Charge and spin transport in strongly correlated one-dimensional quantum systems driven far from equilibrium

We study the charge conductivity in one-dimensional prototype models of interacting particles, such as the Hubbard and the $t\text{−}V$ spinless fermion models, when coupled to some external baths injecting and extracting particles at the boundaries. We show that, if these systems are driven far from equilibrium, a negative differential conductivity regime can arise. The above electronic models can be mapped into Heisenberg-like spin ladders coupled to two magnetic baths, so that charge transport mechanisms are explained in terms of quantum spin transport. The negative differential conductivity is due to oppositely polarized ferromagnetic domains that arise at the edges of the chain and therefore inhibit spin transport: we propose a qualitative understanding of the phenomenon by analyzing the localization of one-magnon excitations created at the borders of a ferromagnetic region. We also show that negative differential conductivity is stable against breaking of integrability. Numerical simulations of nonequilibrium time evolution have been performed by employing a Monte Carlo wave function approach and a matrix product operator formalism.

Figure 1

Schematic drawing of the level structure for a chain with $N=3$ sites. We assume that the lead-chain tunneling rates ${\Gamma }_L,{\Gamma }_R$ are much larger than the intrachain tunneling rates ${\Omega }_1,{\Omega }_2$. The electronic current flows from the right lead (emitter) to the left lead (collector).

Figure 2

(Color online) Spin current for the $\sigma$ (full curves and symbols) and the $\tau$ species (dashed curves and empty symbols) of spin as a function of the driving strength for the Hamiltonian in Eq. (14) with $U=5$. The system-bath coupling is set equal to $\Gamma =0.5$; the simulation time (QT approach) is $T=2\times {10}^5$. Note that curves and symbols for the $\sigma$ and $\tau$ species are nearly superimposed. For the Hubbard model we set $t=1$ as the system’s energy scale.

Figure 3

(Color online) Maximum current minus current at maximum driving strength, ${⟨j⟩}_{f^{\ast }}-{⟨j⟩}_1$, as a function of the on-site repulsion $U$. In the inset we show the spin current as a function of $f$ for a fixed size $N=6$ and different values of $U$: from top to bottom $U=0$ (circles), 0.5 (squares), 1 (diamonds), 1.5 (triangles up), 2 (triangles left), 2.5 (triangles down), 3 (triangles right), and 5 (stars). Data are for $\Gamma =1$, and the simulation time (QT approach) is $T={10}^5$. The current is plotted only for the $\sigma$-spin species; differences with the $\tau$-spin species are negligible in the scales of the figure.

Figure 4

(Color online) Spin current as a function of the driving strength for $\Delta =0.5$. The system-bath coupling is set equal to $\Gamma =1$. The insets show spin magnetization profiles versus the scaled spin index coordinate for two values: $f=0.5,1$. Data are obtained from the QT approach.

Figure 5

(Color online) Spin current as a function of the driving strength $f$ in a chain with anisotropy $\Delta =2$. The system-bath coupling is set equal to $\Gamma =4$. Symbols for $N\le 16$ are obtained using QT after a simulation time $T=2.5\times {10}^3$, while curves display data for a longer integration time $T=7.5\times {10}^4$. Data for $N=40$ are obtained by MPO, with integration times up to $T=4000$. Long-time data for $N\le 16$ are quoted from Ref. 32.

Figure 6

(Color online) Spin magnetization profiles versus scaled spin index coordinate for the same parameters as in Fig. 5, and driving strengths $f=0.3$ (empty symbols) and $f=1$ (filled symbols). Data are obtained from the QT approach. (Figure acknowledged from Ref. 32.)

Figure 7

(Color online) Spin current at a fixed driving strength $f=1$ for the Heisenberg spin model with $\Delta =2$. The system-bath coupling is set equal to $\Gamma =4$; the simulation time (QT approach) is $T=7.5\times {10}^4$. The thick curve displays an exponential fit of numerical data (circles).

Figure 8

(Color online) Spin magnetization profiles versus scaled spin index coordinate in the insulating case $\Delta =2$ with a system-bath coupling $\Gamma =4$ for strong drivings. In the main panel we fix a system size $N=12$ and choose different values of $f$, while in the inset we focus on the case $f=0.9$ and vary the size according to the legend. Data are obtained from the QT approach.

Figure 9

(Color online) Spin magnetization profiles versus scaled spin index coordinate for a chain with $N=8$ spins at maximum driving $\left(f=1\right)$ for different values of $\Delta >1$; the system-bath coupling is set equal to $\Gamma =1$. In the inset we plot an estimate of the thickness of the interface region $\overline{\xi }$ as a function of $\text{ln}\left(\Delta \right)$; the straight line shows the fit $\overline{\xi }\propto {\left(\text{ln}\Delta \right)}^{-1.625}$. Data are obtained from the QT approach.

Figure 10

(Color online) Spin magnetization profiles versus spin index coordinate for a chain with $N=10$ spins and anisotropy $\Delta =2$ at different integration times $T$. Top: symmetric drivings such that $f=1$ and system-bath couplings ${\Gamma }_L=1,{\Gamma }_R=0.1$. Bottom: the system-bath coupling for the left and the right baths is kept fixed and equal to $\Gamma =1$, while the drivings are chosen asymmetrically as $f_L=0,f_R=0.99$. Data are obtained from the QT approach.

