We introduce a construction of symmetry-enriched topological orders on bipartite lattices in which two ${\mathbb{Z}}_{2}$ spin liquids defined on each sublattice are combined, and then anyons are condensed to reduce the topological order. By choosing different anyon condensate structures, one can vary the fractionalization pattern of the resulting spin liquid, some of which cannot be readily constructed from parton-based approaches. We demonstrate the construction for (i) a spin-1/2 honeycomb lattice where we construct a featureless state as well as intermediate states with topological order, (ii) a nonsymmorphic lattice, and (iii) lattices with magnetic translation symmetry. Finally, we discuss constraints on nonchiral topological orders in a bosonic system under magnetic field.We introduce a construction of symmetry-enriched topological orders on bipartite lattices in which two ${\mathbb{Z}}_{2}$ spin liquids defined on each sublattice are combined, and then anyons are condensed to reduce the topological order. By choosing different anyon condensate structures, one can vary the fractionalization pattern of the resulting spin liquid, some of which cannot be readily constructed from parton-based approaches. We demonstrate the construction for (i) a spin-1/2 honeycomb lattice where we construct a featureless state as well as intermediate states with topological order, (ii) a nonsymmorphic lattice, and (iii) lattices with magnetic translation symmetry. Finally, we discuss constraints on nonchiral topological orders in a bosonic system under magnetic field.

arXiv:1802.02155Physical review. B./Physical review. BDec 11, 20184 citations

Fractionalization in a square-lattice model with time-reversal symmetry

Marcel Franz, Conan Weeks, Babak Seradjeh

Bulletin of the American Physical SocietyMar 12, 20080 citations

Non-invertible SPT, gauging and symmetry fractionalization

Yabo Li, Litvinov, Mikhail

We explicitly realize the Rep( $Q_8$ ) non-invertible symmetry-protected topological (SPT) state as a 1+1d cluster state on a tensor product Hilbert space of qubits. Using the Kramers-Wannier operator, we construct the lattice models for the phases of all the symmetries in the Rep( $Q_8$ ) duality web. We further show that we can construct a class of lattice models with Rep( $G$ ) symmetry including non-invertible SPT phases if they have a dual anomalous abelian symmetry. Upon dualizing, there is a rich interplay between onsite symmetries, non-onsite symmetries, non-abelian symmetries, and non-invertible symmetries. We show that these interplay can be explained using the symmetry fractionalization in the 2+1d bulk SET.We explicitly realize the Rep($Q_8$) non-invertible symmetry-protected topological (SPT) state as a 1+1d cluster state on a tensor product Hilbert space of qubits. Using the Kramers-Wannier operator, we construct the lattice models for the phases of all the symmetries in the Rep($Q_8$) duality web. We further show that we can construct a class of lattice models with Rep($G$) symmetry including non-invertible SPT phases if they have a dual anomalous abelian symmetry. Upon dualizing, there is a rich interplay between onsite symmetries, non-onsite symmetries, non-abelian symmetries, and non-invertible symmetries. We show that these interplay can be explained using the symmetry fractionalization in the 2+1d bulk SET.

arXiv:2405.15951May 24, 20245 citations

Symmetry Approach to Solvable Lattice Models

T. Miwa

Springer series in solid-state sciencesJan 1, 19941 citation

Symmetry-protected fractional Chern insulators and fractional topological insulators

