Equivalence of spatial and particle entanglement growth after a quantum quench

We analyze fermions after an interaction quantum quench in one spatial dimension and study the growth of the steady state entanglement entropy density under either a spatial mode or particle bipartition. For integrable lattice models we find excellent agreement between the increase of spatial and particle entanglement entropy, and for chaotic models an examination of two further neighbor interaction strengths suggests similar correspondence. This result highlights the applicability of a statistical ensemble to compute expectation values of local observables after a quantum quench

Figure 1

Two types of reduced density matrices after a quantum quench. A quantum system of interacting fermions in one dimension with periodic boundary conditions can be bipartitioned into a spatial partition of size $\ell$ (left) or a particle partition consisting of $n$ fermions (right). The degrees of freedom which are kept in the reduced density matrix are indicated in blue, while the orange ones are traced out. Interactions between the former and latter are pictured with green lines.

Figure 2

Exact diagonalization results for entanglement. Time dependence of the increase in particle and spatial entanglement entropy after an interaction of strength $V=0.25J$ is turned on ($V^{\prime }=0$). The spatial entanglement entropy (red curve) for $\ell =L/2=13$ sites grows linearly up to a time $tJ\sim \ell /4$, whereas the particle entanglement entropy (purple curve) for $n=\lfloor N/2\rfloor =6$ particles rapidly increases on a scale $tJ\sim 1/2$. The dashed lines show the asymptotic $t\to \infty$ values extracted from the full time dependence (data included in Ref. [31]).

Figure 3

Asymptotic entanglement. Finite size scaling of the $t\to \infty$ entanglement entropy per particle in the maximal subregion corresponding to $n=\lfloor N/2\rfloor$ particles or $\ell =L/2$ sites for different nearest neighbor interactions $V$ and $V^{\prime }=0$. Symbols correspond to exact diagonalization data and lines are fits to the finite size scaling form of Eq. (12). Within the statistical uncertainty (size of shaded region) for $N\to \infty$, particle and spatial entanglement entropy density extrapolate to the same interaction dependent value $\mathfrak{s}$ in the thermodynamic limit.

Figure 4

$n$ dependence of the particle entanglement entropy. Bipartition size ($n$) dependence of the particle entanglement for $N=12$ particles on $L=24$ sites for various nearest neighbor interaction strengths $V$ with $V^{\prime }=0$. The $n$-particle entanglement entropy density is only weakly dependent on the order of the reduced density matrix. Lines on the right-hand side of the figure show the thermodynamic limit value of $\mathfrak{s}$ ($n,N\to \infty$ with $n/N=1/2$) extracted from the fit shown in Fig. 3. Dashed lines are guides to the eye.

Figure 5

Effects of integrability breaking. Finite size scaling of the spatial and particle entanglement entropy density for $t\to \infty$ for a fixed nearest neighbor interaction strength $V=0.25J$ with $V^{\prime }=0.1V$ (left) and $V^{\prime }=1.42V$ (right). Symbols correspond to exact diagonalization and lines are fits to Eq. (12).

Figure 6

Equivalence of spatial mode and particle entanglement. A comparison of the $t\to \infty$ and $N\to \infty$ entanglement density $\mathfrak{s}$ defined in Eq. (12) (extrapolated values in Figs. 3 and 5) as a function of quenched nearest neighbor interaction strength $V$. $V^{\prime }\ne 0$ points with $V=0.25J$ (square and diamond) have been horizontally shifted to better discern their error bars. Here the colors of individual symbols denote the interaction strength (see legend of Fig. 4).