Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:0902.0168 · Journal of Statistical Mechanics Theory and Experiment

Ashkin–Teller universality in a quantum double model of Ising anyons

Charlotte Gils

Published Jul 9, 2009• 20 citations
arXivHTMLPDF

Abstract

We study a quantum double model whose degrees of freedom are Ising anyons. The terms of the Hamiltonian of this system give rise to a competition between single and double topologies. By studying the energy spectra of the Hamiltonian at different values of the coupling constants, we find extended gapless regions which include a large number of critical points described by conformal field theories with central charge c = 1. These theories are part of the orbifold of the bosonic theory compactified on a circle. We observe that the Hilbert space of our anyonic model can be associated with extended Dynkin diagrams of affine Lie algebras, which yields exact solutions at some critical points. In certain special regimes, our model corresponds to the Hamiltonian limit of the Ashkin–Teller model, and hence integrability over a wide range of coupling parameters is established.

DOI
10.1088/1742-5468/2009/07/p07019

Related papers

Boundary critical phenomena in the quantum Ashkin-Teller model

Yifan Liu, Natalia Chepiga, Yoshiki Fukusumi +1 more

We investigate the boundary critical phenomena of the one-dimensional quantum Ashkin-Teller model using boundary conformal field theory and density matrix renormalization group (DMRG) simulations. Based on the $\mathbb{Z}_2$-orbifold of the $c=1$ compactified boson boundary conformal field theory, we construct microscopic lattice boundary terms that renormalize to the stable conformal boundary conditions, utilizing simple current extensions and the underlying $\mathrm{SU}(2)$ symmetry to explicitly characterize the four-state Potts point. We validate these theoretical identifications via finite-size spectroscopy of the lattice energy spectra, confirming their consistency with $D_4$ symmetry and Kramers-Wannier duality. Finally, we discuss the boundary renormalization group flows among these identified fixed points to propose a global phase diagram for the boundary criticality.

cond-mat.str-elCross-listedarXiv:2601.16951•SciPost Physics•Jan 26, 2026•1 citation

Interacting Anyons in Topological Quantum Liquids: The Golden Chain

Adrian E. Feiguin, Simon Trebst, Andreas W. W. Ludwig +4 more

We discuss generalizations of quantum spin Hamiltonians using anyonic degrees of freedom. The simplest model for interacting anyons energetically favors neighboring anyons to fuse into the trivial ("identity") channel, similar to the quantum Heisenberg model favoring neighboring spins to form spin singlets. Numerical simulations of a chain of Fibonacci anyons show that the model is critical with a dynamical critical exponent z=1, and described by a two-dimensional (2D) conformal field theory with central charge c=7/10. An exact mapping of the anyonic chain onto the 2D tricritical Ising model is given using the restricted-solid-on-solid representation of the Temperley-Lieb algebra. The gaplessness of the chain is shown to have topological origin.

arXiv:cond-mat/0612341•Physical Review Letters•Apr 20, 2007•382 citations

Universality and non-universality in the Ashkin-Teller model

Alessandro Giuliani

The Ashkin-Teller model can be formulated as a pair of 2D Ising models, interacting via a four-spin interaction. I consider the case of weak anisotropy (slight a-symmetry between the two Ising layers) and weak coupling. I show that the system admits two critical temperatures whose difference varies continuously with the strength of the coupling, scaling with an anomalous exponent as one let the a-symmetry parameter go to zero. The specific heat diverges logarithmically at the critical points (as for Ising) but the constant in front of the logarithm is renormalized by an anomalous critical exponent. The logarithmic divergence of the specific heat dominates only in an exponentially small interval around the critical temperatures and outside it is modified into an anomalous power law behaviour. The proof is based on an exact mapping of Ashkin-Teller into a model of (1+1)D interacting fermions and on the implementation of constructive Renormalization group methods on the fermionic system. This PhD thesis includes: (1) a review of the exact solution of 2D Ising in terms of Grassmann functional integrals; (2) a review of the Grassmann representation for a class of interacting Ising models, including Ashkin-Teller and the 8 vertex model; (3) a detailed discussion of the multiscale analysis of the Ashkin-Teller model, based on fermionic Renormalization Group methods, including the study of the flow of the effective coupling constants (in particular a proof of vanishing of the beta function for this and similar Luttinger-like models, based on modified approximate Ward identities, is included). This thesis is based on joint work with V. Mastropietro.

