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arXiv:2507.22115 · Physical review. B./Physical review. B

Phases of interacting Fibonacci anyons on a ladder at half-filling

Nico Kirchner, Roderich Moessner, Frank Pollmann, Adam Gammon-Smith

Published Jun 23, 2026• 0 citations
arXivHTMLPDF

Abstract

Two-dimensional many-body quantum systems can exhibit topological order and support collective excitations with anyonic statistics different from the usual fermionic or bosonic ones. With the emergence of these exotic point-like particles, it is natural to ask what phases can arise in interacting many-anyon systems. To study this topic, we consider the particular case of Fibonacci anyons subject to an anyonic tight-binding model with nearest-neighbor repulsion on a two-leg ladder. Focusing on the case of half-filling, for low interaction strengths an “anyonic” metal is found, whereas for strong repulsion, the anyons form an insulating charge-density wave. Within the latter regime, we introduce an effective one-dimensional model up to sixth order in perturbation theory arising from anyonic superexchange processes. We numerically identify four distinct phases of the effective model, which we characterize using matrix product state methods. These include both the ferro- and antiferromagnetic golden chain phases, as well as phases with Z 2 and incommensurate correlations.

DOI
10.1103/c92g-mhly

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Two-dimensional many-body quantum systems can exhibit topological order and support collective excitations with anyonic statistics different from the usual fermionic or bosonic ones. With the emergence of these exotic point-like particles, it is natural to ask what phases can arise in interacting many-anyon systems. To study this topic, we consider the particular case of Fibonacci anyons subject to an anyonic tight-binding model with nearest-neighbor repulsion on a two-leg ladder. Focusing on the case of half-filling, for low interaction strengths an “anyonic” metal is found, whereas for strong repulsion, the anyons form an insulating charge-density wave. Within the latter regime, we introduce an effective one-dimensional model up to sixth order in perturbation theory arising from anyonic superexchange processes. We numerically identify four distinct phases of the effective model, which we characterize using matrix product state methods. These include both the ferro- and antiferromagnetic golden chain phases, as well as phases with Z 2 and incommensurate correlations.
Non-Abelian anyons can exist as pointlike particles in two-dimensional systems and have particle exchange statistics which are neither bosonic nor fermionic. Like in spin systems, the role of fusion (Heisenberg-like) interactions between anyons has been well studied. However, unlike our understanding of the role of bosonic and fermionic statistics in the formation of different quantum phases of matter, little is known concerning the effect of non-Abelian braid statistics. We explore this physics using an anyonic Hubbard model on a two-legged ladder which includes braiding and nearest-neighbor Heisenberg interactions among anyons. We study two of the most prominent non-Abelian anyon models: the Fibonacci and Ising types. We discover rich phase diagrams for both anyon models and show the different roles of their fusion and braid statistics.
Anyons exist as pointlike particles in two dimensions and carry braid statistics, which enable interactions that are independent of the distance between the particles. Except for a relatively few number of models, which are analytically tractable, much of the physics of anyons remains still unexplored. In this paper, we show how U(1) symmetry can be combined with the previously proposed anyonic matrix product states to simulate ground states and dynamics of anyonic systems on a lattice at any rational particle number density. We provide proof of principle by studying itinerant anyons on a one-dimensional chain where no natural notion of braiding arises and also on a two-leg ladder where the anyons hop between sites and possibly braid. We compare the result of the ground-state energies of Fibonacci anyons against hardcore bosons and spinless fermions. In addition, we report the entanglement entropies of the ground states of interacting Fibonacci anyons on a fully filled two-leg ladder at different interaction strength, identifying gapped or gapless points in the parameter space. As an outlook, our approach can also prove useful in studying the time dynamics of a finite number of non-Abelian anyons on a finite two-dimensional lattice.
Quantum spin models have been studied extensively in one and higher dimensions. Furthermore, these systems have been doped with holes to study $t\text{\ensuremath{-}}J$ models of SU(2)\mathrm{SU}(2)SU(2) spin-1/2. Their anyonic counterparts can be built from non-Abelian anyons, such as Fibonacci anyons described by SU(2)3\mathrm{SU}{(2)}_{3}SU(2)3​ theories, which are quantum deformations of the SU(2)\mathrm{SU}(2)SU(2) algebra. Inspired by the physics of SU(2)\mathrm{SU}(2)SU(2) spins, several works have explored ladders of Fibonacci anyons and also one-dimensional (1D) $t\text{\ensuremath{-}}J$ models. Here, we aim to explore the combined effects of extended dimensionality and doping by studying ladders composed of coupled chains of interacting itinerant Fibonacci anyons. We show analytically that in the limit of strong rung couplings these models can be mapped onto effective 1D models. These effective models can either be gapped models of hole pairs, or gapless models described by $t\text{\ensuremath{-}}J$ (or modified $t\text{\ensuremath{-}}J\text{\ensuremath{-}}V$) chains of Fibonacci anyons, whose spectrum exhibits a fractionalization into charge and anyon degrees of freedom. The charge degrees of freedom are described by the hardcore boson spectra while the anyon sector is given by a chain of localized interacting anyons. By using exact diagonalizations for two-leg and three-leg ladders, we show that indeed the doped ladders show exactly the same behavior as that of $t\text{\ensuremath{-}}J$ chains. In the strong ferromagnetic rung limit, we can obtain a new model that hosts two different kinds of Fibonacci particles, which we denote as the heavy $\ensuremath{\tau}$'s and light $\ensuremath{\tau}$'s. These two particle types carry the same (non-Abelian) topological charge but different (Abelian) electric charges. Once again, we map the two-dimensional ladder onto an effective chain carrying these heavy and light $\ensuremath{\tau}$'s. We perform a finite size scaling analysis to show the appearance of gapless modes for certain anyon densities, whereas a topological gapped phase is suggested for another density regime.
We discuss how to construct models of interacting anyons by generalizing quantum spin Hamiltonians to anyonic degrees of freedom. The simplest interactions energetically favor pairs of anyons to fuse into the trivial (“identity") channel, similar to the quantum Heisenberg model favoring pairs of spins to form spin singlets. We present an introduction to the theory of anyons and discuss in detail how basis sets and matrix representations of the interaction terms can be obtained, using non-Abelian Fibonacci anyons as example. Besides discussing the “golden chain”, a one-dimensional system of anyons with nearest neighbor interactions, we also present the derivation of more complicated interaction terms, such as three-anyon interactions in the spirit of the Majumdar-Ghosh spin chain, longer range interactions and two-leg ladders. We also discuss generalizations to anyons with general non-Abelian SU(2)k statistics. The k→∞ limit of the latter yields ordinary SU(2) spin chains.
We study a system of dipolar bosons in a one-dimensional optical lattice using exact diagonalization and density matrix renormalization group methods. In particular, we analyze low energy properties of the system at an average filling of 3/2 atoms per lattice site. We identify the region of the parameter space where the system has non-Abelian Fibonacci anyon excitations that correspond to fractional domain walls between different charge-density waves. When such one-dimensional systems are combined into a two-dimensional network, braiding of Fibonacci anyon excitations has potential application for fault tolerant, universal, topological quantum computation. Contrary to previous calculations, our results also demonstrate that super-solid phases are not present in the phase diagram for the discussed 3/2 average filling. Instead, decreasing the value of the nearest-neighbor tunneling strength leads to a direct, Berezinskii-Kosterlitz-Thouless, superfluid to charge-density-wave quantum phase transition.
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.