10, with ωN = 5 in both the figures).In these temporal evolutions one observes that the local extrema of the curves for C A havingdifferent L/N roughly occur at the same values of t/N . It is insightful to compare thesetemporal evolutions with the corresponding ones characterised by ω = 0 in Fig. 6 and Fig. 7.For PBC, the underlying growth observed when ω = 0 does not occur if ω >
0. For DBC, theplateaux observed in the saturation regime when ω = 0 are replaced by oscillatory behavioursif ω > C A , of ∆ S A and of ∆ S A / C A for the sameglobal quench. The evolutions of C A and of ∆ S A are qualitatively similar when L/N < / t/N : for C A is linear(see also Fig. 11 and the corresponding discussion), while for ∆ S A is quadratic, as highlightedin the insets of the middle panels (the coefficient of this quadratic growth for PBC is twicethe one obtained for DBC) and also observed in [103, 110, 112, 113]. Comparing the bottompanels of Fig. 10 against the bottom panels of Fig. 7, one notices that the similarity observedfor PBC and DBC when ω = 0 does not occur when ω (cid:54) = 0. It is important to perform asystematic analysis considering many other values of ωN and ω N , in order to understandthe effect of a gapped evolution Hamiltonian in the temporal evolution of C A .In Fig. 11 we consider the initial regime of the temporal evolution of C A w.r.t. the initialstate for various choices of ω N and ωN (in particular, ω = 0 in the first and in the secondlines of panels, while ω > t areconsidered with respect to the ones explored in the previous figures. In this regime, data30 ○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○ × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × ○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○ × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × ○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○ × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × ○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○ × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × ×× ○■■■ ■ ○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○ × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × ○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○ × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × ○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○ × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × ○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○ × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × ×× ○■■ ■■ ○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○ × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × ○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○ × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × ○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○ ××××××××××××××××××××××××××××××××××××××××××××××××××× ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × ○■■■■ ○○○○○○○○○○○○○○○○○○○○○○○○○○ ×××××××××××××××××××××××××× ○○○○○○○○○○○○○○○○○○○○○○○○○○ ×××××××××××××××××××××××××× ○○○○○○○○○○○○○○○○○○○○○○○○○○ ×××××××××××××××××××××××××× ○○○○○○○○○○○○○○○○○○○○○○○○○○ × × × × × × × × × × × × × × × × × × × × × × × × × × ○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○ × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × ○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○ × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × ○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○ ××××××××××××××××××××××××××××××××××××××××××××××××××× ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × ○■■■■ ○○○○○○○○○○○○○○○○○○○○○○○○○○ ×××××××××××××××××××××××××× ○○○○○○○○○○○○○○○○○○○○○○○○○○ ×××××××××××××××××××××××××× ○○○○○○○○○○○○○○○○○○○○○○○○○○ ×××××××××××××××××××××××××× ○○○○○○○○○○○○○○○○○○○○○○○○○○ × × × × × × × × × × × × × × × × × × × × × × × × × × ○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○ × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × ○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○ × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × ○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○ × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × ×× ○■■■ ○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○ × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × ○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○ × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × ○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○ × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × ×× ○■■■ Figure 10: Temporal evolution after the global quench with a gapped evolution Hamiltonianof C A in (4.6) (top panels), of ∆ S A in (4.7) (middle panels) and of (cid:112) L/N ∆ S A / C A (bottompanels) for a block A made by L consecutive sites in harmonic chains with either PBC (leftpanels) or DBC (right panels) made by N sites (in the latter case A is adjacent to a boundaryof the segment). When L = N , the complexity (3.7) is shown for N = 100 (solid black lines)and N = 200 (dashed green lines). 31ollapses are observed for different values of L/N when C A / √ ω L is reported as function of t/N . In the special case of L = N , the complexity of pure states (3.7) discussed in Sec. 3 isrecovered, as shown in Fig. 11 by the black solid lines ( N = 100) and by the green dashedlines ( N = 200).Each panel on the left in Fig. 11 is characterised by the same ω N and ωN of the correspond-ing one on the right. From their comparison, one realises that the qualitative behaviour of theinitial growth at very early times is not influenced by the choice of the boundary conditions.Moreover, the linear growth of C A / √ ω L is independent of L/N for very small values of t/N ;hence the slope of the initial growth can be found by considering the case L = N (discussedin Sec. 3) and the approximation described in Sec. 3.4 and in appendix C.1. Combining theseobservations with (C.4) and (C.5), we obtain the initial linear growth a (B) t/N + . . . where thedots represent higher order in t/N and the slope depends on the boundary conditions labelledby B ∈ { P , D } as follows a (P) = (cid:12)(cid:12) ( ωN ) − ( ω N ) (cid:12)(cid:12) ω N (cid:115)(cid:114) m κ ω N coth (cid:18)(cid:114) m κ ω N (cid:19) (4.8) a (D) = (cid:12)(cid:12) ( ωN ) − ( ω N ) (cid:12)(cid:12) √ ω N (cid:115)(cid:114) mκ ω N coth (cid:18)(cid:114) mκ ω N (cid:19) − . (4.9)The grey dashed lines in Fig. 11 represent a (P) t/N (left panels) and a (D) t/N (right panels).Since for DBC and ω = 0 the temporal evolution of C A displays a thermalisation regimeafter the initial growth and the subsequent decrease when the block A with L/N < / A is separated fromthe boundary. Denoting by d L the number of sites separating A from the left boundary ofthe chain (hence d R = N − L − d L sites occur between A and the right boundary), C A mustbe invariant under a spatial reflection w.r.t. the center of the chain, i.e. when d L and d R arereplaced by d R − d L + 1 respectively.In Fig. 12 we show the temporal evolutions of C A and of ∆ S A for this bipartition of thesegment when the evolution Hamiltonian is gapless and ω N = 100, for four different valuesof L/N and fixed values of d L /N given by d L /N = 0 . d L /N = 0 . N = 100and N = 200 nicely collapse on the same curve.When d L (cid:54) = 0, a thermalisation regime where both the curves of C A and ∆ S A are constantoccurs if b/N < /
2, with b = min[ d L + L, d R + L −
1] (see the red and blue curves in Fig. 12).The plateau is observed approximatively for t/N ∈ [ b/N, − b/N ] and its height dependson ω L , on L/N and also on d L /N . A remarkable feature of the temporal evolution of C A when d L > t/N < /
2, while only one maximumis observed when d L = 0 for t/N < / L/N (cid:54) / t/N < /
2. The occurrence of two local maxima inthe temporal evolution of C A when A is separated from the boundary is observed also when N → ∞ . This is shown in Fig. 14 and Fig. 16, where we also highlight the logarithmic nature32 × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × ○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○ × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × ○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○ × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × ○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○■■■ × ○ × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × ○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○ × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × ○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○ × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × ○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○■■■ × ○ × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × ○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○ × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × ○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○ × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × ○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○■■■ × ○ × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × ○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○ × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × ○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○ × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × ○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○■■■ × ○ × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × ○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○ × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × ○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○ × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × ○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○■■■ × ○ × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × ○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○ × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × ○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○ × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × ○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○■■■ × ○ × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × ○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○ × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × ○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○ × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × ○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○■■■ × ○ × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × ○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○ × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × ○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○ × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × ○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○■■■ × ○ Figure 11: Initial growth of C A in (4.6) for a block A made by L consecutive sites in harmonicchains with either PBC (left panels) or DBC (right panels) made by N sites (in the lattercase A is adjacent to a boundary). When L = N , the complexity (3.7) is shown for N = 100(solid black lines) and N = 200 (dashed green lines).33 ○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○ × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × ○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○ × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × ○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○ × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × ○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○ × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × ×× ○ ■■■ ■ ○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○ × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × ○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○ × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × ○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○ × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × ○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○ × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × ×× ○ ■■■ ■ ○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○ × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × ×× × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × ×× × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × ×× × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × ■■■■○ × ○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○ × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × ×× × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × ×× × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × ×× × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × ■■■■○ × Figure 12: Temporal evolution after the global quench with a gapless evolution Hamiltonianof C A in (4.6) (top panels) and of ∆ S A in (4.7) (bottom panels) for a block A made by L consecutive sites and separated by d L sites from the left boundary of harmonic chains withDBC made by N sites. When L = N , the complexity (3.7) is shown for N = 100 (solid blacklines) and N = 200 (dashed green lines).of the growth of C A in the temporal regime between the two local maxima, which can becompared with a logarithmic growth occurring in ∆ S A (see e.g. Fig. 15).Comparing each top panel with the corresponding bottom panel in Fig. 12, we observe thatthe black dashed straight line (it is the same in the two top panels) captures the first localmaximum of C A . The slope of this line is twice the slope of the red dashed straight line in thebottom panels, which identifies the initial linear growth of ∆ S A .34 Subsystem complexity and the generalised Gibbs ensemble
In this section we consider infinite harmonic chains, either on the infinite line or on the semi-infinite line with DBC at the origin, and discuss that the asymptotic value of C A for a blockmade by consecutive sites can be found through the generalised Gibbs ensemble (GGE). An isolated system prepared in a pure state and then suddenly driven out of equilibriumthrough a global quench does not relax. Instead, relaxation occurs for a subsystem [114–116](see also the review [39] and the references therein).Consider a spatial bipartition of a generic harmonic chain given by a finite subsystem A and its complement. Denoting by ˆ ρ ( t ) the density matrix of the entire system and by ˆ ρ A ( t )the reduced density matrix of A , a quantum system relaxes locally to a stationary state if thelimit lim t →∞ lim N →∞ ˆ ρ A ( t ) ≡ ˆ ρ A ( t = ∞ ) exists for any A , where N is the number of sitesin the harmonic chain. This stationary state is described by the time independent densitymatrix ˆ ρ E describing a statistical ensemble if lim N →∞ ˆ ρ E ,A = ˆ ρ A ( t = ∞ ), for any A , whereˆ ρ E ,A is obtained by tracing ˆ ρ E over the degrees of freedom of the complement of A . For theglobal quench of the mass parameter that we are investigating in infinite harmonic chains, thestationary state is described by a GGE [44, 45, 117, 118] (see the review [46] for an extensivelist of references).In terms of the creation and annihilation operators (A.8), the evolution Hamiltonian reads (cid:98) H = N (cid:88) k =1 Ω k (cid:18) ˆ b † k ˆ b k + 12 (cid:19) . (5.1)The GGE that provides the stationary state reads [43]ˆ ρ GGE = e − (cid:80) Nk =1 λ k ˆ b † k ˆ b k Z GGE Z GGE = Tr( ˆ ρ GGE ) = N (cid:89) k =1 − e − λ k (5.2)where ˆ ρ GGE is normalised through the condition Tr( ˆ ρ GGE ) = 1. The conservation of the numberoperators ˆ b † k ˆ b k tells us that the relation between their expectation values and λ k reads [43] n k ≡ Tr (cid:0) ˆ b † k ˆ b k ˆ ρ GGE (cid:1) = 1 e λ k − (cid:104) ψ | ˆ b † k ˆ b k | ψ (cid:105) (5.3)which is strictly positive because λ k > k .Since the GGE in (5.2) is a bosonic Gaussian state, it is characterised by its covariancematrix γ GGE , which can be decomposed as follows γ GGE = (cid:18) Q GGE M GGE M t GGE P GGE (cid:19) (5.4)where (see the appendix A.3)( Q GGE ) i,j = Tr (cid:0) ˆ q i ˆ q j ˆ ρ GGE (cid:1) ( P GGE ) i,j = Tr (cid:0) ˆ p i ˆ p j ˆ ρ GGE (cid:1) ( M GGE ) i,j = Re (cid:2) Tr (cid:0) ˆ q i ˆ p j ˆ ρ GGE (cid:1)(cid:3) . (5.5)35y adapting the computation reported in appendix A.1 to this case, the operators ˆ q andˆ p can be introduced as in (A.7) and for (5.4) one finds Q GGE = (cid:101) V S − phys Tr (cid:0) ˆ q ˆ q t ˆ ρ GGE (cid:1) S − phys (cid:101) V t ≡ (cid:101) V Q GGE (cid:101) V t (5.6) P GGE = (cid:101) V S phys Tr (cid:0) ˆ p ˆ p t ˆ ρ GGE (cid:1) S phys (cid:101) V t ≡ (cid:101) V P GGE (cid:101) V t (5.7) M GGE = (cid:101) V S − phys Re (cid:2) Tr (cid:0) ˆ q ˆ p t ˆ ρ GGE (cid:1)(cid:3) S phys (cid:101) V t ≡ (cid:101) V M GGE (cid:101) V t . (5.8)Then, expressing ˆ q and ˆ p in terms of ˆ b and ˆ b † defined in (A.8), exploiting the fact that thetwo points correlators vanish when the indices of the annihilation and creation operators aredifferent, using (5.3) and Tr (cid:0) ˆ b † k ˆ b † k ˆ ρ GGE (cid:1) = Tr (cid:0) ˆ b k ˆ b k ˆ ρ GGE (cid:1) = 0, we find M GGE = M GGE = and Q GGE ≡ diag (cid:110) Q GGE ,k ; 1 (cid:54) k (cid:54) N (cid:111) P GGE ≡ diag (cid:110) P GGE ,k ; 1 (cid:54) k (cid:54) N (cid:111) (5.9)where Q GGE ,k ≡ n k m Ω k P GGE ,k ≡ m Ω k n k ) . (5.10)Thus, the covariance matrix (5.4) simplifies to γ GGE = Q GGE ⊕ P GGE , where Q GGE and P GGE are given by (5.6) and (5.7).We find it worth writing the Williamson’s decomposition of γ GGE , namely γ GGE = W t GGE D GGE W GGE W GGE = X GGE V t (5.11)where the symplectic spectrum is given by D GGE = 12 + diag (cid:110) n , . . . , n N , n , . . . , n N (cid:111) (5.12)and, like for the 2 N × N symplectic matrix W GGE , we have V = (cid:101) V ⊕ (cid:101) V and that the diagonalmatrix X GGE = X − phys is the inverse of X phys defined in (A.4). We emphasise that γ GGE does notdescribe a pure state. Indeed, since n k (cid:62) k , from (5.12) we have that the symplecticeigenvalues of γ GGE are greater than 1 /
2, as expected for a mixed bosonic Gaussian state.For the global quench in the harmonic chains that we considering, n k in (5.3) can becomputed from the expectation value of ˆ b † k ˆ b k on the initial state obtaining [43] n k = 14 (cid:18) Ω k Ω ,k + Ω ,k Ω k (cid:19) −
12 (5.13)where Ω ,k and Ω k are the dispersion relations of the Hamiltonian defining the initial stateand of the evolution Hamiltonian respectively. Notice that (5.13) is symmetric under theexchange Ω k ↔ Ω ,k . We recall that the boundary conditions defining the harmonic chaininfluence both the dispersion relations and the matrix V .By introducing the reduced covariance matrix γ GGE ,A for A , obtained from (5.4) in theusual way, the entanglement entropy S GGE ,A ≡ − Tr(ˆ ρ GGE ,A log ˆ ρ GGE ,A ) (5.14)can be evaluated from the symplectic spectrum of γ GGE ,A through standard methods [57, 106].36he asymptotic value of the increment of the entanglement entropy ∆ S A when t → ∞ canbe computed as followslim L →∞ lim t →∞ lim N →∞ ∆ S A L = lim L →∞ lim N →∞ S GGE ,A L = lim N →∞ S GGE N (5.15)where the order of the limits is important and in the last step we used that S GGE is an extensivequantity (see the review [119] and the references therein).For the global quench in the harmonic chains that we are considering, the asymptotic value(5.15) for the entanglement entropy reads [63]lim N →∞ S GGE N = (cid:90) π (cid:104) ( n θ + 1) log( n θ + 1) − n θ log n θ (cid:105) dθπ (5.16)= (cid:90) π (cid:40)(cid:20) (cid:18) Ω θ Ω ,θ + Ω ,θ Ω θ (cid:19) + 12 (cid:21) log (cid:20) (cid:18) Ω θ Ω ,θ + Ω ,θ Ω θ (cid:19) + 12 (cid:21) (5.17) − (cid:20) (cid:18) Ω θ Ω ,θ + Ω ,θ Ω θ (cid:19) − (cid:21) log (cid:20) (cid:18) Ω θ Ω ,θ + Ω ,θ Ω θ (cid:19) − (cid:21)(cid:41) dθπ in terms of n θ given in (5.13), where the dispersion relations to employ are (3.24) for PBC and(3.25) for DBC. A straightforward change of integration variable leads to the same expressionfor both the boundary conditions, as already noticed for (3.23). Let us remark that (5.16) isfinite for any choice of the parameter (including ω = 0), both for PBC and DBC. It is alsosymmetric under the exchange Ω θ ↔ Ω ,θ ; hence under ω ↔ ω as well.We study the circuit complexity to construct the GGE (which is a mixed state) starting fromthe (pure) initial state at t = 0, by employing the approach based on the Fisher informationgeometry [74]. The optimal circuit to get γ GGE from the initial covariance matrix γ (0) at t = 0reads [74, 95] G s ( γ (0) , γ GGE ) ≡ γ (0) / (cid:16) γ (0) − / γ GGE γ (0) − / (cid:17) s γ (0) / (5.18)where 0 (cid:54) s (cid:54) C GGE = 12 √ (cid:114) Tr (cid:110)(cid:2) log (cid:0) γ GGE γ (0) − (cid:1)(cid:3) (cid:111) . (5.19)Since M GGE = M (0) = , from (2.5), (5.6) and (5.7) we obtain γ GGE = V (cid:2) Q GGE ⊕ P
GGE (cid:3) V t γ (0) = V (cid:2) Q (0) ⊕ P (0) (cid:3) V t . (5.20)Then, by exploiting (5.9), (2.7) and the fact that the matrix V is the same for both γ GGE and γ (0), we find that the complexity (5.19) reads C GGE = 12 √ (cid:118)(cid:117)(cid:117)(cid:116) N (cid:88) k =1 (cid:26)(cid:20) log (cid:18) Q GGE ,k Q k (0) (cid:19)(cid:21) + (cid:20) log (cid:18) P GGE ,k P k (0) (cid:19)(cid:21) (cid:41) (5.21)= 12 √ (cid:118)(cid:117)(cid:117)(cid:116) N (cid:88) k =1 (cid:40)(cid:20) log (cid:18) Ω ,k Ω k (1 + 2 n k ) (cid:19)(cid:21) + (cid:20) log (cid:18) Ω k Ω ,k (1 + 2 n k ) (cid:19)(cid:21) (cid:41) . (5.22)37 Figure 13: Asymptotic value of C GGE / √ N from (5.24) (left panel) and of S GGE /N from (5.16)(right panel) as functions of ω , for some values of ω .By using (5.13), this expression becomes C GGE = 12 √ (cid:118)(cid:117)(cid:117)(cid:116) N (cid:88) k =1 (cid:40)(cid:20) log (cid:18) Ω ,k k + 12 (cid:19)(cid:21) + (cid:20) log (cid:18) Ω k ,k + 12 (cid:19)(cid:21) (cid:41) (5.23)which is symmetric under the exchange Ω k ↔ Ω ,k , hence under ω ↔ ω as well.The leading order of this expression as N → ∞ is given by C GGE = √ N √ π (cid:118)(cid:117)(cid:117)(cid:116) (cid:90) π (cid:40)(cid:20) log (cid:18) Ω ,θ θ + 12 (cid:19)(cid:21) + (cid:20) log (cid:18) Ω θ ,θ + 12 (cid:19)(cid:21) (cid:41) dθ (5.24)where Ω ,θ and Ω θ are thermodynamic limits of the dispersion relations associated to theHamiltonians before and after the quench respectively. By repeating the argument reportedbelow (3.23), one finds that (5.24) with (3.24) can be employed for both PBC and DBC.Moreover, the resulting expression for C GGE / √ N is finite for