Origin of the slow growth of entanglement entropy in long-range interacting spin systems

Long-range interactions allow far-distance quantum correlations to build up very fast. Nevertheless, numerical simulations demonstrated a dramatic slowdown of entanglement entropy growth after a sudden quench. In this work, we unveil the general mechanism underlying this counterintuitive phenomenon for $d$-dimensional quantum spin systems with slowly-decaying interactions. We demonstrate that the semiclassical rate of collective spin squeezing governs the dynamics of entanglement, leading to a universal logarithmic growth in the absence of semiclassical chaos. In fact, the standard quasiparticle contribution is shown to get suppressed as the interaction range is sufficiently increased. All our analytical results agree with numerical computations for quantum Ising chains with long-range couplings. Our findings thus identify a qualitative change in the entanglement production induced by long-range interactions, and are experimentally relevant for accessing entanglement in highly-controllable platforms, including trapped ions, atomic condensates and cavity-QED systems.

In this work, we identify the qualitative change in the mechanism which governs the growth of entanglement in quantum spin systems as the interaction range is increased.By means of a systematic bosonization of spin excitations, we show that the standard quasiparticle contribution to the von Neumann entanglement entropy S(t) is suppressed for α ≤ d in a prethermal regime.The growth of S(t) is determined by collective spin excitations, directly related to spin squeezing [92][93][94][95][96][97][98] for spin-1/2 systems, as schematically illustrated in Fig. 1.We demonstrate how a slow dynamical rate of spin squeezing after a quench generically leads to a universal logarithmic growth of S(t) as reported in previous numerical studies.The theory further predicts fast entanglement growth for critical quenches.Analytical results are supported by numerical simulations of long-range quantum Ising chains via exact diagonalization (ED) and the matrix-product-state timedependent variational principle (MPS-TDVP) [99,100].< l a t e x i t s h a 1 _ b a s e 6 4 = " W p Q 3 J 7 H q / 4 7 t i I 5 H L p a e v L H F K V I = " > A A A B 9 X i c b V C 7 T s N A E D y H V w i v A C X N i Q i J K r I R E j R I E T S U Q Z C H l F j R + b I J p 5 z P 1 t 0 a F F n 5 B F q o 6 B A t 3 0 P B v 3 A 2 L i B h q t H M r n Z 2 g l g K g 6 7  x e n E f n 1 X l z 3 n 9 G S 0 6 x s 0 / + w P n 4 B n V G k f 4 = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " W p Q 3 J 7 H q / 4 7 t i I 5 H L p a e v L H F K V I = " > A A A B 9 X i c b V C 7 T s N A E D y H V w i v A C X N i Q i J K r I R E j R I E T S U Q Z C H l F j R + b I J p 5 z P 1 t 0 a F F n 5 B F q o 6 B A t 3 0 P B v 3 A 2 L i B h q t H M r n Z 2 g l g K g 6 7  x e n E f n 1 X l z 3 n 9 G S 0 6 x s 0 / + w P n 4 B n V G k f 4 = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " W p Q 3 J 7 H q / 4 7 t i I 5 H L p a e v L H F K V I = " > A A A B 9 X i c b V C 7 T s N A E D y H V w i v A C X N i Q i J K r I R E j R I E T S U Q Z C H l F j R + b I J p 5 z P 1 t 0 a F F n 5 B F q o 6 B A t 3 0 P B v 3 A 2 L i B h q t H M r n Z 2 g l g K g 6 7  x e n E f n 1 X l z 3 n 9 G S 0 6 x s 0 / + w P n 4 B n V G k f 4 = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " W p Q 3 J 7 H q / 4 7 t i I 5 H L p a e v L H F K V I = " > A A A B 9 X i c b V C 7 T s N A E D y H V w i v A C X N i Q i J K r I R E j R I E T S U Q Z C H l F j R + b I J p 5 z P 1 t 0 a F F n 5 B F q o 6 B A t 3 0 P B v 3 A 2 L i B h q t H M r n Z 2 g l g K g 6 7 3: Linear growth in time of the half-system entanglement entropy SN/2 at the dynamical critical point.We compare our general formula (??) with the exact numerical computation for increasing system sizes N = 50÷ 400.Before the Ehrenfest time tEhr ∼ log N , numerical data for SN/2 are accurately reproduced by the analytical result (??) marked by the dotted line with a slope λh c = J.This linear regime is followed by saturation to a value ∼ log N .
1.2 From a paramagnetic initial condition < l a t e x i t s h a 1 _ b a s e 6 4 = " s 6 B w P W T q H x O 7 r 4 8 6 x n e v T a v 2 i a K Z E 9 s k B q R G X n J E 6 u S I N 0 i S c 3 J E n 8 k x e r E f r 1 X q z 3 n 9 G F 6 x i Z 4 / 8 g f X x D R Z Y l k g = < / l a t e x i t > Entanglement entropy in infinite-range spin systems.-Inorder to isolate the contribution of collective degrees of freedom to entanglement dynamics, we start by considering spin-1/2 models with full permutational symmetry.In this limiting situation, the only dynamical spin excitations are collective, i.e., uniform, and quasiparticles at all possible non-vanishing momenta are completely suppressed.This is realized with arbitrary all-to-all multi-body interactions, described by a Hamiltonian of the form where ŝi , i = 1, . . ., N are quantum spins-1/2 (or qubits).
The rescaling factor 1/N p−1 ensures that the energy contribution of all p-body interactions is extensive.These Hamiltonians can be written in terms of the collective spin of the system Ŝ = N i=1 ŝi only, whose magnitude |S| = S(S + 1) is extensive and conserved, and generically maximal (S = N/2) in the ground state [101], leading to an effective eff ∼ /N [102].The behavior of the system (in and out-of-equilibrium) can thus be understood via systematic semiclassical expansions in quantum fluctuations around the average collective spin [103].[104] We aim at understanding entanglement dynamics in spin systems described by Eq. ( 1).For a composite system with Hilbert space H = H A ⊗ H B in a pure state ρ = |ψ ψ|, the entanglement between subsystems A and B can be quantified by the Von Neumann entropy S A = − Tr ρA log ρA of the reduced density matrix ρA = Tr B ρ.In this light, we consider a bipartition of the system into two subsystems A and B with N A = f A N and N B = N − N A = f B N spins, respectively.The collective spin Ŝ can be correspondingly decomposed as Ŝ = ŜA + ŜB (see Fig. 1).We extend to the nonequilibrium setting the two-boson formalism developed in Refs.105 and 106 (see also Ref. [107]) for equilibrium problems.Preparing the system in a fully polarized state, both the instantaneous collective spin configuration Ŝ(t) and its quantum fluctuations evolve in time.This can be described by letting the Z-axis follow the evolution of Ŝ(t) [108,109].Quantum fluctuations of the two spins ŜA,B are bosonized via Holstein-Primakoff transformations around their instantaneous average polarization, and the entanglement between A and B is encoded in the entanglement between these two bosons (q A , pA ) and (q B , pB ).The resulting bosonic Hamiltonian governs the dynamics of the transverse quantum fluctuations ( Q, P ) of the global spin Ŝ around its direction Z(t).The latter encodes the bosonic excitations of ŜA , ŜB via the relations Material [103] for the details.)We may now safely perform a quadratic approximation, as the spin fluctuations are subextensive in ground states [110] as well as out of equilibrium until the so-called Ehrenfest time scale -which diverges with N , see below.The time-evolution of quantum fluctuations ( Q, P ) is then formally equal to the classical evolution of displacements away from the trajectory of the initial state under consideration.
Within this formalism, the Von Neumann entanglement entropy between the two subsystems A and B is computed with standard Gaussian bosonic techniques [103,111,112], obtaining where nexc = ( Q2 + P 2 − 1)/2 represents the number of bosonic excitations of the collective spin Ŝ.The function on the right-hand side vanishes for nexc → 0 and grows as 1 2 log nexc for nexc 1.Hence, equation ( 2) clarifies that the state of subsystem A (or B) is pure only if nexc = 0, i.e., if the spin system is fully polarized (coherent), as occurs in the absence of interactions.Conversely, the state is entangled in the presence of collective spin excitations.This comes via spin squeezing, a concept first introduced in Ref. 113 and usually quantified by the minimal transverse variance of collective spin fluctuations [114,115] as ξ 2 ≡ Min |u|=1,u⊥Z u • Ŝ 2 /(N/4).It is equal to 1 for coherent states, and smaller for squeezed states (see, e.g., Refs. 113 and 115).At the level of Gaussian fluctuations relevant here, one derives that [103, 112] It has long been known [93,94,98] that collective spin squeezing is a witness of many-body quantum entanglement.Equations ( 2) and (3) express the quantitative link -pictorially illustrated in Fig. 1 -between the entanglement entropy S A in collective spin models in and out of equilibrium and the spin-squeezing parameter ξ.
The growth of entanglement entropy out of equilibrium is completely determined by the dynamical generation of the collective excitations nexc (t) , which is in turn induced by the interactions.This rate depends on the nature of the underlying classical dynamics.In fact, the time-dependent quadratic Hamiltonian that governs the quantum evolution of fluctuations ( Q, P ), describes as well the classical evolution of small displacements away from it in the linear approximation [116], see Fig. 2(a) for an illustration.In classical integrable systems, trajectories originating from nearby initial conditions separate linearly in time, which implies Q(t), P (t) ∼ t.Since nexc is quadratic in the spin fluctuations Q, P , the collective excitations grow polynomially in time nexc (t) ∼ t 2 after a generic quench [102].This leads to by Eq. ( 2).A more rigorous analysis [103] further predicts periodic oscillations of S A (t) superimposed to the h l h y E s G J q + y n Z P r z F 7 5 d J q 1 b 1 L L 8 5 q 9 Q v 5 8 0 U 2 R E 7 Z q f M Y + e s z q 5 Z g z W Z Y J o 9 s W f 2 4 j w 6 r 8 6 b 8 / 4 T L T j z m U P 2 B 8 7 H N / p T l s k = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " Z 0 h l h y E s G J q + y n Z P r z F 7 5 d J q 1 b 1 L L 8 5 q 9 Q v 5 8 0 U 2 R E 7 Z q f M Y + e s z q 5 Z g z W Z Y J o 9 s W f 2 4 j w 6 r 8 6 b 8 / 4 T L T j z m U P 2 B 8 7 H N / p T l s k = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " Z 0 h l h y E s G J q + y n Z P r z F 7 5 d J q 1 b 1 L L 8 5 q 9 Q v 5 8 0 U 2 R E 7 Z q f M Y + e s z q 5 Z g z W Z Y J o 9 s W f 2 4 j w 6 r 8 6 b 8 / 4 T L T j z m U P 2 B 8 7 H N / p T l s k = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " Z 0 x p j t b 7 A e 8 t 0 9 I n 6 L P < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = "     logarithmic growth, the period being that of the underlying classical trajectory.In the (non-generic) case of quenches to dynamical critical points [102], the collective spin lies on unstable trajectories (separatrices) in the classical phase space, around which displacements grow exponentially in time, with a rate set by the positive eigenvalue λ of the corresponding unstable fixed point on which they terminate, see Fig. 2(b).The out-of-equilibrium generation of collective excitations is thus exponentially fast, nexc (t) ∼ e 2λt , leading to a linear growth of entanglement entropy with a predicted slope S(t) ∼ λt and without superimposed oscillations.(See also the related discussion in Refs.[117][118][119].)In all cases, entanglement entropy saturates S A ∼ log N A at the Ehrenfest time scale defined by nexc (t Ehr ) ∼ N .Spatially-decaying interactions.-Weare now in a position to understand the effects of having slowly-decaying interactions on entanglement dynamics.For the sake of definiteness, we focus on periodic d-dimensional lattices of spins with arbitrary two-body long-range interactions, described by a Hamiltonian as in Eq. ( 1) with p ≤ 2, where now j = 1, . . ., N = L d label lattice sites at positions denoted r j , and the uniform couplings J µν , with µ, ν = x, y, z, are replaced by J µν /|r i − r j | α [120].The exponent α ≥ 0 characterizes the algebraic decay of spinspin interactions.A Kac rescaling factor 1/N α,N with N α,N = i =j |r i − r j | −α /N replaces the 1/N factor in Eq. ( 1), ensuring the extensivity of the Hamiltonian for α ≤ d [121].The fully-connected limit is recovered by letting α → 0.
When interactions decay with the distance between spins, the full permutational symmetry of the infiniterange Hamiltonian (1) is broken and the finite-wavelength spin modes participate in the dynamics.These excitations now allow the system to explore the full Hilbert space beyond the Dicke manifold, i.e., "inside the Bloch sphere", and the system may be expected to thermalize by accumulating extensive entanglement entropy [5].However, we demonstrate that these quasiparticles are weakly excited in typical quenches, and hence yield only a bounded contribution to entanglement growth for a long temporal regime.In this case, the analysis of dynamical spin squeezing captures the leading behavior of S(t) even for 0 < α < d.
The two-boson approach to entanglement dynamics described above was strictly based on the conservation of the collective spin magnitude.In order to treat a truly many-body problem, one can refine the approach as follows.We rewrite the Hamiltonian in terms of the Fourier spin modes sx,y,z where is proportional to the Fourier transform of the interactions, J's are coefficients depending on the collective spin trajectory, and the collective-mode Hamiltonian H 0 (t) accounts for the infinite-range part f α,0 δ k,0 ≡ δ k,0 of the interaction f α,k .
H 0 (t) is independent of α and describes the dynamics of collective spin fluctuations Q ≡ q0 and P ≡ p0 as detailed above, and conserves the bosonic occupation numbers nk =0 ≡ (q k q−k + pk p−k − 1)/2 of all the spin-wave modes with finite wavelength [123].The dynamical excitation of spin waves with finite wavelengths for α > 0 is responsible for modifications to the spin-squeezing-induced entanglement entropy growth.As is evident in Eq. ( 5), their impact is controlled by the strength of the finite-range part f α,k =0 of the interaction.In fact, the following estimate can be derived for α (|k|L) β , with β ≡ Min d − α, (d + 1)/2 (for α = d the power law is replaced by a logarithm).This bound implies that for all fixed k = 0, the coupling f α,k is vanishingly small in thermodynamic limit L → ∞ whenever α ≤ d, and hence the associated number of bosonic excitations is an approximate constant of motion, Therefore, there exists a long time scale T sw ∼ N β/d , during which the dynamical excitation of spin waves with finite wavelengths is suppressed [124] (note the interesting relation to the prethermalization time in Ref. [125]).On the other hand, permutational symmetry may severely break over large length scales via excitations with |k| ∝ 1/L.Their dynamics governed by the Hamiltonian (5), is equivalent to a discrete set of periodically driven quantum oscillators, the drive being induced by the precession of the collective spin.From a stability analysis, we find that for typical quenches these long-wavelength spin-wave modes are non-resonantly driven and hence weakly excited.Their resulting contribution to entanglement dynamics amounts to bounded oscillations on top of the dominant, spin-squeezing induced logarithmic growth.Near dynamical criticality, however, resonant excitation of these modes may lead to exponentially growing quantum fluctuations (cf.Ref. [125]) and hence linear increase of the entanglement entropy (see the Supplemental Material [103] for details).We thus conclude that long-range interacting spin-1/2 systems with α < d typically exhibit logarithmic growth of entanglement entropy.Numerical simulations.-Wetest our analytical understanding in paradigmatic one-dimensional long-range quantum Ising chains, described by the Hamiltonian where i, j = 1, . . ., N , σx,z i are Pauli matrices, h is a global transverse magnetic field and N α,N is the Kac rescaling factor introduced above.
We compare the numerical computations of entanglement entropy evolution at finite N with the analytical calculation of the spin-squeezing contribution [Eq.( 2)] and with the full spin-wave calculation, obtained from Eq. ( 5) via standard bosonic techniques [103,111,112,119].For the sake of illustration, we focus here on the initial state |ψ 0 = |→ → • • • → , i.e., on quenches in the transverse field from h 0 = 0 to h f .As Figs. 1 and 3 show, in all cases the numerical data are captured by the corresponding analytical curves for t t Ehr (N ).In the fully-connected limit α → 0, equivalent to the Lipkin-Meshov-Glick model [126], Eq. ( 2) is exact in the thermodynamic limit and the finite-size ED data perfectly match it before saturation ations in Eq. ( 30) evolve according to 8 > < > : with ✓ = ✓(t) and = (t) determined by Eq. ( 48).These equations are exact in the limit N ! 1, while finite-size correction occur over the Ehrenfest time scale  46It corresponds to G QP (t = 0) = 0 Eqs.( 48), (49).the Hamiltonian the critical dyna trate, in all case converges onto eral formula for h c , the entangle S A s log t before dynamical critic of the collective as S A s hc t be Hamiltonian, th the unstable fix ations in Eq. ( 30) evolve according to 8 > < > : with ✓ = ✓(t) and = (t) determined by Eq. ( 48).These equations are exact in the limit N ! 1, while finite-size correction occur over the Ehrenfest time scale  ations in Eq. ( 30) evolve according to 8 > < > : with ✓ = ✓(t) and = (t) determined by Eq. ( 48).These equations are exact in the limit N ! 1, while finite-size correction occur over the Ehrenfest time scale  ations in Eq. ( 30) evolve according to 8 > < > : with ✓ = ✓(t) and = (t) determined by Eq. ( 48).These equations are exact in the limit N !Comparison between theory and numerical results for long-range quantum Ising chains.Quenches in the transverse field h0 = 0 → h f > 0 are considered, and the evolution of the half-system entanglement entropy S N/2 (t) is shown.Top: Fully-connected limit with α = 0. Analytical results (black lines) are compared with ED data for increasing system sizes N = 20 ÷ 800.(a.)For a non-critical quench h0 = 0 → h f = 0.2J, the growth of S N/2 (t) is logarithmic up to saturation around 1/2 log N at t Ehr ∼ √ N (b.)For the critical quench h f = hc = J/2, the growth of S N/2 (t) is linear until t Ehr ∼ log N , with a slope λ hc = J.Bottom: Deep quench with h f = 2J in long-range interacting chains with α > 0. The contribution due to collective spin squeezing [Eq.(2)] and the full spin-wave calculation of the time-dependent entanglement (see the main text) are compared with MPS-TDVP data for N = 20 ÷ 80 converged with bond dimension D = 128, for α = 0.1 (c.) and α = 0.7 (d).As α increases, finite corrections due to long-wavelength spin waves appear on top of the dominant spin-squeezing-induced logarithmic growth, see the inset.
at the Ehrenfest time, t Ehr ∼ √ N for generic quenches [h 0 = 0 → h f = 2J in Fig. 1 (bottom) and h f = 0.2J in Fig. 3(a)] and t Ehr ∼ log N for the critical quench [h f = h c ≡ J/2 in Fig. 3(b), cf. the red line in Fig. 2(b)], corresponding to the dynamical phase transition of the model [35,102,127,128].For spatially-decaying interactions with 0 < α < 1, we employ the MPS-TDVP [99,100] with periodic boundary conditions (see the Supplemental Material [103] for details).Upon increasing N , the TDVP data approach the full spin-wave entanglement entropy, for all considered values of α and quench parameters, as shown in the examples in Fig. 3(c),(d).This analysis confirms that the growth of S(t) is logarithmic for typical initial configurations.For further discussion, including varying initial states, bipartition sizes and details on the spin-wave analysis, see the Supplemental Material [103].
Outlook.-The mechanism unveiled in this work complements the available paradigms for entanglement dynamics characterizing systems with local interactions [2][3][4][12][13][14][15]] and, at the same time, improves the current understanding of the efficiency of "classical" simulations of quantum long-range interacting spin chains with matrixproduct-state techniques [99,100].Similar approaches can be used to characterize entanglement entropy dynamics in more general many-body systems [102], including higher-spin systems, whose semiclassical dynamics may display chaos and hence fast entanglement growth for generic quenches.In connection with known instances of ergodicity breaking stemming from long-range interactions [31,44,45,76,77,89,90,125,[128][129][130][131], the suppression of spin-wave excitations clarifies the role of long-range interactions in constraining quantum dynamics to small portions of the many-body Hilbert space in the relevant time regime.Concerning the slow entanglement growth, this is reminiscent of localized or glassy dynamics [18][19][20][21][22].
The bridge between bipartite entanglement entropy and other measures of multi-particle entanglement, most notably spin squeezing [92-98, 132, 133], may help to elucidate the relation between bipartite and multipartite entanglement, making the latter an equivalent metrologically useful measure for systems with long-range interactions.Most importantly, this connection has direct experimental relevance for the detection of entanglement and its dynamics via measurements of collective quantities [134,135], the experimental accessibility of which is well-established with standard techniques and tools of quantum atomic experiments, ranging from spinor condensates and cavity-QED systems to trapped ions [133,136,137].
In particular, When the system is driven out of equilibrium by varying in time some parameter in the Hamiltonian, one has that both the direction of the collective spin configuration θ(t), φ(t) and the collective spin excitations around it G QQ (t), G QP (t), G P P (t) evolve in time.Following Refs. 4 and 5, the motion of the angles θ(t), φ(t) can be accounted for by letting the rotated frame (X, Y, Z) in Eq. ( 3) change in time in such a way that the Z-axis self-consistently follows the evolution of Ŝ(t) .The modified Hamiltonian in this time-dependent frame includes the inertial forces due to the motion of the frame, and reads with ω X = − sin θ φ, ω Y = θ, and ω Z = cos θ φ.The evolution of θ(t) and φ(t) is determined by the vanishing of the linear terms in ŜX and ŜY , and this yields the classical trajectory governed by H cl .
The resulting time-dependent quadratic part of Ĥ(t), denoted h (2) (t) and given by h QQ,P P,sw (t) ≡ h QQ,P P,sw θ(t), φ(t) − cos θ(t) φ(t), h [cf.Eq. ( 8)] determines the dynamical generation of collective bosonic excitations ( Q, P ), which can be monitored thorugh the correlation functions G QQ (t), G QP (t), G P P (t).In order to compute them, one starts from the Heisenberg equations of motion Denoting the solution and collecting the dynamical correlations in the matrix the number of dynamically generated excitations can be expressed as Note that det G(t) ≡ 1/4, which is an exact property of pure Gaussian states [cf.Eq. ( 13)] preserved by the evolution Eq. ( 17).The two-boson formalism outlined in this Section relies on the truncation of the Holstein-Primakoff transformation (2), which is accurate for Gaussian states with a small number of collective excitations nexc N compared to the system size.This assumption is generically valid for ground states, even at quantum critical points 6,7 , as well as in the non-equilibrium setting up to a time scale which diverges with the system size (see below).

