Anyon System Research Articles

Characterizing the entanglement of anyonic systems using the anyonic partial transpose

Characterizing the entanglement of anyonic systems using the anyonic partial transpose

Quantum thermodynamics of small systems: The anyonic otto engine

Recent advances in applying thermodynamic ideas to quantum systems have raised the novel prospect of using non-thermal, non-classical sources of energy, of purely quantum origin, like quantum statistics, to extract mechanical work in macroscopic quantum systems like Bose–Einstein condensates. On the other hand, thermodynamic ideas have also been applied to small systems like single molecules and quantum dots. In this paper, we study the quantum thermodynamics of small systems of anyons, with specific emphasis on the quantum Otto engine which uses, as its working medium, just one or two anyons. Formulae are derived for the efficiency of the Otto engine as a function of the statistics parameter.

Experimental quantum simulation of a topologically protected Hadamard gate via braiding Fibonacci anyons

Topological quantum computation (TQC) is one of the most striking architectures that can realize fault-tolerant quantum computers. In TQC, the logical space and the quantum gates are topologically protected, i.e., robust against local disturbances. The topological protection, however, requires complicated lattice models and hard-to-manipulate dynamics; even the simplest system that can realize universal TQC-the Fibonacci anyon system-lacks a physical realization, letalone braiding the non-Abelian anyons. Here, we propose a disk model that can simulate the Fibonacci anyon system and construct the topologically protected logical spaces with the Fibonacci anyons. Via braiding the Fibonacci anyons, we can implement universal quantum gates on the logical space. Our disk model merely requires two physical qubits to realize three Fibonacci anyons at the boundary. By 15 sequential braiding operations, we construct a topologically protected Hadamard gate, which is to date the least-resource requirement for TQC. To showcase, we implement a topological Hadamard gate with two nuclear spin qubits, which reaches fidelity by randomized benchmarking. We further prove by experiment that the logical space and Hadamard gate are topologically protected: local disturbances due to thermal fluctuations result in a global phase only. As a platform-independent proposal, our work is a proof of principle of TQC and paves the way toward fault-tolerant quantum computation.

Anyonic bound states in the continuum

Bound states in the continuum (BICs), which are spatially localized states with energies lying in the continuum of radiating modes, are discovered both in single- and few-body systems with suitably engineered spatial potentials and particle interactions. Here, we reveal a type of BICs that appear in anyonic systems. It is found that a pair of non-interacting anyons can perfectly concentrate on the boundary of a one-dimensional homogeneous lattice when the statistical angle is beyond a threshold. Such a bound state is embedded into the continuum of two-anyon scattering states, and is called as anyonic BICs. In contrast to conventional BICs, our proposed anyonic BICs purely stem from the statistics-induced correlations of two anyons, and do not need to engineer defect potentials or particle interactions. Furthermore, by mapping eigenstates of two anyons to modes of designed circuit networks, the anyonic BICs are experimentally simulated by measuring spatial impedance distributions and associated frequency responses. Our results enrich the understanding of anyons and BICs, and can inspire future studies on exploring correlated BICs with other mechanisms.

Numerical simulation of non-Abelian anyons

Two-dimensional systems such as quantum spin liquids or fractional quantum Hall systems exhibit anyonic excitations that possess more general statistics than bosons or fermions. This exotic statistics makes it challenging to solve even a many-body system of non-interacting anyons. We introduce an algorithm that allows to simulate anyonic tight-binding Hamiltonians on two-dimensional lattices. The algorithm is directly derived from the low energy topological quantum field theory and is suited for general Abelian and non-Abelian anyon models. As concrete examples, we apply the algorithm to study the energy level spacing statistics, which reveals level repulsion for free semions, Fibonacci anyons, and Ising anyons. Additionally, we simulate nonequilibrium quench dynamics, where we observe that the density distribution becomes homogeneous for large times---indicating thermalization.

Topological quantum computation on supersymmetric spin chains

Quantum gates built out of braid group elements form the building blocks of topological quantum computation. They have been extensively studied in SU(2)k quantum group theories, a rich source of examples of non-Abelian anyons such as the Ising (k = 2), Fibonacci (k = 3) and Jones-Kauffman (k = 4) anyons. We show that the fusion spaces of these anyonic systems can be precisely mapped to the product state zero modes of certain Nicolai-like supersymmetric spin chains. As a result, we can realize the braid group in terms of the product state zero modes of these supersymmetric systems. These operators kill all the other states in the Hilbert space, thus preventing the occurrence of errors while processing information, making them suitable for quantum computing.

