Bipartite Fluctuations and Charge Fractionalization in Quantum Wires

TL;DR

The paper develops a quantum-information framework to detect charge fractionalization in one-dimensional, interacting quantum wires by analyzing bipartite fluctuations of charge and current for chiral quasiparticles. It shows that the fluctuations scale logarithmically with region size, with a prefactor determined by the Luttinger parameter $K$ and the fractional charges $f_R=(1+K)/2$ and $f_L=(1−K)/2$, via a function ${F}_1(x)=\frac{1}{2}\ln(\frac{\alpha^2+x^2}{\alpha^2})$. The approach is validated through a detailed Luttinger-liquid analysis and DMRG studies of a spin-chain analogue, connecting to dephasing in ballistic rings at zero temperature and detecting a Mott transition through a linear term in current fluctuations; it also uncovers a bound state at an interface described by the Jackiw-Rebbi model. Collectively, these results provide a practical route to measure fractional charges in 1D systems, illuminate ground-state energetics, and link topology to entanglement diagnostics in low-dimensional quantum fluids.

Abstract

We introduce a quantum information method for measuring fractional charges in ballistic quantum wires generalizing bipartite fluctuations to the chiral quasiparticles in Luttinger liquids, i.e. analyzing and summing charge and current fluctuations in a region of the wire. Bipartite fluctuations at equilibrium are characterized through a logarithmic scaling with distance encoding the entangled nature of these fractional charges in one-dimensional (1D) fluids. This approach clarifies the physical meaning of the dephasing factor of electronic interferences in a ballistic ring geometry at zero temperature, as a result of charge fractionalization. We formulate an analogy towards ground-state energetics. We show how bipartite current fluctuations represent a useful tool to locate quantum phase transitions associated to Mott physics. We address a spin chain equivalence and verify the fractional charges through an algorithm such as Density Matrix Renormalization Group (DMRG). Adding a potential difference between the two sides (parties) of the wire, bipartite fluctuations can detect a bound state localized at the interface through the Jackiw-Rebbi model coexisting with fractional charges.

Figures (6)

• Figure 1: Current fluctuations from DMRG with PBC for a wire with 300 sites, 20 sweeps. Data are in solid colored lines and the dashed lines represent a fit with a logarithmic term plus a linear term modulo a constant (see Appendix \ref{['AppendixA']}). At weak $U$, results agree with the analytical form in Eq. (\ref{['fluctuationscurrent']}) and when $U$ becomes substantial a linear form develops revealing the Mott transition.
• Figure 2: Prefactor of the logarithmic term in $\pi^2({\cal F}_Q(x)+{\cal F}_{\tilde{J}}(x))$ with DMRG, same protocol as Fig. 1 and fit with the formula revealing the fractional charges in Eq. (\ref{['fluctuations']}) with precisely $\frac{2}{K}(f_R^2+f_L^2)=(\frac{1}{K}+K)$; the function $K$ in terms of $U/t$ is obtained from Bethe Ansatz Nishimoto. (Inset) Increase of linear term at the Mott transition $(U=2t)$.
• Figure 3: Bipartite Current Fluctuations for OBC at $U=0.2t$ with $300$ sites, for various values of $\Delta$. (Inset) Density probability (not normalized) at small $U$ e.g. at $U=0$.
• Figure 4: (Left) Current fluctuations for a subsystem of size $x$ within a system of size $L=300$ with various values of $U/t$. Solid colored lines are datas and fitting curves are the dashed lines. The fitting protocol is done with $f(x)=\frac{1}{\pi^2 K}\log d(x|L)+bx+cst$. The function $d(x|L)$ is the chord distance: $d(x|L)=\frac{L}{\pi}\sin\left(\frac{\pi x}{L}\right)$. (Right) Symmetrized bipartite current fluctuations when summing ${\cal F}_{\tilde{J}}(x)$ and ${\cal F}_{\tilde{J}}(N-x)$, with $N$ being the number of sites or length when fixing the lattice spacing to unity. We verify the fitting form in dashed grey lines, $\frac{1}{\pi^2 K}\log L+cst.$ of the data (solid colored lines).
• Figure 5: Bipartite Current Fluctuations for Open Boundary Conditions (OBC) at $U=t$ with $300$ sites, 20 sweeps, for various values of $\Delta$.