String-net condensation: A physical mechanism for topological phases

By Michael Levin, X. Wen. 2004

1,185 citations recorded by the source corpus.

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.

Subfields: Atomic and Molecular Physics, and Optics, Condensed Matter Physics

View Paper: https://www.semanticscholar.org/paper/fc915e067db18aaf0f7d8e764f59d9e56effe7a4

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In Atomic and Molecular Physics, and Optics

In Condensed Matter Physics

Citations in the corpus

Topological defect lines and renormalization group flows in two dimensions (2018, background): “ollaries. Many of these structures have already been dened and explored in both the mathematics and physics literature, such as in the context of modular tensor category [37{40], and in the works of [42,43,59,21,22,26,41,27]. We hope to rst recast them in a language natural for quantum eld theory, a”
Generalized Symmetries in Condensed Matter (2022, positive): “Gapped edge theories are realized if the bulk theory admits gapped boundary conditions; such a bulk theory has an exactly solvable description as a string-net model (30).”
Measurement as a Shortcut to Long-Range Entangled Quantum Matter (2022, methodology): “This immediately leads to two notable classes of target states achievable with O(logL)-depth local adaptive circuits: (i) gapped topological orders in quantum double models [9] of any finite group and Levin-Wen string-net models [16] (ii) gapless states characterized by conformal field theories (CFT”
Non-Abelian Topological Order on the Surface of a 3D Topological Superconductor from an Exactly Solved Model (2013, methodology): “[6] X. Chen, Z.-C. Gu, and X.-G. Wen.”
Localization-protected quantum order (2013, background): “tries and also topologically ordered systems with discrete gapped excitations such the Levin-Wen models that exhibit non-abelian phases [37].”
Fusion Category Symmetry I: Anomaly In-Flow and Gapped Phases (2019, methodology): “For computing ground state degeneracy, it is also convenient to have a Hamiltonian model of the boundary condition.”
Entanglement entropy and nonabelian gauge symmetry (2014, background): “However a large class of these topological phases contain emergent excitations that have a low-energy description as a gauge theory [14].”
Optimal resources for topological two-dimensional stabilizer codes : Comparative study (2007, background): “Thus, topological orders deviate signiﬁcantly from more standard orders treated within the Landau symmetry-breaking theory [14], [15], [16], [17].”
Braiding Statistics and Link Invariants of Bosonic/Fermionic Topological Quantum Matter in 2+1 and 3+1 dimensions (2016, background): “ /Z[S(3)] exp(− 2πi p q1q2q3 N123 μ̄(γ1, γ2, γ3)) Gauged SPT lattice model [25, 31, 36], Twisted quantum double [18, 20], String-net models [61], D4 discrete gauge theory [22,24,31,36]”
A toolbox for lattice-spin models with polar molecules (2005, background): “Recently there have been exciting theoretical discoveries of models with quasi-local spin interactions with emergent topological orde”
Locality in Quantum Systems (2010, background): “One interesting case is that lattice Hamiltonians are known, the so-called “Levin-Wen models”[31], which realize certain two-dimensional unitary topological quantum field theories (TQFTs), in particular those theories which are quantum doubles.”
Topological quantum order: Stability under local perturbations (2010, methodology): “The above arguments suggest that TQO-2 holds for all 2D models of TQO that can be described by commuting frustration-free such as quantum double models [1] and Levin-Wen string-net models [4].”
Models for Gapped Boundaries and Domain Walls (2011, methodology): “ If all the spins at the corners are stable, then Bk p creates a loop with label k, which is fused with the plaquette boundary using the rules of the tensor category, see Fig. 3 or the original paper [LW04]. The operators Q v and Bk p for diﬀerent vertices and plaquettes commute with each other; Q v”
Three-body interactions with cold polar molecules (2007, neutral): “These ground states can be distinguished by the bond observable in equation (9).”