Matrix-valued Boltzmann equation for the nonintegrable Hubbard chain

The standard Fermi-Hubbard chain becomes nonintegrable by adding to the nearest neighbor hopping additional longer range hopping amplitudes. We assume that the quartic interaction is weak and investigate numerically the dynamics of the chain on the level of the Boltzmann type kinetic equation. Only the spatially homogeneous case is considered. We observe that the huge degeneracy of stationary states in the case of nearest neighbor hopping is lost and the convergence to the thermal Fermi-Dirac distribution is restored. The convergence to equilibrium is exponentially fast. However for small next-nearest neighbor hopping amplitudes one has a rapid relaxation towards the manifold of quasistationary states and slow relaxation to the final equilibrium state.

Figure 1

The dispersion relation $\omega \left(k\right)$ for the next-nearest neighbor model in Eq. (2) with $\eta =\frac{1}{200}$ and $\eta =\frac{1}{2}$ (green solid curves coinciding with the dashed line and with two local maxima, respectively), and for the exponential hopping model in Eq. (3) with $\zeta =\frac{2}{5}$ (upper blue solid curve). All curves are shifted such that $\omega \left(0\right)=0$. The dashed curve shows the dispersion relation for the (reference) nearest neighbor hopping model.

Figure 2

Contour (green straight lines) and gradient (gray vectors) of the next-nearest neighbor energy conservation contour ${\underline{\omega }}_{\eta }=0$ (with $\eta =\frac{1}{2}$) for fixed $k_1=\frac{23}{64}$ and after eliminating $k_2$. The vertical and horizontal lines, ${\gamma }_1$ and ${\gamma }_2$, are the contours $k_3=k_1$ and $k_4=k_1$, respectively. The contour ${\gamma }_{\mathrm{ellip}}$ disappears when $\left|\eta \right|<\frac{1}{4}$.

Figure 3

Contour (blue straight lines) and gradient (gray vectors) of the exponential decay energy conservation ${\underline{\omega }}_{\zeta }=0$ (with $\zeta =\frac{2}{5}$) for fixed $k_1=\frac{23}{64}$ and after eliminating $k_2$. The vertical and horizontal lines, ${\gamma }_1$ and ${\gamma }_2$, are the contours $k_3=k_1$ and $k_4=k_1$, respectively. Compare with Fig. 2 corresponding to the next-nearest neighbor case.

Figure 4

Effect of different mollification parameters ${\epsilon }_1=\frac{1}{50}$ (upper dark gray curve, used for the simulations in Sec. 6), ${\epsilon }_2=\frac{1}{10}$ (middle curve) and ${\epsilon }_3=\frac{1}{2}$ (lower light gray curve). The difference to the initial $W(k,0)$ is quantified by the Hilbert-Schmidt norm.

Figure 5

The initial state $W(k,0)$ used for all simulations in this section. The cyan (upper) curves show the real diagonal entries, and the darker and lighter red curves show the real and imaginary parts of the off-diagonal $|\uparrow \rangle \langle \downarrow |$ entry, respectively.

Figure 6

Three-dimensional shape of the ${\gamma }_{\mathrm{diag}}$ and ${\gamma }_{\mathrm{ellip}}$ collision manifolds for the next-nearest neighbor model with $\eta =\frac{1}{2}$ (as in Fig. 2). Color encodes the Bloch vector of $\mathcal{A}\left[W\right]+\mathcal{A}{\left[W\right]}^{*}$ for the state $W(k,0)$ shown in Fig. 5.

Figure 7

Diagonal matrix entries of the stationary states corresponding to the initial $W(k,0)$ in Fig. 5, for the nearest neighbor hopping model (a), for the exponential hopping model with $\zeta =\frac{2}{5}$ (b), and for the next-nearest neighbor model with $\eta =\frac{1}{200}$ (c) and $\eta =\frac{1}{2}$ (d). The off-diagonal matrix entries are zero.

Figure 8

Entropy increase for the next-nearest neighbor model with small $\eta =\frac{1}{200}$ (green curve). The red curve shows the entropy of the corresponding equilibrium state, and the dashed black curve the entropy of the stationary nearest neighbor state. The entropy grows very slowly after $t\simeq 10$.

Figure 9

(a) Inverse exponential decay rate $1/\kappa$ of the entropy difference in dependence of $\eta$ (next-nearest neighbor model), obtained from a least squares fit as exemplified in (b). The upper dark gray points in (a) correspond to the asymptotic decay rate for large $t$ [dark dot-dashed line in (b)], and the lower light gray points to the initial decay rate at $t=0$ [light dashed line in (b)].