II. ENTANGLEMENT ENTROPY, COLLECTIVE EXCITATIONS AND SPIN SQUEEZING
For a given stationary or time-evolving state of the system, we can compute the entanglement entropy between the two subsystems A and B. This amounts to computing the entanglement entropy between the two bosons (q A , pA ) and (q B , pB ), corresponding to spin excitations localized in A and B respectively.The reduced density matrix of subsystem A is a Gaussian state of the boson (q A , pA ) completely determined by the correlation matrix The Von Neumann entropy of a single boson (q A , pA ) in such a Gaussian state can be expressed in terms of the determinant of G A as 8 On the other hand, the matrix G A can be easily related to the correlation matrix G of collective excitations (Q, P ) in the system by inverting Eqs. ( 5)-( 6).The explicit computation shows that its determinant amounts to Equations ( 23) and ( 24) complemented by the evolution in time of nexc connect entanglement entropy dynamics to the growth of the quantum fluctuations of the collective spin, whose determination represents the main result of this Section.Taking the limits of small or large nexc in Eqs. ( 23), ( 24), one finds Equations ( 23) and ( 24) are valid both in and out of equilibrium.They highlight that the reduced state of subsystems A and B is pure (i.e., det G A = 1/4) only in fully polarized states, in which no collective spin excitations are present in the system, i.e., nexc = 0.
The above equations may be seen as a direct relation between bipartite entanglement entropy and collective spin squeezing, usually quantified by the minimal transverse variance of collective spin fluctuations 9-11 This squeezing parameter ξ is equal to 1 for fully polarized states, while ξ < 1 for squeezed states.The number of collective excitations nexc can be taken as a measure of collective spin squeezing for general Gaussian states: from Eqs. ( 13) and ( 14) one derives ξ as This relation shows that the amount of collective excitations nexc increases from 0 to ∼ N , as the collective spin state is squeezed from a fully polarized configuration with ξ = 1 towards massively squeezed configurations with ξ ∼ 1/ √ N .Besides the number of collective excitations nexc and the spin-squeezing parameter ξ, it is also possible to characterize entanglement entropy via yet another significant quantity, i.e., the effective temperature of the two subsystems.In fact, the reduced density matrices may be written as ρA,B = Z −1 A,B exp(−β eff ĤA,B ), where the state-dependent quadratic operators ĤA,B are usually termed modular or entanglement Hamiltonian.It is straightforward to derive a relation between the effective dimensionless inverse temperature and the other quantities, e.g., This equation makes it explicitly clear that the growth of nexc (t) -which comes via collective spin squeezingis responsible for "heating up" the two subsystems, i.e., for raising their effective temperature, thus continuously accumulating entanglement entropy.

III. NONEQUILIBRIUM GROWTH OF COLLECTIVE EXCITATIONS
The polynomial growth in time of nexc (t) after a typical quench is tightly related to the well-known fact that in classical systems with a single degree of freedom, trajectories originating from nearby initial conditions separate linearly in time.In fact, as discussed above, the dynamics of a fully-connected spin system of the form in Eq. ( 1) of the Letter can be rigorously described in terms of quantum fluctuations G QQ (t), G QP (t), G P P (t) around the classical trajectory of a single spin on the sphere, parameterized by the angles θ(t), φ(t).The time-dependent quadratic Hamiltonian with coefficients h (2) governs the quantum evolution of (Q, P ) in Eq. ( 17) and describes the classical evolution of small displacements around the trajectory in the linear approximation, as well.Linear-in-time growth of these displacements is encoded by U (t) ∼ t in Eq. ( 18) and hence by G(t) ∼ t 2 in Eq. ( 20), which directly leads to S A (t) ∼ log t by Eqs. ( 23) and ( 24), as claimed.
We now make the above argument precise.A time-independent system with a single degree of freedom is integrable, due to the conservation of energy, and hence canonical action-angle variables (A, ϕ) can be introduced, where the action A is a constant of motion related to the area enclosed by a trajectory in phase space, and the angle ϕ sweeps periodically the range [0, 2π] along the trajectory.In these variables, the (classical and quantum) evolution of the system is similar to that of a free particle, with the solution For a given classical trajectory characterized by a value of the action A cl , the evolution of quantum fluctuations around it, is described by The error term follows from the fact that the variables ( Â, φ) parameterize the rescaled collective spin Ŝ/N , and hence their ground state quantum fluctuations are subextensive, i.e., δ Â(0) 2 ∼ 1/N .The time-dependence of correlations can then be derived from the above solution, This t 2 -growth of quantum fluctuations is analogous to the spreading of wavepackets of free quantum particles.Both ( Q, P ) and (δ Â, δ φ) describe quantum fluctuations of the collective spin, hence they must be related via a linear canonical transformation which depends on the instantaneous classical configuration, (θ, φ) or (A cl , ϕ cl ).For a general closed trajectory, the latter varies periodically in time, with a period T cl = 2π/ω(A cl ).Thus, the correlations G QQ (t), G QP (t) and G P P (t) are obtained from those in Eq. ( 33) by a time-periodic linear transformation.We have proved that the time-dependence of correlations G QQ (t), G QP (t) and G P P (t) generically shows a t 2growth after a quench with a periodic modulation superimposed, the periodicity being that of the underlying classical trajectory, i.e., T cl .From Eqs. (33), we see that corrections to this behavior manifest over the Eherenfest time scale t Ehr ∼ √ N , which diverges in the thermodynamic limit.We emphasize that all the quantities above for a given system can in principle be computed analytically, as the system is Liouville-integrable 12 .
A remarkable exception to the behavior described above is represented by isolated trajectories in phase space known as separatrices, which traverse unstable fixed points and divide the sphere into topologically distinct disconnected regions, and which correspond to a singularity of the action-angle variables.These trajectories have a divergent period and are related to the so-called mean-field dynamical criticality 13 .For quenches to such dynamical critical points, the growth of the number of collective excitations is exponential in time rather than polynomial, due to the exponential separation of classical trajectories originating from points near a separatrix.The rate of such an exponential separation is determined by the positive eigenvalue λ of the linearized flow around the unstable fixed point [see Fig. 2(b) of the Letter, for an illustration].In fact, exponential growth of quantum fluctuations is encoded by U (t) ∼ exp(λt) in Eq. ( 18) and hence by G(t) ∼ exp(2λt) in Eq. (20), which directly leads to S A (t) ∼ λt by Eqs. ( 23) and ( 24), as claimed.The rigorous proof of this is essentially analogous to that presented in the previous above for generic quenches.This effect is similar to what is discussed in Ref. 14 in the context of open quantum systems.

