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Field-Space Entanglement Dynamics Between Tunnel-Coupled Luttinger Liquids

T0 review · 1 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read Within a Gaussian approximation, field-space entanglement between two tunnel-coupled Luttinger liquids is fully encoded in the Ermakov factors $\gamma_q(t)$, so mutual information, logarithmic negativity, and Rényi entropies are known…

desk verdict A technically careful Gaussian solution with new early-time and long-time results, but the headline scaling law sits exactly where the harmonic approximation is least controlled; worth a serious referee who pushes on the regime of validity. read the letter →

arxiv 2608.05968 v1 pith:TUHZOT6W submitted 2026-08-06 quant-ph cond-mat.quant-gascond-mat.stat-mechcond-mat.str-el

Léonce Dupays , Taufiq Murtadho , Bi Hong Tiang , Nelly H.Y. Ng , Paola Ruggiero This is my paper · ORCID
classification quant-phcond-mat.quant-gascond-mat.stat-mechcond-mat.str-el
keywords field-spaceentanglementLuttingerliquidErmakovequationlogarithmicnegativitymutualinformationRényientropiestime-dependenttunnelingGaussianapproximation
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Two gapless one-dimensional quantum fluids, initially independent and at zero or finite temperature, are brought into contact by a tunneling term acting uniformly along their whole length; the paper asks how entanglement between the fields themselves grows in time. The answer it establishes is that, within a Gaussian approximation to the tunneling interaction, every information-theoretic measure of field-space entanglement — mutual information, logarithmic negativity, and Rényi entropies — is a known function of the mode-dependent Ermakov factors $\gamma_q(t)$, one nonlinear ordinary differential equation per momentum mode. From that single ingredient the paper derives a zero-temperature early-time power law $I(A{:}B)\propto t^{n+1}$ in the thermodynamic limit, with $n$ the order of the first non-null derivative of the tunneling protocol at $t=0$, and exact long-time averages after the coupling saturates that depend only on the final coupling and the terminal values of $\gamma_q$ and its derivative. The reason to care is that the construction converts a nonequilibrium many-body question into one forced harmonic oscillator per mode, and all predictions are expressed in quantities that 1D Bose-gas simulators can in principle measure.

What carries the argument

The central object is the set of Ermakov factors $\gamma_q(t)$: positive mode-dependent scale factors solving $\ddot{\gamma}_q+\Omega_q(t)^2\gamma_q=\Omega_{q0}^2/\gamma_q^3$, with $\gamma_q(0)=1$ and $\dot{\gamma}_q(0)=0$, where $\Omega_q(t)^2=(vq)^2+4\omega J(t)$ is the instantaneous frequency of the antisymmetric sector. The two coupled fluids are first exchanged for symmetric and antisymmetric phase combinations; the symmetric sector stays a gapless Luttinger liquid, while the antisymmetric sector becomes a time-dependent gapped oscillator at each momentum. The mechanism that generates field-space entanglement is the difference between those two sectors, and the Ermakov factor exactly encodes it: it factorizes the antisymmetric symplectic propagator into a squeezing-and-shearing part built from $(\gamma_q,\dot\gamma_q)$ and a rotation part, and every entropic quantity depends only on $F[\gamma_q]$. Solving one ordinary differential equation per mode therefore replaces the full field-theoretic dynamics.

What would settle it

Solve the full time-dependent sine-Gordon model for a sudden quench at zero temperature and extract the early-time mutual information in the thermodynamic limit; the Gaussian theory predicts $I(A{:}B)\propto t$, so a robustly different power law would falsify the claim that the Ermakov factors control the entanglement dynamics.

