Classical ergodicity and quantum eigenstate thermalization: Analysis in fully connected Ising ferromagnets

We investigate the relation between the classical ergodicity and the quantum eigenstate thermalization in the fully connected Ising ferromagnets. In the case of spin-1/2, an expectation value of an observable in a single-energy eigenstate coincides with the long-time average in the underlying classical dynamics, which is a consequence of the Wentzel-Kramers-Brillouin approximation. In the case of spin-1, the underlying classical dynamics is not necessarily ergodic. In that case, it turns out that, in the thermodynamic limit, the statistics of the expectation values of an observable in the energy eigenstates coincides with the statistics of the long-time averages in the underlying classical dynamics starting from random initial states sampled uniformly from the classical phase space. This feature seems to be a general property in semiclassical systems, and the result presented here is crucial in discussing equilibration, thermalization, and dynamical transitions of such systems.

Figure 1

The equal energy surfaces for $\varepsilon =0$ (solid line) and for $\varepsilon =-0.11$ (dashed line). The parameters are chosen as $h_x=0.2$ and $h_z=0.001$.

Figure 2

The eigenstate expectation values $\langle {\varphi }_n\left|m^z\right|{\varphi }_n\rangle$. The horizontal axis is the energy eigenvalue divided by $N$. The parameters are chosen as $h_x=0.2, h_z=0.001$, and $N=5000$.

Figure 3

Time evolutions of $\langle \mathrm{\Psi }\left(t\right)\left|m^z\right|\mathrm{\Psi }\left(t\right)\rangle$ after the quenches from (top) $h_x^{\left(i\right)}=-0.15$ and (bottom) $h_x^{\left(i\right)}=-0.05$. The solid lines are the solutions of the Schrödinger equation for $N=4000$ and the dashed red lines are the equilibrium values in the TSS.

Figure 4

Blue (gray) solid line: Expectation values of $m^z$ in the steady state after quenches as a function of $\varepsilon$ ($\varepsilon$ is a function of $h_x^{\left(i\right)}$). Red dashed line: Expectation values of $m^z$ in the equilibrium state within the TSS, which is the same one as the curve in Fig. 2. Black solid line: Expectation values of $m^z$ in the true equilibrium state (not restricted to the TSS).

Figure 5

Long-time averages of (a) $m^z$, (b) $m^x$, and (c) $m^0$ in the classical dynamics starting from random initial states sampled uniformly from the phase space (red points) and expectation values of the same quantities in each quantum energy eigenstates for $N=240$ (blue points). The horizontal axis is the energy density $\varepsilon$. The parameters are set as $h_x=0.2, h_z=0.01$, and $D=0.4$. The number of the samples of the random initial states in the classical dynamics is $14000$. The classical dynamics is calculated up to $t=20000$ and the long-time averages are calculated within this time interval.

Figure 6

$P_N\left(m\right)$ for $N=240$ (shaded histogram) and $P_{\mathrm{cl}}\left(m\right)$ (points and line).

Figure 7

${\mathrm{\Delta }}_N$ as a function of $1/\sqrt{N}$. The smallest value and the largest value of $N$ are 40 and 240, respectively.

Figure 8

Time evolutions of $\langle \psi \left(t\right)\left|m^x\right|\psi \left(t\right)\rangle$ for the quenches of (top) $(h_x^{\left(i\right)}=-0.3,h_x^{\left(f\right)}=0.2)$ and (bottom) $(h_x^{\left(i\right)}=-0.39,h_x^{\left(f\right)}=0.2)$. The solid lines are the solutions of the Schrödinger equation for $N=80$. The dashed red lines are the equilibrium values.

Figure 9

Blue (gray) solid line: Expectation values of $m^z$ in the steady state after quenches as a function of $\varepsilon$, the energy density after the quench. Red dashed line: Expectation values of $m^z$ in the equilibrium state within the TSS. Black solid line: Expectation values of $m^z$ in the true equilibrium state (not restricted to the TSS).

Figure 10

Poincare section in the classical dynamics of the spin-1 fully connected Ising ferromagnet. Different colors imply different initial states in each figure. The energy densities are 0.1 (left), 0.22 (middle), and 0.45 (right). The parameters are identical to those in the main text, $h_x=0.2, h_z=0.01$, and $D=0.4$.

Figure 11

Long-time averages of (a) $n=a^{†}a/N$ and (b) $h_{\mathrm{int}}$ in the classical dynamics starting from random initial states sampled uniformly from the phase space (red points) and expectation values of the same quantities in each quantum energy eigenstates for $N=40$ (blue points). The horizontal axis is the energy density $\varepsilon$. The number of the samples of the random initial states in the classical dynamics is 2000. The classical dynamics is calculated up to $t=10000$ and the long-time averages are calculated within this time interval.

Figure 12

The distribution of the expectation values of $n=a^{†}a/N$ in individual energy eigenstates for $N=60$ (shaded histogram) and the distribution of the long-time average of $x$ in the classical dynamics (points and line).

Figure 13

${\mathrm{\Delta }}_N$ as a function of $1/\sqrt{N}$ in the Dicke model. The smallest and the largest values of $N$ are 20 and 70, respectively.