IV. LONG-TIME SATURATION OF ENTANGLEMENT ENTROPY
As remarked above, our approach to the non-equilibrium dynamics is adequate only before the Ehrenfest time scale in correspondence of which the description becomes inaccurate, i.e., nexc (t = t Ehr ) ∼ N : in fact, at such time scale the quantum fluctuations of the collective spin become comparable with the magnitude of the collective spin itself.This time scale diverges with the system size in a way which depends on the nature of the underlying semiclassical trajectories, i.e., t Ehr ∼ √ N for generic quenches and t Ehr ∼ log N for quenches to dynamical critical points, and, accordingly, it sets the limit of validity of semiclassical analyses 15 .At this time scale, the number of excitations reaches its maximal value, implying through Eq. (25b) a saturation value of S A proportional to the logarithm of the number of spins in subsystem A, This is actually related to the usual volume-law scaling of entanglement out of equilibrium.In fact, the stationary states after a quantum quench explore all the allowed Hilbert space, and their entanglement is upper-bounded by S A ≤ log (dimH A ).For generic many-body systems, the dimension of H A is exponentially large with the volume of the subsystem [e.g.dim(H A ) = 2 N A for spins-1/2], causing volume-law scaling.In collective models under consideration here, however, the conservation of the collective spin magnitude | Ŝ| 2 reduces the dimension of the allowed Hilbert space to dim(H A ) = N A + 1.For an illustration with numerical results on the long-range quantum Ising model, see Fig. 3 of the Letter.

V. SPATIALLY-DECAYING INTERACTIONS
We deal with a quantum spin-1/2 system on a d-dimensional cubic lattice of size L governed by a Hamiltonian of the form where the exponent α ≥ 0 characterizes the algebraic decay of spin-spin interactions.The distance |r i − r j | between two sites on the periodic lattice can be taken to be A Kač rescaling factor 1/N α,N with N α,N = i =j |r i − r j | −α /N replaces the 1/N factor in Eq. ( 1) of the Letter, ensuring the extensivity of the Hamiltonian for α ≤ d 16 .The fully-connected limit is recovered by letting α → 0.

A. Nonequilibrium spin-wave expansion
We now refine the two-boson formalism above to make it suitable for many-body problems.We expand the individual spins around the instantaneous direction Z S(t) of the collective spin via Holstein-Primakoff transformations, 4,5 where s = 1/2 and the time-dependent rotated frame (X, Y, Z) parameterized by spherical angles θ, φ was introduced in Eq. ( 3).The rotating-frame Hamiltonian Ĥ(t) = Ĥ − ω(t) • Ŝ [cf.Eq. ( 15)] can then be expanded through Eq. (36) in terms of the spin-wave variables qk = L −d/2 j e −ik•rj qj and pk = L −d/2 j e −ik•rj pj at all possible momenta k.One obtains where J's are coefficients depending on the angles θ(t) and φ(t) and and the collective-mode Hamiltonian accounts for the infinite-range part f α,0 δ k,0 ≡ δ k,0 of the interaction f α,k .The collective-mode Hamiltonian H 0 (t) describes the non-trivial dynamics of collective spin fluctuations Q ≡ q0 and P ≡ p0 , but conserves the bosonic occupation numbers of all the other spin-wave modes at finite wavelength, as nk , Ĥ0 = 0 for all k = 0 (note that this is rigorously true to all orders in the Holstein-Primakoff expansion).
In the above equations, it is assumed that the motion of the angles θ(t) and φ(t) is fixed in such a way that linear terms in the collective quantum fluctuations Q ≡ q0 and P ≡ p0 vanish, which is equivalent to the self-consistency requirement ŜX (t) ≡ ŜY (t) ≡ 0. In this case, the dynamical generation of spin waves can modify the classical trajectory of the collective spin (53), due to the feedback from quantum fluctuations 4,5 .

B. Bounds for fα,k
In view of the above discussion, the corrections due to weak spatial decay of interactions α > 0 to entanglement dynamics associated with collective spin excitations, discussed in Sec.III above, are encoded in the analytic structure of the couplings fα,k in Eq. (37), which are vanishing in the infinite-range model with α = 0. Assuming long-range interactions with α < d, we can safely approximate sums with integrals in their defining expression (38), which captures the leading order exactly.Hence, we can switch to spherical coordinates and integrate over all the angles, obtaining where J ν (x) is the standard Bessel function of order ν.While for small ρ the integral is never singular (due to the assumption of long-range interactions, α < d), for large ρ the integrand is asymptotically oscillating with period 2π/|k| and amplitude decaying as ρ (d−1)/2−α , which yields convergence only for α > (d − 1)/2.In this case, by rescaling |k|ρ = η to obtain a dimensionless integrand and denoting by F (η) its primitive which satisfies F (∞) = 0 and is uniformly bounded, one obtains the asymptotic estimate In the limiting case α = d, one can similarly compute where F is the non-singular (bounded) part of the primitive F .On the other hand, for α ≤ (d − 1)/2, by isolating the purely oscillatory terms and repeatedly integrating by parts, we can obtain an expansion of the primitive of the form with c 1,2,... numerical constants.To sum up, the following estimates have been established, See Fig. 1 for an illustration of the behavior of the function f α,k in d = 1.C. Dynamics of the spin-wave population and entanglement We can now analyze the full contribution of the spin waves to the entanglement entropy growth, which improves on the spin-squeezing-induced entanglement growth expressed by Eq. (23).To this end, we first discuss the evolution of the population of the spin waves after a quench.
Equation (37) shows that, within the linear spin-wave analysis, the system is equivalent to a set of periodically driven quantum harmonic oscillators, labelled by the quasimomentum k.The classical evolution of the collective spin, described by the periodic dynamics of the angles θ(t), φ(t) with a frequency ω cl ≡ 2π/T cl , acts as a drive on these bosonic modes described by the variables (q k (t), pk (t)).The driving frequency ω cl is common to all k's and depends only on the quench, while the driving amplitude depends both on the quench and on k via the coupling strength f α,k .As a consequence of the bounds in Eqs.(45), the driving amplitude is vanishingly small in the thermodynamic limit for all fixed k's when α < d, which implies that the excitation of all finite-wavelength modes is vanishingly weak, so that their effects are generically negligible for large N -and in any case delayed to the divergent time scale T sw ∼ N β/d with β ≡ Min d − α, (d + 1)/2 .(Note that this a posteriori fully validates the Holstein-Primakoff expansion and the absence of corrections to the collective spin trajectory for α < d, see also Refs.4, 5, and 17.)However, long-wavelength modes with |k| ∝ 1/L are driven with a finite intensity.Therefore, in the presence of long-range interactions with α < d, the description of the system dynamics effectively reduces to a discrete set of driven quantum oscillators, corresponding to spin fluctuations with k µ = 0, ±2π/L, ±4π/L, . . ., ±2πn * /L, for µ = 1, . . ., d, where the cutoff n * 1 can be taken independent of L, see Fig. 1, bottom right panel.This is in contrast with systems with shorter-range interactions (i.e., with α > d), in which a continuum of traveling quasiparticles generate linear growth of entanglement entropy through the standard Calabrese-Cardy mechanism 18 .
For a large set of initial conditions, spin waves are non-resonantly driven, and consequently their population remains bounded in time.In this case, the spin-squeezing contribution captures the leading behavior of entanglement entropy, and the contribution of spin waves represents a finite correction to the dominant logarithmic growth characteristic of integrable semiclassical dynamics.However, particular quenches typically near mean-field dynamical critical points, may give rise to a resonant excitation of long-wavelength spin waves, leading to an exponentially growing population thereof.This occurrence is triggered by the well-known mechanism of "parametric resonance" in driven oscillators 12 .This effect is a hallmark of semiclassical chaos induced by the finiteness of the interaction range and is associated with a linear increase of entanglement entropy in time (see below).
A stability analysis of the spin-wave excitations allows one to predict the nature of entanglement growth (logarithmic or linear) for any given quench.It can be performed as follows.For any long-range spin Hamiltonian, one can perform a nonequilibrium spin-wave expansion as explained in Sec.VA above, and compute the time-evolution of each spin-wave mode over one classical period T cl .Hence, one obtains the eigenvalues e ±λ k T cl of the 2 × 2 matrix U k (T cl ).The number λ k , known as the Floquet quasi-frequency (see, e.g., Refs.12 and 19) of the driven oscillator, determines the resonance condition of the driven oscillator.If λ k = iω k is purely imaginary, then the mode is stable and its amplitude remains bounded in time, oscillating at a frequency |ω k |.On the contrary, if λ k is real, the mode in unstable and its amplitude grows exponentially fast in time with a rate |λ k |.Isolated resonances may in principle occur for particular trajectories.It seems to be typically the case (see the examples below) that quenches near dynamical criticality give rise to resonant excitation of spin waves.In other words, the classical separatrix of the mean-field dynamics for α = 0 broadens to a finite layer of instability (chaoticity) for α > 0.
The above discussion allows us to understand the full spin-wave time-dependent entanglement entropy, using known mathematical results for quadratic bosons 20 .Even for α > 0, the entanglement between two subsets of quantum spins is encoded in the entanglement between their respective bosonic fluctuations.For a general system of quadratic bosons, S A (t) can be computed with standard techniques -see, e.g., Refs.[8, 20, and 21].We briefly recall how this computation can be performed for translation-invariant Hamiltonians such as Eq.(37).The time evolution is diagonal in Fourier space, and one integrates N decoupled pairs of equations of motion for (q k (t), pk (t)) to obtain the time-evolved Gaussian state of each Fourier mode, described by the correlations ( Gqq for α, β = q, p.The time-evolved operators qk (t), pk (t) are linearly related to those at time t = 0 via the solution to the Heisenberg equations of motion αk (t) = −i αk (t), Ĥ(t) with the generator Ĥ(t) in Eq. (37).Within the linear spin-wave analysis, the state of a subsystem composed of M < N spins contained in a region A of the lattice is a Gaussian bosonic state determined by the instantaneous correlations between them, which can be expressed in terms of Gαβ k (t) via Fourier antitransform: This set of correlations for i, j ∈ A, collected in a 2M × 2M matrix G A , uniquely identifies the reduced density matrix ρA (t).The von Neumann entropy of this Gaussian bosonic state can be computed as 8,20,21 where ν i are the so-called symplectic eigenvalues of the correlation matrix, defined as follows.From the correlation function in Eq. ( 49), one defines the matrix J = −G Ω, where Ω is the 2N × 2N symplectic unit.The matrix [iJ] A restricted to A can be shown to have pairs of opposite real eigenvalues ±ν i (with ν i > 1 as follows from the Heisenberg relations).The numbers {ν i } i=1,...,M are referred to as the symplectic eigenvalues of G A , and determine the entropy via Eq.( 50).
For long-range interactions with 0 < α < d, the growth of S A (t) turns out to be determined by the stability of the discrete set of long-wavelength excitations, expressed by the Floquet quasi-frequencies λ k with |k| ∝ 1/L: see the above discussion, Fig. 1 and, e.g., Ref. 20.In particular, if all of them are imaginary (i.e., all modes are stable), then S A (t) ∼ log t exhibits a slow growth dominated by the collective spin fluctuations with k = 0. On the other hand, if some of them are real (i.e., some modes are unstable), then S A (t) ∼ h KS t exhibits a fast growth dominated by the unstable quantum fluctuations, inasmuch as h KS = k Re(λ k ) .(The latter quantity coincides with the well-known Kolmogorov-Sinai entropy rate, a standard measure of chaoticity in classical dynamical systems: see, e.g., Ref. 22.) In view of the above discussion of the evolution of the k-resolved spin-wave population after a quench, we conclude that typical quenches in a long-range interacting quantum spin-1/2 system yield a logarithmic growth of the von Neumann entanglement entropy, as argued in the main text.See below for a numerical illustration.

