Zoo of quantum-topological phases of matter

TL;DR

This work surveys quantum-topological phases of matter, connecting long-range entangled topological order with symmetry-protected and invertible phases. It elucidates how topological order transcends Landau symmetry breaking, using invariants from ground-state degeneracy, edge theories, and tensor-category formalisms (e.g., modular tensor categories) to classify 2+1D phases, including FQH states, string-net liquids, and SPT orders. The text also details concrete constructions—string-net models, CZX-based SPT states, and defects as probes—to illustrate how symmetry and topology jointly shape gapped phases, and discusses current progress toward a full 3+1D topological framework. The significance lies in providing a unifying language for highly entangled quantum matter and guiding future classifications and realizations of topological quantum materials and platforms for quantum computation.

Abstract

What are topological phases of matter? First, they are phases of matter at zero temperature. Second, they have a non-zero energy gap for the excitations above the ground state. Third, they are disordered liquids that seem have no feature. But those disordered liquids actually can have rich patterns of many-body entanglement representing new kinds of order. This paper will give a simple introduction and a brief survey of topological phases of matter. We will first discuss topological phases that have topological order (ie with long range entanglement). Then we will cover topological phases that have no topological order (ie with only short-range entanglement).

Figures (8)

• Figure 1: The strings in a spin-1/2 model. In the background of up-spins, the down-spins form closed strings.
• Figure 2: In string liquid, strings can move freely, including reconnecting the strings.
• Figure 3: Deformation of strings and two reconnection moves, plus an exchange of two ends of strings and a $360^\circ$ rotation of one of the end of string, change the configuration (a) back to itself. Note that from (a) to (b) we exchange the two ends of strings, and from (d) to (e) we rotate of one of the end of string by $360^\circ$. The combination of those moves do not generate any phase.
• Figure 4: On a torus, the closed string configurations can be divided into four sectors, depending on even or odd number of strings crossing the x- or y-axes.
• Figure 5: (a) A tensor network representation of the partition function $Z=\text{Tr}\ee^{-\tau H}$ of a 1+1D quantum system obtained from path integral. Each vertex is a rank-4 tensor $T_{abcd}$ where each leg corresponds to an index. The range of the index is the dimension of the tensor $T$. The partition function $Z$ is obtained as a product of all tensors, with the common indices on the edges linking two vertices summed over (which corresponds to the path integral). We can combine several tensors $T$ to form a new tensor $T'$ and obtain a new coarse-grained tensor network that produces the same partition function $Z$. After many coarse-graining iterations, we obtain a fixed-point tensor $T^\text{fix}$ that characterizes a quantum phase. (b) The fixed-point tensor of spin-1 Heisenberg chain has a corner-double-line structure. It gives rise to the fixed-point wave function of an ideal $SO(3)$-SPT state.