Universal properties and dynamical bosonization of strongly interacting one-dimensional anyons

TL;DR

We address a one-dimensional system of strongly interacting anyons (sp-anyons) that interpolate between $p$-wave fermions ($\kappa=1$) and Lieb–Liniger bosons ($\kappa=0$) under confinement. The study reveals exponential decay of correlations with a statistics-dependent phase, a shifted Lorentzian momentum distribution with universal $1/k^2$ tails, and universal ground-state $n$-particle reduced density matrices (RDMs) across confining potentials. In nonequilibrium, release from a harmonic trap induces dynamical bosonization, where the asymptotic momentum distribution matches that of free bosons in the initial trap, while breathing oscillations exhibit trap-dependent dynamics and differ from Tonks–Girardeau behavior. The authors provide an exact solution for the one-particle RDM eigenproblem and show that RDM eigenvalues are confinement-independent, with natural orbitals related by a simple transformation, thereby unifying and extending previous results on $p$-wave fermions and clarifying observed degeneracies.

Abstract

We study a one-dimensional system of strongly interacting anyons with short-range interactions under external confinement. This system, referred to as $p$-wave anyons, interpolates continuously between spin-polarized fermions with $p$-wave interactions and free bosons. At zero temperature, the correlation functions decay exponentially with distance, with oscillations governed by the statistics parameter. The decay rate is maximal for $p$-wave fermions and decreases monotonically as the statistics parameter approaches the bosonic limit, where it vanishes. The momentum distribution is asymmetric, a hallmark of one-dimensional anyons, and takes the form of a shifted Lorentzian with universal power-law tails, $\lim_{k \to \pm \infty} n(k)\sim C/k^2$. We prove analytically that, following release from a harmonic trap, the asymptotic momentum distribution converges to that of free bosons in the same trap, a phenomenon known as dynamical bosonization. We also establish the universality of the groundstate $n$-particle reduced density matrices: their natural occupations are independent of the confining potential, while the associated natural $n$-functions for different confinements are related through a simple analytical transformation. In particular, for the one-particle reduced density matrix, we derive exact expressions for both the natural occupations and the natural orbitals at arbitrary particle number. These results extend and unify earlier partial findings for $p$-wave fermions, and they provide a clear conceptual explanation of the double degeneracy observed in their spectrum.

Figures (3)

• Figure 1: Groundstate momentum distribution of $sp$-anyons for different values of the statistical parameter $\kappa$. Results are shown for a homogeneous system in the thermodynamic limit (top row) and for a harmonically trapped system with parameters $N=50$, $l_0=m=\omega=1$ (bottom row). Note the asymmetry of the momentum distribution, $n(-k)\neq n(k)$, which is a hallmark of one-dimensional anyons for $\kappa \neq 1$.
• Figure 2: Time evolution of the momentum distribution of $sp$-anyons after release from the trap for various statistics parameters. Parameters are $N=10$, $l_0 = m = \omega_0 = 1$ and $\Delta t=2\pi/\omega_0$. Dashed black lines in the last three rows show the momentum distribution of $N$ free bosons in the initial trap [Eq. (\ref{['mombosons']})].
• Figure 3: Density profiles (first column) and momentum distribution oscillations (last three columns) induced by a quench of the trap frequency ($\omega_0=1$, $\omega_1=6\omega_0$, $\alpha \approx -0.972$, $l_0=m=1$) for systems of $N=20$$sp$-anyons with different statistics parameters (top row), and for a comparable system of Tonks-Giradeau anyons described by Eq. (\ref{['swave']}) (bottom row).