Quench dynamics of negativity Hamiltonians

TL;DR

The paper develops a hydrodynamic quasiparticle framework to study the out-of-equilibrium dynamics of negativity in 1D free-fermion quenches, focusing on a tripartite geometry. It derives analytic expressions for both the standard negativity Hamiltonian and the fermionic negativity Hamiltonian, revealing a two-term decomposition into a mixed part tied to the entanglement Hamiltonian and a non-Gaussian pure part (standard) or a quadratic pure part (fermionic). Despite structural differences, the logarithmic negativity and symmetry-resolved negativity are shown to coincide at the ballistic scaling, and the long-time limit approaches a generalized Gibbs ensemble. Numerical checks corroborate the analytic predictions, including light-cone spreading and the characteristic off-diagonal eigenvalues, supporting the robustness of the quasiparticle picture for operatorial entanglement measures in integrable systems.

Abstract

In this paper, we investigate the quench dynamics of the negativity and fermionic negativity Hamiltonians in free fermionic systems. We do this by generalizing a recently developed quasiparticle picture for the entanglement Hamiltonians to tripartite geometries. We obtain analytic expressions for these quantities which are then extensively checked against previous results and numerics. In particular, we find that the standard negativity Hamiltonian contains both non-local hopping terms and four fermion interactions, whereas the fermionic version is purely quadratic. However, despite their marked difference, we show that the logarithmic negativity obtained from either are identical in the ballistic scaling limit, as are their symmetry resolution.

Figures (6)

• Figure 1: Quasiparticle contributions to the reduced density matrix in a tripartite geometry. In the notation of \ref{['eq:densitymatrixdecomposition']}, we see the two main contributions which will determine negativity and entanglement Hamiltonians, namely a pure part coming from pairs which are shared between the two subsystems, $\tilde{\rho}_{\rm pure}$ and a mixed part coming from pairs shared between the full $A$ and its complement, $\rho_{\rm mixed}$. Moreover, we have the fully pure contribution, $\rho_{\rm pure}$ which will have no contribution to the relevant entangling quantities.
• Figure 2: Time dependence of $\log R_n$ for two subsystems of length $\ell=800$ placed at distance $d=800$, in a quench from a dimer state. The symbols are the exact solution, which is obtained by evaluating \ref{['eq:renyinegativitiesdefinition']} using the eigenvalues of the exact negativity Hamiltonian, computed with the methods discussed in section \ref{['sec:numerics']}, while the continuous line represents the theoretical prediction, which corresponds to the first line of \ref{['eq:renyi_negativity']}. As expected, $\log R_2(t)$ is always zero, as $\tilde{s}_2(k) = 0$.
• Figure 3: Mixed part of the negativity Hamiltonian (entanglement Hamiltonian) for two intervals of equal lengths $\ell = 800$ placed at a distance $d=600$ (in lattice sites). The symbols are the exact solution \ref{['eq:entanglement_ham']} evaluated at different times, while the dashed lines are the quasiparticle predictions \ref{['eq:realspaceeh']}. While at small times the result is exactly twice the one for a single interval since there is a maximal speed of propagation $v_{max}=1$, as $t>d$ the presence of the other interval causes an asymmetry which is contained in the counting function of \ref{['eq:realspaceeh']}. Here we consider $j\in A_2$.
• Figure 4: Diagonal part of the negativity Hamiltonian for two intervals of lengths $\ell=1000$ placed at distance $d=600$, with $j\in A_1$ . The dashed lines represent the analytical prediction \ref{['eq:diagpart']}, while the symbols are obtained from the exact solution using \ref{['eq:exactN']} and are normalized subtracting $K_A$, which is itself found exactly from \ref{['eq:ent_Ham']}. We see precisely the expected light cone structure arising as the time crosses the value of $d/2$, with correlations emerging only after the quasiparticle pairs are able to couple the two intervals.
• Figure 5: Sorted eigenvalues of the non-diagonal component of the negativity Hamiltonian, normalized by the factor $i \frac{\pi}{2}$, for intervals of equal length $\ell=600$ placed at distance $d=600$. The numerical analysis confirms the theoretical expectation that the eigenvalues of the off-diagonal part are only 0, $\pm \frac{i\pi}{2}$. As expected, at short times the eigenvalues are all zero, while at different times the only variation is provided by the number of nonzero eigenvalues, which initially grows and then decreases back to zero in the infinite time limit.