VI. DETAILS ON THE NUMERICAL SIMULATIONS
This Section provides the details of our calculations for the long-range quantum Ising chain, described by the Hamiltonian (35) with d = 1 and with non-vanishing Ising coupling J xx ≡ 2J and transverse field h z ≡ 2h, i.e., where σx,z a.The Lipkin-Meshkov-Glick model.We will first focus on its infinite-range version for α = 0, equivalent to the widely-known Lipkin-Meshkov-Glick model 23 .We apply the general scheme and results found in Secs.I and II to show how the entanglement entropy growth is intimately related to the structure of the semiclassical trajectories.
For large values of the transverse field |h| > J the system is paramagnetic, with a single equilibrium configuration of the spins aligned with the field direction, and the non-equilibrium dynamics consist in a precession around it.A quantum phase transition at h = ±J separates this phase from a ferromagnetic one, with a pair of degenerate ground states with spin orientation in the x-z plane, symmetric with respect to flipping the x axis.The out-of-equilibrium behavior has been widely studied 24-26 and, in the case of a quantum quench of the transverse field h 0 → h f , it is characterized by the phenomenon of dynamical phase transitions (DPTs) 13 .The non-equilibrium trajectories of the system may have paramagnetic or ferromagnetic character depending on the initial state and on the final transverse field h f .The two families are distinguished by the time-averaged magnetization S x (t) being vanishing or not, and are separated by a critical trajectory (separatrix ) with a diverging period, see Fig. 2 of the Letter for an illustration.The ground state entanglement entropy of the LMG model has been studied in Refs. 1 and 27, where it is found to be finite away from the quantum critical point and logarithmically divergent with the system size in correspondence of it.More recently, its growth in time after a quench of the transverse field has been numerically found to be consistent with a logarithmic behavior 28,29 .The non-equilibrium evolution governed by the Hamiltonian (51) with α = 0 has been studied with the dynamical approach of Sec.I in Refs.4, 5, and 30.
The expansion (8) of the Hamiltonian in the rotating frame via Eqs.( 15) and ( 16) in this case reads By setting to zero the linear terms h (1) , the classical equations of motion 31 are obtained while the dynamical correlati ons of collective spin fluctuations in Eq. ( 19) evolve according to with θ = θ(t) and φ = φ(t) determined by Eq. ( 53).These equations are exact for N → ∞, while finite-size effects become manifest at the Ehrenfest time scale t Ehr (N ), which depends on the nature of the semiclassical trajectory.As previously discussed, for generic quenches t Ehr ∼ √ N , while at the DPT, corresponding to the separatrix in the classical phase space, it becomes t Ehr ∼ log N .Equations ( 54) are a set of linear time-dependent differential equations and their numerical integration with the appropriate initial conditions [given by Eq. ( 13) for a general quench], determines the time-evolution of the number of collective excitations nexc (t) in Eq. (21).
The analytical treatment is tested against exact-diagonalization (ED) numerical simulations.Due to the conservation of the collective spin | Ŝ| 2 in infinite-range systems, the time-evolving wavefunction is constrained within the maximal-spin Hilbert space sector | Ŝ| = N/2, whose dimension N + 1 grows only linearly with the system size, which allows one to easily simulate the dynamics of large systems.We compute entanglement following the decomposition in Ref. 27.We first focus on quantum quenches from a ferromagnetic ground state.For the sake of definiteness, we consider one of the two ground states of the LMG Hamiltonian (51) with h 0 = 0, e.g., the one fully polarized along the positive x-axis, It corresponds to the initial conditions θ 0 = π/2, φ 0 = 0, G QP (t = 0) = 0 and G QQ (t = 0) = G P P (t = 0) = 1/2 in Eqs. ( 53), (54).The initial state |ψ 0 is then evolved in the presence of a transverse field h = h f , varying above, below and at the dynamical critical point h c = J/2.As Figs. 1 and 3 of the Letter illustrate, in all the three cases the finite-size numerical result perfectly agree with the analytical result based on our general formula for t t Ehr .For quenches above and below h c , the entanglement entropy increases logarithmically after a transient, i.e., S A ∼ log t, before saturation at t Ehr ∼ √ N , see Fig. 1 of the Letter and Fig. 2 below.In turn, at the dynamical critical point, due to the exponential growth of the collective excitations, the entanglement entropy increases linearly in time, i.e., S A ∼ λ hc t, before saturation at t Ehr ∼ log N .The slope λ hc corresponds to the instability rate of the linearized flow, i.e., the imaginary eigenfrequency of the unstable fixed point.For this Hamiltonian and for the unstable point θ = 0, At finite size N , the entanglement entropy is bounded and thus always saturates to a finite value, as in Eq. (34).For N A = N/2 this corresponds to log √ N , as shown in the inset of Fig. 2(left).Conversely, in Fig. 2(right), we plot the      entanglement entropy dynamics for various possible bipartitions, i.e., various fractions f A of spins in subsystem A, and we compare it with the numerically exact results at fixed N .The latter reproduces the former up to t Ehr , when it saturates around the predicted value 1/2 log N A .Let us now discuss the opposite case of a quantum quench from a paramagnetic initial state, i.e., from which corresponds to the initial conditions θ 0 = 0, G QP (t = 0) = 0 and G QQ (t = 0) = G P P (t = 0) = 1/2 in Eqs. ( 53), (54) (due to the singularity of spherical coordinates, the value of φ 0 is immaterial in this case).For h f < J, the initial state lies on top of the unstable trajectory at θ = 0, hence the collective quantum fluctuations grow exponentially in time.The theory then predicts a linear increase S A ∼ λ h f t of entanglement before t Ehr , with λ h f given in Eq. ( 56), see Fig. 3 (left).Conversely, for h f > J, the quantum fluctuations of the initial state undergo oscillations with period T cl , which leads to a periodic dynamics of entanglement entropy as in Fig. 3 (right), with a simple semiclassical interpretation in terms of periodic squeezing of the collective spin.
b. Numerical results for 0 < α ≤ 1.For spatially-decaying interactions, we simulate the entanglement dynamics adopting the matrix-product-state time-dependent variational principle (MPS-TDVP) 32-34 , a well-suited technique for the study of long-range interacting spin systems 35-37 .We approximate the long-range matrix-product-operator (MPO) with a sum of exponentials, as detailed in Ref. 38, with a relative tolerance of 10 −11 .Then, for fixed system size, we increase the MPS bond dimension D up to convergence in the desired time window, as shown in Fig. 4.
In the main text, the converged results are compared with the α-independent collective spin-squeezing contribution in Eq. ( 23) and with the full (α-dependent) spin-wave entanglement entropy converged with respect to increasing N .3 Reply to the referee