any choice of the parameters(including for ω = 0).In Fig. 13 we show C GGE / √ N from (5.24) and S GGE /N from (5.16) as functions of ω , forsome values of ω . The resulting curves are qualitatively similar. At ω = ω they both vanish,but C GGE / √ N is singular at this point, while S GGE /N is smooth.The reduced covariance matrix γ GGE ,A associated to any finite subsystem A is obtained byselecting the rows and the columns in (5.4) corresponding to A . The results of [74] can beapplied again to write the optimal circuit that provides γ GGE ,A from the initial mixed statecharacterised by the reduced covariance matrix γ A (0) at t = 0, obtained from γ (0) throughthe usual reduction procedure. This optimal circuit reads G s ( γ A (0) , γ GGE ,A ) ≡ γ A (0) / (cid:16) γ A (0) − / γ GGE ,A γ A (0) − / (cid:17) s γ A (0) / (5.25)where 0 (cid:54) s (cid:54) C GGE ,A = 12 √ (cid:114) Tr (cid:110)(cid:2) log (cid:0) γ GGE ,A γ A (0) − (cid:1)(cid:3) (cid:111) . (5.26)Since the harmonic chain relaxes locally to the GGE after the quantum quench, for thesubsystem complexity of any finite subsystem A we expectlim t →∞ lim N →∞ C A = lim N →∞ C GGE ,A (5.27)which is confirmed by the numerical results in Fig. 14, Fig. 16, Fig. 19, Fig. 20 and Fig. 21.A numerical analysis shows that (5.27) grows like √ L as L → ∞ for fixed values of ω and ω ; hence, by adapting (5.15) to the subsystem complexity, we expectlim L →∞ lim t →∞ lim N →∞ C A √ L = lim L →∞ lim N →∞ C GGE ,A √ L = lim N →∞ C GGE √ N (5.28)where C GGE is given in (5.24) and the order of the limits is important. Numerical evidencesfor (5.28) are discussed in appendix D (see Fig. 22 and Fig. 23).In the following numerical analysis we show that, for the harmonic chains that we areexploring, the asymptotic limit for t → ∞ of the reduced density matrix after the globalquench is the reduced density matrix obtained from the GGE. This result has been alreadydiscussed for a fermionic chain in [120], where, considering a global quench of the magneticfield in the transverse-field Ising chain and the subsystem given by a finite block made byconsecutive sites in an infinite chain on the line, it has been found that a properly defineddistance between the reduced density matrix at a generic value of time along the evolutionand the asymptotic one obtained from the GGE vanishes as t → ∞ . In order to test (5.26), infinite harmonic chains must be considered. The reference and thetarget states have been described in Sec. 4. In this section we study harmonic chains both onthe line and on the semi-infinite line with DBC imposed at its origin. In the latter case, theblock A made by L consecutive sites is either adjacent to the origin or separated from it.The correlators to employ in the numerical analysis can be obtained from the ones reportedin Sec. 4. For the infinite harmonic chain on the line, we take N → ∞ of (4.4), finding Q i,j ( t ) = 1 π (cid:90) π Q θ ( t ) cos (cid:2) θ ( i − j ) (cid:3) dθP i,j ( t ) = 1 π (cid:90) π P θ ( t ) cos (cid:2) θ ( i − j ) (cid:3) dθM i,j ( t ) = 1 π (cid:90) π M θ ( t ) cos (cid:2) θ ( i − j ) (cid:3) dθ (5.29)where i, j ∈ Z ; while, for the harmonic chain on the semi-infinite line with DBC, the limit39 → ∞ of (4.5) leads to Q i,j ( t ) = 2 π (cid:90) π Q θ ( t ) sin( iθ ) sin( jθ ) dθP i,j ( t ) = 2 π (cid:90) π P θ ( t ) sin( iθ ) sin( jθ ) dθM i,j ( t ) = 2 π (cid:90) π M θ ( t ) sin( iθ ) sin( jθ ) dθ (5.30)where i, j >
0. The functions Q θ ( t ), P θ ( t ) and M θ ( t ) in these integrands are given by (2.6)where Ω ,k and Ω k are replaced respectively by Ω ,θ and Ω θ , which are (3.24) and (3.25) forthe infinite and for the semi-infinite line respectively.Once the proper correlators on the chain are identified, the reduced correlation matrices Q A , P A and M A are the blocks providing the reduced covariance matrix (4.1). These matricesare obtained by restricting the indices of the proper correlators to i, j = 1 , . . . , L when A ison the infinite line and to i, j = 1 + d, . . . , L + d when A is on the semi-infinite line, where d corresponds to its separation from the origin.In the following we discuss numerical data sets obtained for infinite harmonic chains, eitheron the infinite line or on the semi-infinite line, where ωL and ω L are kept fixed. In appendix Dwe report numerical results characterised by fixed values of ω and ω .In Fig. 14 and Fig. 15 we show the temporal evolutions of C A , of ∆ S A and of ∆ S A / C A afterthe quench with ω L = 20 and ωL = 0. In Fig. 16 we display the temporal evolution of C A with ω L = 100 and ωL = 0. The subsystem A is a block made by L consecutive sites eitheron a semi-infinite line, separated by d sites from the origin where DBC are imposed (colouredsymbols), or on the infinite line (black symbols). The black and coloured data points for C A have been found through (4.6) with the reduced correlators obtained from either (5.29) or(5.30) respectively. The coloured horizontal solid lines correspond to either (5.26) or (5.14),with the reduced correlators from (A.34) for the target state and from (5.30) at t = 0 for thereference state, with L = 50. Notice that a black horizontal solid line does not occur becausethe corresponding value is divergent, as indicated also by the left panel in Fig. 17.Considering the block on the semi-infinite line, in Fig. 14 and Fig. 16 we observe that theinitial growth of C A is the same until the first local maximum, for all the values of d/L .After the first local maximum, the temporal evolution of C A depends on whether the block isadjacent to the boundary. If d/L = 0 the curve decreases until it reaches the saturation value.Instead, when d/L >
0, first C A decreases along a different curve (see e.g. Fig. 16) until alocal minimum; then we observe an intermediate growth, followed by a second local maximumand finally by the saturation regime. A fitting procedure shows that the intermediate growthbetween the two local maxima is logarithmic (in the inset of Fig. 16 the grey dashed curvehas been found by fitting the data having L = 40 and d/L = 3 through a logarithm and aconstant). Its temporal duration is approximatively d/L − /
2, for the three values of nonvanishing d/L considered in Fig. 14 and Fig. 16. Fitting the intermediate growth in Fig. 14 andFig. 16, one observes that the coefficient of the logarithmic growth decreases as ω L increases.The first local maximum in the temporal evolution of C A occurs for 0 < t/L <
1. When d > × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ××××××××××××× ◇◇◇◇◇◇◇◇◇◇◇◇◇○○○○○○○○○○○○○ ××××××××××××× ◇◇◇◇◇◇◇◇◇◇◇◇◇○○○○○○○○○○○○○ ××××××××××××× ◇◇◇◇◇◇◇◇◇◇◇◇◇○○○○○○○○○○○○○ ××××××××××××× ◇◇◇◇◇◇◇◇◇◇◇◇◇○○○○○○○○○○○○○ ×××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××× ◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○ ××××××××××××××××××××××××××××× ◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇○○○○○○○○○○○○○○○○○○○○○○○○○○○○○ ○ × ◇ ■■■■ × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ×××××××××××××××××××××××××××××××××××××××××× ◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇○○○○○○○○○○○○○○○○○○○○○○○○○○○○ ×××××××××××××××××××××××××××××××××××××××××× ◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇○○○○○○○○○○○○○○○○○○○○○○○○○○○○○ ×××××××××××××××××××××××××××××××××××××××××× ◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○ × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ×××××××××××××××××××××××××××××××××××××××××× ◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○ × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × ×××××××××××××××××××××××××××××× ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○ × × × ◇ ◇ ◇○ ○ ○ ○ × ◇■■■■ Figure 14: Temporal evolution of C A (top panel) and of ∆ S A (bottom panel) after a globalquantum quench with a gapless evolution Hamiltonian and ω L = 20. The subsystem is ablock A made by L consecutive sites either on the infinite line (black data points) or on thesemi-infinite line, separated by d sites from the origin where DBC hold (coloured data points).The dashed black straight line is the same in both panels.41 ××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××× ◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○ × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × ×× × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × ×× × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × ×× × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ×××××××××××××××××××××××××××× ◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇○○○○○○○○○○○○○○○○○○○○○○○○○○○○ × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇○ ○ ○ ○ ○ ○○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ × ◇■■■■ ×××××××××××××××××××××××××××××××××××× ××××××××××××××××××××××××××××× ×××××××× × ×× × ×××××××××× ×××××××××××××××××× ◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇ ◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇ ◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇ ◇◇◇◇◇◇◇◇◇○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○ ○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○ ○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○ ○○○○○○○○○○ × ×××××××××××××××××× ×× × × ◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇○ ○○○○○○○○○○○○○○ ○○○○○○ ○ ○○○○○ ○ ○○○○ ○○○ Figure 15: Temporal evolution of ∆ S A / C A for the data reported in Fig. 14. The inset zoomsin to highlight the data points having t/L > d/L < t/L < ( d + 1) /L . Notice that these two local maximacan be seen also in the top panels of Fig. 12 for t/N < / d > C A on the infinte line only one localmaximum occurs and the intermediate logarithmic growth mentioned above does not finishwithin the temporal regime that we have considered. This agreement tells us that the secondlocal maximum in the temporal evolution of C A is due to the presence of the boundary.The temporal evolutions of ∆ S A in the bottom panel of Fig. 14 can be explained by em-ploying the quasi-particle picture [41], which provides the different temporal regimes and thecorresponding qualitative behaviour of ∆ S A (for the subsystems where a boundary occurs,the quasi-particle picture has been described e.g. in [68]). The different regimes identifiedby this analysis correspond to the vertical dot-dashed lines in the bottom panel of Fig. 14.Instead, the vertical dashed grey lines in the top panel of Fig. 14 correspond to t/L = 1 + d/L .For d >