Engineering Yang-Lee anyons via Majorana bound states

We propose the platform of a Yang-Lee anyon system which is constructed from Majorana bound states in topological superconductors. Yang-Lee anyons, described by the non-unitary conformal field theory with the central charge $c=-22/5$, are non-unitary counterparts of Fibonacci anyons, obeying the same fusion rule, but exhibiting non-unitary non-Abelian braiding statistics. We consider a topological superconductor junction system coupled with dissipative electron baths, which realizes a non-Hermitian interacting Majorana system. Numerically estimating the central charge, we examine the condition that the non-Hermitian Majorana system can simulate the Ising spin model of the Yang-Lee edge singularity, and confirm that, by controlling model parameters in a feasible way, the Yang-Lee edge criticality is realized. On the basis of this scenario, we present the scheme for the fusion, measurement and braiding of Yang-Lee anyons in our proposed setup.

Path-integral molecular dynamics for anyons, bosons, and fermions.

In this article we develop a general method to numerically calculate physical properties for a system of anyons with path integral molecular dynamics. We provide a unified method to calculate the thermodynamics of identical bosons, fermions, and anyons. Our method is tested and applied to systems of anyons, bosons, and fermions in a two-dimensional harmonic trap. We also consider a method to calculate the energy for fermions as an application of the path integral molecular dynamics to simulate the anyon model.

Quantum teleportation using Ising anyons

Anyons have been extensively investigated as information carriers in topological quantum computation. However, how to characterize the information flow in quantum networks composed of anyons is less understood, which motivates us to study quantum communication protocols in anyonic systems. Here we propose a general topologically protected protocol for quantum teleportation based on the Ising anyon model and prove that with our protocol an unknown anyonic state of any number of Ising anyons can be teleported from Alice to Bob. Our protocol naturally generalizes quantum state teleportation from systems of locally distinguishable particles to systems of Ising anyons, which may promote our understandings of anyonic quantum entanglement as a quantum resource. In addition, our protocol is expected to be realized with the Majorana zero modes, one of the possible physical realizations for the Ising anyon in experiments.

Quantum walks on regular graphs with realizations in a system of anyons

We build interacting Fock spaces from association schemes and set up quantum walks on the resulting regular graphs (distance-regular and distance-transitive). The construction is valid for growing graphs and the interacting Fock space is well defined asymptotically for the growing graph. To realize the quantum walks defined on the spaces in terms of anyons we switch to the dual view of the association schemes and identify the corresponding modular tensor categories from the Bose–Mesner algebra. Informally, the fusion ring induced by the association scheme and a topological twist can be the basis for developing a modular tensor category and thus a system of anyons. Finally, we demonstrate the framework in the case of Grover quantum walk on distance-regular graph in terms of anyon systems for the graphs considered. In the dual perspective interacting Fock spaces gather a new meaning in terms of anyon collisions for the case of distance-regular graphs.

Kohn-Sham density functional theory of Abelian anyons

We develop a density functional treatment of non-interacting abelian anyons, which is capable, in principle, of dealing with a system of a large number of anyons in an external potential. Comparison with exact results for few particles shows that the model captures the behavior qualitatively and semi-quantitatively, especially in the vicinity of the fermionic statistics. We then study anyons with statistics parameter $1+1/n$, which are thought to condense into a superconducting state. An indication of the superconducting behavior is the mean-field result that, for uniform density systems, the ground state energy increases under the application of an external magnetic field independent of its direction. Our density-functional-theory based analysis does not find that to be the case for finite systems of anyons, which can accommodate a weak external magnetic field through density transfer between the bulk and the boundary rather than through transitions across effective Landau levels, but the "Meissner repulsion" of the external magnetic field is recovered in the thermodynamic limit as the effect of the boundary becomes negligible. We also consider the quantum Hall effect of anyons, and show that its topological properties, such as the charge and statistics of the excitations and the quantized Hall conductance, arise in a self-consistent fashion.

