Topological quantum matter

Topological quantum matter has robust low-energy properties controlled by global topology rather than local order parameters.

An Abelian Chern--Simons theory with integer symmetric invertible matrix $K$ describes Abelian anyons whose braiding phases are determined by $K^{-1}$.

The K-matrix is the integral bilinear form multiplying the Abelian Chern--Simons gauge fields. Integer unimodular basis changes send $K$ to $WK{W}^T$ without changing the phase.

An Abelian K-matrix Chern--Simons theory has $|\det K|$ topologically distinct ground states on a torus.

An anyon is a two-dimensional quasiparticle whose exchange or braiding statistics can differ from bosonic and fermionic statistics.

Two particle types are mutual semions when taking either once around the other multiplies the state by $-1$.

An anyon condenses at a boundary when its string operator may terminate there without leaving an excitation, so the boundary can absorb or emit that anyon.

A topological superconductor is a gapped paired-fermion phase supporting protected boundary Majorana modes.

A quadratic fermion Hamiltonian is bilinear in creation and annihilation operators. In a Majorana basis it is determined, up to a constant, by a real antisymmetric matrix.

A Bogoliubov--de Gennes Hamiltonian acts on a Nambu particle-hole spinor and describes quadratic fermion hopping and pairing.

A Majorana fermion operator is self-adjoint and obeys $\{c_j,c_k\}=2{\delta }_{jk}$.

A Majorana zero mode is a spatially localized Majorana operator that commutes with a gapped quadratic Hamiltonian, exactly or up to exponentially small finite-size corrections.

The Kitaev chain is a one-dimensional spinless p-wave superconductor. Its topological phase has a Majorana zero mode at each end of an open chain.

Two local gapped Hamiltonians are topologically equivalent when they are connected by a continuous path of local Hamiltonians with a nonzero bulk gap, equivalently when their ground spaces are related by quasi-local evolution.

For a two-component gapped BdG Hamiltonian constrained to a plane, the winding number counts how many times its coefficient vector encircles the origin as crystal momentum traverses the Brillouin zone.

A symmetry-protected topological phase is short-range entangled but cannot be deformed to a product state without breaking a protecting symmetry or closing the gap.

An on-site symmetry of an injective MPS acts on virtual indices by matrices defined up to phase. Their projective cohomology class is the one-dimensional SPT invariant.

The one-dimensional cluster state is the common positive eigenstate of commuting stabilizers $Z_{j-1}{X}_j{Z}_{j+1}$ and realizes a nontrivial ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$ SPT phase.

Kramers--Wannier duality exchanges order variables with domain-wall or disorder variables. In multiple dimensions it naturally appears as a tensor-network map with a nontrivial symmetry-sector kernel.

A one-form symmetry is generated on closed codimension-one manifolds and acts on line operators rather than local point operators.