Thermalization of local observables in small Hubbard lattices

We present a study of thermalization of a small isolated Hubbard lattice cluster prepared in a pure state with a well-defined energy. We examine how a two-site subsystem of the lattice thermalizes with the rest of the system as its environment. We explore numerically the existence of thermalization over a range of system parameters, such as the interaction strength, system size, and the strength of the coupling between the subsystem and the rest of the lattice. We find thermalization over a wide range of parameters and that interactions are crucial for efficient thermalization of small systems. We relate this thermalization behavior to the eigenstate thermalization hypothesis and quantify numerically the extent to which eigenstate thermalization holds. We also verify our numerical results theoretically with the help of previously established results from random matrix theory for the local density of states, particularly the finite-size scaling for the onset of thermalization.

Figure 1

Schematic diagram of a closed system divided conceptually into a subsystem and a bath.

Figure 2

Schematic diagram of a two-site subsystem in a lattice with nine sites ($L=9$).

Figure 3

The density of states $g\left(E_0\right)$ of the system at composite energy $E_0$ for different coupling strengths, $\lambda$ (as labeled), for an $L=9$ site lattice where $U=J$. $g\left(E_0\right)$ is generated as a histogram by counting eigenstates in a Gaussian window centered on $E_0$ with width $0.5J$.

Figure 4

The density of composite states $g\left(E_0\right)$ of the system at composite energy $E_0$ for different Hubbard interaction strengths, $U$ (as labeled), for an $L=9$ site lattice where $\lambda =0.5$. $g\left(E_0\right)$ is generated as a histogram by counting eigenstates in a Gaussian window centered on $E_0$ with width $0.5J$.

Figure 5

(Left) Time dependence of the initial-state occupation probability ${\rho }_{ss}$ for the initial state ${|s\rangle }_S=|\uparrow ,\uparrow \rangle$ for three coupling strengths $\lambda =0.05$, 0.5, and 0.3. (Right) A measure of the magnitude of the off-diagonal elements ${\Sigma }_{\mathrm{OD}}$, defined by Eq. (11). Interaction strength $U=J$, width of the initial bath state ${\delta }_B=0.5J$, total system energy $E_0=-2J$, size $L=9$.

Figure 6

Memory of initial state $\Delta r$, closeness to the thermal state ${\sigma }_{\omega }$ and the subsystem entropy $S$ as a function of coupling strength $\lambda$ for different composite energies $E_0$. ($U=J$, ${\delta }_B=0.5J$, $L=9$.)

Figure 7

A schematic diagram indicating the range in $\lambda$ where the subsystem reduced density matrix, $r$, is diagonal, and where it is close to the canonical thermal reduced density matrix, $\omega$.

Figure 8

Effective temperature $T_{\mathrm{eff}}$ as a function of coupling strength $\lambda$ in the thermalized regime for different composite energies $E_0$. (Inset) Example of fit of $r_{ss}$ to the Boltzmann form (15). ($U=J$, ${\delta }_B=0.5J$, $L=9$.)

Figure 9

Gaussian width of the density of states, ${\sigma }_{\mathrm{BW}}$ (solid circles), and effective temperature $T_{\mathrm{eff}}$ (hollow squares) as a function of coupling strength $\lambda$. ($E_0=0$, $U=J$, ${\delta }_B=0.5J$, $L=9$.) Both quantities are normalized to their values at ${\lambda }_{\mathrm{th}}$.

Figure 10

Memory of initial state, $\Delta r$, and closeness to the canonical thermal state, ${\sigma }_{\omega }$, as a function of interaction $U/J$ for $\lambda =0.5$. The composite energy $E_0$ is chosen to be fixed on the central maximum in $g\left(E_0\right)$, which lies close to $2U$ (${\delta }_B=0.5J$).

Figure 11

Closeness to canonical thermal state, ${\sigma }_{\omega }$, as a function of coupling strength $\lambda$ for different system sizes $L$, for composite energies $E_0$ in the center of the composite energy spectrum. The number of particles was selected to keep $S^z=0$ with the number of particles equal to $L$ and $L-1$, respectively, for even and odd $L$ ($U=J$, ${\delta }_B=0.5J$).

Figure 12

Closeness to canonical thermal state, ${\sigma }_{\omega }$, as a function of interaction $U/J$ for different system sizes ($\lambda =0.5$).

Figure 13

$\Delta r$, ${\sigma }_{\omega }$, and $T_{\mathrm{eff}}$ as functions of coupling strength $\lambda$ for different bath-window widths ${\delta }_B$ in units of $J$. The composite energy $E_0=-2J$. The average level spacing $\Delta \approx {10}^{-3}J$.