Topological quantum structures from association schemes

Starting from an association scheme induced by a finite group and the corresponding Bose–Mesner algebra, we construct quantum Markov chains, their entangled versions, using the quantum probabilistic approach. Our constructions are based on the intersection numbers and their duals Krein parameters of the schemes. We make the connection for the first time between the fusion rules of anyonic particles evolving on a 2D surface to the Krein parameters of an association scheme. We consider braid group $$B_3$$ that describes the unitary dynamics of the anyons as the automorphism subgroup of the graphs. The dynamics induced by the fusions (and the adjoint splitting operations) may be viewed as the chain evolving on a growing graph and the braiding as automorphisms on a fixed graph. In our quantum probability framework, infinite iterations of the unitaries, which can encode algorithmic content for quantum simulations, can describe asymptotics elegantly if the particles are allowed to evolve coherently for a longer period. We define quantum states on the Bose–Mesner algebra which is also a von Neumann algebra as well as a Frobenius algebra to build the quantum Markov chains providing yet another perspective to topological computation, whereas frameworks such as Unitary Modular Categories can identify and characterize new anyonic systems our framework can build upon them within quantum probabilistic framework that are suitable for asymptotic analysis.

Nonequilibrium dynamics of the anyonic Tonks-Girardeau gas at finite temperature

We derive an exact description of the nonequilibrium dynamics at finite temperature for the anyonic Tonks-Girardeau gas, extending the results of Atas et al. [Phys. Rev. A 95, 043622 (2017)] to the case of arbitrary statistics. The one-particle reduced density matrix is expressed as the Fredholm minor of an integral operator, with the kernel being the one-particle Green's function of free fermions at finite temperature and the statistics parameter determining the constant in front of the integral operator. We show that the numerical evaluation of this representation using Nystr\"om's method significantly outperforms the other approaches present in the literature when there are no analytical expressions for the overlaps of the wave functions. We illustrate the distinctive features and novel phenomena present in the dynamics of anyonic systems in two experimentally relevant scenarios: the quantum Newton's cradle setting and the breathing oscillations initiated by a sudden change of the trap frequency.

Anyonic quasiparticles of hardcore anyons

Strongly interacting topologically ordered many-body systems consisting of fermions or bosons can host exotic quasiparticles with anyonic statistics. This raises the question whether many-body systems of anyons can also form anyonic quasiparticles. Here, we show that one can, indeed, construct many-anyon wavefunctions with anyonic quasiparticles. The braiding statistics of the emergent anyons are different from those of the original anyons. We investigate hole type and particle type anyonic quasiparticles in Abelian systems on a two-dimensional lattice and compute the density profiles and braiding properties of the emergent anyons by employing Monte Carlo simulations.

A programming guide for tensor networks with global [formula omitted] symmetry

This paper is a manual with tips and tricks for programming tensor network algorithms with global SU(2) symmetry. We focus on practical details that are many times overlooked when it comes to implementing the basic building blocks of codes, such as useful data structures to store the tensors, practical ways of manipulating them, and adapting typical functions for symmetric tensors. Here we do not restrict ourselves to any specific tensor network method, but keep always in mind that the implementation should scale well for simulations of higher-dimensional systems using, e.g., Projected Entangled Pair States, where tensors with many indices may show up. To this end, the structural tensors (or intertwiners) that arise in the usual decomposition of SU(2)-symmetric tensors are never explicitly stored throughout the simulation. Instead, we store and manipulate the corresponding fusion trees – an algebraic specification of the symmetry constraints on the tensor – in order to implement basic SU(2)-symmetric tensor operations. This fusion tree approach is readily extensible to anyonic systems, as we demonstrate for a chain of Fibonacci anyons.

Braided distributivity

In category-theoretic models for the anyon systems proposed for topological quantum computing, the essential ingredients are two monoidal structures, ⊕ and ⊗. The former is symmetric but the latter is only braided, and ⊗ is required to distribute over ⊕. What are the appropriate coherence conditions for the distributivity isomorphisms? We came to this question working on a simplification of the category-theoretical foundation of topological quantum computing, which is the intended application of the research reported here.This question was answered by Laplaza when both monoidal structures are symmetric, but topological quantum computation depends crucially on ⊗ being only braided, not symmetric. We propose coherence conditions for distributivity in this situation, and we prove that our conditions are(a)strong enough to imply Laplaza's when the latter are suitably formulated, and(b)weak enough to hold when — as in the categories used to model anyons — the additive structure is that of an abelian category and the braided ⊗ is additive. Working on these results, we found a new redundancy in Laplaza's conditions.

