String-net condensation: A physical mechanism for topological phases

TL;DR

The paper proposes string-net condensation as a universal mechanism for generating doubled topological phases in bosonic systems, linking microscopic string-net dynamics to a tensor-category classification via fixed-point wave functions and exactly soluble Hamiltonians. In $2+1$D, each phase corresponds to a solution of pentagon-like constraints on $F^{ijk}_{lmn}$ and is characterized by a fixed-point wave function and a complete quasiparticle data set (fusion, twists, and $S$-matrix). The authors construct explicit lattice models realizing all such phases, deriving their excitations and illustrating Abelian and non-Abelian cases, including models with potential for fault-tolerant quantum computation. In $3+1$D$ and higher, string-net condensation naturally yields emergent gauge bosons and emergent fermions, with the phase structure tied to symmetric tensor categories, offering a path to unify gauge interactions and Fermi statistics within a common framework. Overall, the work provides a constructive, mathematically grounded approach to topological order beyond Landau theory and suggests broad implications for condensed matter and high-energy physics.

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 the extended objects, called ``string-nets'', become highly fluctuating and condense. We derive exactly soluble Hamiltonians for 2D local bosonic models whose ground states are string-net condensed states. Those ground states correspond to 2D parity invariant topological phases. These 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 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.

Figures (19)

• Figure 1: A schematic phase diagram for the generic string-net Hamiltonian (\ref{['genh']}). When $t/U$ (the ratio of the kinetic energy to the string tension) is small the system is in the normal phase. The ground state is essentially the vacuum with a few small string-nets. When $t/U$ is large the string-nets condense and large fluctuating string-nets fill all of space. We expect a phase transition between the two states at some $t/U$ of order unity. We have omitted string labels and orientations for the sake of clarity.
• Figure 2: The constraint term $\prod_{\text{legs of }\v I} ^x_{\v i}$ and magnetic term $\prod_{\text{edges of }\v p} ^z_{\v j}$ in $Z_2$ lattice gauge theory. In the dual picture, we regard the links with $^x = -1$ as being occupied by a string, and the links with $^x = +1$ as being unoccupied. The constraint term then requires the strings to be closed - as shown on the right.
• Figure 3: Typical string-net configurations in the dual formulation of (a) $Z_2$, (b) $U(1)$, and (c) $SU(2)$ gauge theory. In the case of (a) $Z_2$ gauge theory, the string-net configurations consist of closed (non-intersecting) loops. In (b) $U(1)$ gauge theory, the string-nets are oriented graphs with edges labeled by integers. The string-nets obey the branching rules $E_1 + E_2 + E_3 = 0$ for any three edges meeting at a point. In the case of (c) $SU(2)$ gauge theory, the string-nets consist of (unoriented) graphs with edges labeled by half-integers $1/2,1,3/2,...$. The branching rules are given by the triangle inequality: $\{E_1,E_2,E_3\}$ are allowed to meet at a point if and only if $E_1 \leq E_2 + E_3$, $E_2 \leq E_3 + E_1$, $E_3 \leq E_1 + E_2$, and $E_1+E_2+E_3$ is an integer.
• Figure 4: The orientation convention for the branching rules.
• Figure 5: $i$ and $i^*$ label strings with opposite orientations.