Local gauge symmetry and emergence of topological order in quantum spin systems

Gauging introduces gauge fields in order to localize an existing global symmetry, resulting in a dual global symmetry on the gauge fields that can be gauged again. By iterating the gauging process on spin chains with Abelian group symmetries and arranging the gauge fields in a 2D lattice, the local symmetries become the stabilizer of the $XZZX$ -code for any Abelian group. By twisting the gauging map we obtain new codes that explicitly confine anyons, which violate an odd number of plaquette terms and whose fusion results in mobile dipole excitations. Our construction naturally realizes any gapped boundary by taking different quantum phases of the initial (1+1)D globally symmetric system. Our method establishes a new route to obtain higher dimensional topological codes from lower ones, to identify their gapped boundaries and their tensor network representations.Gauging introduces gauge fields in order to localize an existing global symmetry, resulting in a dual global symmetry on the gauge fields that can be gauged again. By iterating the gauging process on spin chains with Abelian group symmetries and arranging the gauge fields in a 2D lattice, the local symmetries become the stabilizer of the $XZZX$-code for any Abelian group. By twisting the gauging map we obtain new codes that explicitly confine anyons, which violate an odd number of plaquette terms and whose fusion results in mobile dipole excitations. Our construction naturally realizes any gapped boundary by taking different quantum phases of the initial (1+1)D globally symmetric system. Our method establishes a new route to obtain higher dimensional topological codes from lower ones, to identify their gapped boundaries and their tensor network representations.

arXiv:2403.07575Mar 12, 20240 citations

Emergent (2+1)D topological orders from iterative (1+1)D gauging

José Garre-Rubio

Gauging introduces gauge fields in order to localize an existing global symmetry, resulting in a dual global symmetry on the gauge fields that can be gauged again. By iterating the gauging process on spin chains with Abelian group symmetries and arranging the gauge fields in a 2D lattice, the local symmetries become the stabilizer of the XZZX-code for any Abelian group. By twisting the gauging map, we obtain codes that explicitly confine anyons, whose local creating operators violate an odd number of plaquettes. Their fusion results in either mobile dipole excitations twisting only half of the plaquette terms, or complete immobile Sierpiński-like excitations if we twist all the terms. Our construction naturally realizes any gapped boundary by taking different quantum phases of the initial (1+1)D globally symmetric system. In addition, our method also establishes a promising route to obtain higherdimensional topological codes from lower ones and to identify their gapped boundaries and their tensor network representations.Gauging introduces gauge fields in order to localize an existing global symmetry, resulting in a dual global symmetry on the gauge fields that can be gauged again. By iterating the gauging process on spin chains with Abelian group symmetries and arranging the gauge fields in a 2D lattice, the local symmetries become the stabilizer of the XZZX-code for any Abelian group. By twisting the gauging map, we obtain codes that explicitly confine anyons, whose local creating operators violate an odd number of plaquettes. Their fusion results in either mobile dipole excitations twisting only half of the plaquette terms, or complete immobile Sierpiński-like excitations if we twist all the terms. Our construction naturally realizes any gapped boundary by taking different quantum phases of the initial (1+1)D globally symmetric system. In addition, our method also establishes a promising route to obtain higherdimensional topological codes from lower ones and to identify their gapped boundaries and their tensor network representations.

Nature CommunicationsSep 12, 20244 citations

A symmetry principle for topological quantum order

Zohar Nussinov, Gerardo Ortíz

arXiv:cond-mat/0702377Annals of PhysicsDec 11, 2008311 citations

Emergent Gauge Symmetries in Quantum Condensed Matter Systems

WORLD SCIENTIFIC eBooksApr 2, 20250 citations

Mean-Field Theory of Spin Liquids and Quantum Order

Xiao-Gang Wen

Topological order, as a generic phenomenon, not only appears in FQH systems, it can also appear in quantum spin systems. Quantum spin systems even allow the more general quantum order. This chapter develops a mean-field theory for the topological/quantum order in strongly interacting quantum spin systems. The mean-field theory is based on the projective construction (or the slave-particle construction). The chapter introduces the notion of projective symmetry group (PSG) to describe distinct phases that have exactly the same symmetry. PSG allows for the introduction of the notion of quantum order, which is more general then the notion of topological order. Using the mean-field theory, it calculates the phase diagram and phase transitions for quantum spin systems that do not involve change of symmetry. In fact, many phases that are studied do not break any symmetry and correspond to quantum spin liquids. The mean-field theory shows that those spin liquids can have some very exotic properties such as fractionalization, spin-charge separation, emergent gauge bosons and fermions, interacting gapless exactions, etc. The chapter finds that in addition to the symmetry breaking mechanism, PSG and quantum order is another way to produce and protect gapless excitations.Topological order, as a generic phenomenon, not only appears in FQH systems, it can also appear in quantum spin systems. Quantum spin systems even allow the more general quantum order. This chapter develops a mean-field theory for the topological/quantum order in strongly interacting quantum spin systems. The mean-field theory is based on the projective construction (or the slave-particle construction). The chapter introduces the notion of projective symmetry group (PSG) to describe distinct phases that have exactly the same symmetry. PSG allows for the introduction of the notion of quantum order, which is more general then the notion of topological order. Using the mean-field theory, it calculates the phase diagram and phase transitions for quantum spin systems that do not involve change of symmetry. In fact, many phases that are studied do not break any symmetry and correspond to quantum spin liquids. The mean-field theory shows that those spin liquids can have some very exotic properties such as fractionalization, spin-charge separation, emergent gauge bosons and fermions, interacting gapless exactions, etc. The chapter finds that in addition to the symmetry breaking mechanism, PSG and quantum order is another way to produce and protect gapless excitations.

Oxford University Press eBooksSep 6, 20070 citations

XZZX

-code for any Abelian group. By twisting the gauging map we obtain new codes that explicitly confine anyons, which violate an odd number of plaquette terms and whose fusion results in mobile dipole excitations. Our construction naturally realizes any gapped boundary by taking different quantum phases of the initial (1+1)D globally symmetric system. Our method establishes a new route to obtain higher dimensional topological codes from lower ones, to identify their gapped boundaries and their tensor network representations.