Persistent current of Luttinger liquid in one-dimensional ring with weak link: Continuous model studied by configuration interaction and quantum Monte Carlo

TL;DR

This study investigates the persistent current in a one-dimensional Luttinger liquid with a weak link using advanced numerical methods, confirming the power-law decay with ring length and exploring the effects of interaction strength and range.

Contribution

It provides numerical validation of the Luttinger liquid power-law behavior for persistent current in a continuous model, including the influence of interaction range and strength, using CI and DMC methods.

Findings

Persistent current follows a power law with ring length for large L.

The theoretical formula for the power-law exponent holds when the interaction range is small.

Strong interactions lead to Luttinger liquid behavior observable with as few as ten electrons.

Abstract

We study the persistent current of correlated spinless electrons in a continuous one-dimensional ring with a single weak link. We include correlations by solving the many-body Schrodinger equation for several tens of electrons interacting via the short-ranged pair interaction V(x - x'). We solve this many-body problem by advanced configuration-interaction (CI) and diffusion Monte Carlo (DMC) methods. Our CI and DMC results show, that the persistent current (I) as a function of the ring length (L) exhibits for large L the power law typical of the Luttinger liquid, $I \propto L^{-1-\alpha}$, where the power $\alpha$ depends only on the electron-electron (e-e) interaction. For strong e-e interaction the previous theories predicted for $\alpha$ the formula $\alpha = {(1 + 2 \alpha_{RG})}^{1/2} - 1$, where $\alpha_{RG} = [V(0)-V(2k_F)]/2\pi \hbar v_F$ is the renormalisation-group result for weakly interacting electrons, with V(q) being the Fourier transform of V(x-x'). Our numerical data show that this theoretical result holds in the continuous model only if the range of V(x - x') is small (roughly $d \lesssim 1/2k_F$, more precisely $4d^2k_F^2 << 1$). For strong e-e interaction ($\alpha_{RG} > 0.25$) our CI data show the power law $I \propto L^{-1-\alpha}$ already for rings with only ten electrons, i.e., ten electrons are already enough to behave like the Luttinger liquid. The DMC data for $\alpha_{RG} > 0.25$ are damaged by the so-called fixed-phase approximation. Finally, we also treat the e-e interaction in the Hartree-Fock approximation. We find the exponentially decaying I(L) instead of the power law, however, the slope of log(I(L)) still depends solely on the parameter $\alpha_{RG}$ as long as the range of V(x - x') approaches zero.