N
< l a t e x i t s h a 1 _ b a s e 6 4 = " S g u 6 7 + n K L d B Z H I b R h K U e A 6 b U x T M = " > A A A B 8 3 i c b V C 7 S g N B F J 3 1 G e M r a m k z G A S r s C u C l k E b K 0 n A P C B Z w u z k J g 6 Z n V 1 m 7 g g h 5 A t s t b I T W z / I w n 9 x d t 1 C E 0 9 1 O O d e 7 r k n S q U w 6 P u f 3 s r q 2 v r G Z m m r v L 2 z u 7 d f O T h s m 8 R q D i 2 e y E R 3 I 2 Z A C g U t F C i h m 2 p g c S S h E 0 1 u M r / z C N q I R N 3 j N I U w Z m M l R o I z d F L z b l C p + j U / B 1 0 m Q U G q p E B j U P n q D x N u Y 1 D I J T O m F / g p h j O m U X A J 8 3 L f G k g Z n 7 A x 9 B x V L A Y T z v K g c 3 p q D c O E p q C p k D Q X 4 f f G j M X G T O P I T c Y M H 8 y i l 4 n / e T 2 L o 6 t w J l R q E R T P D q G Q k B 8 y X A v X A N C h 0 I D I s u R A h a K c a Y Y I W l D G u R O t q 6 T s + g g W v 1 8 m 7 f N a 4 H j z o l q / L p o p k W N y Q s 5 I Q C 5 J n d y S B m k R T o A 8 k W f y 4 l n v 1 X v z 3 n 9 G V 7 x i 5 4 j 8 g f f x D U U V k V c = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " S g u 6 7 + n K L d B Z H I b R h K U e A 6 b U x T M = " > A A A B 8 3 i c b V C 7 S g N B F J 3 1 G e M r a m k z G A S r s C u C l k E b K 0 n A P C B Z w u z k J g 6 Z n V 1 m 7 g g h 5 A t s t b I T W z / I w n 9 x d t 1 C E 0 9 1 O O d e 7 r k n S q U w 6 P u f 3 s r q 2 v r G Z m m r v L 2 z u 7 d f O T h s m 8 R q D i 2 e y E R 3 I 2 Z A C g U t F C i h m 2 p g c S S h E 0 1 u M r / z C N q I R N 3 j N I U w Z m M l R o I z d F L z b l C p + j U / B 1 0 m Q U G q p E B j U P n q D x N u Y 1 D I J T O m F / g p h j O m U X A J 8 3 L f G k g Z n 7 A x 9 B x V L A Y T z v K g c 3 p q D c O E p q C p k D Q X 4 f f G j M X G T O P I T c Y M H 8 y i l 4 n / e T 2 L o 6 t w J l R q E R T P D q G Q k B 8 y X A v X A N C h 0 I D I s u R A h a K c a Y Y I W l D G u R O t q 6 T s + g g W v 1 8 m 7 f N a 4 H j z o l q / L p o p k W N y Q s 5 I Q C 5 J n d y S B m k R T o A 8 k W f y 4 l n v 1 X v z 3 n 9 G V 7 x i 5 4 j 8 g f f x D U U V k V c = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " S g u 6 7 + n K L d B Z H I b R h K U e A 6 b U x T M = " > A A A B 8 3 i c b V C 7 S g N B F J 3 1 G e M r a m k z G A S r s C u C l k E b K 0 n A P C B Z w u z k J g 6 Z n V 1 m 7 g g h 5 A t s t b I T W z / I w n 9 x d t 1 C E 0 9 1 O O d e 7 r k n S q U w 6 P u f 3 s r q 2 v r G Z m m r v L 2 z u 7 d f O T h s m 8 R q D i 2 e y E R 3 I 2 Z A C g U t F C i h m 2 p g c S S h E 0 1 u M r / z C N q I R N 3 j N I U w Z m M l R o I z d F L z b l C p + j U / B 1 0 m Q U G q p E B j U P n q D x N u Y 1 D I J T O m F / g p h j O m U X A J 8 3 L f G k g Z n 7 A x 9 B x V L A Y T z v K g c 3 p q D c O E p q C p k D Q X 4 f f G j M X G T O P I T c Y M H 8 y i l 4 n / e T 2 L o 6 t w J l R q E R T P D q G Q k B 8 y X A v X A N C h 0 I D I s u R A h a K c a Y Y I W l D G u R O t q 6 T s + g g W v 1 8 m 7 f N a 4 H j z o l q / L p o p k W N y Q s 5 I Q C 5 J n d y S B m k R T o A 8 k W f y 4 l n v 1 X v z 3 n 9 G V 7 x i 5 4 j 8 g f f x D U U V k V c = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " S g u 6 7 + n K L d B Z H I b R h K U e A 6 b U x T M = " > A A A B 8 3 i c b V C 7 S g N B F J 3 1 G e M r a m k z G A S r s C u C l k E b K 0 n A P C B Z w u z k J g 6 Z n V 1 m 7 g g h 5 A t s t b I T W z / I w n 9 x d t 1 C E 0 9 1 O O d e 7 r k n S q U w 6 P u f 3 s r q 2 v r G Z m m r v L 2 z u 7 d f O T h s m 8 R q D i 2 e y E R 3 I 2 Z A C g U t F C i h m 2 p g c S S h E 0 1 u M r / z C N q I R N 3 j N I U w Z m M l R o I z d F L z b l C p + j U / B 1 0 m Q U G q p E B j U P n q D x N u Y 1 D I J T O m F / g p h j O m U X A J 8 3 L f G k g Z n 7 A x 9 B x V L A Y T z v K g c 3 p q D c O E p q C p k D Q X 4 f f G j M X G T O P I T c Y M H 8 y i l 4 n / e T 2 L o 6 t w J l R q E R T P D q G Q k B 8 y X A v X A N C h 0 I D I s u R A h a K c a Y Y I W l D G u R O t q 6 T s + g g W v 1 8 m 7 f N a 4 H j z o l q / L p o p k W N y Q s 5 I Q C 5 J n d y S B m k R T o A 8 k W f y 4 l n v 1 X v z 3 n 9 G V 7 x i 5 4 j 8 g f f x D U U V k V c = < / l a t e x i t > t = 0 7 6 Z S W l l d W 1 8 r r l Y 3 N r e 2 d 6 u 5 e 2 0 S J 5 tD i k Y x 0 N 2 A G p F D Q Q o E S u r E G F g Y S O s H k K v M 7 D 6 C N i N Q d T m P w Q z Z W Y i Q 4 Q y v d 4 o U 7 q N b c u p u D L h K v I D V S o D m o f v W H E U 9 C U M g l M 6 b n u T H 6 K d M o u I R Z p Z 8 Y i B m f s D H 0 L F U s B O O n e d Q Z P U o M w 4 j G o K m Q N B f h 9 0 b K Q m O m Y W A n Q 4 b 3 Z t 7 L x P + 8 X o K j c z 8 V K k 4 Q F M 8 O o Z C Q H z J c C 9 s B 0 K H Q g M iy 5 E C F o p x p h g h a U M a 5 F R N b S s X 2 4 c 1 / v 0 j a J 3 X P 8 p v T W u O y a K Z M D s g h O S Y e O S M N c k 2 a p E U 4 G Z M n 8 k 7 6 Z S W l l d W 1 8 r r l Y 3 N r e 2 d 6 u 5 e 2 0 S J 5 tD i k Y x 0 N 2 A G p F D Q Q o E S u r E G F g Y S O s H k K v M 7 D 6 C N i N Q d T m P w Q z Z W Y i Q 4 Q y v d 4 o U 7 q N b c u p u D L h K v I D V S o D m o f v W H E U 9 C U M g l M 6 b n u T H 6 K d M o u I R Z p Z 8 Y i B m f s D H 0 L F U s B O O n e d Q Z P U o M w 4 j G o K m Q N B f h 9 0 b K Q m O m Y W A n Q 4 b 3 Z t 7 L x P + 8 X o K j c z 8 V K k 4 Q F M 8 O o Z C Q H z J c C 9 s B 0 K H Q g M iy 5 E C F o p x p h g h a U M a 5 F R N b S s X 2 4 c 1 / v 0 j a J 3 X P 8 p v T W u O y a K Z M D s g h O S Y e O S M N c k 2 a p E U 4 G Z M n 8 k 7 6 Z S W l l d W 1 8 r r l Y 3 N r e 2 d 6 u 5 e 2 0 S J 5 tD i k Y x 0 N 2 A G p F D Q Q o E S u r E G F g Y S O s H k K v M 7 D 6 C N i N Q d T m P w Q z Z W Y i Q 4 Q y v d 4 o U 7 q N b c u p u D L h K v I D V S o D m o f v W H E U 9 C U M g l M 6 b n u T H 6 K d M o u I R Z p Z 8 Y i B m f s D H 0 L F U s B O O n e d Q Z P U o M w 4 j G o K m Q N B f h 9 0 b K Q m O m Y W A n Q 4 b 3 Z t 7 L x P + 8 X o K j c z 8 V K k 4 Q F M 8 O o Z C Q H z J c C 9 s B 0 K H Q g M iy 5 E C F o p x p h g h a U M a 5 F R N b S s X 2 4 c 1 / v 0 j a J 3 X P 8 p v T W u O y a K Z M D s g h O S Y e O S M N c k 2 a p E U 4 G Z M n 8 k 7 6 Z S W l l d W 1 8 r r l Y 3 N r e 2 d 6 u 5 e 2 0 S J 5 tD i k Y x 0 N 2 A G p F D Q Q o E S u r E G F g Y S O s H k K v M 7 D 6 C N i N Q d T m P w Q z Z W Y i Q 4 Q y v d 4 o U 7 q N b c u p u D L h K v I D V S o D m o f v W H E U 9 C U M g l M 6 b n u T H 6 K d M o u I R Z p Z 8 Y i B m f s D H 0 L F U s B O O n e d Q Z P U o M w 4 j G o K m Q N B f h 9 0 b K Q m O m Y W A n Q 4 b 3 Z t 7 L x P + 8 X o K j c z 8 V K k 4 Q F M 8 O o Z C Q H z J c C 9 s B 0 K H Q g M i y 5 E C F o p x p h g h a U M a 5 F R N b S s X 2 4 c 1 / v 0 j a J 3 X P 8 p v T W u O y a K Z M D s g h O S Y e O S M N c k 2 a p E U 4 G Z M n 8 k x e n E f n 1 X l z 3 n 9 G S 0 6 x s 0 / + w P n 4 B n V G k f 4 = < / l a t e x i t > t < l a t e x i t s h a 1 _ b a s e 6 4 = " w r V b K 2 d w N o c N T u U I U Z Z 7 S E t P S z 4 = " > A A A B 8 3 i c b V C 7 S g N B F J 2 N r x h f U U u b w S B Y h V 0 R t A z a W C Z g H p A s Y X Z y E 4 f M z i 4 z d 4 Q Q 8 g W 2 W t m J r R 9 k 4 b 8 4 u 2 6 h i a c 6 n H M v 9 9 w T p V I Y 9 P 1 P r 7 S 2 v r G 5 V d 6 u 7 O z u 7 R 9 U D 4 8 6 J r G a Q 5 s n M t G 9 i B m Q Q k E b B U r o p R p Y H E n o R t P b z O 8 + g j Y i U f c 4 S y G M 2 U S J s e A M n d T C Y b X m 1 / 0 c d J U E B a m R A s 1 h 9 W s w S r i N Q S G X z J h + 4 K c Y z p l G w S U s K g N r I G V 8 y i b Q d 1 S x G E w 4 z 4 M u 6 J k 1 D B O a g q Z C 0 l y E 3 x t z F h s z i y M 3 G T N 8 M M t e J v 7 n 9 S 2 O r 8 O 5 U K l F U D w 7 h E J C f s h w L V w D Q E d C A y L L k g M V i n K m G S J o Q R n n Tr S u k o r r I 1 j + f p V 0 L u q B 4 6 3 L W u O m a K Z M T s g p O S c B u S I N c k e a p E 0 4 A f J E n s m L Z 7 1 X 7 8 1 7 / x k t e c X O M f k D 7 + M b g E + R f Q = = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " w r V b K 2 d w N o c N T u U I U Z Z 7 S E t P S z 4 = " > A A A B 8 3 i c bV C 7 S g N B F J 2 N r x h f U U u b w S B Y h V 0 R t A z a W C Z g H p A s Y X Z y E 4 f M z i 4 z d 4 Q Q 8 g W 2 W t m J r R 9 k 4 b 8 4 u 2 6 h i a c 6 n H M v 9 9 w T p V I Y 9 P 1 P r 7 S 2 v r G 5 V d 6 u 7 O z u 7 R 9 U D 4 8 6 J r G a Q 5 s n M t G 9 i B m Q Q k E b B U r o p R p Y H E n o R t P b z O 8 + g j Y i U f c 4 S y G M 2 U S J s e A M n d T C Y b X m 1 / 0 c d J U E B a m R A s 1 h 9 W s w S r i N Q S G X z J h + 4 K c Y z p l G w S U s K g N r I G V 8 y i b Q d 1 S x G E w 4 z 4 M u 6 J k 1 D B O a g q Z C 0 l y E 3 x t z F h s z i y M 3 G T N 8 M M t e J v 7 n 9 S 2 O r 8 O 5 U K l F U D w 7 h E J C f s h w L V w D Q E d C A y L L k g M V i n K m G S J o Q R n n Tr S u k o r r I 1 j + f p V 0 L u q B 4 6 3 L W u O m a K Z M T s g p O S c B u S I N c k e a p E 0 4 A f J E n s m L Z 7 1 X 7 8 1 7 / x k t e c X O M f k D 7 + M b g E + R f Q = = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " w r V b K 2 d w N o c N T u U I U Z Z 7 S E t P S z 4 = " > A A A B 8 3 i c b / l a t e x i t > NB < l a t e x i t s h a 1 _ b a s e 6 4 = " C L H 6 W j a T 3 7 m q N X O q r T C U r q i c z j M = " > A A A B 9 X i c d V D L T g J B E J z F F + I L 9 e h l I j H x R G Z Z B b w R v H g y G O W R A C G z Q 4 M T Z x + Z 6 d U Q w i d 4 1 Z M 3 4 9 X v 8 e C / O C A m a r R O l a r u d H X 5 s Z I G G X t z U g u L S 8 s r 6 d X M 2 v r G 5 l Z 2 e 6 d h o k Q L q I t I R b r l c w N K h l B H i Q p a s Q Y e + A q a / s 3 p 1 G / e g j Y y C q 9 w F E M 3 4 M N Q D q T g a K X L 8 1 6 1 l 8 2 x f K n s e a x A W d 4 7 L h b L n i X s 2 H N P G H X z b I Y c m a P W y 7 5 3 + p F I A g h R K G 5 M 2 2 U x d s d c o x Q K J p l O Y i D m 4 o Y P o W 1 p y A M w 3 f E s 6 o Q e J I Z j R G P Q V C o 6 E + H 7 x p g H x o w C 3 0 4 G H K / N b 2 8 q / u W 1 E x y U u 2 M Z x g l C K K a H U C q Y H T J C S 9 s B 0 L 7 U g M i n y Y H K k A q u O S J o S b k Q V k x s K R n b x 9 f T 9 H / S K O R d y y + O c p X q v J k 0 2 S P 7 5 J C 4 p E Q q 5 I z U S J 0 I M i T 3 5 I E 8 O n f O k / P s v H y O p p z 5 z i 7 5 A e f 1 A / Z 2 k l Q = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " C L H 6 W j a T 3 7 m q N X O q r T C U r q i c z j M = " > A A A B 9 X i c d V D L T g J B E J z F F + I L 9 e h l I j H x R G Z Z B b w R v H g y G O W R A C G z Q 4 M T Z x + Z 6 d U Q w i d 4 1 Z M 3 4 9 X v 8 e C / O C A m a r R O l a r u d H X 5 s Z I G G X t z U g u L S 8 s r 6 d X M 2 v r G 5 l Z 2 e 6 d h o k Q L q I t I R b r l c w N K h l B H i Q p a s Q Y e + A q a / s 3 p 1 G / e g j Y y C q 9 w F E M 3 4 M N Q D q T g a K X L 8 1 6 1 l 8 2 x f K n s e a x A W d 4 7 L h b L n i X s 2 H N P G H X z b I Y c m a P W y 7 5 3 + p F I A g h R K G 5 M 2 2 U x d s d c o x Q K J p l O Y i D m 4 o Y P o W 1 p y A M w 3 f E s 6 o Q e J I Z j R G P Q V C o 6 E + H 7 x p g H x o w C 3 0 4 G H K / N b 2 8 q / u W 1 E x y U u 2 M Z x g l C K K a H U C q Y H T J C S 9 s B 0 L 7 U g M i n y Y H K k A q u O S J o S b k Q V k x s K R n b x 9 f T 9 H / S K O R d y y + O c p X q v J k 0 2 S P 7 5 J C 4 p E Q q 5 I z U S J 0 I M i T 3 5 I E 8 O n f O k / P s v H y O p p z 5 z i 7 5 A e f 1 A / Z 2 k l Q = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " C L H 6 W j a T 3 7 m q N X O q r T C U r q i c z j M = " > A A A B 9 X i c d V D L T g J B E J z F F + I L 9 e h l I j H x R G Z Z B b w R v H g y G O W R A C G z Q 4 M T Z x + Z 6 d U Q w i d 4 1 Z M 3 4 9 X v 8 e C / O C A m a r R O l a r u d H X 5 s Z I G G X t z U g u L S 8 s r 6 d X M 2 v r G 5 l Z 2 e 6 d h o k Q L q I t I R b r l c w N K h l B H i Q p a s Q Y e + A q a / s 3 p 1 G / e g j Y y C q 9 w F E M 3 4 M N Q D q T g a K X L 8 1 6 1 l 8 2 x f K n s e a x A W d 4 7 L h b L n i X s 2 H N P G H X z b I Y c m a P W y 7 5 3 + p F I A g h R K G 5 M 2 2 U x d s d c o x Q K J p l O Y i D m 4 o Y P o W 1 p y A M w 3 f E s 6 o Q e J I Z j R G P Q V C o 6 E + H 7 x p g H x o w C 3 0 4 G H K / N b 2 8 q / u W 1 E x y U u 2 M Z x g l C K K a H U C q Y H T J C S 9 s B 0 L 7 U g M i n y Y H K k A q u O S J o S b k Q V k x s K R n b x 9 f T 9 H / S K O R d y y + O c p X q v J k 0 2 S P 7 5 J C 4 p E Q q 5 I z U S J 0 I M i T 3 5 I E 8 O n f O k / P s v H y O p p z 5 z i 7 5 A e f 1 A / Z 2 k l Q = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " C L H 6 W j a T 3 7 m q N X O q r T C U r q i c z j M = " > A A A B 9 X i c d V D L T g J B E J z F F + I L 9 e h l I j H x R G Z Z B b w R v H g y G O W R A C G z Q 4 M T Z x + Z 6 d U Q w i d 4 1 Z M 3 4 9 X v 8 e C / O C A m a r R O l a r u d H X 5 s Z I G G X t z U g u L S 8 s r 6 d X M 2 v r G 5 l Z 2 e 6 d h o k Q L q I t I R b r l c w N K h l B H i Q p a s Q Y e + A q a / s 3 p 1 G / e g j Y y C q 9 w F E M 3 4 M N Q D q T g a K X L 8 1 6 1 l 8 2 x f K n s e a x A W d 4 7 L h b L n i X s 2 H N P G H X z b I Y c m a P W y 7 5 3 + p F I A g h R K G 5 M 2 2 U x d s d c o x Q K J p l O Y i D m 4 o Y P o W 1 p y A M w 3 f E s 6 o Q e J I Z j R G P Q V C o 6 E + H 7 x p g H x o w C 3 0 4 G H K / N b 2 8 q / u W 1 E x y U u 2 M Z x g l C K K a H U C q Y H T J C S 9 s B 0 L 7 U g M i n y Y H K k A q u O S J o S b k Q V k x s K R n b x 9 f T 9 H / S K O R d y y + O c p X q v J k 0 2 S P 7 5 J C 4 p E Q q 5 I z U S J 0 I M i T 3 5 I E 8 O n f O k / P s v H y O p p z 5 z i 7 5 A e f 1 A / Z 2 k l Q = < / l a t e x i t > NA < l a t e x i t s h a 1 _ b a s e 6 4 = " Z d W / X w P b p J l T D k L 7 a M / D U s 1 o w 4k = " > A A A B 9 X i c d V D L T g J B E J z 1 i f h C P X q Z S E w 8 k V k e A W + o F 0 8 G o z w S I G R 2 a H D C 7 C M z v R p C + A S v e v J m v P o 9 H v w X h x U T N V q n S l V 3 u r q 8 S E m D j L 0 5 C 4 t L y y u r q b X 0 + s b m 1 n Z m Z 7 d h w l g L q I t Q h b r l c Q N K B l B H i Q p a k Q b ue w q a 3 u h s 5 j d v Q R s Z B t c 4 j q D r 8 2 E g B 1 J w t N L V R e + k l 8 m y n F s q u K x M W a 6 Y Z 6 U K s 6 S S L x y 7 J e r m W I I s m a P W y 7x 3 + q G I f Q h Q K G 5 M 2 2 U R d i d c o x Q K p u l O b C D i Y s S H 0 L Y 0 4 D 6 Y 7 i S J O q W H s e E Y 0 g g 0 l Y o m I n z f m H D f m L H v 2 U m f 4 4 3 5 7 c 3 E v 7 x 2 j I N K d y K D K E Y I x O w Q S g X J I S O 0 t B 0 A 7 U s N i H y W H K g M q O C a I 4 K W l A t h x d i W k r Z 9 f D 1 N / y e N f M 6 1 / L K Y r Z 7 O m 0 m R f X J A j o h L y q R K z k m N 1 I k g Q 3 JP H s i j c + c 8 O c / O y + f o g j P f 2 S M / 4 L x + A O h a k k s = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " Z d W / X w P b p J l T D k L 7 a M / D U s 1 o w 4k = " > A A A B 9 X i c d V D L T g J B E J z 1 i f h C P X q Z S E w 8 k V k e A W + o F 0 8 G o z w S I G R 2 a H D C 7 C M z v R p C + A S v e v J m v P o 9 H v w X h x U T N V q n S l V 3 u r q 8 S E m D j L 0 5 C 4 t L y y u r q b X 0 + s b m 1 n Z m Z 7 d h w l g L q I t Q h b r l c Q N K B l B H i Q p a k Q b ue w q a 3 u h s 5 j d v Q R s Z B t c 4 j q D r 8 2 E g B 1 J w t N L V R e + k l 8 m y n F s q u K x M W a 6 Y Z 6 U K s 6 S S L x y 7 J e r m W I I s m a P W y 7x 3 + q G I f Q h Q K G 5 M 2 2 U R d i d c o x Q K p u l O b C D i Y s S H 0 L Y 0 4 D 6 Y 7 i S J O q W H s e E Y 0 g g 0 l Y o m I n z f m H D f m L H v 2 U m f 4 4 3 5 7 c 3 E v 7 x 2 j I N K d y K D K E Y I x O w Q S g X J I S O 0 t B 0 A 7 U s N i H y W H K g M q O C a I 4 K W l A t h x d i W k r Z 9 f D 1 N / y e N f M 6 1 / L K Y r Z 7 O m 0 m R f X J A j o h L y q R K z k m N 1 I k g Q 3 JP H s i j c + c 8 O c / O y + f o g j P f 2 S M / 4 L x + A O h a k k s = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " Z d W / X w P b p J l T D k L 7 a M / D U s 1 o w 4 k = " > A A A B 9 X i c d V D L T g J B E J z 1 i f h C P X q Z S E w 8 k V k e A W + o F 0 8 G o z w S I G R 2 a H D C 7 C M z v R p C + A S v e v J m v P o 9 H v w X h x U T N V q n S l V 3 u r q 8 S E m D j L 0 5 C 4 t L y y u r q b X 0 + s b m 1 n Z m Z 7 d h w l g L q I t Q h b r l c Q N K B l B H i Q p a k Q b u e w q a 3 u h s 5 j d v Q R s Z B t c 4 j q D r 8 2 E g B 1 J w t N L V R e + k l 8 m y n F s q u K x M W a 6 Y Z 6 U K s 6 S S L x y 7 J e r m W I I s m a P W y 7 x 3 + q G I f Q h Q K G 5 M 2 2 U R d i d c o x Q K p u l O b C D i Y s S H 0 L Y 0 4 D 6 Y 7 i S J O q W H s e E Y 0 g g 0 l Y o m I n z f m H D f m L H v 2 U m f 4 4 3 5 7 c 3 E v 7 x 2 j I N K d y K D K E Y I x O w Q S g X J I S O 0 t B 0 A 7 U s N i H y W H K g M q O C a I 4 K W l A t h x d i W k r Z 9 f D 1 N / y e N f M 6 1 / L K Y r Z 7 O m 0 m R f X J A j o h L y q R K z k m N 1 I k g Q 3 J P H s i j c + c 8 O c / O y + f o g j P f 2 S M / 4 L x + A O h a k k s = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " Z d W / X w P b p J l T D k L 7 a M / D U s 1 o w 4 k = " > A A A B 9 X i c d V D L T g J B E J z 1 i f h C P X q Z S E w 8 k V k e A W + o F 0 8 G o z w S I G R 2 a H D C 7 C M z v R p C + A S v e v J m v P o 9 H v w X h x U T N V q n S l V 3 u r q 8 S E m D j L 0 5 C 4 t L y y u r q b X 0 + s b m 1 n Z m Z 7 d h w l g L q I t Q h b r l c Q N K B l B H i Q p a k Q b u e w q a 3 u h s 5 j d v Q R s Z B t c 4 j q D r 8 2 E g B 1 J w t N L V R e + k l 8 m y n F s q u K x M W a 6 Y Z 6 U K s 6 S S L x y 7 J e r m W I I s m a P W y 7 x 3 + q G I f Q h Q K G 5 M 2 2 U R d i d c o x Q K p u l O b C D i Y s S H 0 L Y 0 4 D 6 Y 7 i S J O q W H s e E Y 0 g g 0 l Y o m I n z f m H D f m L H v 2 U m f 4 4 3 5 7 c 3 E v 7 x 2 j I N K d y K D K E Y I x O w Q S g X J I S O 0 t B 0 A 7 U s N i H y W H K g M q O C a I 4 K W l A t h x d i W k r Z 9 f D 1 N / y e N f M 6 1 / L K Y r Z 7 O m 0 m R f X J A j o h L y q R K z k m N 1 I k g Q 3 J P H s i j c + c 8 O c / O y + f o g j P f 2 S M / 4 L x + A O h a k k s = < / l a t e x i t > NA < l a t e x i t s h a 1 _ b a s e 6 4 = " Z d W / X w P b p J l T D k L 7 a M / D U s 1 o w 4 k = " > A A A B 9 X i c d V D L T g J B E J z 1 i f h C P X q Z S E w 8 k V k e A W + o F 0 8 G o z w S I G R 2 a H D C 7 C M z v R p C + A S v e v J m v P o 9 H v w X h x U T N V q n S l V 3 u r q 8 S E m D j L 0 5 C 4 t L y y u r q b X 0 + s b m 1 n Z m Z 7 d h w l g L q I t Q h b r l c Q N K B l B H i Q p a k Q b u e w q a 3 u h s 5 j d v Q R s Z B t c 4 j q D r 8 2 E g B 1 J w t N L V R e + k l 8 m y n F s q u K x M W a 6 Y Z 6 U K s 6 S S L x y 7 J e r m W I I s m a P W y 7 x 3 + q G I f Q h Q K G 5 M 2 2 U R d i d c o x Q K p u l O b C D i Y s S H 0 L Y 0 4 D 6 Y 7 i S J O q W H s e E Y 0 g g 0 l Y o m I n z f m H D f m L H v 2 U m f 4 4 3 5 7 c 3 E v 7 x 2 j I N K d y K D K E Y I x O w Q S g X J I S O 0 t B 0 A 7 U s N i H y W H K g M q O C a I 4 K W l A t h x d i W k r Z 9 f D 1 N / y e N f M 6 1 / L K Y r Z 7 O m 0 m R f X J A j o h L y q R K z k m N 1 I k g Q 3 J P H s i j c + c 8 O c / O y + f o g j P f 2 S M / 4 L x + A O h a k k s = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " Z d W / X w P b p J l T D k L 7 a M / D U s 1 o w 4 k = " > A A A B 9 X i c d V D L T g J B E J z 1 i f h C P X q Z S E w 8 k V k e A W + o F 0 8 G o z w S I G R 2 a H D C 7 C M z v R p C + A S v e v J m v P o 9 H v w X h x U T N V q n S l V 3 u r q 8 S E m D j L 0 5 C 4 t L y y u r q b X 0 + s b m 1 n Z m Z 7 d h w l g L q I t Q h b r l c Q N K B l B H i Q p a k Q b u e w q a 3 u h s 5 j d v Q R s Z B t c 4 j q D r 8 2 E g B 1 J w t N L V R e + k l 8 m y n F s q u K x M W a 6 Y Z 6 U K s 6 S S L x y 7 J e r m W I I s m a P W y 7 x 3 + q G I f Q h Q K G 5 M 2 2 U R d i d c o x Q K p u l O b C D i Y s S H 0 L Y 0 4 D 6 Y 7 i S J O q W H s e E Y 0 g g 0 l Y o m I n z f m H D f m L H v 2 U m f 4 4 3 5 7 c 3 E v 7 x 2 j I N K d y K D K E Y I x O w Q S g X J I S O 0 t B 0 A 7 U s N i H y W H K g M q O C a I 4 K W l A t h x d i W k r Z 9 f D 1 N / y e N f M 6 1 / L K Y r Z 7 O m 0 m R f X J A j o h L y q R K z k m N 1 I k g Q 3 J P H s i j c + c 8 O c / O y + f o g j P f 2 S M / 4 L x + A O h a k k s = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " Z d W / X w P b p J l T D k L 7 a M / D U s 1 o w 4 k = " > A A A B 9 X i c d V D L T g J B E J z 1 i f h C P X q Z S E w 8 k V k e A W + o F 0 8 G o z w S I G R 2 a H D C 7 C M z v R p C + A S v e v J m v P o 9 H v w X h x U T N V q n S l V 3 u r q 8 S E m D j L 0 5 C 4 t L y y u r q b X 0 + s b m 1 n Z m Z 7 d h w l g L q I t Q h b r l c Q N K B l B H i Q p a k Q b u e w q a 3 u h s 5 j d v Q R s Z B t c 4 j q D r 8 2 E g B 1 J w t N L V R e + k l 8 m y n F s q u K x M W a 6 Y Z 6 U K s 6 S S L x y 7 J e r m W I I s m a P W y 7 x 3 + q G I f Q h Q K G 5 M 2 2 U R d i d c o x Q K p u l O b C D i Y s S H 0 L Y 0 4 D 6 Y 7 i S J O q W H s e E Y 0 g g 0 l Y o m I n z f m H D f m L H v 2 U m f 4 4 3 5 7 c 3 E v 7 x 2 j I N K d y K D K E Y I x O w Q S g X J I S O 0 t B 0 A 7 U s N i H y W H K g M q O C a I 4 K W l A t h x d i W k r Z 9 f D 1 N / y e N f M 6 1 / L K Y r Z 7 O m 0 m R f X J A j o h L y q R K z k m N 1 I k g Q 3 J P H s i j c + c 8 O c / O y + f o g j P f 2 S M / 4 L x + A O h a k k s = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " Z d W / X w P b p J l T D k L 7 a M / D U s 1 o w 4 k = " > A A A B 9 X i c d V D L T g J B E J z 1 i f h C P X q Z S E w 8 k V k e A W + o F 0 8 G o z w S I G R 2 a H D C 7 C M z v R p C + A S v e v J m v P o 9 H v w X h x U T N V q n S l V 3 u r q 8 S E m D j L 0 5 C 4 t L y y u r q b X 0 + s b m 1 n Z m Z 7 d h w l g L q I t Q h b r l c Q N K B l B H i Q p a k Q b u e w q a 3 u h s 5 j d v Q R s Z B t c 4 j q D r 8 2 E g B 1 J w t N L V R e + k l 8 m y n F s q u K x M W a 6 Y Z 6 U K s 6 S S L x y 7 J e r m W I I s m a P W y 7 x 3 + q G I f Q h Q K G 5 M 2 2 U R d i d c o x Q K p u l O b C D i Y s S H 0 L Y 0 4 D 6 Y 7 i S J O q W H s e E Y 0 g g 0 l Y o m I n z f m H D f m L H v 2 U m f 4 4 3 5 7 c 3 E v 7 x 2 j I N K d y K D K E Y I x O w Q S g X J I S O 0 t B 0 A 7 U s N i H y W H K g M q O C a I 4 K W l A t h x d i W k r Z 9 f D 1 N / y e N f M 6 1 / L K Y r Z 7 O m 0 m R f X J A j o h L y q R K z k m N 1 I k g Q 3 J P H s i j c + c 8 O c / O y + f o g j P f 2 S M / 4 L x + A O h a k k s = < / l a t e x i t > NB < l a t e x i t s h a 1 _ b a s e 6 4 = " C L H 6 W j a T 3 7 m q N X O q r T C U r q i c z j M = " > A A A B 9 X i c d V D L T g J B E J z F F + I L 9 e h l I j H x R G Z Z B b w R v H g y G O W R A C G z Q 4 M T Z x + Z 6 d U Q w i d 4 1 Z M 3 4 9 X v 8 e C / O C A m a r R O l a r u d H X 5 s Z I G G X t z U g u L S 8 s r 6 d X M 2 v r G 5 l Z 2 e 6 d h o k Q L q I t I R b r l c w N K h l B H i Q p a s Q Y e + A q a / s 3 p 1 G / e g j Y y C q 9 w F E M 3 4 M N Q D q T g a K X L 8 1 6 1 l 8 2 x f K n s e a x A W d 4 7 L h b L n i X s 2 H N P G H X z b I Y c m a P W y 7 5 3 + p F I A g h R K G 5 M 2 2 U x d s d c o x Q K J p l O Y i D m 4 o Y P o W 1 p y A M w 3 f E s 6 o Q e J I Z j R G P Q V C o 6 E + H 7 x p g H x o w C 3 0 4 G H K / N b 2 8 q / u W 1 E x y U u 2 M Z x g l C K K a H U C q Y H T J C S 9 s B 0 L 7 U g M i n y Y H K k A q u O S J o S b k Q V k x s K R n b x 9 f T 9 H / S K O R d y y + O c p X q v J k 0 2 S P 7 5 J C 4 p E Q q 5 I z U S J 0 I M i T 3 5 I E 8 O n f O k / P s v H y O p p z 5 z i 7 5 A e f 1 A / Z 2 k l Q = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " C L H 6 W j a T 3 7 m q N X O q r T C U r q i c z j M = " > A A A B 9 X i c d V D L T g J B E J z F F + I L 9 e h l I j H x R G Z Z B b w R v H g y G O W R A C G z Q 4 M T Z x + Z 6 d U Q w i d 4 1 Z M 3 4 9 X v 8 e C / O C A m a r R O l a r u d H X 5 s Z I G G X t z U g u L S 8 s r 6 d X M 2 v r G 5 l Z 2 e 6 d h o k Q L q I t I R b r l c w N K h l B H i Q p a s Q Y e + A q a / s 3 p 1 G / e g j Y y C q 9 w F E M 3 4 M N Q D q T g a K X L 8 1 6 1 l 8 2 x f K n s e a x A W d 4 7 L h b L n i X s 2 H N P G H X z b I Y c m a P W y 7 5 3 + p F I A g h R K G 5 M 2 2 U x d s d c o x Q K J p l O Y i D m 4 o Y P o W 1 p y A M w 3 f E s 6 o Q e J I Z j R G P Q V C o 6 E + H 7 x p g H x o w C 3 0 4 G H K / N b 2 8 q / u W 1 E x y U u 2 M Z x g l C K K a H U C q Y H T J C S 9 s B 0 L 7 U g M i n y Y H K k A q u O S J o S b k Q V k x s K R n b x 9 f T 9 H / S K O R d y y + O c p X q v J k 0 2 S P 7 5 J C 4 p E Q q 5 I z U S J 0 I M i T 3 5 I E 8 O n f O k / P s v H y O p p z 5 z i 7 5 A e f 1 A / Z 2 k l Q = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " C L H 6 W j a T 3 7 m q N X O q r T C U r q i c z j M = " > A A A B 9 X i c d V D L T g J B E J z F F + I L 9 e h l I j H x R G Z Z B b w R v H g y G O W R A C G z Q 4 M T Z x + Z 6 d U Q w i d 4 1 Z M 3 4 9 X v 8 e C / O C A m a r R O l a r u d H X 5 s Z I G G X t z U g u L S 8 s r 6 d X M 2 v r G 5 l Z 2 e 6 d h o k Q L q I t I R b r l c w N K h l B H i Q p a s Q Y e + A q a / s 3 p 1 G / e g j Y y C q 9 w F E M 3 4 M N Q D q T g a K X L 8 1 6 1 l 8 2 x f K n s e a x A W d 4 7 L h b L n i X s 2 H N P G H X z b I Y c m a P W y 7 5 3 + p F I A g h R K G 5 M 2 2 U x d s d c o x Q K J p l O Y i D m 4 o Y P o W 1 p y A M w 3 f E s 6 o Q e J I Z j R G P Q V C o 6 E + H 7 x p g H x o w C 3 0 4 G H K / N b 2 8 q / u W 1 E x y U u 2 M Z x g l C K K a H U C q Y H T J C S 9 s B 0 L 7 U g M i n y Y H K k A q u O S J o S b k Q V k x s K R n b x 9 f T 9 H / S K O R d y y + O c p X q v J k 0 2 S P 7 5 J C 4 p E Q q 5 I z U S J 0 I M i T 3 5 I E 8 O n f O k / P s v H y O p p z 5 z i 7 5 A e f 1 A / Z 2 k l Q = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " C L H 6 W j a T 3 7 m q N X O q r T C U r q i c z j M = " > A A A B 9 X i c d V D L T g J B E J z F F + I L 9 e h l I j H x R G Z Z B b w R v H g y G O W R A C G z Q 4 M T Z x + Z 6 d U Q w i d 4 1 Z M 3 4 9 X v 8 e C / O C A m a r R O l a r u d H X 5 s Z I G G X t z U g u L S 8 s r 6 d X M 2 v r G 5 l Z 2 e 6 d h o k Q L q I t I R b r l c w N K h l B H i Q p a s Q Y e + A q a / s 3 p 1 G / e g j Y y C q 9 w F E M 3 4 M N Q D q T g a K X L 8 1 6 1 l 8 2 x f K n s e a x A W d 4 7 L h b L n i X s 2 H N P G H X z b I Y c m a P W y 7 5 3 + p F I A g h R K G 5 M 2 2 U x d s d c o x Q K J p l O Y i D m 4 o Y P o W 1 p y A M w 3 f E s 6 o Q e J I Z j R G P Q V C o 6 E + H 7 x p g H x o w C 3 0 4 G H K / N b 2 8 q / u W 1 E x y U u 2 M Z x g l C K K a H U C q Y H T J C S 9 s B 0 L 7 U g M i n y Y H K k A q u O S J o S b k Q V k x s K R n b x 9 f T 9 H / S K O R d y y + O c p X q v J k 0 2 S P 7 5 J C 4 p E Q q 5 I z U S J 0 I M i T 3 5 I E 8 O n f O k / P s v H y O p p z 5 z i 7 5 A e f 1 A / Z 2 k l Q = < / l a t e x i t > Figure 2