Figure 11

(Color online) Spin current as a function of the driving strength for the isotropic case $\Delta =1$. The system-bath coupling is set equal to $\Gamma =2$. The simulation time (QT approach) is $T=5\times {10}^4$. In the inset we show the spin magnetization profiles versus scaled spin index coordinate at maximum driving $f=1$. Different curves stand for various system sizes.

Figure 12

(Color online) Spin current for $\Delta =1$ at a fixed driving strength, $f=0.2$ on the left, while $f=1$ on the right (see Fig. 11). Full curves are fits of numerical data (obtained from QT): while an inverse power-law fit works well at $f⪡1$, for maximum driving an exponential fit seems adequate. The dashed lines indicate a behavior ${⟨j⟩}_f\sim 1/N$ and are plotted as guidelines.

Figure 13

(Color online) Spin magnetization profiles at $\Delta =2$, $\Gamma =4$ for a maximal driving strength: $f=1$. The various panels correspond to different times $T$ (in QT simulation) at which the profiles are plotted. Circles are for a single bath coupled to the first spin $\left(j=1\right)$, while squares are for two baths at the ends of the chain.

Figure 14

(Color online) Integrated level spacing distribution $I\left(S\right)$ for the Hamiltonian in Eq. (22) with $\Delta =1.3$, $B=0.3$, $N=16$; data (full black curve) correspond to the zero magnetization sector. In order to apply the random matrix theory, the system Hamiltonian has to be diagonalized in a subspace in which no symmetries (except the time reversal) are left. For this purpose, in addition to fixing the total magnetization, we applied the staggered magnetic field on all spins except the first one [i.e., we supposed that the last sum in Eq. (22) runs from 2 to $N$]. This breaks the spatial reflection symmetry $j\to N-j$, without affecting the transport properties under investigation. Red dashed (blue dotted) curve indicates Wigner-Dyson (Poissonian) statistics, typical for chaotic (integrable) systems. In the inset (data from QT) we plot the spin current as a function of the driving strength for different chain lengths; we fixed a system-bath coupling $\Gamma =1$.

Figure 15

(Color online) Dependence of the scaled spin current on the chain length in the staggered Heisenberg model, with $\Delta =1.3$, $B=0.3$, and $\Gamma =1$ ($\Delta S$ is the magnetization difference across the chain), at driving strength $f=0.1$ (circles) and $f=0.5$ (squares). The two dashed lines denote the behaviors $\sim 1.3/N^{1.1}$ (for circles) and $\sim 1.85/N^{1.4}$ (for squares), respectively, whereas dotted line indicates $1/N$ behavior found in the linear response regime. In the inset we show magnetization profiles for $f=0.5$ and different system sizes. Data are obtained from MPO.

Figure 16

(Color online) Dependence of the scaled spin current on the chain length for the staggered Heisenberg model, with $\Delta =0.5$, $B=0.5$, and $\Gamma =1$. Both for strong and weak drivings the system shows a normal Ohmic behavior, as it is depicted by the straight dashed line, which corresponds to $\sim 5.2/N$. The magnetization has an approximately linear profile, as shown in the inset, with a slight even-odd oscillation that is due to a staggered magnetic field. Data are obtained from MPO.

Figure 17

(Color online) Spin current as a function of the driving strength for the Hamiltonian in Eq. (B5) with $t=2$, $V=8$ (that corresponds to $J_x=1$, $J_z=2$, so that $\Delta =2$). In contrast to the standard XXZ model, here we added a uniform transverse magnetic field of intensity $V/2$ plus two transverse fields of strength $-V/4$ at the border sites of the Heisenberg chain. The system-bath coupling is set equal to $\Gamma =4$; the simulation time (QT approach) is $T=2.5\times {10}^4$.

Figure 18

(Color online) Spin-spin correlation function $C\left(i,j\right)$ in a chain with 40 spins for $f=0.1$ (top row) and $f=0.9$ (bottom row); here we set $\Delta =2$ and $\Gamma =4$. The code in the left panels denotes ${\text{log}}_{10}\left[C\left(i,j\right)\right]$. The right plots display correlation function $C\left(r\equiv \left|i-j\right|\right)$ along the diagonal denoted by a dashed line in the left plots; the case with $N=20$ spins is also shown. Data are obtained using the MPO ansatz.

Figure 19

(Color online) The one-magnon model for $N=16$ spins. Left: energy spectrum as a function of $\Delta$; large-$N$ analytic result (D12) is shown by thick dashed lines. Right: molecular states $|{\psi }_{\pm }⟩={\sum }_{m=1}^N{\left(c_m\right)}_{\pm }|m⟩$ at $\Delta =2$; the numerically computed coefficients ${\left(c_m\right)}_{+}$ and ${\left(c_m\right)}_{-}$ are shown as circles and triangles; the large-$N$ analytic results are shown as full and dashed curves.