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arXiv:2202.06543 · Physical review. B./Physical review. B

Exact dynamical correlations of hard-core anyons in one-dimensional lattices

Qing-Wei Wang

Published May 31, 2022• 10 citations
arXivHTMLPDF

Abstract

The dynamical correlations of a strongly correlated system are an essential ingredient to describe its nonequilibrium properties. We present a general method to calculate exactly the dynamical correlations of hard-core anyons in one-dimensional lattices, valid for any type of confining potential and any temperature. We obtain exact explicit expressions of the Green's function, the spectral function, and the out-of-time-ordered correlators (OTOCs). We find that the anyonic spectral function displays three main singularity lines which can be explained as a double spectrum in analogy to the Lieb-Liniger gas. The dispersion relations of these lines can be given explicitly and they cross at a hot point $({q}_{m},{\ensuremath{\omega}}_{m})$, which induces a peak in the momentum distribution function at ${q}_{m}$ and a power-law singularity in the local spectral function at ${\ensuremath{\omega}}_{m}$. We also find that the anyonic statistics can induce spatial asymmetry in the Green's function, its spectrum, and the OTOC. Moreover, the information spreading characterized by the OTOCs shows light-cone dynamics, asymmetric for general statistics and low temperatures, but symmetric at infinite temperature. Our results pave the way toward studying the nonequilibrium dynamics of hard-core anyons and experimentally probing anyonic statistics through spectral functions.