Figure
Figure3: Linear growth in time of the half-system entanglement entropy SN/2 at the dynamical critical point.We compare our general formula (??) with the exact numerical computation for increasing system sizes N = 50÷ 400.Before the Ehrenfest time tEhr ∼ log N , numerical data for SN/2 are accurately reproduced by the analytical result (??) marked by the dotted line with a slope λh c = J.This linear regime is followed by saturation to a value ∼ log N .
u 7 P f z p H F s u y e 2 c 3 1 a r l 7 k z R T Y P j t g F e a y M 1 Z l V 6 z G 6 k y w O / b E n t m L 9 W i 9 W m / W + 8 / o g p X v 7 L E / s D 6 + A R A Q l k Q = < / l a t e x i t > (b.) < l a t e x i t s h a 1 _ b a s e 6 4 = " W P d C z M g 4 n j 8 X X o e H M 6 R q Q s Q P O b E = " > A A A B / 3 i c b V C 7 T s N A E D z z D O E V o K Q 5 E S G F x r I B C c o I G s o g k Y e U W N H 5 s g m n n M / m b o 2 I r B R 8 B S 1 U d I i W T 6 H g X 7 C N C 0 i Y a j S z q 5 0 d P 5 L C o O N 8 W g u L S 8 s r q 6 W 1 8 v r G 5 t Z 2 Z W e 3 Z c J Y c 2 j y U I a 6 4 z M D U i h o o k A J n U g D C 3 w J b X 9 8 m f n t e 9 B G h O o G J x F 4 A R s p M R S c Y S p 5 P Y Q H 9 I d J z b e P p v 1 K 1 b G d H H S e u A W p k g K N f u W r N w h 5 H I B C L p k x X d e J 0 E u Y R s E l T M u 9 2 E D E + J i N o J t S x Q I w X p K H n t L D 2 D A M a Q S a C k l z E X 5 v J C w w Z h L 4 6 W T A 8 N b M e p n 4 n 9 e N c X j u J U J F M Y L i 2 S E U E v J D h m u R t g F 0 I D Q g s i w 5 U K E o Z 5 o h g h a U c Z 6 K c V p P O e 3 D n f 1 + n r S O b f f E d q 5 P q / W L o p k S 2 S c H p E Z c c k b q 5 I o 0 S J N w c k e e y D N 5 s R 6 t V + v N e v 8 Z X b C K n T 3 y B 9 b H N x G i l k U = < / l a t e x i t > (c.) < l a t e x i t s h a 1 _ b a s e 6 4 = " A 7 H t L K h o S m W n Z O 0 l f O U n s n n X f / 4 = " > A A A B / 3 i c b V C 7 T s N A E D z z D O E V o K Q 5 E S G F x r I B C c o I G s o g k Y e U W N H 5 s g m n n M / m b o 2 I r B R 8 B S 1 U d I i W T 6 H g X 7 C N C 0 i Y a j S z q 5 0 d P 5 L C o O N 8 W g u L S 8 s r q 6 W 1 8 v r G 5 t Z 2 Z W e 3 Z c J Y c 2 j y U I a 6 4 z M D U i h o o k A J n U g D C 3 w J b X 9 8 m f n t e 9 B G h O o G J x F 4 A R s p M R S c Y S p 5 P Y Q H 9 I d J j d t H 0 3 6 l 6 t h O D j p P 3 I J h g z C T w 0 8 m A 4 a 2 Z 9 T L x P 6 8 b 4 / D c S 4 S K Y g T F s 0 M o J O S H D N c i b Q P o Q G h A Z F l y o E J R z j R D B C 0 o 4 z w V 4 7 S e c t q H O / v 9 P G k d 2 + 6 J 7 V y f V u s X R T M l s k 8 O S I 2 4 5 I z U y R V p k C b h 5 I 4 8 k W f y Y j 1 a r 9 a b 9 f 4 z u m A V O 3 v k D 6 y P b x T G l k c = < / l a t e x i t > (e.) < l a t e x i t s h a 1 _ b a s e 6 4 = " D C y r s o J O I z s Y 4 J 5 o 5 E E 6 w g u 2 N 4 0 = " > A A A B / 3 i c b V C 7 T s N A E D z z D O E V o K Q 5 E S G F x r I B C c o I G s o g k Y e U W N H 5 s g m n n M / m b o 2 I r B R 8 B S 1 U d I i W T 6 H g X 7 C N C 0 i Y a j S z q 5 0 d P 5 L C o O N 8 W g u L S 8 s r q 6 W 1 8 v r G 5 t Z 2 Z W e 3 Z c J Y c 2 j y U I a 6 4 z M D U i h o o k A J n U g D C 3 w J b X 9 8 m f n t e 9 B G h O o G J x F 4 A R s p M R S c Y S p 5 P Y Q H 9 I d J D e y j a b 9 S d W w n B 5 0 n b

