Testing whether all eigenstates obey the eigenstate thermalization hypothesis

We ask whether the eigenstate thermalization hypothesis (ETH) is valid in a strong sense: in the limit of an infinite system, every eigenstate is thermal. We examine expectation values of few-body operators in highly excited many-body eigenstates and search for “outliers,” the eigenstates that deviate the most from ETH. We use exact diagonalization of two one-dimensional nonintegrable models: a quantum Ising chain with transverse and longitudinal fields, and hard-core bosons at half-filling with nearest- and next-nearest-neighbor hopping and interaction. We show that even the most extreme outliers appear to obey ETH as the system size increases and thus provide numerical evidences that support ETH in this strong sense. Finally, periodically driving the Ising Hamiltonian, we show that the eigenstates of the corresponding Floquet operator obey ETH even more closely. We attribute this better thermalization to removing the constraint of conservation of the total energy.

Figure 1

Diagonal elements of a few-body operator in energy eigenbasis vs energy density. The darker, the larger the system size. (a) Ising Hamiltonian [Eq. (7)]. The operator is ${\sigma }_1^x$. (b) Hard-core boson Hamiltonian [Eq. (8)]. The operator is $n_1{n}_2=b_1^{†}{b}_1{b}_2^{†}{b}_2$. For both cases, the fluctuations become smaller as the system size increases.

Figure 2

Distribution of $\left|r\right|$. (a) Ising Hamiltonian [Eq. (7)] and (b) hard-core boson Hamiltonian [Eq. (8)]. As we increase the system size, the distribution becomes sharply peaked near $\left|r\right|=0$. The distribution can be well fitted by a Gaussian distribution (only positive argument) with a standard deviation $\sigma$ decreasing exponentially with the system size $L$.

Figure 3

From top to bottom in each figure: first, second, fourth, and eighth largest value of $\left|r\right|$ and the mean value $\langle \left|r\right|\rangle$. (a) Ising Hamiltonian [Eq. (7)] and (b) hard-core boson [Eq. (8)]. The mean value $\langle \left|r\right|\rangle$ decreases exponentially in $L$ as ETH suggests [14]. The largest “outliers” also decrease, although slower than $\langle \left|r\right|\rangle$, with the system size.

Figure 4

From top to bottom: first, second, fourth, and eighth outliers and the average of absolute value of eigenstate expectation value for the Floquet operator $\widehat{U}$. The few-body operator is ${\sigma }_1^x$. The average value and the outlier values are substantially smaller than those of the corresponding Hamiltonian. Therefore, the Floquet system satisfies ETH more precisely for each system size $L$.

Figure 5

The many-body density of states over the energy ranges we examine. (a) Ising chain [Eq. (7)] with $L=19$. (b) Hard-core bosons [Eq. (8)] with $L=22$. For both Hamiltonians the ratio of maximum to minimum density of states over our energy range is $\sim 1.20$.

Figure 6

The largest outlier values (black solid lines) and the mean of $\left|r\right|$ (red dotted lines) for other few-body operators. (a) Ising chain [Eq. (7)]. Circles, squares, and diamonds are results for ${\sigma }_1^z,{\sigma }_1^x{\sigma }_2^x$, and ${\sigma }_1^y{\sigma }_2^y$, respectively. (b) Hard-core bosons [Eq. (8)]. Circles and squares are results for the real and imaginary parts of $b_1^{†}{b}_2$, respectively. For all cases, the largest outliers decrease with the system size, thus again supporting the strong ETH.