0, when t/L > / S A whose dura-tion depends on d/L according to the quasi-particle picture, until the beginning of a lineardecreases. Considering two sets of data points of ∆ S A having different d/L , they collapseuntil the first linear decrease is reached. 42 × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○○○○○○○○○○ × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○○○○○○○○○○ × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○○○○○○○○○○ × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○○○○○○○○○○ ×××××××××× ◇◇◇◇◇◇◇◇◇◇ ×××××××××× ◇◇◇◇◇◇◇◇ ×××××××××× ◇◇◇◇◇◇◇◇◇◇ ×××××××××× ◇◇◇◇◇◇◇◇◇◇ × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × ×××××××××× ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇◇◇◇◇◇◇◇◇◇ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○○○○○○○ × ○◇■■■■ × × × ◇ ◇○ ○ × × × × × × × × × × × × × × × × × × ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ × × × × × × × × × × × × × × × × × × ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ × × × × × × × × × × × × × × × × × × × × × × × × ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ × × × × × × × × × × × × × × × × × × × × × × × × ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ Figure 16: Temporal evolution of C A after a global quantum quench with a gapless evolutionHamiltonian and ω L = 100, in the same setups of Fig. 14. The inset zooms in on theintermediate temporal regime between the two local maxima for the data having d/L = 3.The initial growths of C A and of ∆ S A in Fig. 14 are very different. For instance, the growthof C A is the same for all the data sets, while for ∆ S A it depends on whether d vanishes.Moreover, while the growth of ∆ S A is linear for t/L < d = 0 and for t/L < / d >
0, the growth of C A is linear only at the very beginning of the temporal evolution andit clearly deviates from linearity within the regime of t/L where ∆ S A grows linearly. Thedashed black straight line passing through the origin in Fig. 14 describes the linear growth of∆ S A when d = 0 and it is the same in both the panels. This straight line intersects the firstlocal maximum of C A . This has been highlighted also for finite systems in Fig. 7 and Fig. 12.In Fig. 15 we show the ratio ∆ S A / C A for the data reported in Fig. 14. We remark that thetwo logarithmic growths occurring in ∆ S A and in C A almost cancel in the ratio; indeed, amild logarithmic decreasing is observed when t/L > < t/L < d/L = 3 (red symbols) that are already collapsed.The curves in Fig. 16 must be compared with the corresponding ones in top panel in Fig. 14in order to explore the effect of ω L . The height of the first local maximum in the temporalevolution of C A and also the saturation values for the data obtained on the semi-infinite lineincrease as ω L increases. Instead, the coefficient of the logarithmic growth after the firstlocal maximum decreases as ω L increases, as already remarked above. Notice that higher43 × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇■■◇ × ○ × × × × × × × × × × × × × × × × × × × × × × × × × × × × ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ × × × × × × × × × × × × × × × × × × × × × × × × × × × × × ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇■■◇ × ○ Figure 17: Asymptotic value of C GGE ,A in (5.26) for a block A made by L consecutive sites ininfinite chains in terms of ωL . The block is either in an infinite chain (left panel) or adjacentto the origin of the semi-infinite line with DBC (right panel).values of L are needed to observe data collapse as ω L increases.From the numerical results reported in the previous figures, we conclude that (5.26) providesthe asymptotic value of the subsystem complexity as t → ∞ ; hence it is worth studying thedependence of this expression on the subsystem size and on the parameters of the quenchprotocol.In Fig. 17 and Fig. 18 we show numerical results for (5.26), obtained by using the reducedcorrelators from (A.31) and (A.34) for the target state and the reduced correlators from (5.29)and (5.30) at t = 0 for the reference state.In Fig. 17 we show (5.26) as function of ωL when the block is either in the infinite line(left panel) or at the beginning of the semi-infinite line with DBC (right panel). The maindifference between the two panels of Fig. 17 is that the limit ωL → ω (cid:54) = 0). This is consistent with the results displayed through the black symbols in the toppanel of Fig. 14 and in Fig. 16.In Fig. 18 we study (5.26) for a block on the semi-infinite line, separated by d sites from theorigin where DBC are imposed. For a given value of ω L , we show C GGE ,A as function of ωL atfixed d/L (top panel) and viceversa (bottom panels). The qualitative behaviour of the curvesin the top panel of Fig. 18 is similar to the one in the right panel of Fig. 17. In the bottomleft panel of Fig. 18, as d/L → ∞ , the data points with ωL > C GGE ,A obtained through (5.26) with the reduced correlators (A.31) forthe target state and (5.29) at t = 0 for the reference state. Instead, when ωL = 0 the data inthe bottom left panel of Fig. 18 do not have a limit as d/L increases. This is consistent withthe divergence of the curves in left panel of Fig. 17 as ωL →
0. In the bottom right panel ofFig. 18 we consider a critical evolution Hamiltonian and large values of ω L . In this regime ofparameters, we highlight the logarithmic growth of C GGE ,A in terms of d/L (the solid lines are44 × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○◇ × ○ ■■ × × × × × × × × × × × × × × × × × × × × × × × × × × △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ × × × × × × × × × × × × × × × × × × × × × × × × × × △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ■■ ◇△ × ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○△ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △△ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △△ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ × × × × × × × × × × × × × × × × × × × × × × × × × ×× × × × × × × × × × × × × × × × × × × × × × × × × ×× × × × × × × × × × × × × × × × × × × × × × × × × × ■■■ × ◇△○ Figure 18: Asymptotic value of C GGE ,A in (5.26) for a block made by L consecutive sites andseparated by d sites from the origin of a semi-infinite line with DBC, in terms of ωL (toppanel) and of d/L (bottom panels).obtained by fitting the data corresponding to L = 40 through the function a log( d/L ) + b ).The numerical data sets discussed in this section are characterised by fixed values of ωL and ω L . In appendix D we report numerical results where ω and ω are kept fixed: besidessupporting further the validity of (5.26), this analysis provides numerical evidences for (5.28).Within the context of the gauge/gravity correspondence, the temporal evolution of the holo-graphic subsystem complexity in the gravitational backgrounds given by Vaidya spacetimeshas been studied numerically through the CV proposal [84–87].We find it worth remarking that the qualitative behaviour of the temporal evolution of C A for an interval in the infinite line shown by the black data points in Fig. 14 and Fig. 16 is inagreement with the results for the temporal evolution of the holographic subsystem complexityreported in [84, 85]. The change of regime occurs at t/L (cid:39) / < t/L < / t/L > / ω L in Fig. 14and Fig. 16, while the holographic subsystem complexity remains constant. However, a similarissue occurs in the corresponding comparison for the entanglement entropy.45 Conclusions
In this manuscript we studied the temporal evolution of the subsystem complexity after aglobal quench of the mass parameter in harmonic lattices, focussing our analysis on harmonicchains with either PBC or DBC and on subsystems given by blocks of consecutive sites. Theinitial state is mainly chosen as the reference state of the circuit. The circuit complexity ofthe mixed states described by the reduced density matrices has been evaluated by employingthe approach based on the Fisher information geometry [74], which provides also the optimalcircuit (see (4.2) and (4.3)).When the entire system is considered (see Sec. 2.2, Sec. 2.3 and Sec. 3), the optimal circuitis made by pure states [17, 18] and for the temporal evolution of the circuit complexity afterthe global quench one obtains the expression given by (2.28) and (2.26), which holds in ageneric number of dimensions. When the reference and the target states are pure states alongthe time evolution of a given quench, we find that the complexity is given by (2.28) and (2.29),which simplifies to (2.33) in the case where the reference state is the initial state. Specialisingthe latter result to the harmonic chains where either PBC or DBC are imposed, one obtains(3.7), where the contribution of the zero mode for PBC is highlighted. The occurrence of thezero mode provides the logarithmic growth of the complexity when the evolution is critical(see (3.15) and Fig. 1). Typical temporal evolutions of the complexity for the entire chainwhen the post-quench Hamiltonian is massive are shown in Fig. 4.The bounds (2.40) and (2.46) are obtained for the temporal evolution of the complexity ofthe entire harmonic lattice. The former ones are simple but not very accurate (see Fig. 2 forharmonic chains with PBC); instead, the latter ones capture the dynamics of the complexityin a very precise way but their analytic expressions are more involved. In the case of har-monic chains, the bounds (2.46) lead to the bounds (3.16) displayed in Fig. 3, which are lessconstraining but easier to deal with.The aim of this manuscript is to investigate the temporal evolution of the subsystem com-plexity C A after a global quench (see Sec. 4 and Sec. 5).For a gapless evolution Hamiltonian, our main results are shown in Fig. 6, Fig. 7, Fig. 8 andFig. 12 for finite chains and in Fig. 14, Fig. 15, and Fig. 16 for infinite chains. In some cases,also the temporal evolutions for the corresponding increment of the entanglement entropy∆ S A are reported, in order to highlight the similar features and the main differences. Thiscomparison allows to observe that the initial growths of C A and ∆ S A are very different, whilethe behaviours in the saturation regime are similar, as highlighted in Fig. 7, Fig. 8 and Fig. 15,where also the temporal evolutions of the ratio ∆ S A / C A are shown. An important differencebetween the temporal evolution of C A and of ∆ S A is that C A displays a local maximum beforethe saturation regime (within a revival for finite systems), as discussed in Sec. 4 and Sec. 5.Interestingly, within the framework of the gauge/gravity correspondence, this feature hasbeen observed also in the temporal evolution of holographic subsystem complexity in Vaidyagravitational backgrounds [84, 85].Some temporal evolutions of C A determined by gapped evolution Hamiltonians have beenreported in Fig. 9 and Fig. 10. However, a more systematic analysis is needed to explore their46haracteristic features.For the infinite harmonic chains that we have considered the asymptotic regime is describedby a GGE; hence in Sec. 5 we have argued that the asymptotic value of the temporal evolutionof C A is given by (5.26). This result has been checked both for ω = 0 (see Fig. 14, Fig. 16 andFig. 19) and for ω > Acknowledgments
We are grateful to Leonardo Banchi, Lucas Hackl, Mihail Mintchev, Nadir Samos S´aenz deBuruaga and Luca Tagliacozzo for useful discussions. ET’s work has been conducted withinthe framework of the Trieste Institute for Theoretical Quantum Technologies.47
Covariance matrix after a global quantum quench
In this appendix we discuss further the covariance matrices after the global quench employedin Sec. 2 and Sec. 3. The explicit expressions for the correlators of the GGE that have beenused in Sec. 5 for some numerical computations are also provided.
