A Class of $P,T$-Invariant Topological Phases of Interacting Electrons

TL;DR

The paper develops a comprehensive framework for parity- and time-reversal-invariant topological phases of interacting electrons in 2+1 dimensions by doubling chiral Chern-Simons theories, notably constructing ${SU(2)_k} imes\overline{SU(2)_k}$ models that host non-Abelian anyons. It provides a detailed, exact solution of the doubled Abelian and non-Abelian theories via Wilson-loop algebras, inner-product structures, and the Jones-Wenzl projector constraints, revealing a finite Hilbert-space structure governed by loop-based combinatorics and the modular functor. A deep connection to 2D conformal field theories via boundary WZW models and edge excitations is established, linking bulk topological data to boundary primary fields and fusion rules. The work further bridges to microscopic realizations through parton constructions, lattice-loop models, and a loop-gas plasma analogy, highlighting pathways toward lattice Hamiltonians and potential applications in topological quantum computation. Overall, it provides a unified, algebraic, and combinatorial route from microscopic mechanisms to robust, non-Abelian topological phases with clear implications for quantum information processing.

Abstract

We describe a class of parity- and time-reversal-invariant topological states of matter which can arise in correlated electron systems in 2+1-dimensions. These states are characterized by particle-like excitations exhibiting exotic braiding statistics. $P$ and $T$ invariance are maintained by a `doubling' of the low-energy degrees of freedom which occurs naturally without doubling the underlying microscopic degrees of freedom. The simplest examples have been the subject of considerable interest as proposed mechanisms for high-$T_c$ superconductivity. One is the `doubled' version (i.e. two opposite-chirality copies) of the U(1) chiral spin liquid. The second example corresponds to $Z_2$ gauge theory, which describes a scenario for spin-charge separation. Our main concern, with an eye towards applications to quantum computation, are richer models which support non-Abelian statistics. All of these models, richer or poorer, lie in a tightly-organized discrete family. The physical inference is that a material manifesting the $Z_2$ gauge theory or a doubled chiral spin liquid might be easily altered to one capable of universal quantum computation. These phases of matter have a field-theoretic description in terms of gauge theories which, in their infrared limits, are topological field theories. We motivate these gauge theories using a parton model or slave-fermion construction and show how they can be solved exactly. The structure of the resulting Hilbert spaces can be understood in purely combinatorial terms. The highly-constrained nature of this combinatorial construction, phrased in the language of the topology of curves on surfaces, lays the groundwork for a strategy for constructing microscopic lattice models which give rise to these phases.

Figures (22)

• Figure 1: The operation ${T_1}{T_2}{T^{-1}_1}{T^{-1}_2}$ results in a phase ${e^{2\pi i/m}}$ because it is equivalent to the braiding operation on the right.
• Figure 2: A surgery operation of this type changes the wavefunction by a factor of $-1$. The shaded area represents an arbitrary multi-curve which completes the $1$-manifold shown.
• Figure 3: When $\alpha$ and $\gamma$ intersect, $\alpha{\cup}_{ R} \gamma$ is defined as shown above.
• Figure 4: The second constraint, which assigns a $-1$ to the operation of cutting and rejoining of two parallel strands requires that a contractible loop change the wavefunction by $-1$.
• Figure 5: Curves which terminate at the quasiparticle (i.e. the inner boundary) or the outer boundary of the annulus must do so at preferred points. The darker lines represent the boundary and the quasiparticle at the origin, while the lighter line represents a curve.