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Extended reading notes

Core claim

The central claim is that field-space entanglement dynamics between two identical tunnel-coupled Luttinger liquids is solvable at the Gaussian level: for each positive momentum mode $q$, everything follows from the Ermakov equation $\ddot{\gamma}_q(t)+\Omega_q(t)^2\gamma_q(t)=\Omega_{q0}^2/\gamma_q^3(t)$ with $\gamma_q(0)=1$, $\dot{\gamma}_q(0)=0$, and $\Omega_q(t)^2=(vq)^2+4\omega J(t)$. Once the Ermakov factors are known, the symplectic eigenvalues that feed all correlation measures are fixed by the composite function $F[\gamma_q(t)]=\frac{1}{4}[(\dot{\gamma}_q/\Omega_{q0})^2+(\gamma_q-1/\gamma_q)^2]$, so mutual information, logarithmic negativity, and Rényi entropies are known at every time and temperature for any tunneling profile. The paper reads three generic behaviours off these expressions: zero-temperature early-time MI grows as $t^{n+1}$ in the thermodynamic limit (after a finite-size transient $t^{2n+2}$ with logarithmic corrections), long-time averages after the ramp saturates are completely fixed by $J_f$ and $\{\gamma_q(t_f),\dot\gamma_q(t_f)\}$, and finite-temperature logarithmic negativity is monotonically suppressed and vanishes above a protocol-dependent threshold. It also gives the adiabatic limit $\gamma_q^{\rm ad}(t)=\sqrt{\Omega_{q0}/\Omega_q(t)}$ for slow ramps. These statements are made within the harmonic expansion of the tunneling cosine, which the paper treats as the defining regime of its analysis.

Load-bearing premise

The load-bearing assumption is that the cosine of the relative phase in the tunneling coupling may be replaced by its quadratic expansion around a phase minimum; if this harmonic/Gaussian approximation fails during the ramp, the Ermakov-factor equations and all derived power laws and long-time averages no longer describe the physical model.

Editorial extensions

If this is right

  • At zero temperature and in the thermodynamic limit, early-time mutual information is a pure power law $I(A{:}B)\propto t^{n+1}$: linear for a sudden quench, quadratic for a linear ramp, cubic for a quadratic ramp, and so on.
  • After the coupling has saturated, the long-time averages of mutual information, logarithmic negativity, and Rényi entropies carry no memory of the ramp interior: only $J_f$ and the terminal pair $\{\gamma_q(t_f),\dot\gamma_q(t_f)\}$ enter.
  • For a zero-temperature sudden quench, the thermodynamic-limit averaged mutual information, logarithmic negativity, and Rényi-2 entropy all scale as $L\sqrt{4\omega J_f}/v$.
  • At finite temperature the averaged logarithmic negativity decreases monotonically and drops to zero at a threshold $T^*$; the averaged mutual information also decreases with temperature but remains positive.
  • For ramp durations much longer than the slowest-mode period $L/v$, mutual information and logarithmic negativity converge to the adiabatic evolution of the same Gaussian equations.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Not claimed by the paper: the power-law exponent $n+1$ offers a direct experimental fingerprint, since protocols whose first non-null derivative has order 0, 1, 2, ... should produce measured early-time MI exponents 1, 2, 3, ... in a 1D Bose-gas simulator.
  • An editorial stress test of the long-time reduction would compare two ramps with identical $J_f$, $\gamma_q(t_f)$, and $\dot\gamma_q(t_f)$ but different interiors; equal time-averaged correlations would confirm that the terminal Ermakov data are the only memory of the protocol.
  • The paper drops the zero mode $q=0$; at short system sizes or under near-uniform initial conditions this single mode could contribute visibly, so a calculation retaining the zero mode would bound the size of that correction.
  • The slow-ramp regime is where the harmonic approximation is least guaranteed; analyzing the same field-space entanglement measures in the full sine-Gordon dynamics would reveal whether the adiabatic predictions survive beyond the Gaussian regime.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

1 major / 4 minor

Summary. The manuscript develops a Gaussian-level theory of entanglement dynamics between two identical Luttinger liquids coupled by a time-dependent tunneling amplitude J(t). After a harmonic expansion of the cosine tunneling term, the model decouples into a gapless symmetric sector and a time-dependent gapped antisymmetric sector, with all mode dynamics encoded in solutions gamma_q(t) of an Ermakov equation. The authors derive analytical expressions for mutual information, logarithmic negativity, and Rényi entropies in terms of gamma_q(t), and from these obtain three main results: an early-time power law I(A:B) ~ t^{n+1} whose exponent is fixed by the first nonvanishing derivative of J(t) at t=0 (Eq. 43); exact long-time averages after saturation controlled by J_f and {gamma_q(t_f), dot_gamma_q(t_f)} (Tables II and III); and a characterization of the adiabatic limit. The analytical results are benchmarked against numerical solutions of the Ermakov equation for several protocols and temperatures.

