Nonequilibrium steady states in the quantum XXZ spin chain

We investigate the real-time dynamics of a critical spin-1/2 chain (XXZ model) prepared in an inhomogeneous initial state with different magnetizations on the left and right halves. Using the time-evolving block decimation method, we follow the front propagation by measuring the magnetization and entanglement entropy profiles. At long times, as in the free fermion case [Antal et al., Phys. Rev. E 59, 4912 (1999)], a large central region develops where correlations become time independent and translation invariant. The shape and speed of the fronts is studied numerically and we evaluate the stationary current as a function of initial magnetic field and as a function of the anisotropy $\Delta$. We compare the results with the conductance of a Tomonaga-Luttinger liquid, and with the exact free-fermion solution at $\Delta =0$. We also investigate the two-point correlations in the stationary region and find a good agreement with the “twisted” form obtained by Lancaster and Mitra [Phys. Rev. E 81, 061134 (2010)] using bosonization. Some deviations are nevertheless observed for strong currents.

Figure 1

Weight of the discarded Schmidt eigenvalues during the TEBD truncation at each time step. The weight is summed over every link of the chain for each time step The figure shows the data for an anisotropy parameter $\Delta =0.4$ and an initial magnetic field of $75\%$ of the saturation value [$h=f_{\mathrm{sat}}(1-\Delta )$, $f_{\mathrm{sat}}=0.75$].

Figure 2

Magnetization as a function of site position $i$ at the initial time (blue) and at $t=40$ (red). Anisotropy parameter $\Delta =0.4$. The different panels correspond to different amplitudes of the initial magnetic field $h$, and hence different heights of the initial magnetization steps $h=f_{\mathrm{sat}}{h}_{\mathrm{sat}}\left(\Delta \right)$, where $h_{\mathrm{sat}}=1-\Delta$. From top to bottom $f_{\mathrm{sat}}=0.1,0.75$, and $0.95$.

Figure 3

Magnetization front in a chain of length $L=200$ sites (keeping respectively 150, 100, 250, and 150 states from top to bottom). Top panel: $\Delta =-0.5$, $h=0.8$ ($m_0\simeq 0.16$, indicated by an horizontal dashed line). Second panel $\Delta =0.6$, $h=0.4$ ($m_0=0.5$). Third panel: $\Delta =0.6$, $h=0.3$ ($m_0=0.30$). Bottom panel: $\Delta =0.6$, $h=0.2$ ($m_0\simeq 0.196$). The position $i$ is scaled with time $t$ to make apparent the emergence of a limiting front shape. A dashed red vertical line indicates the velocity of elementary excitations in an homogeneous XXZ chain at $\langle S^z\rangle =0$ [given by Eq. (9)] and a dashed blue line indicates that velocity for $\langle S^z\rangle =m_0$ [from the numerical solution of the Bethe-Ansatz equations in presence of an external magnetic field (Ref. 31)].

Figure 4

Scaling of the oscillations in the magnetization front in a chain of length $L=400$ sites (keeping 200 states) with $\Delta =0.6$, $h=0.2$ ($m_0\simeq 0.196$). In the main panel the position $i$ has been shifted by $t{v}_{\max }$, where $v_{\max }\simeq 0.56751$ is an estimate of the front velocity from the spinon group velocity in a uniform chain at zero magnetization. The distance to the edge of the front has then been rescaled by a factor $t^{-1/3}$ to make contact with the oscil-lations observed in the noninteracting case (Ref. 35). In the inset the magnetization $\langle S_i^z\rangle$ is shown without any rescaling of the position $i$. The red, green, and blue dashed vertical lines correspond to estimates of the front-edge locations $i=t{v}_{\max }$ at times $t=100$, 150, and 190 respectively.

Figure 5

Entanglement entropy between the left part (sites $j<i$) and the right (sites $j\ge i$) as a function of the cut position $i$ and for different times (curves shifted by $0.35$ for clarity). Bottom panel: magnetization profile in the initial state and at $t=40$. Dotted line: EE in the ground state of the chain, without external magnetic field. Full line: Eq. (11). Parameters: $\Delta =0.0$ and initial magnetization $m_0=0.17$.

Figure 6

Same as Fig. 5 for $\Delta =0.6$ and $m_0=0.19$.

