Matrix-valued Boltzmann equation for the Hubbard chain

We study, both analytically and numerically, the Boltzmann transport equation for the Hubbard chain with nearest-neighbor hopping and spatially homogeneous initial condition. The time-dependent Wigner function is matrix-valued because of spin. The $H$ theorem holds. The nearest-neighbor chain is integrable, which, on the kinetic level, is reflected by infinitely many additional conservation laws and linked to the fact that there are also nonthermal stationary states. We characterize all stationary solutions. Numerically, we observe an exponentially fast convergence to stationarity and investigate the convergence rate in dependence on the initial conditions.

Figure 1

The dispersion relation $\omega \left(k\right)$ of Eq. (16).

Figure 2

Contour (blue straight lines) and gradient (green vectors) of the energy conservation $\underline{\omega }=0$ for fixed $k_1=\frac{23}{64}$ and after eliminating $k_2$. The diagonal blue line ${\gamma }_{\mathrm{diag}}$ is precisely the contour $k_3+k_4=\frac{1}{2}$. The vertical and horizontal blue lines, ${\gamma }_1$ and ${\gamma }_2$, are the contours $k_3=k_1$ and $k_4=k_1$, respectively. $p_i$ marks the intersection of ${\gamma }_i$ with ${\gamma }_{\mathrm{diag}}$ for $i=1,2$.

Figure 3

Contour (blue straight lines) and gradient (green vectors) of the energy conservation for a next-nearest-neighbor model with ${\omega }_{\mathrm{nnn}}\left(k\right)$ of Eq. (38) and fixed $k_1=\frac{23}{64}$.

Figure 4

(a) The term $1/\underline{\omega }$ as a function of $k_3$ and $k_4$, for fixed $k_1=23/64$. (b) The “mollified” version $\underline{\omega }/({\underline{\omega }}^2+{\epsilon }^2)$ with $\epsilon =\frac{1}{2}$ free of singularities.

Figure 5

Diagonal matrix entries of a high-temperature equilibrium state $W_{\mathrm{FD}}$. The state is (almost) independent of $k$.

Figure 6

(a) Initial state $W(k,0)=W_{\mathrm{FD}}\left(k\right)+V\left(k\right)$ with $V\left(k\right)$ defined in Eq. (59). The blue (upper) and green (middle) curves show the real diagonal entries, and the red (lower dark gray) and magenta (lower light gray) curves show the real and imaginary parts of the off-diagonal $|\uparrow \rangle \langle \downarrow |$ entry, respectively. (b) Corresponding eigenvalues of $W(k,0)$ in the interval $[0,1]$.

Figure 7

Convergence of the initial $W\left(0\right)$ (Fig. 6) to the high-temperature equilibrium state $W_{\mathrm{FD}}$ (Fig. 5) as a semilogarithmic plot (blue). The decay rate is the slope of the fitted red (dotted) line.

Figure 8

Bloch sphere representation (dark blue curve) of the initial $W(k,0)$ (Fig. 6), parametrized by $k$ and viewed from two perspectives. The light blue curve shows the corresponding Bloch curve of $W(k,t)$ for $t=1/2$. Finally, the curve for the high-temperature equilibrium state $W_{\mathrm{FD}}\left(k\right)$ is indiscernible from a single point at the tip of the $z$-axis arrow.

Figure 9

Diagonal matrix entries of a low-temperature equilibrium state $W_{\mathrm{FD}}$ ($\beta =7$, ${\mu }_{\uparrow }=\frac{17}{16}$, ${\mu }_{\downarrow }=\frac{15}{16}$).

Figure 10

(a) Initial state $W(k,0)=W_{\mathrm{FD}}\left(k\right)+V\left(k\right)$ with $V\left(k\right)$ defined in Eq. (62). The blue (upper right) and green (upper centered) curves show the real diagonal entries, and the red (lower dark gray) and magenta (lower light gray) curves show the purely imaginary off-diagonal entries. (b) Eigenvalues of $W(k,0)$ in the interval $[0,1]$.

Figure 11

Convergence to the low-temperature equilibrium state $W_{\mathrm{FD}}$ (Fig. 9) as a semilogarithmic plot (blue). The decay rate is the slope of the fitted red (dotted) line. For this example, we observe that the off-diagonal entries converge much faster than the diagonal ones.

Figure 12

Diagonal matrix entries of an equilibrium state $W_{\mathrm{FD}}$ with $\beta =1$ and same chemical potentials ${\mu }_{\uparrow }={\mu }_{\downarrow }=1$.

Figure 13

(a) Initial state $W(k,0)=W_{\mathrm{FD}}\left(k\right)+V\left(k\right)$ with $W_{\mathrm{FD}}\left(k\right)$ proportional to the identity matrix (see Fig. 12) and $V\left(k\right)$ defined in Eq. (59). The blue (center top) and green (center middle) curves show the real diagonal entries, and the red (lower dark gray) and magenta (lower light gray) curves show the real and imaginary parts of the off-diagonal $|\uparrow \rangle \langle \downarrow |$ entry, respectively. (b) The eigenvalues of $W(k,0)$ are nondegenerate, different from the equilibrium state $W_{\mathrm{FD}}\left(k\right)$.

Figure 14

Convergence to the equilibrium state $W_{\mathrm{FD}}$ with degenerate eigenvalues (Fig. 12) as a semilogarithmic plot (blue). The decay rate is the slope of the fitted red (dotted) line.

Figure 15

Two snapshots showing the convergence of the eigenvalues to the equilibrium state $W_{\mathrm{FD}}$ (centered red, same as Fig. 12) with ${\mu }_{\uparrow }={\mu }_{\downarrow }=1$.

Figure 16

(a) Initial state $W(k,0)$ defined in Eq. (64). The blue (two peaks) and green (upper right) curves show the real diagonal entries, and the red (lower dark gray) and magenta (lower light gray) curves show the real and imaginary parts of the off-diagonal $|\uparrow \rangle \langle \downarrow |$ entry, respectively. (b) Eigenvalues of $W(k,0)$ in the interval $[0,1]$.

Figure 17

(a) The $f$ function (solid blue) calculated from $\mathrm{tr}[W(k,0)-W(\frac{1}{2}-k,0)]$ (green dashed) and the fitted “chemical potentials” $a_{\uparrow }=-0.617485$ and $a_{\downarrow }=0.0578622$. The initial $W(k,0)$ is defined in Eq. (64). (b) Resulting stationary state $W_{\mathrm{st}}\left(k\right)$ [Eq. (39)] given by $f$ and $a_{\uparrow }$, $a_{\downarrow }$.

Figure 18

Convergence to the calculated $W_{\mathrm{st}}$ (Fig. 17) as a semilogarithmic plot.