[Paper Review] Transport through Periodically Driven Correlated Quantum Wires

Dante M. Kennes|arXiv (Cornell University)|Jan 9, 2018

Quantum and electron transport phenomena3 citations

TL;DR

This paper investigates transport in one-dimensional, correlated quantum wires subjected to periodic driving, using functional renormalization group (FRG) methods to analyze the steady-state response. It identifies the ratio Δε/Ω as the key control parameter: at small values, the system exhibits a modified Luttinger liquid with multiple power-law suppressions in spectral and conductance responses; at large values, it transitions to effectively higher-dimensional behavior with suppressed power-law features.

ABSTRACT

We study correlated quantum wires subject to harmonic modulation of the onsite-potential concentrating on the limit of large times, where the response of the system has synchronized with the drive. We identify the ratio $Δε/Ω$ of the driving amplitude $Δε$ and the frequency of driving $Ω$ as the scale determining the crossover from a modified Luttinger liquid picture to a system that behaves effectively like a higher dimensional one. We exemplify this crossover by studying the frequency dependency of the boundary density of state $ρ_{ m B}(ω)$ as well as the temperature dependency of the linear conductance $G(T)$ through the wire, if the latter is contacted to leads. Both observables are known to exhibit Luttinger liquid physics without driving given by characteristic power-law suppression as $ω oε_{ m F}$ (with $ε_{ m F}$ the Fermi energy) or $T o 0$, respectively. With driving we find that this suppression is modified from a single power-law to a superposition of an infinite number of power laws. At small $Δε/Ω\ll 1$ only a few terms of this infinite sum are relevant as the prefactors of higher terms are suppressed exponentially. Thus a picture similar to the equilibrium Luttinger liquid one emerges. Increasing $Δε/Ω$ an increasing number of power laws contribute to the sum and approaching $Δε/Ω\gg 1$ the system behaves effectively two dimensional, for which the suppression is wiped out completely.

Motivation & Objective

• To understand the non-equilibrium steady-state transport properties of one-dimensional, correlated quantum wires under periodic driving.
• To investigate how periodic driving modifies the hallmark Luttinger liquid behaviors, such as power-law suppression in the boundary spectral function and temperature-dependent conductance.
• To determine the role of the driving amplitude Δε and frequency Ω in controlling the crossover between Luttinger liquid and higher-dimensional physics.
• To establish whether the weak and strong impurity fixed points in conductance remain continuously connected in the driven case, as in equilibrium.

Proposed method

• Employing functional renormalization group (FRG) with a momentum-space cutoff adapted for driven systems to treat strong correlations and non-equilibrium dynamics.
• Using a time-averaged spectral function and conductance to analyze the system's response in the long-time limit after synchronization with the drive.
• Applying a one-parameter scaling ansatz to the conductance G(T), with the scaling variable y = (T/s)^(K̃−1), where K̃ is the Luttinger liquid parameter for a renormalized interaction Ũ ∝ |Jr(2Δε/Ω)|².
• Deriving analytical expressions for the frequency-dependent boundary spectral function ρ_B(ω), showing it splits into multiple power-law suppressions at ω = ε_F + nΩ.
• Numerically validating the predicted exponents of the power-law components against FRG data up to O(U²), confirming the role of Bessel functions in modulating the exponents.
• Introducing a reservoir model to absorb excess energy from the drive, ensuring the system reaches a stable steady state without heating to infinite temperature.

Experimental results

Research questions

• RQ1 How does periodic driving modify the power-law suppression of the boundary spectral function in a one-dimensional correlated quantum wire?
• RQ2 What is the role of the ratio Δε/Ω in determining the crossover between Luttinger liquid and higher-dimensional effective behavior?
• RQ3 Does the conductance G(T) still exhibit a single power-law suppression at low temperatures in the driven case, or is it replaced by a superposition of multiple power laws?
• RQ4 Are the weak and strong impurity fixed points in the conductance flow continuously connected in the driven system, as in equilibrium?
• RQ5 Can the FRG framework be extended to describe driven correlated systems with analytical control over the frequency-dependent response?

Key findings

• The boundary spectral function ρ_B(ω) exhibits multiple power-law suppressions at frequencies ω = ε_F + nΩ, with each suppression governed by an exponent proportional to |J_r(2Δε/Ω)|² times the equilibrium exponent α_B.
• At small Δε/Ω ≪ 1, only a few terms in the infinite sum of power laws are significant due to exponential suppression of higher-order Bessel function terms, leading to a behavior resembling equilibrium Luttinger liquid physics.
• As Δε/Ω increases, more power-law components contribute, and for Δε/Ω ≫ 1, the system behaves effectively two-dimensional, with the power-law suppression of the spectral function completely wiped out.
• The temperature-dependent conductance G(T) is no longer a single power law but becomes a superposition of infinitely many power laws, each with an exponent modulated by Bessel functions of the driving parameters.
• A one-parameter scaling collapse of G^0(T) is achieved using a modified Luttinger liquid parameter K̃ based on the effective interaction Ũ = |J_r(2Δε/Ω)|² U, confirming continuous connection between weak and strong impurity fixed points.
• Numerical FRG results for G(T) at O(U²) agree quantitatively with the analytical prediction that the exponents of the power-law components are 2|J_r(2Δε/Ω)|² α_B, validating the theoretical framework.