DOI
10.1103/physrevb.105.205143

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The dynamical correlations of a strongly correlated system are an essential ingredient to describe its nonequilibrium properties. We present a general method to calculate exactly the dynamical correlations of hard-core anyons in one-dimensional lattices, valid for any type of confining potential and any temperature. We obtain exact explicit expressions of the Green's function, the spectral function, and the out-of-time-ordered correlators (OTOCs). We find that the anyonic spectral function displays three main singularity lines which can be explained as a double spectrum in analogy to the Lieb-Liniger gas. The dispersion relations of these lines can be given explicitly and they cross at a hot point (qm,\ensuremathωm)({q}_{m},{\ensuremath{\omega}}_{m})(qm​,\ensuremathωm​), which induces a peak in the momentum distribution function at qm{q}_{m}qm​ and a power-law singularity in the local spectral function at \ensuremathωm{\ensuremath{\omega}}_{m}\ensuremathωm​. We also find that the anyonic statistics can induce spatial asymmetry in the Green's function, its spectrum, and the OTOC. Moreover, the information spreading characterized by the OTOCs shows light-cone dynamics, asymmetric for general statistics and low temperatures, but symmetric at infinite temperature. Our results pave the way toward studying the nonequilibrium dynamics of hard-core anyons and experimentally probing anyonic statistics through spectral functions.
We study dynamical correlations of interacting hard-core anyons on a one-dimensional lattice at infinite temperature. This is a setting in which the many-body spectrum is independent of the statistical phase θ , while dynamical correlators remain sensitive to θ through nonlocal Jordan-Wigner strings. We compute single-particle Green's functions, spectral functions, and density-density correlators, thereby separating the effects of fractional statistics on one-body coherence from those on density transport in a maximally mixed ensemble. In the noninteracting case V = 0 , high-temperature averaging leads to inversion-symmetric Green's functions for all θ despite the presence of anyonic strings. Finite nearest-neighbor interactions V generate, however, a pronounced left-right asymmetry in the Green's functions for 0 < θ < π , with the strongest chirality appearing at intermediate couplings V ∼ J where interactions and hopping compete most effectively. In this regime, the Green's function decays exponentially in time with a statistical-angle-dependent decay rate. At strong coupling, the dynamics crosses over to an atomic-limit regime in which the dependence on θ is reduced. Here, the Green's function decays universally as t − 1 and the corresponding spectral function displays a three-band structure. In contrast, density-density correlations are insensitive to statistics and recover the known infinite-temperature transport regimes of the XXZ chain, including ballistic, superdiffusive, and diffusive behaviors. These results identify dynamical correlation functions as direct probes of fractional statistics in high-entropy quantum systems.
We study the nonequilibrium quench dynamics of a one-dimensional anyonic gas. We focus on the integrable anyonic Lieb-Liniger model and consider the quench from noninteracting to hard-core anyons. We study the dynamics of the local properties of the system. By means of integrability-based methods, we compute analytically the one-body density matrix and the density-density correlation function at all times after the quench and in particular at infinite time. Our results show that the system evolves from an initial state where the local momentum distribution function is nonsymmetric to a steady state where it becomes symmetric. Furthermore, while the initial momentum distribution functions (and the equilibrium ones) explicitly depend on the anyonic parameter, the final ones do not. This is reminiscent of the dynamical fermionization observed in the context of free expansions after release from a confining trap.
We address the problem of calculating the correlation functions in a system of one-dimensional hard-core anyons that can be experimentally realized in optical lattices. Using the summation of form factors we have obtained Fredholm determinant representations for the time-, space-, and temperature-dependent Green's functions which are particularly suited to numerical investigations. In the static case we have also derived the large distance asymptotic behavior of the correlators and computed the momentum distribution function at zero and finite temperature. We present extensive numerical results highlighting the characteristic features of one-dimensional systems with fractional statistics.
We investigate the ground-state and dynamical properties of multicomponent interacting anyons confined in a one-dimensional (1D) optical lattice. Adopting the Anyon--Hubbard model, we explore spin and charge correlations as functions of interaction strength and the anyonic statistical parameter. Our results demonstrate that multicomponent effects combined with fractional exchange statistics substantially reshape both charge and spin correlations. For fractional exchange statistics, the canonical symmetric shell structure with prominent singularities in the spinor fermionic momentum distribution undergoes notable structural reconstruction, exhibiting emergent asymmetry, new spectral peaks, and broadened singular features. Unlike the Tonks--Girardeau bosonic gas, pseudobosons, representing a limiting case of anyons, cease to display a dominant zero-momentum peak. Instead, a quasi-fermionic shell structure emerges with finite-momentum peaks, whose locations are modulated by lattice site occupancy and interaction strength. The structure factor of spin correlations uncovers antiferromagnetic ordering in spinor anyon systems, with correlation magnitudes tunable by anyonic exchange statistics. Furthermore, quench dynamics analysis reveals the statistical phase parameter as an effective tuning knob. It enables the switching of dipole oscillations between underdamped and overdamped relaxation regimes and governs overall cloud expansion dynamics. Our findings pave the way for exploring statistics-driven ground-state and dynamical phase transitions in spinor anyon systems.
Anyons, namely particles obeying fractional quantum statistics that interpolate between bosons and fermions, possess a lot of new and exotic physical properties related to the particle exchange statistics. In this work, we explore the few-body quantum dynamics and quantum correlations of indistinguishable anyons with on-site interactions in onedimensional lattices within the scheme of three-body continuous-time quantum walks. By employing a time-evolving block decimation (TEBD) algorithm, we numerically calculate the dynamical evolution process of the local density distribution of anyons among the whole lattice. Numerical simulations shown in the main text mainly focus on a threebody initial state as ψ|(t=0)＞=â†-1â†0â†1|0＞, in which three particles are located on neighbouring sites at lattice centre. This choice of initial state features that the three particles influence one another most strongly, while we have also implemented numerical simulations on other choices of three-body initial states as are discussed in appendix. It is shown that the local density distribution of anyons is dramatically altered by fractional particle statistics with new dynamical structure showing up during the time evolution. For free anyons, an inner cone emerges as the statistical parameter increases, while the outer cone remains robust all along. When the on-site interaction joins in, the structure of the inner cone is further modified with new features. Specifically, for interaction of finite strength, an exotic dynamical asymmetry in real space, is clearly demonstrated during the time evolution of the local density distribution for particles within the fractional statistics regime. However, for boson limit and pseudofermion limit, the time evolution of the local density distribution keeps symmetric as the three-body initial state. And remarkably, the dynamical asymmetry is interactiondependent manifested as the local density distribution of anyons favors opposite side of the lattice for repulsive and attractive interaction, respectively. Moreover, when the on-site interaction is further increased to hard-core limit, the dynamical asymmetry will then be largely suppressed. We also calculate the density-density correlations for anyons before they reach the lattice boundary to reveal the interesting effect of fractional statistics on quantum correlations. It is shown that the inner cone corresponds to co-walking of anyons, while the outer cone is related to individual walking and is immune to the variation of statistical parameter. Furthermore, the exotic real-space asymmetry originated from the interplay of fractional statistics and finite interaction is also shown up in the density-density correlations.
We present a Bethe-ansatz-solvable dissipative model of one-dimensional hard-core anyons under dephasing, where the particles obey fractional statistics. We establish an exact mapping between the Liouvillian and a non-Hermitian Hubbard model with special twisted boundary conditions. With the help of string hypothesis, we rigorously calculate the finite-size corrections and the Liouvillian gap in terms of the statistical parameter, total particle number, lattice size, and dissipation strength. The statistical parameter breaks the $\ensuremath{\eta}$-pairing symmetry of the system, partially lifts out the degeneracy of the spectrum, and contributes to the Liouvillian gap in finite-size corrections. We also derive the general form of the equation of motion for expectation values of the 2w2w2w-point correlator of hard-core anyons and present a proposal for experimentally measuring the correlation function. Our results shed light on the Liouvillians of exactly solvable dissipative models of anyonic particles that obey the fractional statistics.