A.1 Covariance matrix
The matrix H phys defined in (2.1), which characterises the Hamiltonian of the model, reads H phys = Q phys ⊕ P phys (A.1)where P phys = m and Q phys is a N × N real, symmetric and positive definite matrix whoseexplicit expression is not needed for the subsequent discussion.Denoting by (cid:101) V the real orthogonal matrix diagonalising Q phys (for harmonic chains withPBC the matrix (cid:101) V is given in (3.2) and (3.3)), one notices that (A.1) can be diagonalised asfollows H phys = V (cid:20) m diag (cid:0) ( m Ω ) , . . . , ( m Ω N ) , , . . . , (cid:1)(cid:21) V t V ≡ (cid:101) V ⊕ (cid:101) V (A.2)where m Ω k are the real eigenvalues of Q phys . Since (cid:101) V is orthogonal, the 2 N × N matrix V is symplectic and orthogonal. The r.h.s. of (A.2) can be written as H phys = V X phys (cid:104) diag (cid:0) Ω , . . . , Ω N , Ω , . . . , Ω N (cid:1)(cid:105) X phys V t (A.3)where we have introduced the following symplectic and diagonal matrix X phys = diag (cid:16) ( m Ω ) / , . . . , ( m Ω N ) / , ( m Ω ) − / , . . . , ( m Ω N ) − / (cid:17) ≡ S phys ⊕ S − phys . (A.4)From (A.3), the Williamson’s decomposition [96] of the matrix H phys reads H phys = W t phys D phys W phys (A.5)where D phys = diag (cid:0) Ω , . . . , Ω N , Ω , . . . , Ω N (cid:1) W phys = X phys V t . (A.6)The decomposition (A.5) leads to write the Hamiltonian (2.1) in terms of the canonicalvariables defined through W phys as follows (cid:98) H = 12 ˆ s t D phys ˆ s ˆ s ≡ W phys ˆ r ≡ (cid:18) ˆ q ˆ p (cid:19) . (A.7)Following the standard quantisation procedure, the annihilation operators ˆ b k and the cre-ation operators ˆ b † k areˆ b ≡ (cid:0) ˆ b , . . . , ˆ b N , ˆ b † , . . . , ˆ b † N (cid:1) t ≡ Θ − ˆ s ˆ b k ≡ ˆ q k + i ˆ p k √ ≡ √ (cid:18) − i i (cid:19) (A.8)48hich satisfy [ˆ b i , ˆ b j ] = J ij , where J is the standard symplectic matrix J ≡ (cid:18) − (cid:19) (A.9)whose blocks are given by the N × N identity matrix and the matrix filled by zeros. Interms of these operators, the Hamiltonian (A.7) reads (cid:98) H = N (cid:88) k =1 Ω k (cid:18) ˆ b † k ˆ b k + 12 (cid:19) . (A.10)Thus, the symplectic spectrum D phys in (A.6) provides the dispersion relation Ω k , that dependsboth on the dimensionality of the lattice and on the boundary conditions.By applying the above procedure to the Hamiltonian (cid:98) H whose ground state | ψ (cid:105) is theinitial state, one finds (cid:98) H = N (cid:88) k =1 Ω ,k (cid:18) ˆ b † ,k ˆ b ,k + 12 (cid:19) (A.11)where Ω ,k is the dispersion relation of (cid:98) H .To evaluate (2.2) and (2.3), from (A.2), (A.4), (A.6) and (A.7) one obtains (2.5), namely Q ( t ) = (cid:101) V S − phys (cid:104) ψ | e i (cid:98) Ht ˆ q (0) ˆ q t (0) e − i (cid:98) Ht | ψ (cid:105) S − phys (cid:101) V t ≡ (cid:101) V Q ( t ) (cid:101) V t (A.12) P ( t ) = (cid:101) V S phys (cid:104) ψ | e i (cid:98) Ht ˆ p (0) ˆ p t (0) e − i (cid:98) Ht | ψ (cid:105) S phys (cid:101) V t ≡ (cid:101) V P ( t ) (cid:101) V t (A.13) M ( t ) = (cid:101) V S − phys Re (cid:2) (cid:104) ψ | e i (cid:98) Ht ˆ q (0) ˆ p t (0) e − i (cid:98) Ht | ψ (cid:105) (cid:3) S phys (cid:101) V t ≡ (cid:101) V M ( t ) (cid:101) V t . (A.14)In order to find the correlators of the operators ˆ q (0) and ˆ p (0), one first employs (A.8) toexpress all the operators in terms of the creation and annihilation operators. Then, since theinitial state | ψ (cid:105) is annihilated by the operators ˆ b ,k and ˆ b † ,k introduced in (A.11), we have toexpress ˆ b k and ˆ b † k in terms of ˆ b ,k and ˆ b † ,k , as done in [43]. This leads to write the diagonalmatrices Q ( t ), P ( t ) and M ( t ), whose non vanishing elements are given by (2.6). A.2 Complexity through the matrix W TR The Williamson’s decomposition [96] is an important tool to study the circuit complexity ofbosonic Gaussian states [73, 74]. When the reference and the target states are pure states, boththe optimal circuit and the corresponding complexity can be evaluated through the symplecticmatrix W TR ≡ W T W − R , where W R and W T occur in the Williamson’s decomposition of thereference and of the target states respectively [18, 48, 74].In the following we construct the Williamson’s decomposition of the covariance matrix (2.4)after the global quantum quench, that describes a pure state.By using (2.17), we first observe that the block matrix in (2.13) can be decomposed asΓ( t ) = T ( t ) t (cid:18) P ( t ) − P ( t ) (cid:19) T ( t ) T ( t ) ≡ (cid:18) P ( t ) − M ( t ) (cid:19) (A.15)49here the triangular matrix T ( t ) is symplectic and not orthogonal. Then, the symplecticspectrum of the diagonal matrix in (A.15) can be obtained as discussed e.g. in the appendixD of [74], finding (cid:18) P ( t ) − P ( t ) (cid:19) = 12 X ( t ) (A.16)where the symplectic and diagonal matrix X ( t ) can be defined in terms of P k ( t ) in (2.6) as X ( t ) = diag (cid:18) √ P , . . . , √ P N , (cid:112) P , . . . , (cid:112) P N (cid:19) . (A.17)Plugging (A.16) into (A.15), one finds the Williamson’s decomposition of the covariancematrix (2.4) γ ( t ) = 12 W ( t ) t W ( t ) W ( t ) = X ( t ) T ( t ) V t (A.18)which tells us also that all the symplectic eigenvalues of γ ( t ) are equal to 1 /
2, as expected forpure states.By using this decomposition for both the reference and the target states, with the samematrix V (see (2.19)), we find that W TR ≡ W T W − R becomes W TR = X T T T T − R X − R . (A.19)For the sake of simplicity, let us focus on the complexity w.r.t. the initial state, which isalso the case mainly explored throughout this manuscript (hence t R = 0 and t T = t ).From (A.15), it is straightforward to check that T ( t ) − = (cid:18) −P − ( t ) M ( t ) (cid:19) . (A.20)Then, since M R = when t R = 0, using (A.17) we obtain W TR = (cid:32) (cid:112) P − T P R (cid:112) P − T P R M T (cid:112) P − R P T (cid:33) (A.21)which gives W t TR W TR = (cid:18) P − T P R (cid:0) + 4 M T (cid:1) M T M T P − R P T (cid:19) (A.22)whose eigenvalues provide the circuit complexity. Indeed, by employing the Williamson’sdecomposition (A.18) for the covariance matrices of the reference and of the target states intothe expression (2.8), one finds that it can be written as follows C = 12 √ (cid:114) Tr (cid:110)(cid:2) log (cid:0) W t TR W TR (cid:1)(cid:3) (cid:111) . (A.23)Since the matrix (A.22) is a special case of (2.22), its eigenvalues can be found by applying(2.23). The resulting spectrum is given by the pairs (cid:0) χ TR (cid:1) k and (cid:0) χ TR (cid:1) − k , labelled by 1 (cid:54) k (cid:54) , with (cid:0) χ TR (cid:1) k = P T ,k + P R ,k (cid:0) M T ,k (cid:1) + (cid:113)(cid:2) P T ,k + P R ,k (cid:0) M T ,k (cid:1)(cid:3) − P T ,k P R ,k P T ,k P R ,k (A.24)= 12 (cid:20) Q T ,k Q R ,k + P T ,k P R ,k (cid:21) + (cid:115) (cid:20) Q T ,k Q R ,k + P T ,k P R ,k (cid:21) − k wehave 1 + 4 M T ,k = 4 Q T ,k P T ,k and 1 = 4 Q R ,k P R ,k .Comparing (A.25) with (2.25) and (2.30), we conclude that, for any 1 (cid:54) k (cid:54) N , we have (cid:0) χ TR (cid:1) k = g (+) TR ,k . (A.26)Thus, the complexity (2.28) can be written in terms of (cid:0) χ TR (cid:1) k . Since this result is expectedfor the circuit complexity of bosonic Gaussian pure states, (A.26) provides a non-trivial con-sistency check of the entire procedure. Furthermore, by extending this analysis to the case t R (cid:54) = 0 in the straightforward way, (A.26) is recovered.In the space of covariance matrices and after a proper change of basis, the optimal circuitmade by pure states that connects the reference state to the target state reads [18] G s ( γ R , γ T ) = 12 (cid:0) X TR (cid:1) s (A.27)where the symplectic diagonal matrix X TR is defined as follows X TR ≡ diag (cid:110)(cid:0) χ TR (cid:1) , . . . (cid:0) χ TR (cid:1) N , (cid:0) χ TR (cid:1) − , . . . , (cid:0) χ TR (cid:1) − N (cid:111) (A.28)in terms of the eigenvalues of the matrix (A.22), given in (A.25). A.3 GGE correlators
