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:cond-mat/0404617 · Physical Review B

String-net condensation: A physical mechanism for topological phases

Michael Levin, Xiao-Gang Wen

Published Jan 12, 2005• 1,469 citations
arXivHTMLPDF

Abstract

We show that quantum systems of extended objects naturally give rise to a large class of exotic phases---namely topological phases. These phases occur when extended objects, called ``string-nets,'' become highly fluctuating and condense. We construct a large class of exactly soluble 2D spin Hamiltonians whose ground states are string-net condensed. Each ground state corresponds to a different parity invariant topological phase. The models reveal the mathematical framework underlying topological phases: tensor category theory. One of the Hamiltonians---a spin-$1∕2$ system on the honeycomb lattice---is a simple theoretical realization of a universal fault tolerant quantum computer. The higher dimensional case also yields an interesting result: we find that 3D string-net condensation naturally gives rise to both emergent gauge bosons and emergent fermions. Thus, string-net condensation provides a mechanism for unifying gauge bosons and fermions in 3 and higher dimensions.

DOI
10.1103/physrevb.71.045110

Related papers

Topological Phases and Exactly Soluable Lattice Models

Chien-Hung Lin

In the past few years, the discovery of topological insulators has resurrected the study of exotic phases of matter beyond the Landau symmetry breaking paradigm. In this thesis, we focus on two particular classes of exotic phases, known as "topological phases" and "symmetry protected topological phases (SPT phases)" in 2D and 3D. Topological phases are generalizations of fractional quantum Hall liquids while SPT phases are generalizations of topological insulators. We investigate topological phases and SPT phases by constructing and analyzing appropriate exactly soluble lattice spin models. We address the following three questions: The first question involves a large class of exactly soluble lattice spin models called string-net models. It is known that string-net models can realize a large class of 2D topological phases, but it is unclear what topological phases can and can not be realized by these models. Thus our first question is: what are the most general topological phases that can be realized by string-net models? We find that a topological phase can be realized by a string-net model if and only if it supports a gapped edge, i.e. the edge can be gapped by suitable local interactions. In the second question we ask: how can we distinguish different 3D topological phases? While it is well known that 3D topological phases can be distinguished by the braiding statistics between particle and loop-like excitations, we construct two exactly soluble lattice models that demonstrate that we also have to consider the braiding statistics of loops with other loops. In the final part of the thesis, we derive a bulk-boundary correspondence for 3D SPT phases. More specifically, we prove that a particular exactly soluble lattice spin model has protected surface states using the fact that the vortex loop excitations in the bulk have non-trivial braiding statistics. We prove the result for a particular 3D exactly soluble lattice model, but our arguments apply more generally.

•Dec 1, 2015•0 citations

Exact topological quantum order in D=3 and beyond: Branyons and brane-net condensates

Héctor Bombín, Miguel Angel Martin-Delgado

We construct an exactly solvable Hamiltonian acting on a three-dimensional lattice of spin-$\frac{1}{2}$ systems that exhibits topological quantum order. The ground state is a string net and a membrane-net condensate. Excitations appear in the form of quasiparticles and fluxes, as the boundaries of strings and membranes, respectively. The degeneracy of the ground state depends on the homology of the 3-manifold. We generalize the system to $D\ensuremath{\geqslant}4$, where different topological phases may occur. The whole construction is based on certain special complexes that we call colexes.

arXiv:cond-mat/0607736•Physical Review B•Feb 7, 2007•138 citations

Colloquium : Photons and electrons as emergent phenomena

Michael Levin, Xiao-Gang Wen

Recent advances in condensed-matter theory have revealed that new and exotic phases of matter can exist in spin models (or more precisely, local bosonic models) via a simple physical mechanism, known as ``string-net condensation.'' These new phases of matter have the unusual property that their collective excitations are gauge bosons and fermions. In some cases, the collective excitations can behave just like the photons, electrons, gluons, and quarks in our vacuum. This suggests that photons, electrons, and other elementary particles may have a unified origin---string-net condensation in our vacuum. In addition, the string-net picture indicates how to make artificial photons, artificial electrons, and artificial quarks and gluons in condensed-matter systems.

arXiv:cond-mat/0407140•Reviews of Modern Physics•Sep 7, 2005•162 citations

Fermion condensation and super pivotal categories

David Aasen, Ethan Lake, Kevin Walker

We study fermionic topological phases using the technique of fermion condensation. We give a prescription for performing fermion condensation in bosonic topological phases that contain a fermion. Our approach to fermion condensation can roughly be understood as coupling the parent bosonic topological phase to a phase of physical fermions and condensing pairs of physical and emergent fermions. There are two distinct types of objects in the resulting fermionic fusion categories, which we call “m-type” and “q-type” objects. The endomorphism algebras of q-type objects are complex Clifford algebras, and they have no analogs in bosonic theories. We construct a fermionic generalization of the tube category, which allows us to compute the quasiparticle excitations arising from the condensed theories. We prove a series of results relating data in fermionic theories to data in their parent bosonic theories; for example, if C is a modular tensor category containing a fermion, then the tube category constructed from the condensed theory satisfies Tube(C/ψ)≅C×(C/ψ). We also study how modular transformations, fusion rules, and coherence relations are modified in the fermionic setting, prove a fermionic version of the Verlinde dimension formula, construct a commuting projector lattice Hamiltonian for fermionic theories, and write down a fermionic version of the Turaev-Viro-Barrett-Westbury state sum. A large portion of this work is devoted to three detailed examples of performing fermion condensation to produce fermionic topological phases: we condense fermions in the Ising theory, the SO(3)6 theory, and the 12E6 theory and compute the quasiparticle excitation spectrum in each of the condensed theories.