Figure 1 .
Figure 1.Entanglement dynamics and collective spin squeezing in long-range quantum spin systems.(a) The system is partitioned into two blocks of NA and NB spins-1/2, initially fully polarized.(b) Collective spins of the two blocks.(c) Collective spin in the factorized initial state, represented on a sphere of radius N/2.The shaded area represents the quantum uncertainty of transverse components.(d) Nonlinear interactions determine spin squeezing, which makes the two blocks increasingly correlated (entangled).The slow rate of squeezing after non-critical quenches determines the slow growth of entanglement.(e) The analytical formula for spin-squeezing dynamics derived in this work captures the growth of entanglement entropy until saturation (here, quantum quench in a fully-connected quantum Ising model).

6 t 1 C
/ y Z o r k g B y S K v H I G a m T K 9 I g T c K J I k / k m b w 4 j 8 6 r 8 + a 8 / 4 w W n H x n n / y B 8 / E N F Z O V L w = = < / l a t e x i t > hnexc(t)i ⇠ < l a t e x i t s h a 1 _ b a s e 6 4 = " I U T O i D j d 4 u b 9 G p 3 x p j t b 7 A e 8 t 0 9 I n 6 L P < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " I U T O i D j d 4 u b s 5 x N b e x p G b j I G G 9 r d X i P 9 5 n Y z 6 R 9 1 c 6 j Q j 1 K I 4 R N L F L w 5 Z Y a T r C v m V N E g E x e f I p e Y C D B C h k R y E c G L m y i u 7 P o L f 6 f + S 1 v 5 e 4 P j 5 Q f X 4 Z N p M i W 2 z H V Z j A T t k x + y M N V i T C X b H H t g j e / L u v W f v x X v 9 G p 3 x p j t b 7 A e 8 t 0 9 I n 6 L P < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " I U T O i D j d 4 u b G G L K b F 3 w B t z t p u 8 I 5 H G f D e z Q A l P 0 X C p + E T E 7 x s 5 x N b e x p G b j I G G 9 r d X i P 9 5 n Y z 6 R 9 1 c 6 j Q j 1 K I 4 R N L F L w 5 Z Y a T r C v m V N E g E x e f I p e Y C D B C h k R y E c G L m y i u 7 P o L f 6 f + S 1 v 5 e 4 P j 5 Q f X 4 Z N p M i W 2 z H V Z j A T t k x + y M N V i T C X b H H t g j e / L u v W f v x X v 9 G p 3 x p j t b 7 A e 8 t 0 9 I n 6 L P < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " I U T O i D j d 4 u b G G L K b F 3 w B t z t p u 8 I 5 H G f D e z Q A l P 0 X C p + E T E 7 x s 5 x N b e x p G b j I G G 9 r d X i P 9 5 n Y z 6 R 9 1 c 6 j Q j 1 K I 4 R N L F L w 5 Z Y a T r C v m V N E g E x e f I p e Y C D B C h k R y E c G L m y i u 7 P o L f 6 f + S 1 v 5 e 4 P j 5 Q f X 4 Z N p M i W 2 z H V Z j A T t k x + y M N V i T C X b H H t g j e / L u v W f v x X v 9 G p 3 x p j t b 7 A e 8 t 0 9 I n 6 L P < / l a t e x i t > e 2 t < l a t e x i t s h a 1 _ b a s e 6 4 = "