Significance. Within the quadratic model, the paper is thorough and internally consistent: the appendices contain complete derivations, the symplectic-eigenvalue formulas are explicit, and Figs. 3-6 show good agreement between the approximate closed forms and numerical integration of the Ermakov equations. The reduction of all information-theoretic quantities to the single function F[gamma_q(t)] is elegant, and the early-time exponent and the L sqrt(4 omega J_f/v) scaling of the sudden-quench averages are concrete, falsifiable predictions of the Gaussian model. The central caveat, acknowledged in Sec. VII but not resolved there, is that the mapping from the physical sine-Gordon tunneling term to the quadratic Hamiltonian of Eq. (6) is not controlled in some regimes where predictions are made; if that mapping can be justified or tested, the framework would be a valuable benchmark for field-space entanglement in 1D quantum simulators.

major comments (1)
  1. [Sec. II A and Sec. IV, Eqs. (6) and (43)] The early-time power law is the headline physical prediction, but it is derived exactly in the regime where the harmonic approximation of the tunneling cosine is least controlled. Equation (6) is obtained by expanding cos(phi_A - phi_B) to quadratic order around a phase minimum; this is standard when the tunneling-induced mass is large relative to fluctuations. However, the scaling result (43) follows from Taylor-expanding J(t) around t=0 (Eq. (36)), i.e. for J(t) arbitrarily small, where the relative phase is essentially unpinned and the omitted non-Gaussian terms of the cosine are not small. Section VII concedes a possible breakdown in the adiabatic slow-ramp regime, but not in the early-time window. Since Eq. (43) is the central claim, the manuscript should either (i) provide a quantitative condition under which the quadratic expansion remains valid at early times, (ii) test the prediction against the full time-dependent sine-Gordon dynamics using, for example, the truncated-Wigner or self-consistent harmonic methods cited in Sec. VII, or (iii) explicitly limit the claim to the Gaussian model and avoid presenting the exponent as a property of tunnel-coupled Luttinger liquids.
minor comments (4)
  1. [Table III] The definition of M_q in Table III appears garbled and dimensionally inconsistent; it should read M_q = 1/2 [gamma_q^2(t_f) + (dot_gamma_q(t_f)/Omega_qf)^2 + (zeta_q/gamma_q(t_f))^2], consistent with Eq. (51), but the printed expression contains an extra factor of gamma_q^2(t_f).
  2. [Sec. IV (after Eq. (43))] The crossover time t_cut^(0) is written in a way that looks dimensionful; the statement should be made as the dimensionless ratio t_cut^(0)/t_0^* = v pi/(sqrt(omega J_f) L), as in the caption of Fig. 3.
  3. [Sec. II B] The removal of the zero mode q=0 is justified only heuristically by saying it is one out of a continuum of modes; for the finite-size expressions such as Eq. (42), an estimate of the q=0 contribution would be needed before the finite-L quantitative claims can be taken as physical.
  4. [Sec. VI] The convergence statements 'tf ~ 10 L/v for MI' and 'tf ~ L/v for LN' are based on visual overlap with the adiabatic solution; a quantitative error measure or a bound on the deviation would make these claims more robust.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the analytical results follow from the stated harmonic model; self-citations are auxiliary, not load-bearing.

full rationale

The paper's derivation chain is explicit: the harmonic approximation of the tunneling term (Eq. 6) decouples the symmetric and antisymmetric sectors into time-dependent oscillators; the Ermakov equation (Eq. 13) governs the mode dynamics; symplectic eigenvalues (Eqs. 23 and 31) then determine the mutual information, logarithmic negativity, and Rényi entropies (Eqs. 25 and 30). Each step is a stated mathematical consequence of the preceding one, and no parameter is fitted to the quantities being predicted. The early-time scaling I(A:B) ~ tau_n ~ t^{n+1} (Eq. 43) follows by Taylor expanding the input protocol J(t) (Eq. 36) and solving the Ermakov equation in that regime (Appendix C); the exponent is a derived consequence of the chosen protocol class, not an output used to define the protocol. The long-time averages depend on {gamma_q(tf), dot_gamma_q(tf)} through the exact post-ramp Ermakov solution (Eqs. 50-55 and Appendix D), again a theorem rather than an imposed equivalence. Self-citations appear in the paper: Ref. [44] is cited for experimental reach of field-space entanglement detection, Refs. [48] for experimental access to phase fields, and Refs. [66,67] for known Ermakov solutions used in numerical examples. These are auxiliary and do not carry the central scaling or averaging claims; the main derivations are self-contained within the stated Gaussian model and standard mathematical tools. The principal caveat, acknowledged in Sec. VII, is that the harmonic approximation of cos(phi_A - phi_B) may be uncontrolled when J(t) is small or in slow adiabatic ramps; that is a physical validity concern about the model-to-system mapping, not a circularity in the derivation. Accordingly, no specific circular step is identified; the score of 1 simply reflects the presence of minor self-citations that are not load-bearing.

