We find a series of non-Abelian topological phases that are separated from the deconfined phase of ${Z}_{N}$ gauge theory by a continuous quantum phase transition. These non-Abelian states, which we refer to as the ``twisted'' ${Z}_{N}$ states, are described by a recently studied $U(1)\ifmmode\times\else\texttimes\fi{}U(1)\ensuremath{\rtimes}{Z}_{2}$ Chern-Simons (CS) field theory. The $U(1)\ifmmode\times\else\texttimes\fi{}U(1)\ensuremath{\rtimes}{Z}_{2}$ CS theory provides a way of gauging the global ${Z}_{2}$ electric-magnetic symmetry of the Abelian ${Z}_{N}$ phases, yielding the twisted ${Z}_{N}$ states. We introduce a parton construction to describe the Abelian ${Z}_{N}$ phases in terms of integer quantum Hall states, which then allows us to obtain the non-Abelian states from a theory of ${Z}_{2}$ fractionalization. The non-Abelian twisted ${Z}_{N}$ states do not have topologically protected gapless edge modes and, for $N>2$ , break time-reversal symmetry.We find a series of non-Abelian topological phases that are separated from the deconfined phase of ${Z}_{N}$ gauge theory by a continuous quantum phase transition. These non-Abelian states, which we refer to as the ``twisted'' ${Z}_{N}$ states, are described by a recently studied $U(1)\ifmmode\times\else\texttimes\fi{}U(1)\ensuremath{\rtimes}{Z}_{2}$ Chern-Simons (CS) field theory. The $U(1)\ifmmode\times\else\texttimes\fi{}U(1)\ensuremath{\rtimes}{Z}_{2}$ CS theory provides a way of gauging the global ${Z}_{2}$ electric-magnetic symmetry of the Abelian ${Z}_{N}$ phases, yielding the twisted ${Z}_{N}$ states. We introduce a parton construction to describe the Abelian ${Z}_{N}$ phases in terms of integer quantum Hall states, which then allows us to obtain the non-Abelian states from a theory of ${Z}_{2}$ fractionalization. The non-Abelian twisted ${Z}_{N}$ states do not have topologically protected gapless edge modes and, for $N>2$, break time-reversal symmetry.

arXiv:1012.2417Physical Review BAug 13, 201232 citations

Topological phase transitions

M. A. Contínentino

Physica B Condensed MatterNov 3, 201625 citations

Topological classification of phase transitions

Lucien Gilles Benguigui, Lawrence S. Schulman

Physics Letters AOct 1, 197319 citations

Topological BF Theory construction of twisted dihedral quantum double phases from spontaneous symmetry breaking

Zhi-Qiang Gao, Chunxiao Liu, Joel E. Moore

Nonabelian topological orders host exotic anyons central to quantum computing, yet established realizations rely on case-by-case constructions that are often conceptually involved. In this work, we present a systematic construction of nonabelian dihedral quantum double phases based on a continuous $O(2)$ gauge field. We first formulate a topological $S[O(2)\times O(2)]$ BF theory, and by identifying the Wilson loops and twist operators of this theory with anyons, we show that our topological BF theory reproduces the complete anyon data, and can incorporate all Dijkgraaf--Witten twists. Building on this correspondence, we present a microscopic model with $O(2)$ lattice gauge field coupled to Ising and rotor matter whose Higgsing yields the desired dihedral quantum double phase. A perturbative renormalization group analysis further suggests a direct transition from this phase to a $U(1)$ Coulomb or chiral topological phase at a stable multicritical point with emergent $O(3)$ symmetry. Our proposal offers an alternative route to nonabelian topological order with promising prospects in synthetic gauge field platforms.Nonabelian topological orders host exotic anyons central to quantum computing, yet established realizations rely on case-by-case constructions that are often conceptually involved. In this work, we present a systematic construction of nonabelian dihedral quantum double phases based on a continuous $O(2)$ gauge field. We first formulate a topological $S[O(2)\times O(2)]$ BF theory, and by identifying the Wilson loops and twist operators of this theory with anyons, we show that our topological BF theory reproduces the complete anyon data, and can incorporate all Dijkgraaf--Witten twists. Building on this correspondence, we present a microscopic model with $O(2)$ lattice gauge field coupled to Ising and rotor matter whose Higgsing yields the desired dihedral quantum double phase. A perturbative renormalization group analysis further suggests a direct transition from this phase to a $U(1)$ Coulomb or chiral topological phase at a stable multicritical point with emergent $O(3)$ symmetry. Our proposal offers an alternative route to nonabelian topological order with promising prospects in synthetic gauge field platforms.