Yuan-Ming Lu, Ying Ran

In this paper we construct fully symmetric wave functions for the spin-polarized fractional Chern insulators (FCIs) and time-reversal-invariant fractional topological insulators (FTIs) in two dimensions using the parton approach. We show that the lattice symmetry gives rise to many different FCI and FTI phases even with the same filling fraction $\ensuremath{\nu}$ (and the same quantized Hall conductance ${\ensuremath{\sigma}}_{xy}$ in the FCI case). They have different symmetry-protected topological orders, which are characterized by different projective symmetry groups. We mainly focus on FCI phases which are realized in a partially filled band with Chern number 1. The low-energy gauge groups of a generic ${\ensuremath{\sigma}}_{xy}=1/m\ifmmode\cdot\else\textperiodcentered\fi{}{e}^{2}/h$ FCI wave function can be either $\text{SU}(m)$ or the discrete group ${Z}_{m}$ , and in the latter case the associated low-energy physics are described by Chern-Simons-Higgs theories. We use our construction to compute the ground-state degeneracy. Examples of FCI/FTI wave functions on honeycomb lattice and checkerboard lattice are explicitly given. Possible non-Abelian FCI phases which may be realized in a partially filled band with Chern number 2 are discussed. Generic FTI wave functions in the absence of spin conservation are also presented whose low-energy gauge groups can be either $\text{SU}(m)\ifmmode\times\else\texttimes\fi{}\text{SU}(m)$ or ${Z}_{m}\ifmmode\times\else\texttimes\fi{}{Z}_{m}$. The constructed wave functions also set up the framework for future variational Monte Carlo simulations.In this paper we construct fully symmetric wave functions for the spin-polarized fractional Chern insulators (FCIs) and time-reversal-invariant fractional topological insulators (FTIs) in two dimensions using the parton approach. We show that the lattice symmetry gives rise to many different FCI and FTI phases even with the same filling fraction $\ensuremath{\nu}$ (and the same quantized Hall conductance ${\ensuremath{\sigma}}_{xy}$ in the FCI case). They have different symmetry-protected topological orders, which are characterized by different projective symmetry groups. We mainly focus on FCI phases which are realized in a partially filled band with Chern number 1. The low-energy gauge groups of a generic ${\ensuremath{\sigma}}_{xy}=1/m\ifmmode\cdot\else\textperiodcentered\fi{}{e}^{2}/h$ FCI wave function can be either $\text{SU}(m)$ or the discrete group ${Z}_{m}$, and in the latter case the associated low-energy physics are described by Chern-Simons-Higgs theories. We use our construction to compute the ground-state degeneracy. Examples of FCI/FTI wave functions on honeycomb lattice and checkerboard lattice are explicitly given. Possible non-Abelian FCI phases which may be realized in a partially filled band with Chern number 2 are discussed. Generic FTI wave functions in the absence of spin conservation are also presented whose low-energy gauge groups can be either $\text{SU}(m)\ifmmode\times\else\texttimes\fi{}\text{SU}(m)$ or ${Z}_{m}\ifmmode\times\else\texttimes\fi{}{Z}_{m}$. The constructed wave functions also set up the framework for future variational Monte Carlo simulations.

arXiv:1109.0226Physical Review BApr 20, 201257 citations

We introduce a construction of symmetry-enriched topological orders on bipartite lattices in which two

{\mathbb{Z}}_{2}

spin liquids defined on each sublattice are combined, and then anyons are condensed to reduce the topological order. By choosing different anyon condensate structures, one can vary the fractionalization pattern of the resulting spin liquid, some of which cannot be readily constructed from parton-based approaches. We demonstrate the construction for (i) a spin-1/2 honeycomb lattice where we construct a featureless state as well as intermediate states with topological order, (ii) a nonsymmorphic lattice, and (iii) lattices with magnetic translation symmetry. Finally, we discuss constraints on nonchiral topological orders in a bosonic system under magnetic field.

We explicitly realize the Rep(

Q_8

) non-invertible symmetry-protected topological (SPT) state as a 1+1d cluster state on a tensor product Hilbert space of qubits. Using the Kramers-Wannier operator, we construct the lattice models for the phases of all the symmetries in the Rep(

Q_8

) duality web. We further show that we can construct a class of lattice models with Rep(

G

) symmetry including non-invertible SPT phases if they have a dual anomalous abelian symmetry. Upon dualizing, there is a rich interplay between onsite symmetries, non-onsite symmetries, non-abelian symmetries, and non-invertible symmetries. We show that these interplay can be explained using the symmetry fractionalization in the 2+1d bulk SET.

\text{SU}(m)

or the discrete group

{Z}_{m}

, and in the latter case the associated low-energy physics are described by Chern-Simons-Higgs theories. We use our construction to compute the ground-state degeneracy. Examples of FCI/FTI wave functions on honeycomb lattice and checkerboard lattice are explicitly given. Possible non-Abelian FCI phases which may be realized in a partially filled band with Chern number 2 are discussed. Generic FTI wave functions in the absence of spin conservation are also presented whose low-energy gauge groups can be either $\text{SU}(m)\ifmmode\times\else\texttimes\fi{}\text{SU}(m)$ or ${Z}_{m}\ifmmode\times\else\texttimes\fi{}{Z}_{m}$. The constructed wave functions also set up the framework for future variational Monte Carlo simulations.