In the following we report the explicit expressions of the correlators for the harmonic chains inthe GGE state which have been employed to construct the reduced covariance matrix γ GGE ,A from the covariance matrix γ GGE defined in (5.4). The matrix γ GGE ,A occurs in the expression(5.26) for the complexity C GGE ,A .The harmonic chains where either PBC or DBC are imposed must be treated separately.For PBC, by using (3.2) or (3.3), (5.9) and (5.13) into (5.6) and (5.7), we obtainTr (cid:0) ˆ q i ˆ q j ˆ ρ GGE (cid:1) = 1 N N (cid:88) k =1 Ω ,k + Ω k k Ω ,k cos (cid:2) ( i − j ) 2 πk/N (cid:3) (A.29)Tr (cid:0) ˆ p i ˆ p j ˆ ρ GGE (cid:1) = 1 N N (cid:88) k =1 Ω ,k + Ω k ,k cos (cid:2) ( i − j ) 2 πk/N (cid:3) (A.30)in terms of the dispersion relations (3.4). Notice that the correlators (A.29) diverge when ω = 0; hence for PBC the massless limit must be studied by taking ω very small, but non51anishing. In the thermodynamic limit N → ∞ , these correlators become respectively (cid:90) π Ω ,θ + Ω θ Ω θ Ω ,θ cos (cid:2) θ ( i − j ) (cid:3) dθ π (cid:90) π Ω ,θ + Ω θ Ω ,θ cos (cid:2) θ ( i − j ) (cid:3) dθ π (A.31)where the dispersion relations are given in (3.24).When DBC are imposed, by using the matrix (cid:101) V in (3.5), we obtainTr (cid:0) ˆ q i ˆ q j ˆ ρ GGE (cid:1) = 2 N N − (cid:88) k =1 Ω ,k + Ω k k Ω ,k sin (cid:0) πki/N (cid:1) sin (cid:0) πkj/N (cid:1) (A.32)Tr (cid:0) ˆ p i ˆ p j ˆ ρ GGE (cid:1) = 2 N N − (cid:88) k =1 Ω ,k + Ω k ,k sin (cid:0) πki/N (cid:1) sin (cid:0) πkj/N (cid:1) (A.33)in terms of the dispersion relations (3.6). Notice that, in this case, all these correlators arefinite when ω = 0. The thermodynamic limit N → ∞ of these correlators gives respectively (cid:90) π Ω ,θ + Ω θ Ω θ Ω ,θ sin( iθ ) sin( jθ ) dθ π (cid:90) π Ω ,θ + Ω θ Ω ,θ sin( iθ ) sin( jθ ) dθ π (A.34)where (3.25) must be employed.These correlators have been used to construct γ GGE ,A , that occurs in C GGE ,A defined in(5.26). In particular, the expressions (A.31) have been exploited to draw the horizontal linesin the left panels of Fig. 19, Fig. 20 and Fig. 21 and in the bottom left panel of Fig. 18. Theyhave also provided the data points in the left panels of Fig. 17 and Fig. 22. Instead, thehorizontal lines in Fig. 14, Fig. 16 and in the right panels of Fig. 19, Fig. 20 and Fig. 21 havebeen obtained through the correlators (A.34), which have provided also the data points inFig. 18 and in the right panels of Fig. 17 and of Fig. 22.As consistency check of the fact that the GGE describes the limit t → ∞ after the globalquench, one observes that the correlators in (A.31) and (A.34) are recovered by taking (5.29)and (5.30) respectively and replacing all the oscillatory functions with their averages (i.e.[sin(Ω θ t )] and [cos(Ω θ t )] by 1 / θ t ) by 0). B Complexity w.r.t. the unentangled product state
In this appendix we consider the temporal evolution of the complexity between the tar-get state defined as the state at time t after the quench, characterised by the parameters( κ T , m T , ω T , ω , T ) ≡ ( κ, m, ω, ω ), and the reference state defined as the state at t = 0, whenthe system is prepared in the unentangled product state, characterised by the parameters( κ R , m R , ω R ) ≡ (0 , m, µ ). This unentangled product state has been largely employed as refer-ence state to explore the circuit complexity [17–19, 73], also in time-dependent settings [47],hence we find it worth providing a brief discussion for the complexity when this state is chosenas reference state.For circuits made by pure states, the complexity is (2.28) with C TR ,k given by (2.31).When the reference state is the unentangled product state, from (3.4) and (3.6) one observes52hat Ω , R ,k = µ for any k , independently of whether PBC or BDC are imposed. Thus, theexpression of C TR ,k simplifies to C TR ,k = 12 µ Ω ,k (cid:32) Ω ,k + µ + (Ω k − Ω ,k )(Ω k − µ )Ω k [sin(Ω k t )] (cid:33) (B.1)where we have defined Ω T ,k ≡ Ω k and Ω , T ,k ≡ Ω ,k to enlighten the expression. Isolating thezero mode contribution, as done also in Sec. 2.3.1, the complexity (2.28) reads C = η (cid:26) arccosh (cid:20) ω + µ µ ω + ( ω − ω )( ω − µ )2 µ ω ω [sin( ωt )] (cid:21)(cid:27) (B.2)+ 14 N − (cid:88) k =1 (cid:26) arccosh (cid:20) Ω ,k + µ µ Ω ,k + ( ω − ω )(Ω k − µ )2 µ Ω ,k Ω k [sin(Ω k t )] (cid:21)(cid:27) where η has been introduced in (3.7) and the dispersion relations (3.4) or (3.6) must beemployed, depending respectively on whether PBC or DBC hold.When ω = 0, the zero mode term having k = N in (B.1) simplifies to C TR ,N = ω + µ µ ω + µ ω t (B.3)which is divergent when t → ∞ , while C TR ,k with k (cid:54) = N is bounded for any value of t . Thus,for the critical evolution the complexity (B.2) becomes C = η (cid:20) arccosh (cid:18) ω + µ µ ω + µ ω t (cid:19)(cid:21) (B.4)+ 14 N − (cid:88) k =1 (cid:26) arccosh (cid:20) Ω ,k + µ µ Ω ,k + ω ( µ − Ω k )2 µ Ω ,k Ω k [sin(Ω k t )] (cid:21)(cid:27) . We remark that, when PBC are imposed (hence η = 1), the complexity (B.4) diverges loga-rithmically as t → ∞ because of the occurrence of the zero mode contribution. This featureis observed also when the reference state is the initial state (see Sec. 2.3.1).When the reference state is the unentangled product state, the complexity is non vanishingat t = 0. In particular, from (B.2) we obtain C (cid:12)(cid:12) t =0 = η (cid:20) arccosh (cid:18) ω + µ µ ω (cid:19)(cid:21) + 14 N − (cid:88) k =1 (cid:20) arccosh (cid:18) Ω ,k + µ µ Ω ,k (cid:19)(cid:21) (B.5)Specialising this expression to PBC and DBC, one recovers the results found in [17] and [22]respectively. The expression (B.5) provides the leading term in the expansion of (B.2) as t →
0, that reads C = (cid:0) C| t =0 (cid:1) + ( ω − ω )2 (cid:34) N − η (cid:88) k = 1 Ω k − µ (cid:12)(cid:12) Ω ,k − µ (cid:12)(cid:12) arccosh (cid:32) Ω ,k + µ µ Ω ,k (cid:33)(cid:35) t + O ( t ) . (B.6)From this expansion it is straightforward to realise that C − C| t =0 = O ( t ) as t →
0, wherethe sign of the r.h.s. is not well defined. This quadratic behaviour in t as t → C| t =0 is non vanishing. Furthermore, the sign of the O ( t )term in (B.6) determines whether the complexity increases or decreases with respect to itsinitial value during the early time regime. A similar feature has been observed also in thetemporal evolution of the complexity considered in [47]. C Technical details about some limiting regimes
In this appendix we report some technical details about the large N regimes discussed inSec. 3.4 for the temporal evolution of the complexity of the entire harmonic chain. C.1 Approximation for small kN at finite N In the following we provide some details about the derivation of the expressions given by(3.20) and (3.21) for the complexity and by (4.8) and (4.9) for the slope of its linear initialgrowth, obtained in the approximation introduced at the beginning of Sec. 3.4.When DBC hold and therefore the dispersion relations (3.6) are employed, the argumentof the sum in (3.7) is a function of kN whose main contribution comes from the regime where kN (cid:28)
1. This suggests to introduce the approximation [sin (cid:0) πk N (cid:1) ] (cid:39) (cid:0) πk N (cid:1) in (3.6), whichleads to the approximate expression for (3.7) given in (3.21), which depends only on ωN , ω N and t/N . The argument of the sum in (3.21) decreases very rapidly as k increases; henceincreasing N does not change significantly the value of C approx .When PBC hold, in the expression (3.10) for the complexity let us observe that c definedin (3.8) can be written as the following function of ωN , ω N and t/Nc ( t ) = (cid:20) arcsinh (cid:18) ( ωN ) − ( ω N ) ωN ) ( ω N ) sin (cid:18) ωN tN (cid:19)(cid:19)(cid:21) (C.1)without any approximation. If we restrict 1 (cid:54) k (cid:54) N/
2, the argument of the sum in (3.10)is non vanishing when k/N (cid:28)
1. This suggests to approximate [sin (cid:0) πkN (cid:1) ] (cid:39) (cid:0) πkN (cid:1) in (3.4)which leads (3.10) to become the following function of ωN , ω N and t/N C approx = (cid:118)(cid:117)(cid:117)(cid:117)(cid:116) c ( t ) + 2 [ N − ] (cid:88) k =1 (cid:34) arcsinh (cid:32) ( ωN ) − ( ω N ) (cid:101) Ω (P) k (cid:101) Ω (P) ,k sin (cid:18)(cid:101) Ω (P) k tN (cid:19)(cid:33)(cid:35) + c approx N/ ( t ) (C.2)where c approx N/ ( t ) ≡ (cid:34) arcsinh (cid:32) ( ωN ) − ( ω N ) (cid:101) Ω (P) N/ (cid:101) Ω (P) ,N/ sin (cid:18)(cid:101) Ω (P) N/ tN (cid:19)(cid:33)(cid:35) even N N (C.3)and (cid:101) Ω (P) is defined in (3.22). The expression (C.2) does not grow with N because the terms ofthe sum in (C.2) become negligible from a certain value of k . Let us observe that, consistentlywith this approximation, the term c approx N/ ( t ) in (C.2) can be neglected and (3.20) is obtained.54n this approximation N is kept finite, both for PBC and DBC, as long as it is large enough.The initial growth within this approximation can be obtained by applying the steps dis-cussed above (3.29), finding C approx = t N (cid:118)(cid:117)(cid:117)(cid:116) N − (cid:88) k =1 (cid:2) ( ωN ) − ( ω N ) (cid:3) ( ω N ) + π k κ/m + O ( t ) DBC (C.4) C approx = t N (cid:118)(cid:117)(cid:117)(cid:117)(cid:116) (cid:2) ( ωN ) − ( ω N ) (cid:3) ( ω N ) + 2 [ N − ] (cid:88) k =1 (cid:2) ( ωN ) − ( ω N ) (cid:3) ( ω N ) + 4 π k κ/m + O ( t ) PBC . (C.5)Since the arguments of the sums in (C.4) and (C.5) are negligible from a certain value of k ,we are allowed to extend the sums up to infinite. Then, using (cid:80) ∞ k =1 1 k + a = aπ coth( aπ ) − a , wefinally get C approx = a (B) t/N + . . . for (C.4) and (C.5) with B ∈ { P , D } , where the dots representhigher orders in t/N and the slopes a (P) and a (D) are given in (4.8) and (4.9) respectively. C.2 Thermodynamic limit