Figure 7

Same as Fig. 5 but with an initial state constructed from a “step” magnetic field profile [Eq. (12)]. Parameters: $\Delta =0.0$ and initial magnetization $m_0=0.17$.

Figure 8

Classical evolution of the density $n(p,r,t)$. The occupied region of phase space, where $n(p,r,t)=1$ is colored in blue (zero otherwise). The red curve represents the position of a particle with momentum $p$ located initially at $r=0$. For $k_F^{-}\le \left|p\right|\le k_F^{+}$, this curve separates the occupied ($n=1$) from the empty region ($n=0$) of phase space. Top: Short time $t^{\prime }$. The locations of front of the fronts and back of the fronts are indicated by dashed red lines. Center: time $t^{\prime }$ after the first bounce at the boundaries. Bottom: In the long-time limit $L\ll t$ the red curve spans the interval $[-L/2,L/2]$ many times and the the striped area therefore corresponds to an alternation of thin occupied and empty regions, with an average density $n(p,r,t=\infty )=\frac{1}{2}$ for $k_F^{-}\le \left|p\right|\le k_F^{+}$.

Figure 9

Density profile for $\Delta =0$, $L=200$, and $h_0=\sqrt{2}/2$ (corresponding to densities 0.25 and 0.75 at $t=0$). Crosses: exact result. Red: classical result (thermodynamic limit).

Figure 10

Entanglement entropy between the two halves of the chain as a function of time (scale by $L$). The vertical lines correspond to the times where the two fronts cross each other in the center of the chain. The limit $t/L\ll 1$ for $m_0=\frac{1}{2}$ corresponds to the logarithmic increase of the entropy studied in Ref. 37.

Figure 11

Time evolution of the spin current $J$ measured in the center of a chain of length $L=1000$ and for $\Delta =0$ (free fermions). Green curve: step initial magnetic field. Blue: $\tanh$ initial magnetic field.

Figure 12

Spin current $J=\mathrm{Im}\langle S_0^{+}{S}_1\rangle$ measured in the center of the chain in the steady-state regime. Inset: The slope $G=dJ/d{h}_{|h=0}$ (conductance) is plotted as a function of $\Delta$, and matches the Tomonaga-Luttinger liquid prediction [Eq. (18)].

Figure 13

Crosses: various correlations measured at time $t=40$ in a chain of length 80 for $\Delta =0$ and $h/h_{\mathrm{sat}}=0.25$ (TEBD with $\chi =100$ states), and $m_0\simeq 0.08$. From top to bottom: (i) Modulus of $\langle c_{i0}^{†}{c}_{i0+r}\rangle$, (ii) argument of $\langle c_{i0}^{†}{c}_{i0+r}\rangle$, (iii) modulus of $\langle S_{i0}^{+}{S}_{i0+r}^{-}\rangle$, (iv) argument of $\langle S_{i0}^{+}{S}_{i0+r}^{-}\rangle$, (v) magnetization. For (i)–(iv), the circles represent the same correlator measured in the zero-magnetization ground state and multiplied by a phase factor $e^{i\theta r}$. $\theta$ is determined numerically by a fit over five sites in the middle of the chain (from 41 to 45).

Figure 14

Same as Fig. 13 for $\Delta =0.4$, $h/h_{\mathrm{sat}}=0.25$.

Figure 15

Same as Fig. 13 for $\Delta =0.6$, $h/h_{\mathrm{sat}}=0.75$.

Figure 16

Same as Fig. 13 for $\Delta =-0.6$, $h/h_{\mathrm{sat}}=0.6$.

Figure 17

The twist angle $\theta$ describing the steady-state correlation [Eq. (5)]. The blue line corresponds to bosonization result of Ref. 10: $\theta /\pi =\frac{m_0}{K\left(\Delta \right)}$ with $K\left(\Delta \right)$ given in Eq. (8).

Figure 18

Schematic representation of the distribution of the occupations numbers as a function of the momentum $p$ in the region where the nonequilibrium steady state is well established, ${\langle c_p^{†}{c}_p\rangle }_{\mathrm{NESS}}$. The occupation ${\langle c_p^{†}{c}_p\rangle }_L$ on the far left side of the chain, which acts as a reservoir, contributes to the distribution of the right movers on the NESS region, whereas ${\langle c_p^{†}{c}_p\rangle }_R$ contributes to the distribution of left movers.