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arXiv:1201.5323 · Physical Review B

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

Malcolm Einhellinger, Alex Cojuhovschi, Eric Jeckelmann

Published Jun 21, 2012• 24 citations
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Abstract

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 $eV\ensuremath{\ll}4{t}_{\text{H}}$ 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.

DOI
10.1103/physrevb.85.235141

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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 (LCLCLC 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 III-VVV characteristic of the spinless fermion model with nearest-neighbor hopping tH{t}_{\text{H}}tH​ and interaction VH{V}_{\text{H}}VH​ using two different setups to generate currents (turning on/off a potential bias). Our numerical results agree with exact results for noninteracting fermions (VH=0{V}_{\text{H}}=0VH​=0). For interacting fermions we find that in the linear regime eV\ensuremath≪4tHeV\ensuremath{\ll}4{t}_{\text{H}}eV\ensuremath≪4tH​ the current III 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 VVV the steady-state current depends on the current-generating setup and as VVV 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.
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Nonlinear transport in the one dimensional Hubbard model at half-filling under a finite bias voltage is investigated by the adaptive time-dependent density matrix renormalization group method. For repulsive on-site interaction, dielectric breakdown of the Mott insulating ground state to a current-carrying nonequilibrium steady state is clearly observed when the voltage exceeds the charge gap. It is found that by increasing the voltage further the current-voltage characteristics are scaled only by the charge gap and the scaling curve exhibits almost linear dependence on the voltage whose slope is suppressed by the electron correlation. In the case of attractive interaction the linear conductance is the perfect one 2e2/h2e^2/h2e2/h which agrees with the prediction by the Luttinger liquid theory.
We consider electron transport in a model of a spinless superconductor described by a Kitaev-type lattice Hamiltonian where the electron interactions are modeled through a superconducting pairing term. The superconductor is sandwiched between two normal metals that are kept at different temperatures and chemical potentials and that are themselves modeled as noninteracting spinless fermions. For this setup we compute the exact steady-state properties of the system using the quantum-Langevin-equation approach. Closed-form exact expressions for particle current, energy current, and other two-point correlations are obtained in the Landauer-type forms and involve two nonequilibrium Green's functions. The current expressions are found to be the sum of three terms having simple physical interpretations. We then discuss a numerical approach where we construct the time evolution of the two-point correlators of the system from the eigenspectrum of the complete quadratic Hamiltonian describing the system and leads. By starting from an initial state corresponding to the leads in thermal equilibrium and the system in an arbitrary state, the long-time solution for the correlations, before recurrence times, gives us steady-state properties. We use this independent numerical method for verifying the results of the exact solution. We also investigate analytically the presence of high-energy bound states and obtain expressions for their contributions to two-point correlators. As applications of our general formalism we present results on thermal conductance and on the conductance of a Kitaev chain with next-nearest-neighbor interactions which allows topological phases with different winding numbers.
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