Assumptions & free parameters 1 free parameters · 7 assumptions · 0 invented entities

The central claims rest on standard mathematical tools (Ermakov equation, Gaussian symplectic formalism) and on the physical modeling assumptions listed above. No new particles or forces are postulated. The only free parameter counted is the UV cutoff, which controls the threshold temperature; the other parameters (density, speed of sound, Luttinger parameter, temperature, coupling strength) are inputs from the experimental setup rather than fitted values.

free parameters (1)
  • UV momentum cutoff Lambda = Lambda = xi^{-1} (about 4.2 inverse micrometers for the 87Rb parameters)
    Introduced to regularize momentum sums and define the threshold temperature T* in Eq. (59). The choice is physically motivated by the healing length, but T* is cutoff dependent, as the paper acknowledges in Sec. V C.
assumptions (7)
  • standard math Ermakov equation with unit Wronskian and Pinney solution (Ref. [53])
    Used to solve the quadrature dynamics in Eq. (13) and to express post-ramp solutions in Appendix D.
  • standard math Gaussian state covariance matrix and symplectic eigenvalue formalism (Ref. [54])
    Underlies all entropy, mutual information, and logarithmic negativity formulas in Secs. II C and III.
  • domain assumption Luttinger liquid hydrodynamic description is valid at low energies (Refs. [45-47])
    Justifies the harmonic Hamiltonian of Eq. (2) for the phase and density fluctuations.
  • domain assumption Harmonic approximation of the sine-Gordon coupling replaces cos(phi_A - phi_B) by a quadratic term (Eq. (6))
    The central modeling step. It is expected to be accurate when tunneling dominates fluctuations, but the authors note in Sec. VII that it may break down for slow adiabatic ramps.
  • domain assumption The two Luttinger liquids share identical speed of sound v and Luttinger parameter K
    Required for the A-B exchange symmetry that decouples the symmetric and antisymmetric sectors in Eq. (6). The authors discuss the broken-symmetry case as future work in Sec. VII.
  • domain assumption Initial state is thermal and diagonal in the energy basis, with J(0) = 0
    Sets the initial covariance matrix in Eq. (18) and the tensor-product structure in the plus/minus basis.
  • domain assumption Zero mode q = 0 is dropped from the mode expansion
    Sec. II B argues it is one mode among a continuum and should not significantly affect entanglement, but no quantitative estimate is given.

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Pith. "Pith review of Field-Space Entanglement Dynamics Between Tunnel-Coupled Luttinger Liquids." pith.science (2026). https://pith.science/paper/TUHZOT6W

@misc{pith2026260805968,
  author       = {Pith},
  title        = {Pith review of: Field-Space Entanglement Dynamics Between Tunnel-Coupled Luttinger Liquids},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/TUHZOT6W}},
  note         = {Machine review of arXiv:2608.05968}
}
read the original abstract

Entanglement dynamics depend not only on how a quantum system is partitioned, but critically on how interactions across that partition are structured. For a spatial bipartition of a locally interacting system, entanglement is generated near the boundary and then propagates into the bulk. By contrast, when two extended quantum fields are coupled locally along their entire length, the interaction crosses the field-space partition everywhere, and this generates correlations throughout the system. Here, we study the entanglement dynamics between two gapless one-dimensional quantum many-body systems described by Luttinger liquid theory. The systems are initially decoupled and prepared at zero or finite temperature, after which a time-dependent tunneling interaction is activated uniformly along their length. Within a Gaussian approximation, we derive general analytical expressions for the logarithmic negativity, mutual information, and R\'enyi entropies under arbitrary coupling protocols. At zero temperature, entanglement displays an early-time power-law growth whose exponent is fixed solely by the first non-null derivative of the tunneling protocol. Once the coupling saturates, we obtain exact long-time averages of the information-theoretic quantities and characterise how the correlations scale with temperature and the final coupling strength. We also analyse how mutual information and logarithmic negativity approach the adiabatic limit for a very slow protocol with respect to intrinsic system timescale. This work extends the study of entanglement dynamics in nonequilibrium field theory to field-space partitions and mixed initial states.