arXiv:cond-mat/0412176•Dec 7, 2004•0 citations

Stacked quantum Ising systems and quantum Ashkin-Teller model

Davide Rossini, Ettore Vicari

We analyze the quantum states of an isolated composite system consisting of two stacked quantum Ising (SQI) subsystems, coupled by a local Hamiltonian term that preserves the $Z_2$ symmetry of each subsystem. The coupling strength is controlled by an intercoupling parameter $w$, with $w=0$ corresponding to decoupled quantum Ising systems. We focus on the quantum correlations of one of the two SQI subsystems, $S$, in the ground state of the global system, and study their dependence on both the state of the weakly-coupled complementary part $E$ and the intercoupling strength. We concentrate on regimes in which $S$ develops critical long-range correlations. The most interesting physical scenario arises when both SQI subsystems are critical. In particular, for identical SQI subsystems, the global system is equivalent to the quantum Ashkin-Teller model, characterized by an additional $Z_2$ interchange symmetry between the two subsystem operators. In this limit, one-dimensional SQI systems exhibit a peculiar critical line along which the length-scale critical exponent $ν$ varies continuously with $w$, while two-dimensional systems develop quantum multicritical behaviors characterized by an effective enlargement of the symmetry of the critical modes, from the actual $Z_2\oplus Z_2$ symmetry to a continuous O(2) symmetry.

cond-mat.stat-mechCross-listedarXiv:2601.18922•Physical review. B./Physical review. B•Jan 28, 2026•1 citation

Z n clock models and chains of so ( n ) 2 non-Abelian anyons: symmetries, integrable points and low energy properties

Peter E. Finch, Michael Flohr, Holger Frahm

Abstract We study two families of quantum models which have been used previously to investigate the effect of topological symmetries in one-dimensional correlated matter. Various striking similarities are observed between certain Z n quantum clock models, spin chains generalizing the Ising model, and chains of non-Abelian anyons constructed from the so ( n ) 2 fusion category for odd n , both subject to periodic boundary conditions. In spite of the differences between these two types of quantum chains, e.g. their Hilbert spaces being spanned by tensor products of local spin states or fusion paths of anyons, the symmetries of the lattice models are shown to be closely related. Furthermore, under a suitable mapping between the parameters describing the interaction between spins and anyons the respective Hamiltonians share part of their energy spectrum (although their degeneracies may differ). This spin-anyon correspondence can be extended by fine-tuning of the coupling constants leading to exactly solvable models. We show that the algebraic structures underlying the integrability of the clock models and the anyon chain are the same. For n = 3,5,7 we perform an extensive finite size study—both numerical and based on the exact solution—of these models to map out their ground state phase diagram and to identify the effective field theories describing their low energy behaviour. We observe that the continuum limit at the integrable points can be described by rational conformal field theories with extended symmetry algebras which can be related to the discrete ones of the lattice models.

arXiv:1710.09620•Journal of Statistical Mechanics Theory and Experiment•Feb 1, 2018•11 citations

N -color Ashkin-Teller model

Gary S. Grest, Michael Widom

We study a model Hamiltonian consisting of $N$ Ising models coupled pairwise through a four-spin interaction ${K}_{4}$. When $N=2$ this model is the well-known Ashkin-Teller model which shows nonuniversal critical behavior in two dimensions in the neighborhood of ${K}_{4}=0$. To see if this behavior persists for $N\ensuremath{\ne}2$ we perform a first-order-perturbation expansion around the decoupling point in two dimensions. As an aid in interpreting the results of this perturbation expansion we have determined the phase diagram of the system through mean-field theory and Monte Carlo studies in both two and three dimensions for $N=3$. The results show that $N=2$ is special because the coupling between Ising models is marginal over a range of values of ${K}_{4}$. We discuss the effect of the coupling ${K}_{4}$ for $N\ensuremath{\ne}2$.

