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## "A class of P, T-invariant topological phases of interacting electrons", Michael Freedman et al., 2004

*Reviewed March 22, 2024*

*Citation:* Freedman, Michael, et al. "A class of P, T-invariant topological phases of interacting electrons." Annals of Physics 310.2 (2004): 428-492.

*Web:* https://arxiv.org/abs/cond-mat/0307511

*Tags:* Foundational, Temperley-Lieb, Abelian-anyons, TQFT, Quantum-groups

This paper introduces a family of topological phases of matter
corresponding to doubled SU(2)_k quantum group models. This
is done from the Chern-Simons theory. The authors use the Chern-Simons
theory QFT formalism to derive a loop algebra for excitations, which
gives a graphical calculus for determining braiding and fusion rules.
The graphical language for describing doubled SU(2)_k is shown to be
exactly the same as the graphical language of the k-strand Temperley-Lieb
algebra.

One of the big takeaways from this paper is the loop algebra
method of determining the properties
of quasiparticles. It has since become a staple of the field,
and is a standard way of thinking about topological order.
It emphasizes that standard CFT methods like Chern-Simons
theory can be connected anyon diagrammatics in special
cases without going through general machinery such as
Turaev-Viro TQFT.

One interesting feature of this paper
is that despite having strong control
of the properties of these doubled phases,
the authors are not able to construct explicit
lattice Hamiltonians to realize the phases.
Deriving such a Hamiltonian had to wait until

> Freedman, Michael, Chetan Nayak, and Kirill Shtengel. "Non-Abelian topological phases in an extended Hubbard model." arXiv preprint cond-mat/0309120 (2003).

Nowadays, we would simply view the
Hamiltonian as a special case of the Levin-Wen model.

This paper is amazingly physically minded and
physically grounded. It touches on
a large number of very interesting
topics. At times this can
make it hard to read, but it is
very much worth it.