One-dimensional Anyons Research Articles

Quantum Alchemy and Universal Orthogonality Catastrophe in One-Dimensional Anyons

Many-particle quantum systems with intermediate anyonic exchange statistics are supported in one spatial dimension. In this context, the anyon-anyon mapping is recast as a continuous transformation that generates shifts of the statistical parameter κ. We characterize the geometry of quantum states associated with different values of κ, i.e., different quantum statistics. While states in the bosonic and fermionic subspaces are always orthogonal, overlaps between anyonic states are generally finite and exhibit a universal form of the orthogonality catastrophe governed by a fundamental statistical factor, independent of the microscopic Hamiltonian. We characterize this decay using quantum speed limits on the flow of κ, illustrate our results with a model of hard-core anyons, and discuss possible experiments in quantum simulation.

Quantum entanglement of anyon composites

Studying quantum entanglement in systems of indistinguishable particles, in particular anyons, poses subtle challenges. Here, we investigate a model of one-dimensional anyons defined by a generalized algebra. This algebra has the special property that fermions in this model are composites of anyons. A Hubbard-like Hamiltonian is considered that allows hopping between nearest neighbor sites not just for the fundamental anyons, but for the fermionic anyon composites. Some interesting results regarding the quantum entanglement of these particles are obtained.

Topological exchange statistics in one dimension

The standard topological approach to indistinguishable particles formulates exchange statistics by using the fundamental group to analyze the connectedness of the configuration space. Although successful in two and more dimensions, this approach gives only trivial or near trivial exchange statistics in one dimension because two-body coincidences are excluded from configuration space. Instead, we include these path-ambiguous singular points and consider configuration space as an orbifold. This orbifold topological approach allows unified analysis of exchange statistics in any dimension and predicts novel possibilities for anyons in one-dimensional systems, including non-abelian anyons obeying alternate strand groups. These results clarify the non-topological origin of fractional statistics in one-dimensional anyon models.

Large-time and long-distance asymptotics of the thermal correlators of the impenetrable anyonic lattice gas

We study thermal correlation functions of the one-dimensional impenetrable lattice anyons. These correlation functions can be presented as a difference of two Fredholm determinants. To describe the large-time and long-distance behavior of these objects, we use the effective form-factor approach. The asymptotic behavior is different in the spacelike and timelike regions. In particular, in the timelike region we observe the additional power factor on top of the exponential decay. We argue that this result is universal as it is related to the discontinuous behavior of the phase shift function of the effective fermions. At particular values of the anyonic parameter, we recover the asymptotics of the spin-spin correlation functions in the $XX$ quantum chain.

Gaussian optical networks for one-dimensional anyons

We study the dynamics of bosonic and fermionic anyons defined on a one-dimensional lattice, under the effect of Hamiltonians quadratic in creation and annihilation operators, also called Gaussian Hamiltonians. These anyonic models are obtained from deformations of the standard bosonic or fermionic commutation relations via the introduction of a nontrivial exchange phase between different lattice sites. We study the effects of the anyonic exchange phase on the usual bosonic and fermionic bunching behaviors. We show how to exploit the inherent Aharonov-Bohm effect exhibited by these particles to build a deterministic, entangling two-qubit gate and prove quantum computational universality in these systems. We define coherent states for bosonic anyons and study their behavior under two-mode passive, Gaussian devices. In particular we prove that, for a particular value of the exchange factor, an anyonic mirror can generate cat states, an important resource in quantum information processing with continuous variables.

Bose-Fermi dualities for arbitrary one-dimensional quantum systems in the universal low-energy regime

I consider general interacting systems of quantum particles in one spatial dimension. These consist of bosons or fermions, which can have any number of components, arbitrary spin or a combination thereof, featuring low-energy two- and multiparticle interactions. The single-particle dispersion can be Galilean (non-relativistic), relativistic, or have any other form that may be relevant for the continuum limit of lattice theories. Using an algebra of generalized functions, statistical transmutation operators that are genuinely unitary are obtained, putting bosons and fermions in a one-to-one correspondence without the need for a short-distance hard core. In the non-relativistic case, low-energy interactions for bosons yield, order by order, fermionic dual interactions that correspond to the standard low-energy expansion for fermions. In this way, interacting fermions and bosons are fully equivalent to each other at low energies. While the Bose-Fermi mappings do not depend on microscopic details, the resulting statistical interactions heavily depend on the kinetic energy structure of the respective Hamiltonians. These statistical interactions are obtained explicitly for a variety of models, and regularized and renormalized in the momentum representation, which allows for theoretically and computationally feasible implementations of the dual theories. The mapping is rewritten as a gauge interaction, and one-dimensional anyons are also considered.

