Solvable lattice Hamiltonians with fractional Hall conductivity

Fractional Hall conductivity and spin-c structure in solvable lattice Hamiltonians

“…However, similar to what we have discussed in detail in [13], there are serious problems with such "prototype" Hamiltonian H:…”

“…In terms of JHEP02(2023)130 the Hamiltonian, the U a terms in the Hamiltonian (3.11) at links separated by r 0 anticommute rather than commute, ruining the "almost commuting" nice property of H (2) . (Following the idea of appendix F of [13], it is not hard to argue that such a Hamiltonian has a topologically trivial low energy sector, which is indeed expected for a theory whose supposed "emergent gauge field" turns out anomalous and hence ceases to be a gauge field.) Below we will explain how to suitably couple the odd k Hamiltonian to fermions in order to obtain the desired fermionic topological order.…”

“…In our previous work [13], using the intuition provided by the continuum double U(1) Chern-Simons theory (2.1), the untwisted theories with k = 0 have been constructed and solved, showing that controllably solvable lattice Hamiltonians can indeed be constructed for theories with Hall conductivity. (The lattice Hamiltonian has also been suggested in [28] based on different reasoning, although the parameters considered there are different from the solvable limit in [13].) Here we show the same can be completed for general values of k, including the particularly interesting odd values that are associated with fermionic theories.…”

“…the electromagnetic field A, must be presented as an ordinary U(1) on the lattice -we cannot "choose" it to take the Villainized form in our construction because it is a background data; to put in other words, the U(1) global symmetry is not a [U(1)/R] 2-group global symmetry. This difference might seem subtle, but it will, in the end, lead to the introduction of noncommuting terms in the lattice Hamiltonian [13,29], thereby circumventing the Kapustin-Fidkowski no-go theorem. As we proceed, we will see how this comes about in detail.…”

“…Our main result is that, for any bosonic or fermionic abelian topological order in (2+1)d that intrinsically admits a commuting projector Hamiltonian realization, even if such realization becomes prohibited after a U(1) enrichment that generates Hall conductivity, we can still construct a lattice Hamiltonian that can be controllably solved at low energies, and show the Hamiltonian realizes the desired topological order with the desired Hall conductivity. (In our previous work [13], we studied the simplest among the bosonic cases.) Moreover, for the fermionic cases, our construction leads to a simple and natural notion of spin-c structure on the lattice.…”