Figures

Figures reproduced from arXiv: 2608.05968 by the authors.

Figure 1
Figure 1. FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figures from the paper (5 more)
Figure 4
Figure 4. Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p010_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5 [PITH_FULL_IMAGE:figures/full_fig_p012_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6 [PITH_FULL_IMAGE:figures/full_fig_p013_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7 [PITH_FULL_IMAGE:figures/full_fig_p015_7.png]
Figure 8
Figure 8. Figure 8: shows the time evolution of the Rényi-2 entropy following a sudden quench. Similar to the MI and LN, the Rényi-2 entropy exhibits oscillatory behavior. As the temperature increases, its time-averaged value is shifted upward. This increase can be attributed to the enhan…

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    in the main text for the propagator in the antisymmetric sec tor G− q (t) = ( γ−1 q (t) − ˙γq(t)/ω 0 γq(t) ) /bracehtipupleft /bracehtipdownright/bracehtipdownleft/bracehtipupright Uq(t) ( cosθq(t) (Ω q0/ω) sinθq(t) −(ω/Ωq0) sinθq(t) cos θq(t) ) /bracehtipupleft /bracehtipdown...

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    at t = 0 . Consider the decomposition of dimensionless mode quadrat ures for q >0 in terms of bosonic creation and annihilation operators ˆφσ q = √ ω 2Ωq0 [ˆbσ q + h.c.], δ ˆnσ q = √ Ωq0 2ω [iˆbσ q + h.c.], (B16) withσ ∈ {+, −} and the bosonic operators satisfy the commutation...

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    of the main text Γ++ q (0) = Γ−− q (0) = ( Ωq0/ω 0 0 ω/Ωq0 ) Cq(β). (B20) Using the above thermal initial state and the propagators ( B13) and ( B14) derived in the previous section, we observe that the symplectic rotation part of the evolution leaves the initia l covariance m...

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    in the main text] Γq(t) = 1 2 ( 1 1 1 −1 )(Γ++ q (t) 0 0 Γ−− q (t) )( 1 1 1 −1 ) = 1 2 (Γ++ q (t) + Γ−− q (t) Γ++ q (t) − Γ−− q (t) Γ++ q (t) − Γ−− q (t) Γ++ q (t) + Γ−− q (t) ) . (B22) By matching the elements, we find ΓAA q (t) = ΓBB q (t) = 1 2 [Γ++ q (t) + Γ−− q (t)] = 1 2 ...

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    Symplectic eigenvalues and the dynamics of the mutual inf ormation The dynamics of quantum information-theoretic quantities for Gaussian states can be extracted from the symplectic eig en- values of the covariance matrix. For a single-mode covarian ce matrix, the symplectic ei...

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    V B of the main text

    Dynamics of the logarithmic negativity Here we derive the general result on the dynamics of entangle ment as quantified by the logarithmic negativity (LN), prese nted in Subsec. V B of the main text. The LN is determined by the symplectic eigen values of the partially transpose...

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    Denesting two nested square roots : One can check that V1 ≥ 0 and 0 ≤V2 ≤V 2 1 . Hence, applying the denesting identity for two nested square roots, we obtain √ V1 ± √ V2 = √ V1 + √ V 2 1 −V2 2 ± √ V1 − √ V 2 1 −V2 2 , (B35) where we use the denesting identity for two nested s...

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    Evaluation of V 2 1 −V2: Directly evaluating the terms give V 2 1 −V2 = 1 4 [ 4 + 4 ( ˙γqγq Ωq0 )2] − ( ˙γqγq Ωq0 )2 = 1. (B36)

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    ( B36), we haveV1 ± √ V 2 1 −V2 = 1 2 [ ( 1 γq ±γq )2 + ( ˙γq Ωq0 )2]

    Evaluation of V1 ± √ V 2 1 −V2: Using Eq. ( B36), we haveV1 ± √ V 2 1 −V2 = 1 2 [ ( 1 γq ±γq )2 + ( ˙γq Ωq0 )2] . Combining all results gives rise to ν± q (t) = Cq(β) √ V1 ± √ V2 =Cq(β)   √ V1 + √ V 2 1 −V2 2 ± √ V1 − √ V 2 1 −V2 2   (B37) = Cq(β) 2   √ ( ˙γq(t) Ωq0 )2 +...

