Spin transport through a spin-$\frac{1}{2}$ XXZ chain contacted to fermionic leads

We employ matrix-product state techniques to numerically study the zero-temperature spin transport in a finite spin-$\frac{1}{2}$ XXZ chain coupled to fermionic leads with a spin bias voltage. Current-voltage characteristics are calculated for parameters corresponding to the gapless XY phase and the gapped Néel phase. In both cases, the low-bias spin current is strongly suppressed unless the parameters of the model are fine tuned. For the XY phase, this corresponds to a conducting fixed point where the conductance agrees with the Luttinger-liquid prediction. In the Néel phase, fine tuning the parameters similarly leads to an unsuppressed spin current with a linear current-voltage characteristic at low bias voltages. However, with increasing the bias voltage, there occurs a sharp crossover to a region where the current-voltage characteristic is no longer linear and a smaller differential conductance is observed. We furthermore show that the parameters maximizing the spin current minimize the Friedel oscillations at the interface, in agreement with the previous analyses of the charge current for inhomogeneous Hubbard and spinless fermion chains.

Figure 1

Schematic depiction of the setup defined by the Hamiltonian ${\widehat{H}}_0+{\widehat{H}}_{\mathrm{V}}$ according to Eqs. (1) and (6). Blue (green) circles indicate the spin chain (left and right leads). The red dashed line denotes the spin bias potential, which linearly decreases inside the spin chain.

Figure 2

Time evolution of the spin current $j_j^z\left(\tau \right)$ in a junction composed of an isotropic spin chain ($\mathrm{\Delta }=1$) of $N_{\mathrm{S}}=8$ sites (shaded region) coupled to tight-binding leads of $N_{\mathrm{L}}=100$ sites for spin bias $V/t=1$ and several values of $J/t$. (a) Spin current profile at three different times $\tau t=10$, 20, and 30. (b) Time dependence of the spin current $j_0^z\left(\tau \right)$ between the spin chain and the left lead (solid lines) and estimated steady-state value (dashed lines). The result for $J/t=2.4$ with a larger size of the leads $N_{\mathrm{L}}=500$ is indicated by the green line. In the inset, the amplitude $a_1$ of the current oscillations [see Eq. (9)] is shown for several different lead sizes $N_{\mathrm{L}}$. The solid line is a fit to $a_1\propto 1/N_{\mathrm{L}}$.

Figure 3

Steady-state current $j^z$ as a function of $J/t$ for four different values of $\mathrm{\Delta }$. Other parameters are $N_{\mathrm{S}}=8, N_{\mathrm{L}}=500$, and $V/t=0.2$. The dashed line shows the current $V/\left(4\pi \right)$ expected for a Luttinger liquid with smooth interfaces.

Figure 4

Magnetization profile $\langle {\widehat{S}}_j^z\rangle$ around the interface for an applied magnetic field $h/J=0.05$. The dashed line indicates the interface between the tight-binding lead $(j\le 0)$ and the spin chain $(j>0)$. The systems sizes are $N_{\mathrm{L}}=400$ and $N_{\mathrm{S}}=400$ for $\mathrm{\Delta }=1$ and $N_{\mathrm{L}}=400$ and $N_{\mathrm{S}}=800$ for $\mathrm{\Delta }=2$. The inset in (f) is a magnified view of the region close to the interface, highlighting the Friedel oscillations with wave number $\pi$. Solid black lines in (a) and (c) are fits of the data in the form of Eq. (10). The results are obtained by finite temperature calculations at the inverse temperature $\beta$.

Figure 5

Strength of the Friedel oscillations $O_{\mathrm{F}}$ defined in Eq. (11) around the interface inside the spin chain at the inverse temperature $\beta$. The system sizes are the same as in Fig. 4.

Figure 6

Current-voltage curve in the two-lead setup described in Eq. (1) for different parameters in the gapless phase. The dashed line is the conductance $G=1/\left(4\pi \right)$ of a Luttinger liquid smoothly connected to noninteracting leads.

Figure 7

Current-voltage curve for a spin chain with $N_{\mathrm{S}}=8$, anisotropy $\mathrm{\Delta }=2$, and different values of $J/t$. The dashed line corresponds to the ideal conductance $G=1/\left(4\pi \right)$ obtained in the Luttinger-liquid regime.

Figure 8

Same as in Fig. 7 but for a spin chain with $J/t=1.7$ and various chain lengths.