Thermal states of random quantum many-body systems

We study a distribution of thermal states given by random Hamiltonians with a local structure. We show that the ensemble of thermal states monotonically approaches the unitarily invariant ensemble with decreasing temperature if all particles interact according to a single random interaction and achieves a state $t$-design at temperature $O(1/\log \left(t\right))$. For the system where the random interactions are local, we show that the ensemble achieves a state 1-design. We then provide numerical evidence indicating that the ensemble undergoes a phase transition at finite temperature.

Figure 1

(a) Shows upper bounds on $T_t^{\left(n\right)}\left(\beta \right)$ for $D=4$, where $t=11,8,5,2$ from top to bottom. The inset shows the scaling of $T_t^{\left(n\right)}\left(\beta \right)/t$ with $\beta$ for $t=3,6,9,12$ from top to bottom, which indicates that $T_t^{\left(n\right)}\left(\beta \right)$ scales as $t{e}^{-c\beta }$ for large $\beta$. (b) Shows a sufficient temperature for thermal states of ${\mathfrak{H}}_n$ to be an $\epsilon$-approximate $t$-design for $\epsilon =0.5,0.4,0.3,0.2$ from top to bottom.

Figure 2

(a) Shows the distance $T_t^{\left(k\right)}\left(\beta \right)$ for $n=5,t=2$, and neighboring interactions. (b) Shows its derivative ${\partial }_{\beta }{T}_t^{\left(k\right)}\left(\beta \right)$ in terms of $\beta$. The purple (up-triangle), orange (square), green (circle) represent $k=2,3,4$, respectively, and the blue solid line is for random global Hamiltonians ${\mathfrak{H}}_n$. The expectation of ${\mathbb{E}}_{{\mathfrak{H}}_k}\left[{\left({\rho }_H\left(\beta \right)\right)}^{\otimes t}\right]$ is taken by sampling $2\times {10}^4$ Hamiltonians. It is observed that ${\beta }_{\mathrm{c}}^{\left(2\right)}\sim 0.8,{\beta }_{\mathrm{c}}^{\left(k\right)}\sim 0.85$, and ${\beta }_{\mathrm{c}}^{\left(k\right)}\sim 1.05$.