Partial breakdown of quantum thermalization in a Hubbard-like model

We study the possible breakdown of quantum thermalization in a model of itinerant electrons on a one-dimensional chain without disorder, with both spin and charge degrees of freedom. The eigenstates of this model exhibit peculiar properties in the entanglement entropy, the apparent scaling of which is modified from a “volume law” to an “area law” after performing a partial, site-wise measurement on the system. These properties and others suggest that this model realizes a new, nonthermal phase of matter, known as a quantum disentangled liquid (QDL). The putative existence of this phase has striking implications for the foundations of quantum statistical mechanics.

Figure 1

Doublon occupancy of the itinerant electron model [Eq. (3)] obtained from exact diagonalization at system size $L=12$. Plotted are all eigenstates at half filling $\left(N_{\uparrow }=N_{\downarrow }=L/2\right)$, with total spin singlets emphasized in red. The top panels show the pure Hubbard model $\left(V=0\right)$, while the bottom panels show the model with an additional nearest-neighbor repulsion term $\left(V=3/4\right)$, which breaks integrability. The left panels show a “large” value of $U$, while the right panels show results for “small” $U$. In each plot, the Heisenberg ferromagnet (the unique state with maximum total spin) is plotted in dark gray and labeled “FM.”

Figure 2

Putative sketch of the doublon occupancy versus eigenstate energy density for large $U$ in the thermodynamic limit, for both (a) the Hubbard model $\left(V=0\right)$ and (b) the nonintegrable model $\left(V\ne 0\right)$. In each case, eigenstates at half filling which are total spin singlets are considered.

Figure 3

(a) Doublon occupancy for $U=4,V=3/4$ at system size $L=14$, as calculated using ARPACK's iterative eigensolver, which returns a portion of the spectrum. Included are all eigenstates that are total spin singlets at half filling. The “spin band” states are emphasized in blue. States in all momentum, spin-flip, and particle-hole sectors are combined in this plot. (b) Logarithmic histogram plot of same quantity.

Figure 4

Putative sketch of the doublon occupancy versus eigenstate energy density for small $U$ in the thermodynamic limit, for both (a) the Hubbard model $\left(V=0\right)$ and (b) the nonintegrable model $\left(V\ne 0\right)$. In each case, eigenstates at half filling which are total spin singlets are considered.

Figure 5

Numerical half-cut (a) entanglement entropy density $S_{L/2}/L$ and (b) QDL diagnostic density $S_{L/2}^{c/s}/L$ for the model in Eq. (3) at $L=12,U=4,V=3/4$, and $t=1$, the same nonintegrable parameters as Figs. 1 and 3. Here, all eigenstates that are total spin singlets are plotted, and the states identified to be in the spin band are colored in blue, while charge band states are in red. The QDL diagnostic shown is the average entanglement entropy after a partial measurement of the spin on each site, as detailed in Sec. 2. Starred are three states that are explored in detail in Fig. 8.

Figure 6

Proposed sketches of (a) the entanglement entropy density and (b) the QDL diagnostic density in the thermodynamic limit, for singlets in the large-$U$ nonintegrable model, as based on Figs. 5 and 8.

Figure 7

Numerical half-cut entanglement entropy density results for system size $L=14$ at the nonintegrable point $U=4,V=3/4$. The “spin band” states, as identified in Fig. 3, are plotted in blue.

Figure 8

Scaling of the entanglement entropy and QDL diagnostics with subsystem cut size $x$, for the three starred states in Fig. 5. The left column is the overall entanglement entropy. The middle and right columns plot the QDL diagnostic entanglement entropy after a partial measurement of the charge and spin on each site, respectively. The top row (in black) shows each quantity plotted for the ground state, each of which scales subthermally. The middle row (in red) shows the quantities for a highly excited state in the charge band, each of which appears to scale as a volume law. Finally, the bottom row (in blue) shows the three quantities for a highly excited state in the spin band. Here, $S$ and $S^{s/c}$ both scale as a volume law, but the entanglement entropy after a spin measurement $S^{c/s}$ scales as an area law, thus fulfilling the criteria for a quantum disentangled eigenstate as defined in Sec. 2.