We construct and study lattice realizations of finite 2-group symmetries in ${2+1}$ d quantum lattice systems with finite-dimensional tensor-product Hilbert spaces. We focus on two broad classes of 2-groups with 0-form symmetry group $G$ and 1-form symmetry group $A$ : split 2-groups with trivial Postnikov class ${[β]\in\mathcal{H}^3(G,A)}$ , and central 2-groups with trivial action ${ρ\colon G\to\text{Aut}(A)}$ . In both cases, we construct symmetry operators on the full tensor-product Hilbert space that become 2-group symmetry operators when restricted to the topological subspace of the lattice $A$ 1-form symmetry. While the lattice split 2-group symmetry operators are onsite, the lattice central 2-group symmetry operators are not, and can only be made onsite after introducing ancillae. We extensively explore various manifestations of $ρ$ and $[β]$ for these lattice 2-group symmetry operators and demonstrate their agreement with expectations from quantum field theory. These manifestations arise in the transformation of operators carrying symmetry charge, the structure of lattice 2-group symmetry defects, and the dual fusion 2-category symmetries obtained by gauging the lattice 2-group symmetries. We further propose families of local symmetric Hamiltonians for both classes of lattice 2-group symmetries and identify exactly solvable limits lying in phases with spontaneous 2-group symmetry breaking and nontrivial symmetry-enriched topological order. In one such limit, the gauged Hamiltonians are exactly solvable lattice realizations of the corresponding 2-group gauge theories, whose ground-state degeneracies we calculate.We construct and study lattice realizations of finite 2-group symmetries in ${2+1}$d quantum lattice systems with finite-dimensional tensor-product Hilbert spaces. We focus on two broad classes of 2-groups with 0-form symmetry group $G$ and 1-form symmetry group $A$: split 2-groups with trivial Postnikov class ${[β]\in\mathcal{H}^3(G,A)}$, and central 2-groups with trivial action ${ρ\colon G\to\text{Aut}(A)}$. In both cases, we construct symmetry operators on the full tensor-product Hilbert space that become 2-group symmetry operators when restricted to the topological subspace of the lattice $A$ 1-form symmetry. While the lattice split 2-group symmetry operators are onsite, the lattice central 2-group symmetry operators are not, and can only be made onsite after introducing ancillae. We extensively explore various manifestations of $ρ$ and $[β]$ for these lattice 2-group symmetry operators and demonstrate their agreement with expectations from quantum field theory. These manifestations arise in the transformation of operators carrying symmetry charge, the structure of lattice 2-group symmetry defects, and the dual fusion 2-category symmetries obtained by gauging the lattice 2-group symmetries. We further propose families of local symmetric Hamiltonians for both classes of lattice 2-group symmetries and identify exactly solvable limits lying in phases with spontaneous 2-group symmetry breaking and nontrivial symmetry-enriched topological order. In one such limit, the gauged Hamiltonians are exactly solvable lattice realizations of the corresponding 2-group gauge theories, whose ground-state degeneracies we calculate.

cond-mat.str-elCross-listedarXiv:2609.10682Sep 11, 20260 citations

Staggered lattice models

David Hansel, Th. Jolicœur, J.-M. Maillard

Some staggered models are studied along three different lines: exact solvability (Yang-Baxter equations); the existence of other exact solutions called disorder solutions; their group of symmetries. The analysis, based on the commutation of row to row transfer matrices of arbitrary size N, suggests that the exactly solvable models are either of non-staggered type or of free-fermion staggered type. In contrast, disorder solutions can easily be exhibited for staggered models: the example of the Ashkin-Teller model is analysed. Finally, the symmetry group of these models is seen to be a straightforward generalisation of the one of homogeneous models.Some staggered models are studied along three different lines: exact solvability (Yang-Baxter equations); the existence of other exact solutions called disorder solutions; their group of symmetries. The analysis, based on the commutation of row to row transfer matrices of arbitrary size N, suggests that the exactly solvable models are either of non-staggered type or of free-fermion staggered type. In contrast, disorder solutions can easily be exhibited for staggered models: the example of the Ashkin-Teller model is analysed. Finally, the symmetry group of these models is seen to be a straightforward generalisation of the one of homogeneous models.

Journal of Physics A Mathematical and GeneralOct 1, 19873 citations

On the algebraic approach to solvable lattice models

Andrei Babichenko, Doron Gepner

arXiv:hep-th/0703006Physics Letters BMay 26, 20072 citations

Space-group symmetry fractionalization in a family of exactly solvable models with Z2 topological order