In order to study the thermodynamic limit of the complexity discussed in Sec. 3, let us recallsome basic facts about the Euler-Maclaurin formula.The Euler-Maclaurin formula quantifies the discrepancy between the sum S = (cid:80) bn = a +1 f ( n )and the integral I = (cid:82) ba f ( x ) dx . It reads [98] S − I = f ( b ) − f ( a )2 + p (cid:88) j =1 B j (2 j )! (cid:2) f (2 j − ( b ) − f (2 j − ( a ) (cid:3) + R p +1 (C.6)where B j are the Bernoulli numbers, f ( j ) ≡ ∂ jx f and the remainder R p +1 ≡ − p + 1)! (cid:90) ba P p +1 ( x ) f (2 p +1) ( x ) dx (C.7)where P k ( x ) = B k ( x − (cid:98) x (cid:99) ) are expressed in terms of the Bernoulli polynomials B k ( x ). Thereminder R p +1 is bounded as follows | R p +1 | < e π (2 π ) p +1 (cid:90) ba | f (2 p +1) ( x ) | dx . (C.8)Let us consider the cases where p = 0 in (C.6) and (C.8), which leads to S − I = f ( b ) − f ( a )2 + R | R | < e π π (cid:90) ba | f (cid:48) ( x ) | dx . (C.9)By applying (C.9) for S = C and the extrema a = 0 and b = N − η (where η = 1 forPBC and η = 0 for DBC), for the complexity (3.7) we find I = (cid:90) N − η f B ( k ) dk ≡ I ( B ) N B ∈ (cid:8) P , D (cid:9) (C.10)55here, for PBC and DBC, we have respectively f P ( k ) ≡ (cid:20) arcsinh (cid:18) ω − ω k Ω ,k sin(Ω k t ) (cid:19)(cid:21) f D ( k ) ≡ f P ( k/ . (C.11)Since in f P ( k ) the dependence on k occurs only through (cid:112) κ/m sin( πk/N ), we find it con-venient to introduce F t ( y ) ≡ (cid:20) arcsinh (cid:18) ω − ω (cid:112) ω + y (cid:112) ω + y sin (cid:0)(cid:112) ω + y t (cid:1)(cid:19)(cid:21) (C.12)which leads to write (C.11) as f P ( k ) = F t ( s k ) s k ≡ (cid:112) κ/m sin( πk/N ) . (C.13)By introducing the integration variable θ = πk/N in (C.10) and taking N → ∞ , theexpression (3.23) is obtained independently of the boundary conditions.The remainder (C.7) for p = 0, which depends on the boundary conditions, is denoted by R (P) ,N for PBC and by R (D) ,N for DBC, where we have indicated explicitely the dependence onthe size N of the chain. In order to investigate the behaviour of R (B) ,N , with B ∈ (cid:8) P , D (cid:9) , atlarge N , we approximate its expression through its bound given in (C.9). Thus, for S = C ,we find that (C.9) becomes S − I ( B ) N = C − I ( B ) N = f B ( N − η ) − f B (0)2 + R (B) ,N B ∈ (cid:8) P , D (cid:9) . (C.14)When PBC are imposed, the expression (C.14) becomes C − I ( P ) N = f P ( N ) − f P (0)2 + R (P) ,N = R (P) ,N . (C.15)In order to estimate R (P) ,N we consider its bound in (C.9) and therefore we have R (P) ,N = 2 e π π (cid:90) N (cid:12)(cid:12)(cid:12)(cid:12) df P ( k ) dk (cid:12)(cid:12)(cid:12)(cid:12) dk = 2 e π N (cid:114) κm (cid:90) N (cid:12)(cid:12) F (cid:48) t ( s k ) cos( πk/N ) (cid:12)(cid:12) dk (C.16)where (C.13) has been used. By introducing θ = πk/N , one can take N → ∞ , obtaining R (P) , ∞ ≡ lim N →∞ R (P) ,N = 2 e π π (cid:114) κm (cid:90) π (cid:12)(cid:12) F (cid:48) t (cid:0)(cid:112) κ/m sin θ (cid:1) cos θ (cid:12)(cid:12) dθ (C.17)where F (cid:48) t can be computed from (C.12). This calculation provides a complicated expression inthe integrand of (C.17), hence we evaluate R (P) , ∞ numerically. Since I ( P ) N → C TD when N → ∞ ,in this limit (C.15) gives (3.26) with B = P.In the case of DBC, the expression in (C.14) becomes C − I ( D ) N = f D ( N − − f D (0)2 + R (D) ,N = f P (( N − / − f P (0)2 + R (D) ,N (C.18)56here in the last step we have emplyed the relation between f D and f P . Using (C.11), thelimit N → ∞ of the first term in the r.h.s. of (C.18) gives ζ ≡ lim N →∞ f P (( N − / − f P (0)2 (C.19)= 12 (cid:26)(cid:20) arcsinh (cid:18) ( ω − ω ) sin (cid:0)(cid:112) ω + 4 κ/m t (cid:1) (cid:112) ω + 4 κ/m (cid:112) ω + 4 κ/m (cid:19)(cid:21) − (cid:20) arcsinh (cid:18) ω − ω ω ω sin( ωt ) (cid:19)(cid:21) (cid:27) . Now we estimate R (D) ,N by approximating it through its bound in (C.9). From the relationbetween f D and f P and (C.13), we get R (D) ,N = 2 e π π (cid:90) N − | f (cid:48) D ( k ) | dk = 2 e π N (cid:114) κm (cid:90) N − (cid:12)(cid:12)(cid:12)(cid:12) cos (cid:18) πk N (cid:19) F (cid:48) t (cid:18)(cid:114) κm sin (cid:18) πk N (cid:19)(cid:19)(cid:12)(cid:12)(cid:12)(cid:12) dk . (C.20)Changing the integration variable to θ = πk N and taking the limit N → ∞ , we get R (D) , ∞ ≡ lim N →∞ R (D) ,N = 2 e π π (cid:114) κm (cid:90) π/ (cid:12)(cid:12)(cid:12)(cid:12) F (cid:48) t (cid:18)(cid:114) κm sin θ (cid:19) cos θ (cid:12)(cid:12)(cid:12)(cid:12) dθ = R (P) , ∞ θ → π − θ .Thus, (C.18) becomes (3.26) with B = D as N → ∞ , given that I ( D ) N → C TD . When ω = 0,from (C.19) we get ζ → − (log t ) as t → ∞ . Since C is finite for any value of time when ω = 0 and DBC are imposed, from (C.18) we have C TD + R (D) , ∞ → (log t ) for t → ∞ . We arenot able to identify the asymptotic behaviour of C TD and R (D) , ∞ separately.As for the initial growth, transforming the sum in (3.29) into an integral as shown inSec. 3.4, we can write an explicit expression for the slope of the initial growth. The same limitcan be done for the higher order terms in (2.36), obtaining C TD √ N = t | ω − ω | (cid:113) ω (cid:0) κm + ω (cid:1) (cid:20) − t B + O ( t ) (cid:21) (C.22)where B ≡ (cid:113) ω (cid:0) κm + ω (cid:1)(cid:0) ω κm + 4 ω (cid:1) − ( ω − ω ) (cid:0) ω (3 ω + ω ) + 2 κm (7 ω + ω ) (cid:1) ω (cid:0) κm + ω (cid:1) . (C.23)Let us stress that the formula (C.22) for the initial growth does not distinguish betweenPBC or DBC, differently from (C.4) and (C.5).The thermodynamic limit discussed above can be easily applied also to the case discussedin appendix B, where the reference state is the unentangled product state. From (B.2), atleading order in N we obtain C TD = N π (cid:90) π (cid:26) arccosh (cid:20) Ω ,θ + µ µ Ω ,θ + ( ω − ω )(Ω θ − µ )2 µ Ω ,θ Ω θ [sin(Ω θ t )] (cid:21)(cid:27) dθ (C.24)where the dispersion relations are given by (3.24).57 .3 Continuum limit In this appendix we report some details on the continuum limit procedure that leads to (3.27)which is valid for both PBC and DBC.Starting from PBC, we can exploit the identity sin( x ) = sin( π − x ) to rewrite the complexity(3.7) as follows C = N/ (cid:88) k = − N/ (cid:20) arcsinh (cid:18) ω − ω k Ω ,k sin(Ω k t ) (cid:19)(cid:21) even N (C.25)and C = ( N − / (cid:88) k = − ( N − / (cid:20) arcsinh (cid:18) ω − ω k Ω ,k sin(Ω k t ) (cid:19)(cid:21) odd N . (C.26)In the continuum limit N → ∞ and the lattice spacing a ≡ (cid:112) m/κ → N a ≡ (cid:96) is keptfixed. In this limit the dispersion relation (3.4) becomes (cid:112) ω + (2 πk/(cid:96) ) = (cid:112) ω + p = Ω p ,where Ω p has been defined in (3.28) and p ≡ kπaN ∈ R , because of the range of k in (C.25) and(C.26). The resulting dispersion relation identifies the frequency of the harmonic chain withthe mass of the underlying continuum field theory, which is the Klein-Gordon field theory.Replacing the sum over the integers k with the integral (cid:96) (cid:82) ∞−∞ dp π over the momenta p weobtain (3.27) at leading order in (cid:96) .When DBC are imposed we cannot exploit the identity for sin x mentioned above. In thiscase, we first observe that the dispersion relation (3.6) in this limit becomes (cid:112) ω + ( πk/(cid:96) ) = (cid:112) ω + p = Ω p , with Ω p given by (3.28) and p = πkaN ∈ ( π/(cid:96), ∞ ) because k (cid:62) ,p = (cid:112) ω + p .At leading order in (cid:96) , we have that π/(cid:96) vanishes and therefore p ∈ [0 , ∞ ). By substitutingthe sum over k in (3.7) with (cid:96) (cid:82) ∞ dpπ , we get C cont = (cid:114) (cid:96)π (cid:115)(cid:90) ∞ (cid:20) arcsinh (cid:18) ω − ω p Ω ,p sin(Ω p t ) (cid:19)(cid:21) dp . (C.27)The expression (3.27) is easily recovered by using that the integrand in (C.27) is even in p .This procedure can be applied also to study the continuum limit of (B.2) where the referencestate is the unentangled product state. For both PBC and DBC, for the leading term we find C cont = (cid:96) π (cid:90) ∞−∞ (cid:26) arccosh (cid:20) Ω ,p + µ µ Ω ,p + ( ω − ω )(Ω p − µ )2 µ Ω ,p Ω p [sin(Ω p t )] (cid:21)(cid:27) dp (C.28)where the dispersion relations are given by (3.28). This result does not coincide with the onereported in [47] for the temporal evolution of the complexity because of the different choiceof gates. The role of the set of allowed gates in the determination of the temporal evolutionof the complexity deserves further future analyses.Let us remark that, while the expression (3.27), obtained by choosing the initial state asreference state, is UV finite, (C.28) is UV divergent. This UV divergence can be regularised58 × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○ × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○ × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○ × ◇■■■ × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○ × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○ × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○ × ○◇■■■ Figure 19: Temporal evolution of C A after a global quantum quench with a gapless evolutionHamiltonian for a block A made by L consecutive sites adjacent to a boundary of harmonicchains with DBC made by N sites. The data corresponding to N → ∞ are obtained througha chain on the semi-infinite line. The horizontal dashed grey lines correspond to (5.26).by introducing a cutoff | p | (cid:54) Λ on the momenta. Alternatively, since the UV divergence comesfrom C cont | t =0 , it is natural to introduce the following UV finite quantity∆ C cont = C cont − C cont (cid:12)(cid:12) t =0 (C.29)whose sign is not definite for t > t = 0, when the result of[17] is recovered, and they both display a UV divergence that can be regularised as done in(C.29). For t >
0, after an initial growth both the expressions show persistent oscillations butthey do not coincide. For instance, while the initial growth of the result of [47] is linear, thenext term after the constant in the expansion of (C.28) as t → D Further numerical results on the relaxation to the GGE
