Complexity of thermal states in quantum spin chains

We study the quantum correlations and complexity of simulation, characterized by quantum mutual information and entanglement entropy in operator space, respectively, for thermal states in critical, noncritical, and quantum chaotic spin chains. A simple general relation between the two quantities is proposed. We show that in all cases mutual information and entanglement entropy saturate with system size, whereas as a function of the inverse temperature, we find logarithmic divergences for critical cases and uniform bounds in noncritical cases. A simple efficient quasiexact method for computation of arbitrary entropy-related quantities in thermalized $XY$ spin chains is proposed.

Figure 1

(Color online) Logarithmic growth of OSEE $S^{♯}$ (upper curves) and QMI $I_{A:B}$ (lower curves) for thermal states of critical models: (a) $XX$ chain $H_{XY}(\gamma =0,h=0)$, (b) transverse Ising chain $H_{XY}(\gamma =1,h=1)$, both for $n=1024,256$, and (c) $XXZ$ chain $H_{XXZ}(\Delta =0.5,h_l\equiv 0)$ for $n=256,32,8$. In case (c) we show mutual purity $I_P$ instead of $I_{A:B}$. Dotted lines indicate conjectures (9, 10) combined with (9) (see text).

Figure 2

(Color online) Saturation of OSEE $S^{♯}$ (upper curves) and mutual purity $I_P$ (lower curves) for quantum chaotic noncritical systems—namely, (a) for the $XXZ$ model in a staggered field $H_{XXZ}(\Delta =0.5,h_l=-[1+{\mathbf{(}-1)}^l]∕2\mathbf{)}$ and (b) for tilted Ising $H_I(h_x=1,h_z=1)$, for $n=8,32,128$. Note that the curves for $n=32$ and $n=128$ are practically indistinguishable.

Figure 3

(Color online) OSEE $S^{♯}$ (solid curves) and QMI $I_{A:B}$ (dashed curves) for a noncritical $XY$ chain with $\gamma =0.5$, $h=0.9$, and $n=20,40,60$. Note a crossover between two saturation plateaus due to a single exponentially close (in $n$) excited state. The inset shows the $h$ dependence of the gap decay length $\xi$, computed from the fit to $\Delta E\propto e^{-n∕\xi }$, for $n=30,\dots ,60$. The vertical line indicates $h^{*}=\sqrt{1-{\gamma }^2}$.