Entanglement Entropy dynamics in Heisenberg chains

TL;DR

This study analyzes the dynamics of block entanglement entropy in Heisenberg spin chains using time-dependent DMRG, focusing on both homogeneous and disordered cases. In the clean chain, static entropy follows CFT scaling with central charge $c\approx 1$, and post-quench dynamics show linear growth with a saturation time tied to the block length and open boundary reflections; in the disordered chain, the static entropy scales with an effective central charge $c_{eff}=\ln 2$, and the dynamics become logarithmic in time, suggesting entanglement localization. The work combines t-DMRG with exact XX-model checks to validate the numerical results and highlights how disorder qualitatively alters entanglement spreading. Overall, the paper elucidates how boundaries and randomness shape both the static and dynamical entanglement properties in quantum spin chains, with implications for quantum information processing in disordered quantum systems.

Abstract

By means of the time-dependent density matrix renormalization group algorithm we study the zero-temperature dynamics of the Von Neumann entropy of a block of spins in a Heisenberg chain after a sudden quench in the anisotropy parameter. In the absence of any disorder the block entropy increases linearly with time and then saturates. We analyze the velocity of propagation of the entanglement as a function of the initial and final anisotropies and compare, wherever possible, our results with those obtained by means of Conformal Field Theory. In the disordered case we find a slower (logarithmic) evolution which may signals the onset of entanglement localization.

Figures (9)

• Figure 1: The Block entropy $S_\ell$ for $N=200$ for a critical value $\Delta=0.0$ (circles) and non-critical value $\Delta=1.8$ (squares) and $m=120$. The critical data compared with the CFT prediction Eq. (\ref{['eq:CFT-static']}) (dashed line). Lower inset: central charge extrapolated by fitting the numerical data $S_\ell$ with Eq. (\ref{['eq:CFT-static']}) for different values of $\Delta$. The data are for $N=1000$ and $m=120$. Upper inset: scaling of $c$ extrapolated as a function of $1/N$ for the worst case $\Delta=0.5$ and compared to a quadratic fit (dashed line).
• Figure 2: Evolution of the entropy $S_{6}$ with various quenches. $\Delta_0=1.5$ while $\Delta_1=0.0, 0.2, 0.4, 0.6, 0.8$ as a function of $v(\Delta_1) t$. Inset: initial slope value of $S_{6}$ as a function of $\Delta_1$ and comparison to a linear fit with slope $-0.85\pm0.02$ (dashed line).
• Figure 3: Evolution of the entropy $S_{6}$ with various quenches. $\Delta_0=1.2,1.5,3.0, \infty$ while $\Delta_1=0.0$ as a function of $v(\Delta_1)t$ and shifted so to coincide in $t=0$. For $\Delta_0=\infty$ we show also the exact result obtained by diagonalization (circles ). Inset: initial slope value of $S_{6}$ as a function of $\Delta_1$.
• Figure 4: Evolution of the entropy $S_{6}$ with fixed quench as a function of $v(\Delta_1) t$. Fixed quench $\Delta_0-\Delta_1=1.5$ for $\Delta_0=1.5, 1.7, 1.9,2.1$. Inset: initial slope value of $S_{6}$ as a function of $\Delta_0$.
• Figure 5: Derivative $s$ of the entropy with respect to time as a function of $\Delta_0$ and $\Delta_1$ (surface) and compared to a linear fit (grid lines): $s = 1.50 \Delta_0 -0.84 \Delta _1 -0.90$. The deviation with the linear fit is less than $10\%$ for $(\Delta_0,\Delta_1)\in[1.4,2.4]\times[0.4,0.8]$.