Figure 14

Histograms (plotted as color scale) of eigenstate projections $p_1=\langle A|P_1|A\rangle$ on to the two-site subsystem ground state for different subsystem-bath coupling $\lambda$ ($U=J$, ${\delta }_B=0.5J$).

Figure 15

The same histogram of eigenstate projections $p_1=\langle A|P_1|A\rangle$ from Fig. 14 with $\lambda =0.5$ is shown with a modified color scale, comparing the thermal values ${\omega }_{11}$ (black line) with the mean eigenstate projection from each vertical array of histogram bins (green line). Positions of one standard deviation (${\sigma }_{\mathrm{EP}}$) either side of the mean of the eigenstate projection are also shown (blue lines).

Figure 16

The spread of projection values, ${\sigma }_{\mathrm{EP}}$, as a function of subsystem-bath coupling strength $\lambda$ (hollow squares). The values of ${\sigma }_{\mathrm{EP}}$ were found by averaging over composite energies in a window of width $0.5J$ centered on $E_0=1.77J$. (Solid line) Theoretical estimate (27) for small $\lambda$ illustrating the scaling ${\sigma }_{\mathrm{EP}}\propto {\lambda }^{-1}$.

Figure 17

Local density of states, ${\sigma }_{Asb}^2$, as a function of $\Delta E=E_A-E_{sb}$ for the Hubbard model with $U=J$, at weak coupling $\lambda =0.1$. The averaging uses the overlaps of all eigenstates $|A\rangle$, with ${|s\rangle }_S=|\uparrow ,\uparrow \rangle$ and ${|b\rangle }_B$ selected within energy $J$ from the center of the bath spectrum. (Solid line) Lorentzian with width $W_L$ as given by Eq. (20).

Figure 18

Two-dimensional histogram showing the distribution of the overlaps $\langle A|sb\rangle$ as a function of the energy difference $E_A-E_{sb}$ for the Hubbard model with $U=J$ at weak coupling $\lambda =0.1$. The histogram includes the overlaps of all eigenstates $|A\rangle$, with ${|s\rangle }_S=|\uparrow ,\uparrow \rangle$ and ${|b\rangle }_B$ selected within energy $J$ from the center of the bath spectrum. The histogram bin widths are 0.02$J$ and 0.002$J$ on the energy and overlap axes, respectively.

Figure 19

Distribution of overlaps $X=\langle A|sb\rangle$ at weak coupling $\lambda =0.1$, scaled by the width ${\sigma }_{Asb}(\Delta E=E_A-E_{sb})$ at different values of $\Delta E$. (Same system parameters as in Fig. 18.) (Solid line) Hyperbolic secant distribution with zero mean and unit variance. (Inset) Log plot of the same data.

Figure 20

Coupling matrix elements linking the $n_s=3,s_s^z=\frac{1}{2}$ and $n_s=2,s_s^z=1$ subsectors for the Hubbard model with $U=J$. Color scale indicates the magnitude of the matrix elements. The banded diagonal structure is typical for the coupling of all subsystem states. Sizeable matrix elements lie within a band of width $4J$, with some very small matrix elements lying outside of the band due to the finite interaction strength $U$.

Figure 21

A comparison of ${\lambda }_{\mathrm{th}}$, ${\lambda }_{\mathrm{np}}$, and ${\lambda }_{\mathrm{ETH}}$ for different system sizes $L$ when $U=J$. ${\lambda }_{\mathrm{th}}$ is set as the coupling strength for which ${\sigma }_{\omega }$ falls to 0.25 for composite energies $E_0$ in the center of the band. ${\lambda }_{\mathrm{np}}$ is found using (32) with (34). ${\lambda }_{\mathrm{ETH}}$ is obtained from Eq. (33).

Figure 22

Coupling matrix elements linking the $n_s=3,s_s^z=\frac{1}{2}$ and $n_s=2,s_s^z=1$ subsectors for the Hubbard model with $U=0.01J$. Color scale indicates the magnitude of the matrix elements. The banded diagonal structure is typical for the coupling of all subsystem states. The finite matrix elements lie within a band of width $4J$ but, in contrast to Fig. 20, the matrix appears striped.

Figure 23

A diagram showing four possible occupations of two sites (across the coupling link at $i=2$ and 3) by spin-up fermions, irrespective of the configuration of spin-down fermions on these sites.