Thermalization rates in the one-dimensional Hubbard model with next-to-nearest neighbor hopping

We consider a fermionic Hubbard chain with an additional next-to-nearest neighbor hopping term. We study the thermalization rates of the quasimomentum distribution function within a quantum Boltzmann equation approach. We find that the thermalization rates are proportional to the square of the next-to-nearest neighbor hopping: Even weak next-to-nearest neighbor hopping in addition to nearest neighbor hopping leads to thermalization in a two-particle scattering quantum Boltzmann equation in one dimension. We also investigate the temperature dependence of the thermalization rates, which away from half filling become exponentially small for small temperature of the final thermalized distribution.

Figure 1

Third eigenfunction ${\chi }_3\left(k\right)$ (arbitrary vertical scale) for different final temperatures. The curves are normalized such that the slope at $k=\frac{1}{4}$ is always the same. The relative strength of the next-to-nearest-neighbor hopping is set to $\varepsilon =5\times {10}^{-3}$.

Figure 2

From left to right: Fourth, fifth, and sixth eigenfunction (arbitrary vertical scale) for $\varepsilon =5\times {10}^{-3}$ and $\beta$ ranging from 0.01 to 12. All eigenfunctions satisfy the approximate symmetry given by Eq. (18). The curves $sin\left[2\pi \right(n-1\left)k\right]$ describe the high-temperature limit with very good accuracy.

Figure 3

Rescaled eigenvalues ${\lambda }_n{t}_0/{\gamma }^2$ as a function of the relative strength of the next-to-nearest-neighbor hopping $\varepsilon =J^{\prime }/J$. Left plot $\beta =0.1$, right plot $\beta =10$. The lines are fits to quadratic behavior, ${\lambda }_n\propto {\varepsilon }^2$.

Figure 4

${\tilde{\lambda }}_n\left(\beta \right)$ for $\mu =0,0.1,0.9$ from top to bottom as a function of final inverse temperature $\beta$. $\mu =0$ corresponds to half filling. The first nonzero eigenvalue ${\tilde{\lambda }}_3\left(\beta \right)$ decreases exponentially as a function of inverse temperature for all fillings. The higher eigenvalues $\left(n\ge 4\right)$ also decay exponentially away from half filling, but are asymptotically constant for $T\to 0$ at half filling.

Figure 5

Umklapp process at half filling near the Fermi edge $k_{\text{F}}\approx \frac{1}{4}$.

Figure 6

This plot shows that ${\tilde{\chi }}_3\left(k\right)$ becomes a very good approximation for ${\chi }_3\left(k\right)$ at low temperatures $T$.

Figure 7

First, second, and third eigenvalue for different $N$-point interpolations $N_I$ for a reasonable set of parameters $\mu ,\beta ,\varepsilon ,N_k$.

Figure 8

We show the interpolation to $N_k\to \infty$ for $\mu =0$. The symbols are the data for $\beta =0.1\wedge \varepsilon =0.01$ (A). $\beta =0.1\wedge \varepsilon =0.001$ (B) and $\beta =10\wedge \varepsilon =0.001$ (C). The largest eigenvalue is ${\lambda }_{50}$. The lower eigenvalues are ${\lambda }_3$ to ${\lambda }_{10}$ from bottom to top. The areas around the straight lines mark the error of the fit.

Figure 9

The crosses mark the momenta of forward (A), backward (B), and umklapp (C) scattering processes for $\varepsilon =0.05$ conserving momentum and energy. There is only a small window allowed for backscattering processes.