Bloch oscillations in the spin-$\frac{1}{2}$ XXZ chain

Under a perfect periodic potential, the electric current density induced by a constant electric field may exhibit nontrivial oscillations, so-called Bloch oscillations, with an amplitude that remains nonzero in the large system size limit. Such oscillations have been well studied for nearly noninteracting particles and observed in experiments. In this work we revisit Bloch oscillations in strongly interacting systems. By analyzing the spin-$\frac{1}{2}$ XXZ chain, we demonstrate that the current density at special values of the anisotropy parameter $\mathrm{\Delta }=-cos(\pi /p)$ $(p=3,4,5,\cdots )$ in the ferromagnetic gapless regime behaves qualitatively the same as in the noninteracting case $(\mathrm{\Delta }=0)$ even in the weak electric field limit. When $\mathrm{\Delta }$ deviates from these values, the amplitude of the oscillation under a weak electric field is suppressed by a factor of the system size. We estimate the strength of the electric field required to observe such a behavior using the Landau-Zener formula.

Figure 1

(a) One-dimensional ring pierced by a flux $\mathrm{\Phi }\left(t\right)=A\left(t\right)L$, which induces an electric field $E\left(t\right)={\partial }_{t}A\left(t\right)$. (b) Oscillations of the energy density $\varepsilon \left(t\right)$ and the current density $j\left(t\right)$ for the model (1) with $A=2\pi t/T$ and $L=24$ in the noninteracting case $\mathrm{\Delta }=0$. We set $T=1$ in this plot but results are identical up to $T={10}^6$, the largest period we computed. Both $j\left(t\right)$ and $\varepsilon \left(t\right)$ show a single oscillation over the period $T$ where $A$ increases by $2\pi$. Solid curves represent the analytic expressions $\varepsilon \left(t\right)=-L^{-1}cosA\left(t\right)/sin(\pi /L)$ and $j\left(t\right)=L^{-1}sinA\left(t\right)/sin(\pi /L)$. The inset shows the amplitude $j_{\text{max}}$ as a function of $L$, suggesting that it is of $O\left(1\right)$ in the large $L$ limit.

Figure 2

The many-body energy levels of $\widehat{H}\left(A\right)$, obtained by the exact diagonalization, for $L=12$ as a function of the gauge field $A$ in the case of (a) a generic value $\mathrm{\Delta }=+0.5$ and (b) a special value $\mathrm{\Delta }=-cos(\pi /3)$ ($p=3$). Only the states in the $S_z=0$ sector are shown. The adiabatic ground state is highlighted with red color. Because $\mathrm{\Delta }\leftrightarrow -\mathrm{\Delta }$ corresponds to $\widehat{H}\left(A\right)\leftrightarrow -\widehat{H}\left(A\right)$, (b) can be seen as the inverted version of (a) in this example.

Figure 3

The energy density of the adiabatic ground state ${\varepsilon }_{\text{AGS}}\left(A\right)$ as a function of $A$ for (a) $\mathrm{\Delta }=-cos(\pi /3)=-0.5$, (b) $\mathrm{\Delta }=-cos(\pi /4)=-0.707107$, and (c) $\mathrm{\Delta }=-cos(\pi /5)=-0.809017$. The black dotted curve is the analytic expression for $L\to \infty$ up to the $A^6$ order in (4). The red dotted lines are the analytic expressions for the minimum and the maximum of ${\varepsilon }_{\text{AGS}}\left(A\right)$ (see the main text).

Figure 4

Oscillations of the energy density $\varepsilon \left(t\right)$ and the current density $j\left(t\right)$ in for (a) $\mathrm{\Delta }=-cos(\pi /3)=-0.5$, (b) $\mathrm{\Delta }=-cos(\pi /4)$, (c) $\mathrm{\Delta }=-cos(\pi /5)$, and (d) $\mathrm{\Delta }=+0.5$. We set $L=24$, $T={10}^6$, and $A\left(t\right)=2\pi t/T$ in these calculations. The number of peaks is $p-1$ for (a)–(c) and $L/2$ for (d) as predicted. The inset shows the amplitude $j_{\text{max}}$ as a function of $L$, which remains nonzero in the large $L$ limit for (a)–(c), while it is inversely proportional to $L$ and vanishes in the large $L$ limit for (d).

Figure 5

(a) The zoom-up of the orange dashed box in Fig. 2, focusing on the adiabatic ground state and the next level near $A=A_0=2\pi /L$. The dashed curve represents $L{\varepsilon }_{\text{AGS}}\left(A\right)$ in the first line of (12). (b) The coefficient $C$ in (14) obtained by fitting with numerical data. The dotted curve represents the analytic expression in Appendix pp1. The vertical lines represent $\mathrm{\Delta }=-cos(\pi /p)$ with $p=3,4,\cdots ,7$. (c) The power $a$ in (14) obtained by fitting. The dotted curve represents $a=4K-1$. In (b) and (c) the blue open dots use data for $L=12,14,\cdots ,32$ and the yellow filled dots use data for $L=20,22,\cdots ,32$ only. (d) Numerical verification of the Landau-Zener formula for $\mathrm{\Delta }=0.5$ and $L=12$. The blue dots represent $p_{\text{AGS}}\left(t\right)$ in (15) at $t=2t_0$ and the solid curve represents $1-p_{\text{LZ}}$ given by (13).

Figure 6

(a) $1-p_{\text{AGS}}\left(t\right)$ at $t=T/2$ for $\mathrm{\Delta }=0.5$ ($p=3$) as a function of $T=2\pi /E$. Fitting lines are obtained by using data within $0.02<1-p_{\text{AGS}}\left(t\right)<0.7$. (b) The slope of the fitting lines in (a) against the system size $L$. The dotted line represents the linear fitting $s\left(L\right)=c{L}^{-2.096}$.

Figure 7

Same as Fig. 4 but for $T=1$, which corresponds to a strong electric field $E=2\pi$ in the unit of $J/\left(ea\right)$. Solid curves represent the analytic expressions for $\mathrm{\Delta }=0$ for comparison (see the caption of Fig. 1).

Figure 8

The plot of the energy density $\varepsilon \left(t\right)$ (red) and the current density $j\left(t\right)$ (blue) under the electric field $E=2\pi /T$ for various values of $T$ and $\mathrm{\Delta }$: $T=1$, 10, ${10}^2$, $\cdots$, and ${10}^6$ and $\mathrm{\Delta }=0$, $-cos(\pi /3)$, $-cos(\pi /4)$, $-cos(\pi /5)$, 0.5, and $-0.14$. The system size is $L=24$. While the gap above the adiabatic ground state is maximized at $\mathrm{\Delta }=-0.14$ for $\mathrm{\Delta }<0$ and $L=24$, $T={10}^6$ is not large enough to see $O\left(L\right)$ oscillations as observed for $\mathrm{\Delta }=0.5$.