arXiv:1709.01941•Journal of Mathematical Physics•Dec 1, 2019•86 citations

Landau-Ginzburg Paradigm of Topological Phases

Y. B. Zhao, Yidun Wan

Topologically ordered matter phases have been regarded as beyond the Landau-Ginzburg symmetry breaking paradigm of matter phases. Recent studies of anyon condensation in topological phases, however, may fit topological phases back in the Landau-Ginzburg paradigm. To truly do so, we realized that the string-net model of topological phases is in fact an effective lattice gauge theory coupled with anyonic matter once two modifications are made: (1) We reinterpret anyons as matter fields coupled to lattice gauge fields, thus extending the HGW model to a genuine Hamiltonian lattice gauge theory. (2) By explicitly incorporating the internal degrees of freedom of anyons, we construct an enlarged Hilbert space that supports well-defined gauge transformations and covariant coupling, restoring the analogy with conventional lattice gauge field theory. In this modified string-net model, topological phase transitions induced by anyon condensation and their consequent phenomena, such as order parameter fields, coherent states, Goldstone modes, and gapping gauge degrees of freedom, can be formulated exactly as Landau's effective theory of the Higgs mechanism. To facilitate the understanding, we also compare anyon condensation to/with the Higgs boson condensation in the electroweak theory and the Cooper pair condensation.

arXiv:2506.05319•Jun 5, 2025•1 citation

String Condensation—An Unification of Light and Fermions

Xiao-Gang Wen

This book has three types of theories: boson, fermion, and gauge theories. It asks which theories describe the elementary particles in our universe. Surprisingly, elementary particles are described by the two more complicated and non-local theories: fermion theory and gauge theory. In the last chapter and in this chapter it is demonstrated that fermion and gauge theory are not really needed. Boson theory may be able to explain everything, since both fermion theory and gauge theory can emerge from local bosonic models (or qbit models), if bosons (qbits) are organized to have a string-net condensed order. This chapter studies soluble quantum spin models to explain string-net condensation. It finds that gauge bosons are simply collective excitations associated with string density wave (with the electric field corresponding to the string density). The fermons can appear as the ends of open strings. Thus the string-net order explains the emergence of gauge bosons and fermions. It unifies gauge interaction and Fermi statistics. The chapter shows that non-orientable strings give rise to Z_2 gauge theory while the orientable string give rise to U(1) gauge theory.