Hao Song, Michael Hermele

We study square lattice space-group symmetry fractionalization in a family of exactly solvable models with ${\mathbb{Z}}_{2}$ topological order in two dimensions. In particular, we have obtained a complete understanding of which distinct types of symmetry fractionalization (symmetry classes) can be realized within this class of models, which are generalizations of Kitaev's ${\mathbb{Z}}_{2}$ toric code to arbitrary lattices. This question is motivated by earlier work of Essin and one of us (M. H.) [Phys. Rev. B 87, 104406 (2013)], where the idea of symmetry classification was laid out, and which, for square lattice symmetry, produces 2080 symmetry classes consistent with the fusion rules of ${\mathbb{Z}}_{2}$ topological order. This approach does not produce a physical model for each symmetry class, and indeed there are reasons to believe that some symmetry classes may not be realizable in strictly two-dimensional systems, thus raising the question of which classes are in fact possible. While our understanding is limited to a restricted class of models, it is complete in the sense that for each of the 2080 possible symmetry classes, we either prove rigorously that the class cannot be realized in our family of models, or we give an explicit model realizing the class. We thus find that exactly 487 symmetry classes are realized in the family of models considered. With a more restrictive type of symmetry action, where space-group operations act trivially in the internal Hilbert space of each spin degree of freedom, we find that exactly 82 symmetry classes are realized. In addition, we present a single model that realizes all ${2}^{6}=64$ types of symmetry fractionalization allowed for a single anyon species ( ${\mathbb{Z}}_{2}$ charge excitation), as the parameters in the Hamiltonian are varied. The paper concludes with a summary and a discussion of two results pertaining to more general bosonic models.We study square lattice space-group symmetry fractionalization in a family of exactly solvable models with ${\mathbb{Z}}_{2}$ topological order in two dimensions. In particular, we have obtained a complete understanding of which distinct types of symmetry fractionalization (symmetry classes) can be realized within this class of models, which are generalizations of Kitaev's ${\mathbb{Z}}_{2}$ toric code to arbitrary lattices. This question is motivated by earlier work of Essin and one of us (M. H.) [Phys. Rev. B 87, 104406 (2013)], where the idea of symmetry classification was laid out, and which, for square lattice symmetry, produces 2080 symmetry classes consistent with the fusion rules of ${\mathbb{Z}}_{2}$ topological order. This approach does not produce a physical model for each symmetry class, and indeed there are reasons to believe that some symmetry classes may not be realizable in strictly two-dimensional systems, thus raising the question of which classes are in fact possible. While our understanding is limited to a restricted class of models, it is complete in the sense that for each of the 2080 possible symmetry classes, we either prove rigorously that the class cannot be realized in our family of models, or we give an explicit model realizing the class. We thus find that exactly 487 symmetry classes are realized in the family of models considered. With a more restrictive type of symmetry action, where space-group operations act trivially in the internal Hilbert space of each spin degree of freedom, we find that exactly 82 symmetry classes are realized. In addition, we present a single model that realizes all ${2}^{6}=64$ types of symmetry fractionalization allowed for a single anyon species (${\mathbb{Z}}_{2}$ charge excitation), as the parameters in the Hamiltonian are varied. The paper concludes with a summary and a discussion of two results pertaining to more general bosonic models.

arXiv:1409.1504Physical Review BJan 6, 201522 citations

We construct and study lattice realizations of finite 2-group symmetries in

{2+1}

d quantum lattice systems with finite-dimensional tensor-product Hilbert spaces. We focus on two broad classes of 2-groups with 0-form symmetry group

G

and 1-form symmetry group

A

: split 2-groups with trivial Postnikov class

{[β]\in\mathcal{H}^3(G,A)}

, and central 2-groups with trivial action

{ρ\colon G\to\text{Aut}(A)}

. In both cases, we construct symmetry operators on the full tensor-product Hilbert space that become 2-group symmetry operators when restricted to the topological subspace of the lattice

A

1-form symmetry. While the lattice split 2-group symmetry operators are onsite, the lattice central 2-group symmetry operators are not, and can only be made onsite after introducing ancillae. We extensively explore various manifestations of

ρ

and

[β]

for these lattice 2-group symmetry operators and demonstrate their agreement with expectations from quantum field theory. These manifestations arise in the transformation of operators carrying symmetry charge, the structure of lattice 2-group symmetry defects, and the dual fusion 2-category symmetries obtained by gauging the lattice 2-group symmetries. We further propose families of local symmetric Hamiltonians for both classes of lattice 2-group symmetries and identify exactly solvable limits lying in phases with spontaneous 2-group symmetry breaking and nontrivial symmetry-enriched topological order. In one such limit, the gauged Hamiltonians are exactly solvable lattice realizations of the corresponding 2-group gauge theories, whose ground-state degeneracies we calculate.

We study square lattice space-group symmetry fractionalization in a family of exactly solvable models with

{\mathbb{Z}}_{2}

topological order in two dimensions. In particular, we have obtained a complete understanding of which distinct types of symmetry fractionalization (symmetry classes) can be realized within this class of models, which are generalizations of Kitaev's

{\mathbb{Z}}_{2}

toric code to arbitrary lattices. This question is motivated by earlier work of Essin and one of us (M. H.) [Phys. Rev. B 87, 104406 (2013)], where the idea of symmetry classification was laid out, and which, for square lattice symmetry, produces 2080 symmetry classes consistent with the fusion rules of

{\mathbb{Z}}_{2}

topological order. This approach does not produce a physical model for each symmetry class, and indeed there are reasons to believe that some symmetry classes may not be realizable in strictly two-dimensional systems, thus raising the question of which classes are in fact possible. While our understanding is limited to a restricted class of models, it is complete in the sense that for each of the 2080 possible symmetry classes, we either prove rigorously that the class cannot be realized in our family of models, or we give an explicit model realizing the class. We thus find that exactly 487 symmetry classes are realized in the family of models considered. With a more restrictive type of symmetry action, where space-group operations act trivially in the internal Hilbert space of each spin degree of freedom, we find that exactly 82 symmetry classes are realized. In addition, we present a single model that realizes all

{2}^{6}=64

types of symmetry fractionalization allowed for a single anyon species (

{\mathbb{Z}}_{2}

charge excitation), as the parameters in the Hamiltonian are varied. The paper concludes with a summary and a discussion of two results pertaining to more general bosonic models.