cond-mat.str-elCross-listedarXiv:2511.19589Nov 26, 20250 citations

Mott transition and magnetism in a fragile topological insulator

Ashish Joshi, Robert Peters

We study the effects of electronic correlations on fragile topology using dynamical mean-field theory. Fragile topological insulators (FTIs) offer obstruction to the formation of exponentially localized Wannier functions, but they can be trivialized by adding certain trivial degrees of freedom. For the same reason, FTIs do not host symmetry-protected flow of edge states between bulk bands in cylindrical boundary conditions but are expected to have a spectral flow between the fragile bands and other bands under certain twisted boundary conditions. We here analyze commonly observed effects of strong correlations, such as the Mott insulator transition and magnetism, on a known model hosting fragile topology. We show that in the nonmagnetic case, fragile topology, along with the twisted boundary states, is stable with interactions below a critical interaction strength. Above this interaction strength, a transition to the Mott insulating phase occurs, and the twisted boundary states disappear. Furthermore, by applying a homogeneous magnetic field, the fragile topology is destroyed. However, we show that a magnetic field can induce a topological phase transition which converts a fragile topological insulator to a Chern insulator. Finally, we study ferromagnetic solutions of the fragile topological model.We study the effects of electronic correlations on fragile topology using dynamical mean-field theory. Fragile topological insulators (FTIs) offer obstruction to the formation of exponentially localized Wannier functions, but they can be trivialized by adding certain trivial degrees of freedom. For the same reason, FTIs do not host symmetry-protected flow of edge states between bulk bands in cylindrical boundary conditions but are expected to have a spectral flow between the fragile bands and other bands under certain twisted boundary conditions. We here analyze commonly observed effects of strong correlations, such as the Mott insulator transition and magnetism, on a known model hosting fragile topology. We show that in the nonmagnetic case, fragile topology, along with the twisted boundary states, is stable with interactions below a critical interaction strength. Above this interaction strength, a transition to the Mott insulating phase occurs, and the twisted boundary states disappear. Furthermore, by applying a homogeneous magnetic field, the fragile topology is destroyed. However, we show that a magnetic field can induce a topological phase transition which converts a fragile topological insulator to a Chern insulator. Finally, we study ferromagnetic solutions of the fragile topological model.

arXiv:2103.07203Physical review. B./Physical review. BApr 22, 20212 citations

We find a series of non-Abelian topological phases that are separated from the deconfined phase of

{Z}_{N}

gauge theory by a continuous quantum phase transition. These non-Abelian states, which we refer to as the ``twisted''

{Z}_{N}

states, are described by a recently studied $U(1)\ifmmode\times\else\texttimes\fi{}U(1)\ensuremath{\rtimes}{Z}_{2}$ Chern-Simons (CS) field theory. The $U(1)\ifmmode\times\else\texttimes\fi{}U(1)\ensuremath{\rtimes}{Z}_{2}$ CS theory provides a way of gauging the global

{Z}_{2}

electric-magnetic symmetry of the Abelian

{Z}_{N}

phases, yielding the twisted

{Z}_{N}

states. We introduce a parton construction to describe the Abelian

{Z}_{N}

phases in terms of integer quantum Hall states, which then allows us to obtain the non-Abelian states from a theory of

{Z}_{2}

fractionalization. The non-Abelian twisted

{Z}_{N}

states do not have topologically protected gapless edge modes and, for

N>2

, break time-reversal symmetry.

O(2)

gauge field. We first formulate a topological

S[O(2)\times O(2)]

BF theory, and by identifying the Wilson loops and twist operators of this theory with anyons, we show that our topological BF theory reproduces the complete anyon data, and can incorporate all Dijkgraaf--Witten twists. Building on this correspondence, we present a microscopic model with

O(2)

lattice gauge field coupled to Ising and rotor matter whose Higgsing yields the desired dihedral quantum double phase. A perturbative renormalization group analysis further suggests a direct transition from this phase to a

U(1)

Coulomb or chiral topological phase at a stable multicritical point with emergent

O(3)

symmetry. Our proposal offers an alternative route to nonabelian topological order with promising prospects in synthetic gauge field platforms.