Oxford University Press eBooks•Sep 6, 2007•0 citations
We show that quantum systems of extended objects naturally give rise to a large class of exotic phases---namely topological phases. These phases occur when extended objects, called ``string-nets,'' become highly fluctuating and condense. We construct a large class of exactly soluble 2D spin Hamiltonians whose ground states are string-net condensed. Each ground state corresponds to a different parity invariant topological phase. The models reveal the mathematical framework underlying topological phases: tensor category theory. One of the Hamiltonians---a spin-1∕21∕21∕2 system on the honeycomb lattice---is a simple theoretical realization of a universal fault tolerant quantum computer. The higher dimensional case also yields an interesting result: we find that 3D string-net condensation naturally gives rise to both emergent gauge bosons and emergent fermions. Thus, string-net condensation provides a mechanism for unifying gauge bosons and fermions in 3 and higher dimensions.
In the past few years, the discovery of topological insulators has resurrected the study of exotic phases of matter beyond the Landau symmetry breaking paradigm. In this thesis, we focus on two particular classes of exotic phases, known as "topological phases" and "symmetry protected topological phases (SPT phases)" in 2D and 3D. Topological phases are generalizations of fractional quantum Hall liquids while SPT phases are generalizations of topological insulators. We investigate topological phases and SPT phases by constructing and analyzing appropriate exactly soluble lattice spin models. We address the following three questions: The first question involves a large class of exactly soluble lattice spin models called string-net models. It is known that string-net models can realize a large class of 2D topological phases, but it is unclear what topological phases can and can not be realized by these models. Thus our first question is: what are the most general topological phases that can be realized by string-net models? We find that a topological phase can be realized by a string-net model if and only if it supports a gapped edge, i.e. the edge can be gapped by suitable local interactions. In the second question we ask: how can we distinguish different 3D topological phases? While it is well known that 3D topological phases can be distinguished by the braiding statistics between particle and loop-like excitations, we construct two exactly soluble lattice models that demonstrate that we also have to consider the braiding statistics of loops with other loops. In the final part of the thesis, we derive a bulk-boundary correspondence for 3D SPT phases. More specifically, we prove that a particular exactly soluble lattice spin model has protected surface states using the fact that the vortex loop excitations in the bulk have non-trivial braiding statistics. We prove the result for a particular 3D exactly soluble lattice model, but our arguments apply more generally.
We construct an exactly solvable Hamiltonian acting on a three-dimensional lattice of spin-12\frac{1}{2}21​ systems that exhibits topological quantum order. The ground state is a string net and a membrane-net condensate. Excitations appear in the form of quasiparticles and fluxes, as the boundaries of strings and membranes, respectively. The degeneracy of the ground state depends on the homology of the 3-manifold. We generalize the system to $D\ensuremath{\geqslant}4$, where different topological phases may occur. The whole construction is based on certain special complexes that we call colexes.
Recent advances in condensed-matter theory have revealed that new and exotic phases of matter can exist in spin models (or more precisely, local bosonic models) via a simple physical mechanism, known as ``string-net condensation.'' These new phases of matter have the unusual property that their collective excitations are gauge bosons and fermions. In some cases, the collective excitations can behave just like the photons, electrons, gluons, and quarks in our vacuum. This suggests that photons, electrons, and other elementary particles may have a unified origin---string-net condensation in our vacuum. In addition, the string-net picture indicates how to make artificial photons, artificial electrons, and artificial quarks and gluons in condensed-matter systems.
We study fermionic topological phases using the technique of fermion condensation. We give a prescription for performing fermion condensation in bosonic topological phases that contain a fermion. Our approach to fermion condensation can roughly be understood as coupling the parent bosonic topological phase to a phase of physical fermions and condensing pairs of physical and emergent fermions. There are two distinct types of objects in the resulting fermionic fusion categories, which we call “m-type” and “q-type” objects. The endomorphism algebras of q-type objects are complex Clifford algebras, and they have no analogs in bosonic theories. We construct a fermionic generalization of the tube category, which allows us to compute the quasiparticle excitations arising from the condensed theories. We prove a series of results relating data in fermionic theories to data in their parent bosonic theories; for example, if C is a modular tensor category containing a fermion, then the tube category constructed from the condensed theory satisfies Tube(C/ψ)≅C×(C/ψ). We also study how modular transformations, fusion rules, and coherence relations are modified in the fermionic setting, prove a fermionic version of the Verlinde dimension formula, construct a commuting projector lattice Hamiltonian for fermionic theories, and write down a fermionic version of the Turaev-Viro-Barrett-Westbury state sum. A large portion of this work is devoted to three detailed examples of performing fermion condensation to produce fermionic topological phases: we condense fermions in the Ising theory, the SO(3)6 theory, and the 12E6 theory and compute the quasiparticle excitation spectrum in each of the condensed theories.
Topologically ordered matter phases have been regarded as beyond the Landau-Ginzburg symmetry breaking paradigm of matter phases. Recent studies of anyon condensation in topological phases, however, may fit topological phases back in the Landau-Ginzburg paradigm. To truly do so, we realized that the string-net model of topological phases is in fact an effective lattice gauge theory coupled with anyonic matter once two modifications are made: (1) We reinterpret anyons as matter fields coupled to lattice gauge fields, thus extending the HGW model to a genuine Hamiltonian lattice gauge theory. (2) By explicitly incorporating the internal degrees of freedom of anyons, we construct an enlarged Hilbert space that supports well-defined gauge transformations and covariant coupling, restoring the analogy with conventional lattice gauge field theory. In this modified string-net model, topological phase transitions induced by anyon condensation and their consequent phenomena, such as order parameter fields, coherent states, Goldstone modes, and gapping gauge degrees of freedom, can be formulated exactly as Landau's effective theory of the Higgs mechanism. To facilitate the understanding, we also compare anyon condensation to/with the Higgs boson condensation in the electroweak theory and the Cooper pair condensation.
This book has three types of theories: boson, fermion, and gauge theories. It asks which theories describe the elementary particles in our universe. Surprisingly, elementary particles are described by the two more complicated and non-local theories: fermion theory and gauge theory. In the last chapter and in this chapter it is demonstrated that fermion and gauge theory are not really needed. Boson theory may be able to explain everything, since both fermion theory and gauge theory can emerge from local bosonic models (or qbit models), if bosons (qbits) are organized to have a string-net condensed order. This chapter studies soluble quantum spin models to explain string-net condensation. It finds that gauge bosons are simply collective excitations associated with string density wave (with the electric field corresponding to the string density). The fermons can appear as the ends of open strings. Thus the string-net order explains the emergence of gauge bosons and fermions. It unifies gauge interaction and Fermi statistics. The chapter shows that non-orientable strings give rise to Z_2 gauge theory while the orientable string give rise to U(1) gauge theory.