arxiv: 2605.01595 · v1 · submitted 2026-05-02 · ❄️ cond-mat.stat-mech · cond-mat.quant-gas· cond-mat.str-el· hep-th· quant-ph

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Entanglement dynamics after quenches with inhomogeneous Hamiltonians

Andrea Di Pasquale , Federico Rottoli , Vincenzo Alba

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Pith reviewed 2026-05-09 17:34 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech cond-mat.quant-gascond-mat.str-elhep-thquant-ph

keywords entanglement entropyquantum quenchesinhomogeneous Hamiltoniansquasiparticle picturetransmission coefficientXX chainIsing chainXXZ chain

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The pith

The long-time entanglement entropy after quenches in inhomogeneous Hamiltonians is determined by the quasiparticle transmission coefficient at the interface.

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

This paper derives exact analytical expressions for the entanglement entropy between two regions in systems with inhomogeneous Hamiltonians after a quench, in the long-time hydrodynamic limit. The formulas come from a quasiparticle picture in which particles scatter at the boundary between the left and right homogeneous regions, with the amount of entanglement set by how much they transmit across. A reader might care because this shows how spatial variations in the Hamiltonian, common in real materials, limit the spread of quantum correlations. The derivation uses the transmission coefficient solved from the stationary lattice Schrödinger equation, and the bounded dispersion relation explains why strong inhomogeneity reduces both particle transport and entanglement production.

Core claim

In the hydrodynamic limit of long times, fermions or spins incident on the interface between two regions with different homogeneous Hamiltonians scatter, and the entanglement entropy is controlled by the transmission coefficient obtained analytically from the stationary lattice Schrödinger equation. This quasiparticle scattering picture applies to the XX chain and the transverse-field Ising chain, leading to suppressed entanglement growth under strong inhomogeneity due to the finite bandwidth of the dispersion. Numerical checks confirm the analytics, and tDMRG simulations of the interacting XXZ chain show similar suppression of entanglement growth when transport is blocked, though some time-

What carries the argument

The transmission coefficient obtained by solving the stationary lattice Schrödinger equation, which dictates the fraction of quasiparticles that transmit and thereby generate entanglement between reflected and transmitted modes.

Load-bearing premise

That the entanglement can be fully described by independent quasiparticles scattering at the interface according to the transmission coefficient, even in the long-time limit of the inhomogeneous system.

What would settle it

A numerical time evolution of the XX chain with a strong inhomogeneity where the measured entanglement entropy growth rate differs from the value predicted by the analytical transmission coefficient formula at large times.

Figures

Figures reproduced from arXiv: 2605.01595 by Andrea Di Pasquale, Federico Rottoli, Vincenzo Alba.

**Figure 1.** Figure 1: (a) Cartoon of the setup employed in this work. The system is divided into two parts L, R, which are prepared in two different initial states. The system then evolves in time under an inhomogeneous Hamiltonian H = HL +HR, with HL and HR acting nontrivially in the left and the right parts of the system, respectively. Here HL/R describe the XXZ chain and the transverse-field Ising chain. We consider initial … view at source ↗

**Figure 2.** Figure 2: Transmission coefficients T(k) for the inhomogeneous Ising and XX chains (left and right panels, respectively) as a function of the momentum kL. We consider the setup of view at source ↗

**Figure 3.** Figure 3: Growth of the entanglement entropy after a quench to an inhomogeneous Hamiltonian. In the main plot we report the evolution of the half chain entanglement entropy after a quench from the ground state of the Ising chain with transverse field h0 = 3.0 to the inhomogeneous Ising chain with magnetic field hL = 5.0 on the left j ⩽ 0 and hR = 4.0 on the right j > 0 in a system of total size L = 1000. The half-sy… view at source ↗

**Figure 4.** Figure 4: Slope of the linear growth of entanglement entropy after a quench from ground state of the Ising model with h0 = 3.0. (a) Quench to inhomogeneous Ising model with hL = 3.0, as a function of hR. The symbols are the numerical results for the slopes obtained through a scaling analysis and the solid line is the theoretical prediction in Eq. (9). (b) Quench to inhomogeneous Ising model with hL = 5.0. Again the … view at source ↗

**Figure 5.** Figure 5: Slopes of the linear growth of the entanglement entropy after a quench from the state (58) in the XX chain with hL = 0 as a function of hR. The symbols are the numerical results for the slopes and the solid line is the theoretical prediction in Eq. (9). We observe a perfect agreement between the prediction and the numerics. 4 Conclusions We have investigated entanglement dynamics under inhomogeneous Hamilt… view at source ↗

**Figure 6.** Figure 6: Entanglement entropy as a function of time for a quench from the domain wall Néel state (58) to the inhomogeneous XXZ chain with ∆ = 0.7 and magnetic field on the left hL = 0. The symbols are the numerical results for the entropies for different values of the magnetic fields on the right hR = 1.6, 2.0, 2.4, and 3.0. The simulations have been performed with TEBD algorithm with maximum bond dimension χ = 800… view at source ↗

**Figure 7.** Figure 7: Entanglement entropy as a function of time for a quench from the domain wall Néel state to the inhomogeneous XXZ chain with ∆ = 2.0 and magnetic field on the left hL = 0. The symbols are the numerical results for the entropies for two values of the magnetic field on the right hR = 3.0 and 5.0 The simulations have been performed with TEBD algorithm with different values of the maximum bond dimension χ depen… view at source ↗

read the original abstract

We investigate entanglement dynamics in bipartite systems governed by inhomogeneous Hamiltonians of the form $H = H_L + H_R$, where $H_{L/R}$ acts only on the left or right region and is homogeneous within each region. Focusing on the XX chain and the transverse-field Ising chain, we derive analytical formulas for the entanglement entropy between the two regions in the hydrodynamic limit of long times. In this regime, fermions incident on the interface undergo scattering, generating entanglement between reflected and transmitted modes. The resulting quasiparticle picture is controlled by the transmission coefficient, which we obtain analytically by solving the stationary lattice Schr\"odinger equation. Due to the bounded dispersion, strong inhomogeneity suppresses both transport and entanglement growth. We benchmark our analytical predictions against numerical simulations in paradigmatic setups. Finally, we extend the analysis to the interacting XXZ chain using tDMRG. The numerical data show qualitative agreement with the quadratic case: entanglement growth remains suppressed in the strongly inhomogeneous limit. Notably, however, entanglement continues to increase even when transport is suppressed, at least at intermediate times.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper derives usable analytical formulas for long-time entanglement entropy in free inhomogeneous quenches, controlled by a lattice-derived transmission coefficient, with only qualitative support for the interacting case.

read the letter

The main point is that for quadratic models like the XX chain and transverse-field Ising model with a sharp left-right interface, you can write closed expressions for the entanglement entropy between the two sides in the long-time limit. The formulas are built around a transmission coefficient T(k) obtained by solving the stationary single-particle Schrödinger equation on the lattice, then fed into the usual quasiparticle counting. This is a direct extension of the homogeneous-quench literature and gives a concrete way to see how strong inhomogeneity suppresses both transport and entanglement growth because of the bounded dispersion relation. The derivations look straightforward and the numerical benchmarks for the free cases line up well, which is the strongest part of the work. The physical picture is that reflected and transmitted modes generate the entanglement at the interface and then propagate ballistically on each side. For the interacting XXZ extension they switch to tDMRG and report only qualitative agreement: entanglement still grows, albeit slowly, even when transport is suppressed. That observation is interesting but remains at the level of trends rather than precise matching. On the question of whether the single transmission coefficient misses extra interface correlations or multiple-scattering effects, the free-fermion numerics do not show obvious deviations, so the reduction appears to work for those models. The interacting data are too coarse to settle the same question there. This is aimed at people who study entanglement dynamics in inhomogeneous or engineered quantum systems, especially those who want analytical handles on the free case before turning to numerics. The analytical content and the clean benchmarks are enough to justify sending it to referees.

Referee Report

0 major / 3 minor

Summary. The paper investigates entanglement dynamics after quenches in bipartite spin chains with inhomogeneous Hamiltonians of the form H = H_L + H_R, where each half is homogeneous internally. For quadratic models (XX and transverse-field Ising chains), analytical formulas for the long-time entanglement entropy between left and right regions are derived in the hydrodynamic limit via a quasiparticle picture in which scattering at the interface is controlled by the transmission coefficient T(k) obtained from the exact solution of the stationary lattice Schrödinger equation. These predictions are benchmarked against numerical simulations; the analysis is extended to the interacting XXZ chain via tDMRG, where entanglement growth is found to remain suppressed for strong inhomogeneity (though it continues at intermediate times even when transport is suppressed).

Significance. If the results hold, the work supplies a controlled analytical framework for entanglement spreading across interfaces in inhomogeneous integrable systems, directly linking the transmission coefficient to the hydrodynamic entanglement growth and explaining its suppression by bounded dispersion. The derivation of explicit analytical expressions for free models together with direct numerical benchmarks constitutes a clear strength; the qualitative consistency observed in the interacting case broadens the potential relevance to non-integrable dynamics.

minor comments (3)

The hydrodynamic limit is invoked repeatedly for the validity of the quasiparticle formula; a brief explicit statement of the time and length scales (relative to the inhomogeneity strength) at which the reduction to T(k) becomes accurate would improve clarity.
In the XXZ section, the statement that entanglement 'continues to increase even when transport is suppressed, at least at intermediate times' would benefit from a quantitative indication of the time window relative to system size or light-cone arrival.
Figure captions and the main text should consistently label the inhomogeneity parameter (e.g., the ratio of couplings or fields across the interface) so that readers can directly map the plotted curves to the analytic T(k) expressions.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive and constructive report, which accurately summarizes our main results on entanglement suppression due to interface scattering in inhomogeneous quenches. We appreciate the recommendation for minor revision and will incorporate any suggested clarifications or minor improvements in the revised manuscript.

Circularity Check

0 steps flagged

No significant circularity; derivation uses independent single-particle scattering solution

full rationale

The central derivation computes the transmission coefficient T(k) by directly solving the stationary lattice Schrödinger equation at the interface, an independent single-particle problem unrelated to the many-body entanglement formula. The hydrodynamic entanglement entropy is then expressed via the standard quasiparticle picture in terms of this T(k), with explicit benchmarking against numerical tDMRG and exact diagonalization data. No parameters are fitted from entanglement observables, no self-citations form a load-bearing chain, and no step reduces to another by definition or renaming. The approach remains self-contained for the free models and shows only qualitative extension to the interacting case.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the applicability of the quasiparticle picture to the inhomogeneous case and on the validity of the long-time hydrodynamic limit; no free parameters or new entities are introduced.

axioms (2)

domain assumption The quasiparticle picture with scattering of fermions into reflected and transmitted modes at the interface remains valid for inhomogeneous Hamiltonians.
Invoked to link the transmission coefficient to entanglement generation.
domain assumption The hydrodynamic limit of long times applies, allowing entanglement to be determined by the transmission coefficient alone.
Used to obtain the analytical formulas for entanglement entropy.

pith-pipeline@v0.9.0 · 5500 in / 1415 out tokens · 61998 ms · 2026-05-09T17:34:27.122210+00:00 · methodology

discussion (0)

Reference graph

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