Solvable models for unitary and nonunitary topological phases

Abstract

We introduce a broad class of simple models for quantum Hall states based on the expansion of their parent Hamiltonians near the one-dimensional limit of "thin cylinders", i.e., when one dimension Ly of the Hall surface becomes comparable to the magnetic length ℓB. Formally, the models can be viewed as topological generalizations of the 1D Hubbard model with center-of-mass-preserving hopping of multiparticle clusters. In some cases, we show that the models can be exactly solved using elementary techniques, and yield simple wave functions for the ground states as well as the entire neutral excitation spectrum. We study a large class of Abelian and non-Abelian states in this limit, including the Read-Rezayi Z k series, as well as states deriving from non-unitary or irrational conformal field theories -the "Gaffnian", "Haffnian", Haldane-Rezayi, and the "permanent" state. We find that the thin-cylinder limit of unitary (rational) states is "classical": their effective Hamiltonians reduce to only Hartree-type terms, the ground states are trivial insulators, and excitation gaps result from simple electrostatic repulsion. In contrast, for states deriving from non-unitary or irrational conformal field theories, the thin-cylinder limit is found to be intrinsically quantum -it contains hopping terms that play an important role in the structure of the ground states and in the energetics of the low-lying neutral excitations.

Geometric Construction of Quantum Hall Clustering Hamiltonians

“…[39]. The fermionic Haffnian occurs at the filling fraction ν ¼ 1=3, and, up to COM translation, has the following root patterns [61] on the sphere: 110000110000… and 100100100100… The latter pattern is identical to that of the Laughlin state.…”

“…Its thin-torus limit was recently studied in Ref. [39]. The latter state, Φ 221 , has been addressed by Ardonne and Regnault [64], who computed some of its properties by identifying its root configuration on the sphere and deriving its "squeezing" properties.…”

“…Accessing the set of topologically degenerate ground states further allows us to extract the modular S; T matrices that encode all topological information about the quasiparticles [32,33]. Furthermore, parent Hamiltonians on a cylinder or torus can be used to derive solvable models for quantum Hall states when one of the spatial dimensions becomes comparable to the magnetic length [34][35][36][37][38][39]. It has been shown that such models can be used to construct "matrix-product state" representations for quantum Hall states, and in some cases, they can be used to study the physics of "nonunitary" states and classify their gapless excitations [39][40][41].…”

Fractional charge pumping of interacting bosons in one-dimensional superlattice

“…They prevent the occupancy patterns · · · 3 · · · , · · · 21 · · · , therefore the admissible patterns are just the three-fold root patterns of MR-like states 49 .…”

“…Most notable examples include Moore-Read (MR) states at ν = 1 (k = 2) and Read-Rezayi (RR) states at ν = 3/2 (k = 3) 45,46 . Remarkably, the corresponding topological nature is captured by the analysis of "root configuration" or generalized Pauli principle in "thin-torus limit": No more than k bosons in two consecutive orbitals [47][48][49] . By squeezing the geometry into the thin-torus limit, these quantum Hall states are adiabatically connected to a charge density wave characterized by the root configuration 22,23,50 .…”