Work extraction from a single energy eigenstate

Work extraction from the Gibbs ensemble by a cyclic operation is impossible, as represented by the second law of thermodynamics. On the other hand, the eigenstate thermalization hypothesis (ETH) states that just a single energy eigenstate can describe a thermal equilibrium state. Here, we attempt to unify these two perspectives and investigate the second law at the level of individual energy eigenstates, by examining the possibility of extracting work from a single energy eigenstate. Specifically, we performed numerical exact diagonalization of a quench protocol of local Hamiltonians and evaluated the number of work-extractable energy eigenstates. We found that it becomes exactly zero in a finite system size, implying that a positive amount of work cannot be extracted from any energy eigenstate, if one or both of the prequench and the postquench Hamiltonians are nonintegrable. We argue that the mechanism behind this numerical observation is based on the ETH for a nonlocal observable. Our result implies that quantum chaos, characterized by nonintegrability, leads to a stronger version of the second law than the conventional formulation based on the statistical ensembles.

Figure 1

The average work $W_j/N$ versus the eigenenergy $E_j/\left(JN\right)$ in the full spectrum for $N=12$ (upper panels) and $N=18$ (lower panels). The waiting time after the quench is $J\tau ={10}^4$ in all the cases. The vertical dashed line indicates the energy at ${\beta }_j=0$, and the horizontal one indicates $W_j/\left(JN\right)=0$. ${\beta }_j>0$ in the left side of the vertical line, while ${\beta }_j<0$ in the right side. Quench from (a) nonintegrable to nonintegrable $[J=J^{\prime }=1,g_{\mathrm{I}}=g_{\mathrm{Q}}=(\sqrt{5}+5)/8,h_{\mathrm{I}}=0\to h_{\mathrm{Q}}=1.0]$, (b) integrable to nonintegrable $[J=1,J^{\prime }=0,g_{\mathrm{I}}=g_{\mathrm{Q}}=(\sqrt{5}+5)/8,h_{\mathrm{I}}=0\to h_{\mathrm{Q}}=1.0]$, (c) nonintegrable to integrable $[J=1,J^{\prime }=0,g_{\mathrm{I}}=g_{\mathrm{Q}}=(\sqrt{5}+5)/8,h_{\mathrm{I}}=1.0\to h_{\mathrm{Q}}=0]$, and (d) integrable to integrable ($J=1,J^{\prime }=0,h=0,g_{\mathrm{I}}=0.5\to g_{\mathrm{Q}}=1.5$).

Figure 2

The system-size dependence of the ratio $D_{\mathrm{out}}^U/D_{{\beta }_{\mathrm{s}}}$. The lack of a data point means that $D_{\mathrm{out}}^U$ is exactly zero there. The waiting time after the quench is $J\tau ={10}^4$ in all the cases. We set the thresholds as $J{\beta }_{\mathrm{s}}=0.02$ and $\delta /J=0.001$. Quench from (a) nonintegrable to nonintegrable $[J=J^{\prime }=1,g_{\mathrm{I}}=g_{\mathrm{Q}}=(\sqrt{5}+5)/8,h_{\mathrm{I}}=0\to h_{\mathrm{Q}}=0.5,1.0$, and $1.5]$, (b) integrable to nonintegrable $[J=1,J^{\prime }=0,g_{\mathrm{I}}=g_{\mathrm{Q}}=(\sqrt{5}+5)/8,h_{\mathrm{I}}=0\to h_{\mathrm{Q}}=0.5,1.0$, and $1.5]$, (c) nonintegrable to integrable $[J=1,J^{\prime }=0,g_{\mathrm{I}}=g_{\mathrm{Q}}=(\sqrt{5}+5)/8,h_{\mathrm{I}}=0.5,1.0,$ and $1.5\to h_{\mathrm{Q}}=0]$, and (d) integrable to integrable ($J=1,J^{\prime }=0,h=0,g_{\mathrm{I}}=0.5\to g_{\mathrm{Q}}=1.0,1.5$, and $2.0$).

Figure 3

Numerical validation of the ETH of the nonlocal observable $H_{\mathrm{I}}\left(\tau \right)$ with respect to the Hamiltonian $H_{\mathrm{I}}$. The system-size dependencies of the average, the largest, the fifth largest, and the tenth largest values of $r_j$ are shown for the four cases of the integrability. The parameters of Hamiltonians are the same as those used in Fig. 1. The weak ETH is true for all the cases, while the strong ETH is true only for (a), (b), and (c).

Figure 4

The $h$ dependence of the distance $\alpha$ between the distribution of the ratio of consecutive energy gaps and that of the GOE prediction [Eq. (A2)]. The other parameters are fixed to $J=1,J^{\prime }=0,g=(\sqrt{5}+5)/8,N=18$. Our calculation was performed using the central half of the spectrum, where the width of the bin is $\mathrm{\Delta }r={10}^{-2}$.

Figure 5

The $\tau$ dependence of $W_j/N$ with $N=18$ and $\tau ={10}^{-1},{10}^0,{10}^1,{10}^2,{10}^3,{10}^4,{10}^5$. Quench from (a) nonintegrable to nonintegrable $[J=J^{\prime }=1,g_{\mathrm{I}}=g_{\mathrm{Q}}=(\sqrt{5}+5)/8,h_{\mathrm{I}}=0\to h_{\mathrm{Q}}=1.0]$, (b) integrable to nonintegrable $[J=1,J^{\prime }=0,g_{\mathrm{I}}=g_{\mathrm{Q}}=(\sqrt{5}+5)/8,h_{\mathrm{I}}=0\to h_{\mathrm{Q}}=1.0]$, (c) nonintegrable to integrable $[J=1,J^{\prime }=0,g_{\mathrm{I}}=g_{\mathrm{Q}}=(\sqrt{5}+5)/8,h_{\mathrm{I}}=1.0\to h_{\mathrm{Q}}=0]$, and (d) integrable to integrable ($J=1,J^{\prime }=0,h=0,g_{\mathrm{I}}=0.5\to g_{\mathrm{Q}}=1.5$).

Figure 6

The $N$ dependence of $D_{\mathrm{out}}^U/D_{{\beta }_{\mathrm{s}}}$ for various ${\beta }_{\mathrm{s}}$. Quench from (a) nonintegrable to nonintegrable $[J=J^{\prime }=1,g_{\mathrm{I}}=g_{\mathrm{Q}}=(\sqrt{5}+5)/8,h_{\mathrm{I}}=0\to h_{\mathrm{Q}}=1.0]$, (b) integrable to nonintegrable $[J=1,J^{\prime }=0,g_{\mathrm{I}}=g_{\mathrm{Q}}=(\sqrt{5}+5)/8,h_{\mathrm{I}}=0\to h_{\mathrm{Q}}=1.0]$, (c) nonintegrable to integrable $[J=1,J^{\prime }=0,g_{\mathrm{I}}=g_{\mathrm{Q}}=(\sqrt{5}+5)/8,h_{\mathrm{I}}=1.0\to h_{\mathrm{Q}}=0]$, and (d) integrable to integrable ($J=1,J^{\prime }=0,h=0,g_{\mathrm{I}}=0.5\to g_{\mathrm{Q}}=1.5$).

Figure 7

The $N$ dependence of $D_{\mathrm{out}}^U/D_{{\beta }_{\mathrm{s}}}$ for various $\delta$. The parameters of the Hamiltonians are the same as those used in Fig. 6.