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s z q 5 Z g z W Z Y J o 9 s W f 2 4 j w 6 r 8 6 b 8 / 4 T L T j z m U P 2 B 8 7 H N / p T l s k = < / l a t e x i t > t l a t e x i t s h a 1 _ b a s e 6 4 = " j g e O K E z z Y T D B C 0 o 4 9 y K i S 2 l b P v w 5 r 9 f J O 1 6 z b P 8 + r T a u C i a K Z F D c k R O i E f O S I N c k S Z p E U 7 G 5 I k 8 k x f n 0 X l 1 3 p z 3 n 9 E l p 9 g 5 I H / g f H w D q / S S I Q = = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " j g e O K E z z Y T D B C 0 o 4 9 y K i S 2 l b P v w 5 r 9 f J O 1 6 z b P 8 + r T a u C i a K Z F D c k R O i E f O S I N c k S Z p E U 7 G 5 I k 8 k x f n 0 X l 1 3 p z 3 n 9 E l p 9 g 5 I H / g f H w D q / S S I Q = = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " j g e O K E z z Y T D B C 0 o 4 9 y K i S 2 l b P v w 5 r 9 f J O 1 6 z b P 8 + r T a u C i a K Z F D c k R O i E f O S I N c k S Z p E U 7 G 5 I k 8 k x f n 0 X l 1 3 p z 3 n 9 E l p 9 g 5 I H / g f H w D q / S S I Q = = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " j g e O K E z z Y T D B C 0 o 4 9 y K i S 2 l b P v w 5 r 9 f J O 1 6 z b P 8 + r T a u C i a K Z F D c k R O i E f O S I N c k S Z p E U 7 G 5 I k 8 k x f n 0 X l 1 3 p z 3 n 9 E l p 9 g 5 I H / g f H w D q / S S I Q = = < / l a t e x i t > hnexc(t)i ⇠ < l a t e x i t s h a 1 _ b a s e 6 4 = " I U T O i D j d 4 u b G G L K b F 3 w B t z t p u 8 I 7 x s 5 x N b e x p G b j I G G 9 r d X i P 9 5 n Y z 6 R 9 1 c 6 j Q j 1 K I 4 R N L F L w 5 Z Y a T r C v m V N E g E x e f I p e Y C D B C h k R y E c G L m y i u 7 P o L f 6 f + S 1 v 5 e 4 P j 5 Q f X 4 Z N p M i W 2 z H V Z j A T t k x + y M N V i T C X b H H t g j e / L u v W f v x X v 9 G p 3 x p j t b 7 A e 8 t 0 9 I n 6 L P < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " I U T O i D j d 4 u b G G L K b F 3 w B t z t p u 8 I 0 j B a 6 r R z K 5 2 Z 6 J U S U u + / + 7 N z M 7 N L y y W l s r L K 6 t r 6 5W N z Z Z N M i O w K R K V m M s I L C q p s U m S F F 6 m B i G O F L a j 6 9 P C b 9 + g s T L R F 3 S b Y j e G g Z Z 9 K Y C c 1 K v U Q w V 6 o J C H Q y C u e 3 l I O K I c R 2 I 8 r l E 9 N B M 3 t D L u V a r + n j 8 B / 0 u C K a m y K R q 9 y k d 4 l Y g s R k 1 C g b W d w E + p m 4 M h K R S O y 2 F m M Q V x D Q P s O K o h R t v N J5 H G f D e z Q A l P 0 X C p + E T E 7 x s 5 x N b e x p G b j I G G 9 r d X i P 9 5 n Y z 6 R 9 1 c 6 j Q j 1 K I 4 R N L F L w 5 Z Y a T r C v m V N E g E x e f I p e Y C D B C h k R y E c G L m y i u 7 P o L f 6 f + S 1 v 5 e 4 P j 5 Q f X 4 Z N p M i W 2 z H V Z j A T t k x + y M N V i T C X b H H t g j e / L u v W f v x X v 9 G p 3 x p j t b 7 A e 8 t 0 9 I n 6 L P < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " I U T O i D j d 4 u b G G L K b F 3 w B t z t p u 8 I 0 j B a 6 r R z K 5 2 Z 6 J U S U u + / + 7 N z M 7 N L y y W l s r L K 6 t r 6 5 W N z Z Z N M i O w K R K V m M s I L C q p s U m S F F 6 m B i G O F L a j 6 9 P C b 9 + g s T L R F 3 S b Y j e G g Z Z 9 K Y C c 1 K v U Q w V 6 o J C H Q y C u e 3 l I O K I c R 2 I 8 r l E 9 N B M 3 t D L u V a r + n j 8 B / 0 u C K a m y K R q 9 y k d 4 l Y g s R k 1 C g b W d w E + p m 4 M h K R S O y 2 F m M Q V x D Q P s O K o h R t v N J 5 H G f D e z Q A l P 0 X C p + E T E 7 x s 5 x N b e x p G b j I G G 9 r d X i P 9 5 n Y z 6 R 9 1 c 6j Q j 1 K I 4 R N L F L w 5 Z Y a T r C v m V N E g E x e f I p e Y C D B C h k R y E c G L m y i u 7 P o L f 6 f + S 1 v 5 e 4 P j 5 Q f X 4 Z N p M i W 2 z H V Z j A T t k x + y M N V i T C X b H H t g j e / L u v W f v x X v9 G p 3 x p j t b 7 A e 8 t 0 9 I n 6 L P < / l a t e x i t > (a) < l a t e x i t s h a 1 _ b a s e 6 4 = " w F X 7 y K 7 C R i n Z W A m e 2 8 7 G C e f x 5 a o = " > A A A B / H i c b V A 9 T w J B E N 3 D L 8 Q v 1 N J m I z H B h t w Z E y 2 J N p a Y y E c E Q u a W A T f s 7 V 1 2 5 4 z k g r / C V i s 7 Y + t / s f C / e I c U C r 7 q 5 b 2 Z z J v n R 0 p a c t 1 P J 7 e 0 v L K 6 l l 8 v b G x u b e 8 U d / c a N o y N w Lo I V W h a P l h U U m O d J C l s R Q Y h 8 B U 2 / d F l 5 j f v 0 V g Z 6 h s a R 9 g N Y K j l Q A q g V L r t E D 5 Q U o b j S a 9 Y c i v u F H y R e D N S Y j P U e s W v T j 8 U c Y C a h A J r 2 5 4 b U T c B Q 1 I o n B Q 6 s c U I x A i G 2 E 6 p h g B t N 5 k m n v C j 2 A K F P E L D p e J T E X 9 v J B B Y O w 7 8 d D I A u r P z X i b + 5 7 V j G p x 3 E 6 m j m F C L 7 B B J h d N D V h i Z V o G 8 L w 0 S Q Z Y c u d R c g A E i N J K D E K k Y p 9 0 U 0 j 6 8 + e 8 X S e O k 4 q X 8 + r R U v Z g 1 k 2 c H 7 J C V m c f O W J V d s R q r M 8 E 0 e 2 L P 7 M V 5 d F 6 d N + f 9 Z z T n z H b 2 2 R 8 4 H 9 8 U A p U u < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " w F X 7 y K 7 C R i n Z W A m e 2 8 7 G C e f x 5 a o = " > A A A B / H i c b V A 9 T w J B E N 3 D L 8 Q v 1 N J m I z H B h t w Z E y 2 J N p a Y y E c E Q u a W A T f s 7 V 1 2 5 4 z k g r / C V i s 7 Y + t / s f C / e I c U C r 7 q 5 b 2 Z z J v n R 0 p a c t 1 P J 7 e 0 v L K 6 l l 8 v b G x u b e 8 U d / c a N o y N w L o I V W h a P l h U U m O d J C l s R Q Y h 8 B U 2 / d F l 5 j f v 0 V g Z 6 h s a R 9 g N Y K j l Q A q g V L r t E D 5 Q U o b j S a 9 Y c i v u F H y R e D N S Y j P U e s W v T j 8 U c Y C a h A J r 2 5 4 b U T c B Q 1 I o n B Q 6 s c U I x A i G 2 E 6 p h g B t N 5 k m n v C j 2 A K F P E L D p e J T E X 9 v J B B Y O w 7 8 d D I A u r P z X i b + 5 7 V j G p x 3 E 6 m j m F C L 7 B B J h d N D V h i Z V o G 8 L w 0 S Q Z Y c u d R c g A E i N J K D E K k Y p 9 0 U 0 j 6 8 + e 8 X S e O k 4 q X 8 + r R U v Z g 1 k 2 c H 7 J C V m c f O W J V d s R q r M 8 E 0 e 2 L P 7 M V 5 d F 6 d N + f 9 Z z T n z H b 2 2 R 84 H 9 8 U A p U u < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " w F X 7 y K 7 C R i n Z W A m e 2 8 7 G C e f x 5 a o = " >A A A B / H i c b V A 9 T w J B E N 3 D L 8 Q v 1 N J m I z H B h t w Z E y 2 J N p a Y y E c E Q u a W A T f s 7 V 1 2 5 4 z k g r / C V i s 7 Y + t / s f C / e I c U C r 7 q 5 b 2 Z z J v n R 0 p a c t 1 P J 7 e 0 v L K 6 l l 8 v b G x u b e 8 U d / c a N o y N w L o I V W h a P l h U U m O d J C l s R Q Y h 8 B U 2 / d F l 5 j f v 0 V g Z 6 h s a R 9 g N Y K j l Q A q g V L r t E D 5 Q U o b j S a 9 Y c i v u F H y R e D N S Y j P U e s W v T j 8 U c Y C a h A J r 2 5 4 b U T c B Q 1 I o n B Q 6 s c U I x A i G 2 E 6 p h g B t N 5 k m n v C j 2 A K F P E L D p e J T E X 9 v J B B Y O w 7 8 d D I A u r P z X i b + 5 7 V j G p x 3 E 6 m j m F C L 7 B B J h d N D V h i Z V o G 8 L w 0 S Q Z Y c u d R c g A E i N J K D E K k Y p 9 0 U 0 j 6 8 + e 8 X S e O k 4 q X 8 + r R U v Z g 1 k 2 c H 7 J C V m c f O W J V d s R q r M 8 E0 e 2 L P 7 M V 5 d F 6 d N + f 9 Z z T n z H b 2 2 R 8 4 H 9 8 U A p U u < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " w F X 7 y K 7 C R i n Z W A m e 2 8 7 G C e f x 5 a o = " > A A A B / H i c b V A 9 T w J B E N 3 D L 8 Q v 1 N J m I z H B h t w Z E y 2 J N p a Y y E c E Q u a W A T f s 7 V 1 2 5 4 z k g r / C V i s 7 Y + t / s f C / e I c U C r 7 q 5 b 2 Z z J v n R 0 p a c t 1 P J 7 e 0 v L K 6 l l 8 v b G x u b e 8 U d / c a N o y N w L o I V W h a P l h U U m O d J C l s R Q Y h 8 B U 2 / d F l 5 j f v 0 V g Z 6 h s a R 9 g N Y K j l Q A q g V L r t E D 5 Q U o b j S a 9 Y c i v u F H y R e D N S Y j P U e s W v T j 8 U c Y C a h A J r 2 5 4 b U T c B Q 1 I o n B Q 6 s c U I x A i G 2 E 6 p h g B t N 5 k m n v C j 2 A K F P E L D p e J T E X 9 v J B B Y O w 7 8 d D I A u r P z X i b + 5 7 V j G p x 3 E 6 m j m F C L 7 B B J h d N D V h i Z V o G 8 L w 0 S Q Z Y c u d R c g A E i N J K D E K k Y p 9 0 U 0 j 6 8 + e 8 X S e O k 4 q X 8 + r R U v Z g 1 k 2 c H 7 J C V m c f O W J V d s R q r M 8 E 0 e 2 L P 7 M V 5 d F 6 d N + f 9 Z z T n z H b 2 2 R 8 4 H 9 8 U A p U u < / l a t e x i t > Z < l a t e x i t s h a 1 _ b a s e 6 4 = " k a p D + P O o h M c x c k s k d S y o v A l / r 9 A = " > A A A B 8 3 i c b M b 0 P T d G P 2 E a B Z c w r w y s g Z j x K Z t A P 6 W K h W D 8 J A 8 6 p y f WM I x o D J o K S X M R f m 8 k L D R m F g b p Z M j w w S x 6 m f i f 1 7 c 4 v v I T o W K L o H h 2 C I W E / J D h W q Q N A B 0 J D Y g s S w 5 U K M q Z Z o i g B W W c p 6 J N K 6 m k f X i L 3y + T z l n d O 6 + 7 r Y t a 4 7 p o p k y O y D E 5 J R 6 5 J A 1 y S 5 q k T T g B 8 k S e y Y t j n V f n z X n / G S 0 5 x c 4 h + Q P n 4 x t Y c Z F l < / l a t e x i t > X < l a t e x i t s h a 1 _ b a s e 6 4 = " R + b o q J y R U Q v E L 0 v 0 + m j w x o N 4 P 8 k = " > A A A B 8 3 i c b V C 7 T s N A E D z z D O E V o K Q 5 E S F R R T Y g Q R l B Q 5 l I 5 C E l V r S + b M I p 5 4 f u 9 p C i K F 9 A C x U d o u W D K P g X b O M C E q Y a z e x q Z y d I l D T k u p / O y u r a + s Z m a a u 8 v b O 7 t 1 8 5 O G y b 2

Figure 2 .
Figure 2. Schematic illustration of spin-squeezing dynamics.The classical phase space of a ferromagnetic fully-connected Ising model [cf.Eq. (7) with α = 0] is pictured on the left.It features ferromagnetic (green) and paramagnetic (blue) periodic trajectories, separated by a critical trajectory (separatrix, red).Initial spin-coherent (non-squeezed) states (a) and (b) at t = 0 are represented by points surrounded by small grey circles representing the quantum fluctuations of transverse spin components.Due to nonlinear interactions, the spin state undergoes squeezing, quantified by the parameter ξ(t) [cf.Eq. (3)].The rate of squeezing is governed by the separation of nearby semiclassical trajectories, and by Eq. (2) it determines the rate of growth of entanglement entropy.Right panels: (a) For generic (non-critical) quenches, nearby trajectories separate linearly in time, leading to a polynomially fast squeezing.(b) For a critical quench, the collective spin lies on the stable manifold of an unstable fixed point in phase space.In this case, nearby trajectories separate exponentially fast in time at a rate λ set by the eigenvalue of the linearized flow.

Figure 3 .
Figure 3. Logarithmic growth in time of the half-system entanglement entropy S N/2 after a quantum quench above (top) and below (bottom) the dynamical critical point.We compare our general formula(34)  with the exact numerical computation for increasing system sizes N = 50 ÷ 800.The exact diagonalization results follow the logarithmic growth up to tEhr s p N , where they saturate to S N/2 s log N .The inset shows the same data with S N/2 rescaled by log N and time by p N .

Figure 3 .
Figure 3. Logarithmic growth in time of the half-system entanglement entropy S N/2 after a quantum quench above (top) and below (bottom) the dynamical critical point.We compare our general formula(34)  with the exact numerical computation for increasing system sizes N = 50 ÷ 800.The exact diagonalization results follow the logarithmic growth up to t Ehr s p N , where they saturate to S N/2 s log N .The inset shows the same data with S N/2 rescaled by log N and time by p N .

Figure 3 .
Figure 3. Logarithmic growth in time of the half-system entanglement entropy S N/2 after a quantum quench above (top) and below (bottom) the dynamical critical point.We compare our general formula(34)  with the exact numerical computation for increasing system sizes N = 50 ÷ 800.The exact diagonalization results follow the logarithmic growth up to tEhr s p N , where they saturate to S N/2 s log N .The inset shows the same data with S N/2 rescaled by log N and time by p N .

Figure 3 .
Figure 3. Logarithmic growth in time of the half-system entanglement entropy S N/2 after a quantum quench above (top) and below (bottom) the dynamical critical point.We compare our general formula(34)  with the exact numerical computation for increasing system sizes N = 50 ÷ 800.The exact diagonalization results follow the logarithmic growth up to tEhr s p N , where they saturate to S N/2 s log N .The inset shows the same data with S N/2 rescaled by log N and time by p N .

Figure 7 :
Figure 7: Entanglement dynamics from quenches starting from a paramagnetic

Figure 7 :
Figure 7: Entanglement dynamics from quenches starting from a paramagnetic

Figure 1 .
Figure 1.Plots of the function f α,k for d = 1.Top panel: f α,k is shown for several values of α, for N = L = 400.One recognizes a function squeezed towards k = 0 (0 ≤ α ≤ 1), a finite function with a cusp behavior for small k (1 < α < 2), and a cosine-like function (α 2).Bottom left panel: f α,k is shown for α = 0.7 and increasing values of N = L.A qualitatively similar behavior occurs for 0 ≤ α ≤ 1. Squeezing towards a delta function as L → ∞ occurs with a speed N −(1−α) for α < 1 and 1/ log N for α = 1.Bottom right panel: a "zoom" of the plot in the bottom left panel is shown, for larger values of N = L.The rescaled function in the vicinity of k = 0 converges to a finite limiting curve as L → ∞.This discrete structure approaches a continuum as α 1.The blue solid line illustrates the bound obtained in Eq. (45).

are
Pauli matrices on sites i = 1, . . ., N of the chain, h is the global transverse magnetic field and N α,N = i =j |i − j| −α /N is the Kač rescaling, ensuring the extensivity of the Hamiltonian for α ≤ d.

Figure 1 :
Figure1: Logarithmic growth in time of the half-system entanglement entropy SN/2 after a quantum quench above (top) and below (bottom) the dynamical critical point.We compare our general formula (??) with the exact numerical computation for increasing system sizes N = 50 ÷ 800.The exact diagonalization results follow the logarithmic growth up to tEhr ⇠ p N , where they saturate to SN/2 ⇠ log N .The inset shows the same data with SN/2 rescaled by log N and time by p N .

Figure 1 :. 2 Figure 2 .
Figure 1: Logarithmic growth in time of the half-system entanglement entropy SN/2 after a quantum quench above (top) below (bottom) the dynamical critical point.We compare our general formula (??) with the exact numerical computation increasing system sizes N = 50 ÷ 800.The exact diagonalization results follow the logarithmic growth up to tEhr ⇠ p where they saturate to SN/2 ⇠ log N .The inset shows the same data with SN/2 rescaled by log N and time by p N .

Figure 5 : 4 Figure 3 .
Figure 5: Logarithmic growth in time of the half-system entanglement entropy S N/2 for ↵ = 0.1, 0.5, 0.7.We compare our general formula (??) valid in the infinite range case, with the numerical computation for increasing system sizes N = 20 ÷ 50.Numerical results obtained with TDVP with bond dimension D = 64 for N = 20 , 50 and D = 128 for N = 65.

3. 1
Comparison with spin waves for standard quenches

Figure 9 :
Figure 9: Spin waves population in time: the zero mode grows polynomially, whereas the long-wavelength modes, after a quadratic growth, oscillate periodically in time and are bounded.(Left) ↵ = 0.1.(Right) ↵ = 0.7

Figure 6 .
Figure 6.Comparison between finite-size MPS-TDVP numerical data (light-to-dark blue curves for increasing N ), the spinsqueezing contribution (grey) and full spin-wave entanglement (black), for α = 0.1 (left panel) and 0.7 (right panel), for the quench h0 = 0 → h f = 2J.Numerical data exhibit convergence to the spin-wave result as N → ∞, increasingly more slowly as α is raised from 0 to 1 (cf.Fig.1, bottom right panel).

3
Reply to the referee 3.1 Comparison with spin waves for standard quenches

Figure 9 :
Figure 9: Spin waves population in time: the zero mode grows polynomially, whereas the long-wavelength modes, after a quadratic growth, oscillate periodically in time and are bounded.(Left) ↵ = 0.1.(Right) ↵ = 0.7

Figure 7 .
Figure 7. Time-dependent k-resolved spin-wave population for α = 0.1 (left panel) and α = 0.7 (right panel) after a quench from h0 = 0 to h f = 2J.The blue color gradient for the spin-wave populations in Fourier modes follows the quasimomentum |k| from the darkest (k = ±2π/L) down to smaller-wavelength modes with larger |k| (only the first 20 modes are shown).