Numerical method for nonlinear steady-state transport in one-dimensional correlated conductors

We present a method for investigating the steady-state transport properties of one-dimensional correlated quantum systems. Using a procedure based on our analysis of finite-size effects in a related classical model ($LC$ line) we show that stationary currents can be obtained from transient currents in finite systems driven out of equilibrium. The nonequilibrium dynamics of correlated quantum systems is calculated using the time-evolving block decimation method. To demonstrate our method we determine the full $I$-$V$ characteristic of the spinless fermion model with nearest-neighbor hopping $t_{\text{H}}$ and interaction $V_{\text{H}}$ using two different setups to generate currents (turning on/off a potential bias). Our numerical results agree with exact results for noninteracting fermions ($V_{\text{H}}=0$). For interacting fermions we find that in the linear regime $eV\ll 4t_{\text{H}}$ the current $I$ is independent from the setup and our numerical data agree with the predictions of the Luttinger liquid theory combined with the Bethe Ansatz solution. For larger potentials $V$ the steady-state current depends on the current-generating setup and as $V$ increases we find a negative differential conductance with one setup while the currents saturate at finite values in the other one. Both effects are due to finite renormalized bandwidths.

Figure 1

One-dimensional conductor consisting of two coupled leads. A bias between its right- and left-hand halves is applied and the current is measured in the middle of the system at the junction between both sides (dashed connection).

Figure 2

Setup (I). The charge reservoirs (two halves: source and drain) have different potentials but are coupled for $t=0$ while for $t>0$ the potential difference is set to $\Delta \epsilon =0$ instantaneously.

Figure 3

Setup (II). The charge reservoirs (two halves: source and drain) are in equilibrium and coupled with $t_{\text{H}}$ and $V_{\text{H}}$ for $t=0$. For $t>0$ a potential difference $\Delta \epsilon >0$ is applied.

Figure 4

Discarded weight $D_{N/2}$ for different parameters. Solid black curve, $\Delta \epsilon =0.5t_{\text{H}},V_{\text{H}}=0.8t_{\text{H}}$; dashed blue curve, $\Delta \epsilon =1t_{\text{H}},V_{\text{H}}=1.6t_{\text{H}}$; dotted red curve, $\Delta \epsilon =2t_{\text{H}},V_{\text{H}}=0$. The vertical lines indicate the runaway times described in the text.

Figure 5

Current for noninteracting fermions for setup (I). Shown are TEBD (dots on top of the dashed line) and exact results (dashed and solid lines) obtained with the equation-of-motion method for the single-particle reduced density matrix (13) for different system sizes $N$.

Figure 6

Classical $LC$ line with on-site applied potentials ${\phi }_i$.

Figure 7

Current $I_{250}\left(t\right)$ for a classical wire of size $N=500$ through the inductor at position $N/2$. $L=C=1$, $Q_L=Q_{0,i\le N/2}=1$, and $Q_R=Q_{0,i>N/2}=-1$.

Figure 8

(Solid curve) Current $I_{\infty }\left(t\right)$ for a classical infinite wire through the inductor at the middle with $L=C=1$, $Q_L=1$, and $Q_R=-1$. The dashed curve is the current for a finite system with size $N=30$. The constant line denotes the stationary value 1 against which the solid line converges.

Figure 9

Exact current $I_{\infty }\left(t\right)$ (solid line) and approximative current $I_{\text{app}}\left(t\right)$ (dashed line) with $L=C=1$, $Q_L=1$, and $Q_R=-1$.

Figure 10

(Solid curve) Current $I_{\text{m,}\infty }\left(t\right)$ through two adjacent inductors in the middle; Eq. (51). (Dashed curve) Current $I_{\infty }\left(t\right)$ from Eq. (44) through a single inductor. $L=C=1$, $Q_L=1$, and $Q_R=-1$.

Figure 11

Constructed signal $I_{\mathcal{R}}(t,t_s)$ from simulation results together with the Fourier series representation of a square wave ${\mathcal{R}}_m\left(t\right)$ from (55) with $m=8$ and an ideal square wave $R\left(t\right)$. The Fourier transformation of $I_{\mathcal{R}}(t,t_s)$ and expression (62) was used to calculate the amplitude of $R\left(t\right)$.

Figure 12

Current in the tight-binding model for a long time scale, calculated using the one-particle equation of motion. The initial state was prepared with a completely filled left half, a completely empty right half, and $t_{\text{H}}=0$ for all bonds.

Figure 13

Current in the tight-binding model for $\Delta \epsilon =2t_{\text{H}}$, calculated using the one-particle equation of motion. The dashed lines are currents through a single bond in the chain while the solid ones represent the mean value of the currents through two adjacent bonds in the middle.

Figure 14

(Solid line) $J_{\text{F}}\left(t\right)$ from Eq. (68); (dashed line) corresponding approximation $J_{\text{F,app}}\left(t\right)$ from (69).

Figure 15

Current-voltage curve of the spinless fermion model with setup (I) for several positive $V_{\text{H}}$. For $V_{\text{H}}=0$ the current has been computed using TEBD and the one-particle equation of motion (eom). The lines are guides for the eyes.

Figure 16

Magnification of Fig. 15. The lines indicate the linear response according to the TLL theory (10) for small $\Delta \epsilon$.

Figure 17

Current-voltage curve of the spinless fermion model with setup (I) for several negative $V_{\text{H}}$. The dashed lines indicate the theoretical beginning of the current plateaus according to (70). The solid lines are guides for the eyes.

Figure 18

Magnification of Fig. 17. The lines indicate the linear response according to the TLL theory (10) for small $\Delta \epsilon$.

Figure 19

Current-voltage curve of the spinless fermion model with setup (II) for several positive $V_{\text{H}}$. For $V_{\text{H}}=0$ the current has been computed using TEBD and the one-particle equation of motion (eom). The lines are guides for the eyes.

Figure 20

Magnification of Fig. 19. The lines indicate the linear response according to the TLL theory (10) for small $\Delta \epsilon$.

Figure 21

Current-voltage curve of the spinless fermion model with setup (II) for several negative $V_{\text{H}}$. The lines are guides for the eyes.

Figure 22

Magnification of Fig. 21. The lines indicate the linear response according to the TLL theory (10) for small $\Delta \epsilon$.

Figure 23

Period $T^{\text{max}}$ of the rectangular current oscillation in the finite spinless fermion model as a function of the inverse renormalized Fermi velocity $v^{-1}$ from expression (2). The dashed lines are described by Eq. (71).