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    Early-time expansion of the symplectic eigenvalues for a sudden quench For a sudden quench, we have the exact solution of the Ermakov equation (cf. Eq. ( 35)):γq(t) = √ 1 − ( 1 −ζ2q ) sin2(Ωqft), whereζq ≡ Ωq0/Ωqf . At early times Ωqft ≪ 1, Taylor expansion yields γq(t) ≈ √ 1 ...

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    ( 13), useful to obtain the early-time expansion of the Ermakov factor for general protocols

    Early-time expansion of the symplectic eigenvalues for a smooth protocol In this section, we provide an integral representation to th e Ermakov Eq. ( 13), useful to obtain the early-time expansion of the Ermakov factor for general protocols. We first rearrange Eq. (13) into ¨γq...

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    Early-time expansion of the mutual information (MI) at ze ro initial temperature In the early-time regime, the symplectic eigenvalue at zero -temperature is approximatelyλq(t,β → ∞) ≈ 1 2 √ 1 + ( εn(t) q )2 , as obtained in the β → ∞ limit of Eq. ( C19). This form allows us to...

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    Indeed, the discretisation of the sum is performed with the spacing ∆q =qmin =π/L

    Phase 1: The time interval for which εn(t) ≪qmin In this case, we cannot replace the discrete sum with an integ ral. Indeed, the discretisation of the sum is performed with the spacing ∆q =qmin =π/L. So that, to go from the sum to integral representation ∑ q>0 f (εn q ) → L π ...

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    I(A :B,t ) = 2L π ∫ ∞ 0 S [λq(t,β )] dq ≈ 2L π ∫ Λ 0 S [ 1 2 √ 1 + ε2 n(t) q2 ] dq + 2L π ∫ ∞ Λ S [λq(t,β )] dq (C29) ≈ 2L π ∫ ∞ 0 S [ 1 2 √ 1 + ε2 n(t) q2 ] dq

    Phase 2: Thermodynamic-Limit Behaviour ( εn(t) ≫qmin) To compute the MI, we can use the approximate expression for t he symplectic eigenvalues, i.e. I(A :B,t ) = 2L π ∫ ∞ 0 S [λq(t,β )] dq ≈ 2L π ∫ Λ 0 S [ 1 2 √ 1 + ε2 n(t) q2 ] dq + 2L π ∫ ∞ Λ S [λq(t,β )] dq (C29) ≈ 2L π ∫ ∞...

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    We estimate the regime in which τn = εn(t)/qmin ≪ 1, with qmin = π/L

    Crossover from ultra-early time to early-time regime We now provide an estimate of the time at which the crossover f rom one regime to the other happens for different system sizes, in the case of the sudden quench and the linear protoco l. We estimate the regime in which τn = ...

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    For a sudden quench, ε0(t) = 2ωJf t v , t∗ 0 = (4ωJf )−1/2

    The time t(n) cut corresponds to the value of t for which the approximation τn ≪ 1 breaks down. For a sudden quench, ε0(t) = 2ωJf t v , t∗ 0 = (4ωJf )−1/2. The condition τ0 =ε0(t)/qmin ≪ 1, givest(0) cut = vπ √ ωJf Lt∗ 0. System size linear protocol Sudden quench L t(1) cut/t∗...

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    Letv,q,ω,J f > 0, and suppose that J(t) = Jf for allt ≥tf

    Exact solution to the Ermakov equation after saturation Theorem 1. Letv,q,ω,J f > 0, and suppose that J(t) = Jf for allt ≥tf . Define Ωq0 =vq, Ωqf = √ (vq)2 + 4ωJf, ζ q = Ωq0 Ωqf . (D1) F or eachq, letγq(t) solve ¨γq(t) + Ω2 qfγq(t) = Ω2 q0 γ3q (t) , withγq(0) = 1 and ˙γq(0) = ...

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    ( 23), that is expressed in terms of the Ermakov factor Eq

    The expression for the symplectic eigenvalues In this subsection, from the knowledge of γq(t ≥ tf ) for the finite-ramp, we derive an exact expression for the sym plectic eigenvalue given by Eq. ( 23), that is expressed in terms of the Ermakov factor Eq. ( 24). From the previou...

