Topological states of matter have been an important area of study since the discovery of the quantum Hall effect in the 1980s, in which phases are distinguished not by which symmetries (like magnetization) are broken, as in Landau's theory, but by topological properties. Landau's theory is essentially an effective theory of the order parameter. One reason for the breakdown of the Landau paradigm is the need to understand phase transitions and phases of matter, which are classified by non-local parameters such as topological orders. We will discuss these in this section. The Landau theory was the standard language of phases and the criticality of transitions [senthil2006quantum]. But some gapped quantum phases cannot be characterized by any local order parameter and instead are described by long-range entanglement and topology.
Soon after the discovery of the fractional Hall effect (in (integral) quantum Hall effect (see here [klitzing1980new]), the Hall conductance is quantised in the sense that it is given by $\sigma_H = n \frac{e^2}{h}$ where $n\in \mathbb{Z}$, and if $n \in \mathbb{Q}$ then it is called the fractional quantum Hall effect. The quantized value of Hall conductance is independent of the material.) (FQHE) in 1982 [tsui1982two], several intriguing observations emerged that were not explained by Landau's symmetry-breaking theory. They were 1) Ground State Degeneracy and its dependence on the topology, and 2) there were observations about quasi-particles (an example which we will soon see as topological excitation in a quantum toric model) having fractional charges and fractional statistics. But these only happen in two-dimensional models [wu1984general]. There can even be non-Abelian statistics in these quasiparticles, also called anyons. Anyon's statistics reveal the deep structure of mathematics, centered on topological order.
Ground-state degeneracy (GSD) has been an important tool for studying quantum phases. Beyond the Landau paradigm, the GSD is determined by long-range entanglement, and the degeneracy changes as we change the topology (e.g., sphere, torus, and so on). This degeneracy is not due to the Hamiltonian symmetry [wen1990topological] but is protected by the gapped symmetry and locality. An intuitive picture of Landau phases and topological orders can be found in an excellent review by Wen [Wen:2012hm].
Similar to the classifications of Landau's phases, which can be described using some order parameters (like magnetization for ferromagnetic order, zero-vorticity for superfluids, and zero-resistivity for superconductors), the classification of topological orders can be completely described by
- Topological ground state degeneracies, which have been discussed above.
- Non-Abelian geometric phases of degenerate ground states.
The orthonormal ground state degeneracy is not sufficient, as two different topological orders can have the same topological degeneracy for a topology. We need some invariant (or quantum numbers) to distinguish topological orders.
When there is only a single ground state below the gap, it picks up a phase $e^{i \gamma}$ when it adiabatically transports around a closed loop. Hence, it picks up a $U(1)$ scalar phase. However, if we have n-fold degeneracy, then we have a matrix $n \times n$ which acts non-commutatively within points to generate a ground-state subspace. The matrix is calculated from a one-parameter space of ground-state Hamiltonian $H_\lambda, \lambda\in [0,1]$ given $H_0 = H_1$. Provided that the ground states are kept separated from the excited states, then the adiabatic transport of othornormal basis of states $\{ |u_a \rangle \}_{a=1}^{n}$ gives a non-Abelian Berry connection or Wilczek–Zee connection $$\mathcal{A}_{ab} = i \langle u(\lambda)| d u(\lambda) \rangle$$ and a closed cycle $C$ in parameter space gives the holonomy $$ U_C = \mathcal{P} \exp{i \oint \mathcal{A}}$$ where $\mathcal{P}$ denotes the path-ordering. The path-order exponential is written instead of an ordinary exponential since matrices at different points on the closed loop do not commute. This holonomy is also called the non-Abelian geometric phase.
(Parts of this article have been extracted (and briefly edited) from my writing in an upcoming Revering Musings review on Topological Orders with Vaibhav Kalvakota.)
Anveshanā: The July 2026 issue of Anveshanā is out at https://anveshanamagazine.github.io, with the following description: This issue brings together conversations with Rukmini Dey, Partha Sarathi Chakraborty, and K.K. Suresh Kumar, alongside articles exploring mathematical finance and the mathematics of knots.