Physical review. B, Condensed matter•Dec 1, 1981•75 citations
We study a quantum double model whose degrees of freedom are Ising anyons. The terms of the Hamiltonian of this system give rise to a competition between single and double topologies. By studying the energy spectra of the Hamiltonian at different values of the coupling constants, we find extended gapless regions which include a large number of critical points described by conformal field theories with central charge c = 1. These theories are part of the orbifold of the bosonic theory compactified on a circle. We observe that the Hilbert space of our anyonic model can be associated with extended Dynkin diagrams of affine Lie algebras, which yields exact solutions at some critical points. In certain special regimes, our model corresponds to the Hamiltonian limit of the Ashkin–Teller model, and hence integrability over a wide range of coupling parameters is established.
We investigate the boundary critical phenomena of the one-dimensional quantum Ashkin-Teller model using boundary conformal field theory and density matrix renormalization group (DMRG) simulations. Based on the Z2\mathbb{Z}_2Z2​-orbifold of the c=1c=1c=1 compactified boson boundary conformal field theory, we construct microscopic lattice boundary terms that renormalize to the stable conformal boundary conditions, utilizing simple current extensions and the underlying SU(2)\mathrm{SU}(2)SU(2) symmetry to explicitly characterize the four-state Potts point. We validate these theoretical identifications via finite-size spectroscopy of the lattice energy spectra, confirming their consistency with D4D_4D4​ symmetry and Kramers-Wannier duality. Finally, we discuss the boundary renormalization group flows among these identified fixed points to propose a global phase diagram for the boundary criticality.
We discuss generalizations of quantum spin Hamiltonians using anyonic degrees of freedom. The simplest model for interacting anyons energetically favors neighboring anyons to fuse into the trivial ("identity") channel, similar to the quantum Heisenberg model favoring neighboring spins to form spin singlets. Numerical simulations of a chain of Fibonacci anyons show that the model is critical with a dynamical critical exponent z=1, and described by a two-dimensional (2D) conformal field theory with central charge c=7/10. An exact mapping of the anyonic chain onto the 2D tricritical Ising model is given using the restricted-solid-on-solid representation of the Temperley-Lieb algebra. The gaplessness of the chain is shown to have topological origin.
The Ashkin-Teller model can be formulated as a pair of 2D Ising models, interacting via a four-spin interaction. I consider the case of weak anisotropy (slight a-symmetry between the two Ising layers) and weak coupling. I show that the system admits two critical temperatures whose difference varies continuously with the strength of the coupling, scaling with an anomalous exponent as one let the a-symmetry parameter go to zero. The specific heat diverges logarithmically at the critical points (as for Ising) but the constant in front of the logarithm is renormalized by an anomalous critical exponent. The logarithmic divergence of the specific heat dominates only in an exponentially small interval around the critical temperatures and outside it is modified into an anomalous power law behaviour. The proof is based on an exact mapping of Ashkin-Teller into a model of (1+1)D interacting fermions and on the implementation of constructive Renormalization group methods on the fermionic system. This PhD thesis includes: (1) a review of the exact solution of 2D Ising in terms of Grassmann functional integrals; (2) a review of the Grassmann representation for a class of interacting Ising models, including Ashkin-Teller and the 8 vertex model; (3) a detailed discussion of the multiscale analysis of the Ashkin-Teller model, based on fermionic Renormalization Group methods, including the study of the flow of the effective coupling constants (in particular a proof of vanishing of the beta function for this and similar Luttinger-like models, based on modified approximate Ward identities, is included). This thesis is based on joint work with V. Mastropietro.
We analyze the quantum states of an isolated composite system consisting of two stacked quantum Ising (SQI) subsystems, coupled by a local Hamiltonian term that preserves the Z2Z_2Z2​ symmetry of each subsystem. The coupling strength is controlled by an intercoupling parameter www, with w=0w=0w=0 corresponding to decoupled quantum Ising systems. We focus on the quantum correlations of one of the two SQI subsystems, SSS, in the ground state of the global system, and study their dependence on both the state of the weakly-coupled complementary part EEE and the intercoupling strength. We concentrate on regimes in which SSS develops critical long-range correlations. The most interesting physical scenario arises when both SQI subsystems are critical. In particular, for identical SQI subsystems, the global system is equivalent to the quantum Ashkin-Teller model, characterized by an additional Z2Z_2Z2​ interchange symmetry between the two subsystem operators. In this limit, one-dimensional SQI systems exhibit a peculiar critical line along which the length-scale critical exponent ννν varies continuously with www, while two-dimensional systems develop quantum multicritical behaviors characterized by an effective enlargement of the symmetry of the critical modes, from the actual Z2⊕Z2Z_2\oplus Z_2Z2​⊕Z2​ symmetry to a continuous O(2) symmetry.
Abstract We study two families of quantum models which have been used previously to investigate the effect of topological symmetries in one-dimensional correlated matter. Various striking similarities are observed between certain Z n quantum clock models, spin chains generalizing the Ising model, and chains of non-Abelian anyons constructed from the so ( n ) 2 fusion category for odd n , both subject to periodic boundary conditions. In spite of the differences between these two types of quantum chains, e.g. their Hilbert spaces being spanned by tensor products of local spin states or fusion paths of anyons, the symmetries of the lattice models are shown to be closely related. Furthermore, under a suitable mapping between the parameters describing the interaction between spins and anyons the respective Hamiltonians share part of their energy spectrum (although their degeneracies may differ). This spin-anyon correspondence can be extended by fine-tuning of the coupling constants leading to exactly solvable models. We show that the algebraic structures underlying the integrability of the clock models and the anyon chain are the same. For n = 3,5,7 we perform an extensive finite size study—both numerical and based on the exact solution—of these models to map out their ground state phase diagram and to identify the effective field theories describing their low energy behaviour. We observe that the continuum limit at the integrable points can be described by rational conformal field theories with extended symmetry algebras which can be related to the discrete ones of the lattice models.
We study a model Hamiltonian consisting of NNN Ising models coupled pairwise through a four-spin interaction K4{K}_{4}K4​. When N=2N=2N=2 this model is the well-known Ashkin-Teller model which shows nonuniversal critical behavior in two dimensions in the neighborhood of K4=0{K}_{4}=0K4​=0. To see if this behavior persists for $N\ensuremath{\ne}2$ we perform a first-order-perturbation expansion around the decoupling point in two dimensions. As an aid in interpreting the results of this perturbation expansion we have determined the phase diagram of the system through mean-field theory and Monte Carlo studies in both two and three dimensions for N=3N=3N=3. The results show that N=2N=2N=2 is special because the coupling between Ising models is marginal over a range of values of K4{K}_{4}K4​. We discuss the effect of the coupling K4{K}_{4}K4​ for $N\ensuremath{\ne}2$.