Topological order (two macroscopic definitions/characterizations)

September 9, 2020

For a long time, people thought Landau symmetry breaking theory can describe all phase transitions and all phases of matter. When encounter a new phase of matter, people first first try to identify the symmetry breaking pattern and the associated order parameter.

In a study of High Tc superconductor, a spin liquid was discovered [1]. Influenced by Landau symmetry breaking theory, we tried to identify which symmetry is broken, and found that time-reversal and parity symmetries are broken [2]. So the new spin liquid was named chiral spin liquid. But we found several seemingly different chiral spin liquids with exactly the same symmetry breaking pattern and exactly the same order parameter. This implies that symmetry breaking is not enough to fully characterize those chiral spin liquids.

In order to distinguish them, those chiral spin liquids were put on closed spaces with different topologies, such as sphere, torus, etc. This led to the discovery of topology-dependent ground state degeneracies [3]. Such ground state degeneracies can be interpreted as quantum numbers (similar to order parameters) or topological invariants that partially characterize the new order (macroscopically).

Here a macroscopic characterization/definition is a characterization/definition using macroscopic measureable quantities (also known as topological invariants in mathemtics). For example, we can use zero conductance to partially characterize the notion of superconductivity. Here we use ground state degeneracies to partially characterize/define topological order.

A few months earlier, topological quantum field theory (TQFT) was discovered [4], which revealed a connection between 2+1D Chern-Simons theory and 1+1D conformal field theory (CFT). Since the low energy effective theories of chiral spin liquids are a Chern-Simons theories, the new order beyond symmetry breaking description was named topological order [5]. In general, topological order describes gapped liquid phases [6] of matter which may not have any symmetry.

Soon after, a full macroscopic characterization of topological order was proposed via mapping class group representations [5] (which were phrased as non-Abelian geometric phases of the degenerate ground states, i.e. the homology on the moduli spaces). This full characterization can be viewed as a macroscopic definition of topological order in terms of ground states, which actually work in all the dimensions.

Now we know that topological order can also be partially characterized (macroscopically) by fractional statistics of quasi-particles [7] [8]. But at that time, the fractional statistics was regarded as coming from the dynamics of excitations, such as charge-flux binding, rather than from a ground-state order beyond symmetry breaking.

At that time, there was a concern that the ground state degeneracies are finite-size effects which do not reflect the intrinsic order that characterize a phase of matter. In fact the same fractional quantum Hall state was put on sphere and torus, and different ground state degeneracies were found [9] [10] [11]. The difference was attributed to the different boundary conditions of finite systems not related to intrinsic order of matter [12]. To address this concern, it was shown that the ground state degeneracies are robust against any local perturbations [13], that may break any symmetry. Thus, the ground state degeneracies are "topological invariants" that can indeed characterize an intrinsic order and a phase of matter. The close connection between the fractional statistics and the ground state degeneracy of topological order was also uncovered.

One may wonder whether the experimentally measured quantized Hall conductance characterizes the topological order. The answer is No and Yes. No because topological order does not need any symmetry. Without the U(1) symmetry (the conservation of electron number), the Hall conductance is not even well defined. But without the U(1) symmetry, the ground state degeneracies are still the same and topological order can still be defined. Yes because topological order can coexist with symmetry, which is called symmetry enriched topological order [14]. Quantum Hall states correspond to symmetry enriched topological orders and quantized Hall conductance characterizes such U(1)-symmetry enriched topological orders. In fact, a close connection [15] between quantized Hall conductance and ground state degeneracy was discovered in 1985 for quantum Hall states. We note that two different topological orders can have the same quantized Hall conductance, and two different quantized Hall conductances can correspond to the same topological order (but different symmetry enriched topological orders).

The gauge charge in non-Abelain Chern-Simons theory, forming a higher dimensional representation [16] of braid group [17], has non-Abelian statistics [18]. The discrete non-Abelian gauge theory also has charge-flux bound state with non-Abelian statistics [19]. In early 1990's, the non-Abelian topological order in electron and quantum spin systems was discovered [20]  [21], which realize non-Abelain Chern-Simons theory as low energy effective theory and supports non-Abelian statistics. The edge state of those topologically ordered states are described by CFT's [22], realizing the connection between TQFT and CFT. In particular, chiral Majorana fermion edge state was discovered for one of the constructed non-Abelian topological order [23], which implies the existance of non-Abelian statistics in the bulk. Sign of such chiral Majorana fermion edge state was observed in a recent experiment [24].

In early 2000's, it was realized [25] that non-Abelian statistics of a full set of topological excitations can be described by modular tensor category [26], in a study of topological quantum computation. This leads to the second way to macroscopically characterize 2+1D topological order, in terms of the full set of topological excitations [27] [28] (rather than in terms of the full set of degenerate ground states [5]). Physically, it is harder to obtain the full set of topological excitations than the full set of ground states. So the first way is more convenient. The second way is also not totally complete since it misses invertible topological orders which have only trivial topological excitations. But the second way is more developed mathematically.

To use topological excitations (the second way) to characterize higher dimensional topological orders (up to invertible topological orders), one needs to use braided fusion higher categories [29]. In recent years, a lot of progresses were made in this direction [30] [31], such as classification of 3+1D topological orders [32] [33] [34].

Two kinds fo topology