Statistically related many-body localization in the one-dimensional anyon Hubbard model

Many-body localization (MBL) has been widely investigated for both fermions and bosons, it is, however, much less explored for anyons. Here we numerically calculate several physical characteristics related to MBL of a one-dimensional disordered anyon-Hubbard model in both localized and delocalized regions. We figure out a logarithmically slow growth of the half-chain entanglement entropy and an area-law rather than volume-law obedience for the highly excited eigenstates in the MBL phase. The adjacent energy level gap-ratio parameter is calculated and is found to exhibit a Poisson-like probability distribution in the deep MBL phase. By studying a hybridization parameter, we reveal an intriguing effect that the statistics can induce localization-delocalization transition. Several physical quantities, such as the half-chain entanglement, the adjacent energy level gap-ratio parameter, {\color{black} the long-time limit of the particle imbalance}, and the critical disorder strength, are shown to be non-monotonically dependent on the anyon statistical angle. Furthermore, a feasible scheme based on the spectroscopy of energy levels is proposed for the experimental observation of these statistically related properties.

Quantum entanglement in one-dimensional anyons

Anyons in one spatial dimension can be defined by correctly identifying the configuration space of indistinguishable particles and imposing Robin boundary conditions. This allows an interpolation between the bosonic and fermionic limits. In this paper, we study the quantum entanglement between two one-dimensional anyons on a real line as a function of their statistics.

Probing the exchange statistics of one-dimensional anyon models

We propose feasible scenarios for revealing the modified exchange statistics in one-dimensional anyon models in optical lattices based on an extension of the multicolor lattice-depth modulation scheme introduced in [{Phys. Rev. A 94, 023615 (2016)}]. We show that the fast modulation of a two-component fermionic lattice gas in the presence a magnetic field gradient, in combination with additional resonant microwave fields, allows for the quantum simulation of hardcore anyon models with periodic boundary conditions. Such a semi-synthetic ring set-up allows for realizing an interferometric arrangement sensitive to the anyonic statistics. Moreover, we show as well that simple expansion experiments may reveal the formation of anomalously bound pairs resulting from the anyonic exchange.

Three-body-interaction effects on the ground state of one-dimensional anyons

A quantum phase transition driven by the statistics was observed in an anyon-Hubbard model with local three-body interactions. Using a fractional Jordan-Wigner transformation, we arrived at a modified Bose-Hubbard model, which exhibits Mott insulator and superfluid phases. The absence of a Mott insulator state with one particle per site depends on the anyonic angle, and a quantum phase transition from a superfluid to a Mott insulator state is obtained for a fixed value of the hopping. The critical points were estimated with the von Neumann block entropy and increase as the hopping grows. The statistics modify the ground state, and three different superfluid regions were observed for larger values of the anyonic angle.

One-dimensional hard-core anyon gas in a harmonic trap at finite temperature

We investigate the strongly interacting hard-core anyon gases in a one dimensional harmonic potential at finite temperature by extending thermal Bose-Fermi mapping method to thermal anyon-ferimon mapping method. With thermal anyon-fermion mapping method we obtain the reduced one-body density matrix and therefore the momentum distribution for different statistical parameters and temperatures. At low temperature hard-core anyon gases exhibit the similar properties as those of ground state, which interpolate between Bose-like and Fermi-like continuously with the evolution of statistical properties. At high temperature hard-core anyon gases of different statistical properties display the same reduced one-body density matrix and momentum distribution as those of spin-polarized fermions. The Tan's contact of hard-core anyon gas at finite temperature is also evaluated, which take the simple relation with that of Tonks-Girardeau gas $C_b$ as $C=\frac12(1-cos\chi\pi)C_b$.

Floquet Realization and Signatures of One-Dimensional Anyons in an Optical Lattice.

We propose a simple scheme for mimicking the physics of one-dimensional anyons in an optical-lattice experiment. It relies on a bosonic representation of the anyonic Hubbard model to be realized via lattice-shaking-induced resonant tunneling against potential offsets, which are created by a combination of a lattice tilt and strong on-site interactions. No lasers additional to those used for the creation of the optical lattice are required. We also discuss experimental signatures of the continuous interpolation between bosons and fermions when the statistical angle θ is varied from 0 to π. Whereas the real-space density of the bosonic atoms corresponds directly to that of the simulated anyonic model, this is not the case for the momentum distribution. Therefore, we propose to use Friedel oscillations in the density as a probe for continuous fermionization of the bosonic atoms.

One-body reduced density matrix of trapped impenetrable anyons in one dimension

We study the one-body reduced density matrix of a system of N one-dimensional impenetrable anyons trapped by a harmonic potential. To this purpose we extend two methods developed to tackle related problems, namely the determinant approach and the replica method. While the former is the basis for exact numerical computations at finite N, the latter has the advantage of providing an analytic asymptotic expansion for large N. We show that the first few terms of such expansion are sufficient to reproduce the numerical results to an excellent accuracy even for relatively small N, thus demonstrating the effectiveness of the replica method.