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    We conclude this appendix by collecting the inequalities ob eyed by the coefficients parametrizing the post-ramp symple ctic eigenvalues

    Properties of Mq, Nq, Dq and Eq. We conclude this appendix by collecting the inequalities ob eyed by the coefficients parametrizing the post-ramp symple ctic eigenvalues. These inequalities are used in the next two sec tions to satisfy the assumptions required by the integral t...

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    LetA ≥ 1 2 andB ≥ 0

    Proof of the core integral formula for computing averaged MI Theorem 3. LetA ≥ 1 2 andB ≥ 0. Define the function S[x] ≡ ( x + 1 2 ) ln ( x + 1 2 ) − ( x − 1 2 ) ln ( x − 1 2 ) . Then ∫ π/2 0 S ( A √ 1 +B sin2u ) du =πA √ 1 +B ( F [ arcsin ( 1 2A ) , 1√ 1 +B ] −E [ arcsin ( 1 2A...

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    Averaged MI formula for a general protocol at finite and zer o temperature In this section, we derive the expression for the MI followin g a finite-ramp protocol at finite temperature. In the next sec tion, we will recover the expression for the MI after a general prot ocol in th...

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    (E25) From Theorem

    Observe that the first term is of the form: ∫ π/2 0 S [λq(u)] du = ∫ π/2 0 S [ C′ q × √ 1 + Eq Dq sin2u ] du. (E25) From Theorem. 2, we know that Dq ≥ 1. So, C′ q ≥Cq ≥ 1 2 . Combining this observation with Theorem. 3, we obtain ∫ π/2 0 S [λq(u)] du =πC ′ q √ 1 + Eq Dq  F  ...

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    The second term is a time-independent constant, which is g iven by S[λ+ q (0)] = Cq ln ( 2Cq+1 2Cq−1 ) + 1 2 ln (4C 2 q −1 4 ) . Combining these two terms, we get I(A :B) = 4 ∑ q>0 { C′ q √ 1 + Eq Dq ( F [ arcsin ( 1 2C′q ) , √ Dq Dq +Eq ] −E [ arcsin ( 1 2C′q ) , √ Dq Dq +Eq ...

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    Averaged MI formula for a sudden quench and finite and zero t emperature For a sudden quench at finite temperature, we simply set Dq = 1,E q =Aq. So, Eq. ( E27) reduces to I(A :B,β ) = 4 ∑ q>0 { Cq √ 1 +Aq ( F [ arcsin ( 1 2Cq ) , 1√ 1 +Aq ] −E [ arcsin ( 1 2Cq ) , 1√ 1 +Aq ]) +...

  94. [102]

    For simplicity, we define q = √ 4ωJf v x, so that Aq = 1 4x2(1 +x2)

    The scaling law of averaged MI for sudden quench at zero tem perature We now compute the scaling constant of the averaged MI after a sudden quench at zero-temperature limit and thermodynamic limit. For simplicity, we define q = √ 4ωJf v x, so that Aq = 1 4x2(1 +x2) . Thus, the ...

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    Letx> 1 2 and definef (x) = x ln ( x+1/2 x−1/2 )

    Decay of averaged MI with temperature for arbitrary proto cols Lemma 4. Letx> 1 2 and definef (x) = x ln ( x+1/2 x−1/2 ) . Then f (x) is a strictly decreasing function on [ 1 2, ∞). Proof. We calculate the derivative off (x) asf ′(x) = ln ( 2x+1 2x−1 ) − 4x 4x2−1 . Since x> 1/2...

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    (F1) 38 Lemma 6

    Proof of the core integral formula for computing averaged LN after a sudden quench Before we prove the key integral formula, let’s first prove two useful integral identities, which (including many other quantities in this section) are directly dependent on the dilogarithm f unc...

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    (F16) Next, using the definition of x, we have arcsinh ( sinx √ A ) = arcsinh(B)

    The second term: Direct calculation yields dx dA = 1√ 1 −B2/A ( − 1 2 B A3/2 ) = −B 2A √ A −B2. (F16) Next, using the definition of x, we have arcsinh ( sinx √ A ) = arcsinh(B). So, − [ arcsinh ( sinx √ A )] dx dA = B 2A √ A −B2 arcsinh(B). (F17)

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    (F18) Lett = cosφ anddt = − sinφdφ

    The first term: The integral is ∫ π/2 x ∂ ∂A [ arcsinh ( sinφ √ A )] dφ = ∫ π/2 x 1 2 √ A sinφ√ 1 +A sin2φ dφ. (F18) Lett = cosφ anddt = − sinφdφ . Then it becomes ∫ π/2 x [ 1 2 √ A sinφ√ 1 +A sin2φ ] dφ = ∫ cos x 0 1 2 √ A √ 1 +A(1 −t2) dt. (F19) Letk = √ A 1 +A . Then ∫ cos x...

