The moduli space of gapped Hamiltonians that are in the same topological phase is an intrinsic object that is associated with the topological order. The topology of these moduli spaces has been used recently in the construction of Floquet codes. We propose a systematical program to study the topology of these moduli spaces. In particular, we use effective field theory to study the cohomology classes of these spaces, which includes and generalizes the Berry phase. We discuss several applications for studying phase transitions. We show that a nontrivial family of gapped systems with the same topological order can protect isolated phase transitions in the phase diagram, and we argue that the phase transitions are characterized by screening of topological defects. We argue that the family of gapped systems obeys bulk-boundary correspondence. We show that a family of gapped systems in the bulk with the same topological order can rule out a family of gapped systems on the boundary with the topological order given by the topological boundary condition, constraining phase transitions on the boundary.
A genus is a pair of an anyon model with a non-negative rational number c such that . The word genus comes from the case of abelian anyon models and lattice conformal field theories. In this case, c is the chiral central charge of the lattice CFT, and the abelian anyon model of the lattice CFT is given by the genus of the lattice.
More generally, one can vary the parameter over an auxiliary bulk such that there is no bulk dependence; this produces the integer class Berry phase,15,17 which is the analog of the Wess–Zumino term in the nonlinear sigma model.
By Hamiltonian schema, we mean a well-defined procedure to write down Hamiltonians from some celluations of a space manifold, potentially with some extra background information such as orientation, branching, and framing.
The equalities hold under suitable choices of topology, using and BG = K(G, 1) for G = U(1) with the discrete topology.
More generally, we can couple the system to particles in gauge theory that can have non-trivial statistics.
For given n1, ω2 there can be no solution for n2. For instance, if we take M = S1 × S2, and n1 is the integral volume form on S1 modulo 2, and ω2 is the integral volume form on S2 modulo 2, then n1 ∪ ω2 + c−ω2 ∪1ω2 is the integral volume form on S1 × S2 modulo 2, which is closed but not exact in the cohomology with the coefficient, and thus the solution for n2 does not exist.
Here it is an untwisted quantum double, corresponding to “minimally” gauging the symmetry without stacking extra symmetry protected topological phases.
One can also compute the higher Berry curvature from the formula in Ref. 15, which we will leave for future work.
It would also be interesting if the bulk-boundary correspondence could rule out isolated gapless theories without relevant deformations.
For N = 0 mod 4, the center of Spin(N) has three subgroups, and the three quotients for these three subgroups give SO(N), Sc(N) and Ss(N).