Yangian-invariant spin models and Fibonacci numbers

We study a wide class of finite-dimensional su(m|n)-supersymmetric models closely related to the representations of the Yangian Y(gl(m|n)) labeled by border strips. We quantitatively analyze the degree of degeneracy of these models arising from their Yangian invariance, measured by the average degeneracy of the spectrum. We compute in closed form the minimum average degeneracy of any such model, and show that in the non-supersymmetric case it can be expressed in terms of generalized Fibonacci numbers. Using several properties of these numbers, we show that (except in the simpler su(1|1) case) the minimum average degeneracy grows exponentially with the number of spins. We apply our results to several well-known spin chains of Haldane–Shastry type, quantitatively showing that their degree of degeneracy is much higher than expected for a generic Yangian-invariant spin model. Finally, we show that the set of distinct levels of a Yangian-invariant spin model is described by an effective model of quasi-particles. We study this effective model, discussing its connections to one-dimensional anyons and properties of generalized Fibonacci numbers.

Effect of Inter-particle Interactions on Pair Correlations of One-Dimensional Anyon Gases

The pair correlation function of the one-dimensional interacting anyonic system in its ground state is investigated based on the exact Bethe ansatz solution for arbitrary coupling constant (\(0\le c\le \infty \)) and statistics parameter (\(0\le \kappa \le \pi \)). We discuss the effects of the inter-particle interactions and the fractional statistics on the pair correlations in both position and momentum spaces. The pair correlations of anyons with coupling constant c and statistical parameter \(\kappa \) in position space are identical to that of the Lieb–Liniger Bose model with effective coupling constant \(c^{^{\prime }}=c/\cos \left( \kappa /2\right) \). Besides the effect of renormalized coupling, the correlations in momentum space reveal more effects induced by the statistics parameter. The anyonic statistics results in the nonsymmetric correlation when the statistics parameter deviates from 0 (Bose statistics) and \(\pi \) (Fermi statistics) for any coupling constant c. The correlations display peaks and dips, representing the bunching and antibunching of atoms, respectively. The correlations show crossover from bunching behavior of bosons to antibunching behavior of fermions as \(\kappa \ \) varies from 0 to \(\pi \) for arbitrary coupling constant. Besides the fractional effect, we also observe the effects induced by the inter-particle interactions in the momentum correlations. With the increase of the coupling constant, the bunching effect between particles weakens and the antibunching points in the correlations shift.

Dynamics and noise correlations in the expansion of one-dimensional anyon gases from a regular array

The dynamics and the noise correlations of one-dimensional anyons are investigated. We solve the time-dependent Schrodinger equation for the one-dimensional anyons interacting via a repulsive delta-function potential based on the Bethe Ansatz. Then the exact solutions are used to derive the explicit formulae of the noise correlations for both the non-interacting and interacting anyons released from a regular array. For the non-interacting anyons, the noise correlations exhibit striking peaks, where the location, the value and the sign of those peaks differ as the anyonic statistical parameter varies, interpolating between the known results of Bose and Fermi gas. For the interacting anyons, apart from the anyonic statistics, the Hamiltonian also exhibits dynamical interactions. Thus the noise correlations for the interacting anyons have a number of features that distinguish it from the non-interacting problem. A set of line-shapes appear in the noise correlations due to the distribution of scattering phases. The width, the sign and location of the line-shape depend strongly on the interaction strength and the statistical parameter. In the Tonks-Giradeau limit, the anyonic correlations reverse the noise correlations of the non-interacting cases due to the \( \pi\) phase shift in the scattering phase.

Correlation functions and momentum distribution of one-dimensional hard-core anyons in optical lattices

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.

From spin to anyon notation: the XXZ Heisenberg model as a D3 (or su(2)4) anyon chain

We discuss a relationship between certain one-dimensional quantum spin chains and anyon chains. In particular, we show how the XXZ Heisenberg chain is realized as a D3 (alternatively su(2)4) anyon model. We find that the difference between the models lies primarily in the choice of boundary conditions.

Quantum phase transition in a multicomponent anyonic Lieb-Liniger model

We study a one-dimensional multicomponent anyon model that reduces to a multicomponent Lieb-Liniger gas of impenetrable bosons (Tonks-Girardeau gas) for vanishing statistics parameter. At fixed component densities, the coordinate Bethe ansatz gives a family of quantum phase transitions at special values of the statistics parameter. We show that the ground-state energy changes extensively between different phases. Special regimes are studied and a general classification for the transition points is given. An interpretation in terms of statistics of composite particles is proposed.

Statistical Interaction Term of One-Dimensional Anyon Models

A form of statistical interaction term of one-dimensional anyons is introduced, based on which one-dimensional anyon models are theoretically realized, and the statistical transmutation between bosons (or fermions) and anyons is established in quantum mechanics formalism. Two kinds of anyon models which are being studied are recovered and reexplained naturally in our formalism.