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    The second term of Eq. (F22): Consider the integral ∫ A B2 B 2α √ α −B2 arcsinh(B)dα = arcsinh(B) ∫ θq 0 B 2(B2 sec2θ)(B tanθ) (2B2 sec2θ tanθ)dθ (F23) = arcsinh(B)θq , (F24) where we use the substitution α = B2 sec2θ anddα = 2B2 sec2θ tanθdθ . Note that we also define cosθq = ...

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    (F22): Define θ = arcsin (√ α−B2 1+α ) , and dv = 1 2αdα,v = 1 2 lnα

    The first term of Eq. (F22): Define θ = arcsin (√ α−B2 1+α ) , and dv = 1 2αdα,v = 1 2 lnα. Using this definition of θ,v and integration by parts, we get ∫ A B2 1 2α arcsin (√ α −B2 1 +α ) dα = [ 1 2 lnα arcsin (√ α −B2 1 +α )] A B2 − 1 2 ∫ D 0 lnα(θ)dθ (F26) = 1 2 lnA arcsin (√ ...

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    Similar to the construction of Eq

    Integral representation of long-time averaged LN for a fin ite-ramp at finite temperature First, the LN at time t is given by EN (t) = ∑ q>0 max{0, − ln[2νq(t,β )]}, where νq(t,β ) = Cq (√ 1 + F [γq(t)] − √ F [γq(t)] ) . Similar to the construction of Eq. ( E24), the time averag...

  102. [110]

    The time-averaged LN then reduces to EN = 2 π ∑ q>0 ∫ π/2 0 max { 0, arcsinh (√ Aq sinu ) − ln(2Cq) } du

    Exact expression of averaged LN for sudden quench at finite temperature In the sudden quench limit, we set Dq = 1 andEq =Aq. The time-averaged LN then reduces to EN = 2 π ∑ q>0 ∫ π/2 0 max { 0, arcsinh (√ Aq sinu ) − ln(2Cq) } du. (F40) To contribute nonzero entanglement, we re...

  103. [111]

    Consequently, this momentum mode gives zero contribution to the entanglement, i.e

    Aq <V 2 q : In this case, we have √ Aq sinu ≤ √ Aq <V q for allu ∈ [0,π/ 2]. Consequently, this momentum mode gives zero contribution to the entanglement, i.e. ∫ π/2 0 max { 0, arcsinh (√ Aq sinu ) − ln(2Cq) } du = 0. (F42) 42

  104. [112]

    To exhibit entanglement, the time parameteru for momentum mode q must exceed this threshold

    Aq ≥V 2 q : We define the threshold time for each mode as u∗ q = arcsin ( Vq √ Aq ) . To exhibit entanglement, the time parameteru for momentum mode q must exceed this threshold. Therefore, we have ∫ π/2 0 max { 0, arcsinh (√ Aq sinu ) − ln(2Cq) } du = ∫ π/2 u∗ q [ arcsinh (√ A...

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    In the zero temperat ure limitβ → ∞, the time-averaged LN reduces to EN (β → ∞) = 2 π ∑ q>0 ∫ π/2 0 max { 0, arcsinh (√ Aq sinu )} du

    Exact expression of averaged LN for sudden quench in the ze ro-temperature limit Let us consider the sudden quench limit. In the zero temperat ure limitβ → ∞, the time-averaged LN reduces to EN (β → ∞) = 2 π ∑ q>0 ∫ π/2 0 max { 0, arcsinh (√ Aq sinu )} du. (F47) Since√ Aq is p...

  106. [114]

    From the previous subsectio n, we have EN = 2 π ∑ q>0 ∫ π/2 0 max { 0, arcsinh (√ Aq sinu ) − ln coth (βℏΩq0 2 )} du

    Exact expression for entanglement threshold temperatur e for a sudden quench In this section, we would like to derive an expression for the threshold temperature at which the LN is null, signature of t he vanishing of the entanglement. From the previous subsectio n, we have EN...

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