Microscopic explanation for black hole phase transitions via Ruppeiner geometry: Two competing factors–the temperature and repulsive interaction among BH molecules

Charged dilatonic black hole (BH) has rather rich phase diagrams which may contain zeroth-order, first-order as well as reentrant phase transitions (RPTs) depending on the value of the coupling constant α between the electromagnetic field and the dilaton. We try to give a microscopic explanation for these phase transitions by adopting Ruppeiner's approach. By studying the behaviors of the Ruppeiner invariant R along the co-existing lines, we find that the various phase transitions may be qualitatively well explained as a result of two competing factors: the first one is the low-temperature effect which tends to shrink the BH and the second one is the repulsive interaction between the BH molecules which, on the contrary, tends to expand the BH. In the standard phase transition without RPT, as temperature is lowered, the first kind of factor dominates over the second one, so that large black hole (LBH) tends to shrink and thus transits to small black hole (SBH); While in the RPT, after the LBH-SBH transition, as temperature is further decreased, the strength of the second factor increases quickly and finally becomes strong enough to dominate over the first factor, so that SBH tends to expand to release the high repulsion and thus transits back to LBH. Moreover, by comparing the behavior of R versus the temperature T with fixed pressure to that of ordinary two-dimensional thermodynamical systems but with fixed specific volume, it is interesting to see that SBH behaves like a Fermionic gas system in cases with RPT, while it behaves oppositely to an anyon system in cases without RPT. And in all cases, LBH behaves like a nearly ideal gas system.

Corrections to thermodynamics of the system of magnetically charged anyons

In this paper, we calculate the thermodynamics of the system of anyons with magnetic charges in the magnetic field. We demonstrate how the contribution of the energy spectrum correction due to magnetic charges affects the second virial coefficient and the magnetic susceptibility. Dependences of the respective corrections as functions of temperature and the anyonic parameter are presented.

The spinful large charge sector of non-relativistic CFTs: from phonons to vortex crystals

We study operators in Schrödinger invariant field theories (non-relativistic conformal field theories or NRCFTs) with large charge (particle number) and spin. Via the state-operator correspondence for NRCFTs, such operators correspond to states of a superfluid in a harmonic trap with phonons or vortices. Using the effective field theory of the Goldstone mode, we compute the dimensions of operators to leading order in the angular momentum L and charge Q. We find a diverse set of scaling behaviors for NRCFTs in both d = 2 and d = 3 spatial dimensions. These results apply to theories with a superfluid phase, such as unitary fermions or critical anyon systems.

Anyonic exclusions statistics on surfaces with gapped boundaries

An anyonic exclusion statistics, which generalizes the Bose-Einstein and Fermi-Dirac statistics of bosons and fermions, was proposed by Haldane [1]. When fusion of anyons is involved, certain ‘pseudo-species’ anyons appear in the exotic statistical weights of non-Abelian anyon systems; however, the meaning and significance of pseudo-species remains an open problem. The relevant past studies had considered only anyon systems without any physical boundary but boundaries often appear in real-life materials. In this paper, we propose an extended anyonic exclusion statistics on surfaces with gapped boundaries, introducing mutual exclusion statistics between anyons as well as the boundary components. Motivated by refs. [2, 3], we present a formula for the statistical weight of many-anyon states obeying the proposed statistics. Taking the (doubled) Fibonacci topological order as an example, we develop a systematic basis construction for non-Abelian anyons on any Riemann surfaces with gapped boundaries. The basis construction offers a standard way to read off a canonical set of statistics parameters and hence write down the extended statistical weight of the anyon system being studied. The basis construction reveals the meaning of pseudo-species. A pseudo-species has different ‘excitation’ modes, each corresponding to an anyon species. The ‘excitation’ modes of pseudo-species corresponds to good quantum numbers of subsystems of a non-Abelian anyon system. This is important because often (e.g., in topological quantum computing) we may be concerned about only the entanglement between such subsystems.