In this appendix we report further numerical results supporting (5.26) and (5.28).In Fig. 19, Fig. 20. Fig. 21 and Fig. 23 we show some temporal evolutions of C A for a block A made by L consecutive sites in harmonic chains with N sites where N is either finite orinfinite, with the aim to check that (5.26) provides the correct asymptotic value as t → ∞ .Each set of data corresponds to a choice of N , L , ω and ω . The data represented bycoloured markers have been found through (4.6), with the reduced correlators obtained ei-ther from (4.4) or from (4.5) when N is finite (for PBC and DBC respectively) and eitherfrom (5.29) or from (5.30) when N → ∞ (on the infinite line and on the semi-infinite linerespectively). The horizontal dashed lines show the subregion complexity between the initialstate and the GGE given by (5.26), obtained by reducing the correlators (A.31) and (A.34)59 × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○ × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○ × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○ × ○◇■■■ × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○ × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○ × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○ × ○◇◇■■■ × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○ × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○ × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○ ×× ◇◇○○ × ○◇■■■ × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○ × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○ × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○ ×× ◇◇○○ × ○◇■■■ Figure 20: Temporal evolution of C A after a global quantum quench with a gapped evolutionHamiltonian. In the left panels the chains are either on the circle or on the infinite line, whilein the right panels the chains are either on the segment or on the semi-infinite line with DBCand A is adjacent to a boundary. The dashed grey lines correspond to (5.26).for the target state and the correlators (5.29) and (5.30) at t = 0 for the reference state (forthe infinite line and the semi-infinite line respectively).In Fig. 19 we consider the temporal evolution of C A when ω = 0, DBC are imposed andthe block is adjacent to a boundary. The data obtained for finite N are compared againstthe ones for N → ∞ , found for a block at the beginning of the semi-infinite chain. Themain feature to highlight in these temporal evolutions are the plateaux occurring both forfinite N and for N → ∞ . The horizontal dashed lines in Fig. 19 correspond to the subregioncomplexity (5.26). These agreements support the assumption that the target state relaxesto an asymptotic state locally described by the GGE in (5.2) as t → ∞ . For a given set ofparameters, the height of the plateaux is independent of N , while it increases as either L or ω increases. Comparing the two panels of Fig. 19, where different values of ω are considered,one observes that the local maxima occur (when N → ∞ there is only the first one) for largeenough ω . We also highlight the absence of oscillations in the formation of the plateaux whenthe evolution Hamiltonian is gapless. 60 × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ × ○◇■■■ × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ × ○◇■■■ Figure 21: Temporal evolution of C A after a global quantum quench with a gapped evolutionHamiltonian for a block made by L consecutive sites in harmonic chains either with PBC oron the infinite line. The dashed grey lines correspond to (5.26).In Fig. 20 and Fig. 21 the evolution Hamiltonians are gapped with ω = 0 .
05. We show dataobtained for harmonic chains either with PBC or on the infinite line in the left panels andfor harmonic chains either with DBC or on the semi-infinite line (with the block adjacentto a boundary) in the right panels. In these evolutions, data corresponding to the same ω and ω collapse for t < N/ t < N in the right panels. The maindifference with respect to the gapless evolutions in Fig. 19 are the oscillations after the initialgrowth around the asymptotic value, which is evaluated through (5.26) and corresponds tothe horizontal dashed grey lines, whose height depends on ω and ω . Fig. 21 highlights thefact that, for harmonic chains with PBC or on the infinite line, very long time is needed toreach the asymptotic value given by (5.26). Comparing the two panels in Fig. 21, one noticesthat the amplitude of the oscillations decreases as | ω − ω | increases.The numerical results in Fig. 22 provide the main outcome of this appendix. Since (5.24)tells us that C GGE / √ N is finite as N → ∞ (see the left panel of Fig. 13) let us consider C GGE ,A / √ L in the limit L → ∞ , where C GGE ,A is given by (5.26). The data reported inFig. 22 for this quantity support the validity of the last equality in (5.28). The coloured datapoints have been obtained from (5.26), by employing the reduced correlators from (A.31)and (A.34) for the target state and the reduced correlators from (5.29) e (5.30) at t = 0 forthe reference state (like in Fig. 17 and Fig. 18). The horizontal dashed lines represent theasymptotic values obtained from (5.24), which depend only on ω and ω . Comparing the twopanels in Fig. 22, one realises that larger L ’s are needed to reach the asymptotic value whenthe evolution Hamiltonian is gapless. Considering the red data points in the left panel ofFig. 22, notice that the asymptotic value (5.24) is symmetric under the exchange ω ↔ ω , asalready remarked in the text above (5.24), while the sets of data points converging to it donot display this symmetry.In Fig. 23 we consider harmonic chains on the infinite line and gapless evolution Hamil-61 × × × × × × × × × ×× × × × × × × × × × × × × × × × × × ×× × × × × × × × × × × × × × × × × × × × × × × × × × × × × ×× × × × × × × × × × × × × × × × × × × × × × × × × × × × × ×× × × × × × × × × × × × × × × × × × × × × × × × × × × × × × ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ×× ×× ○ × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × ×× × × ×× × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × ×× × × × × × × × × × × × × × × × × × × × × × × × × × × × × × ×× × × × × × × × × × × × × × × × × × × ×× × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × ×× × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × ××× ××× Figure 22: Asymptotic value of C GGE ,A in (5.26) for a block made by L consecutive sites whichis either in the infinite chain (left panel) or adjacent to the origin of a semi-infinite chain withDBC (right panel). The horizontal dashed lines show C GGE / √ N as N → ∞ , from (5.24).These results support the last equality in (5.28).tonians. In particular, we study the temporal evolutions of C A / √ L (from (4.6)), of ∆ S A /L and of the ratio ∆ S A / ( √ L C A ) in terms of t/L , for various L ’s and two values of ω . Thereduced covariance matrices have been obtained from the correlators (5.29). The growths of C A and of ∆ S A from t/L (cid:39) t/L (cid:39)
25 have been fitted through the function a log( t/L ) + b (coloured solid lines in Fig. 23), finding that the coefficient of the logarithmic term is positiveand decreases as L increases. In the top panels of Fig. 23, after the initial growth, C A reachesa local maximum, then it decreases until t/L (cid:39) / ω increases, ishighlighted in the insets. It would be interesting to explore higher value of L in order to checkwhether, in the limit of L → ∞ , a saturation is observed to the value given by (5.24), whichprovides the horizontal dashed lines in the top panels of Fig. 23. The horizontal dashed linesin the middle panels are obtained from (5.16). In the bottom panels of Fig. 23, we show thetemporal evolutions of the ratio ∆ S A / ( √ L C A ), which exhibit a mild logarithmic decreasingfor large values of t/L . This tells us that the logarithmic growths of C A and ∆ S A are verysimilar. However, the values of L are not large enough to determine whether the numericaldata points asymptote to a constant value as t/L → ∞ .62 ○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○ ○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○ ○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○ ○○○○○ ○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○ ○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○ ○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○ ○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○ ○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○ ○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○ ○○○○○ ○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○ ○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○ ○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○ ○○○○○ ○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○ ○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○ ○○○○○ Figure 23: Temporal evolutions after a global quantum quench with a gapless evolutionHamiltonian and either ω = 0 . ω = 0 . 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