Effective Abelian theory from a non-Abelian topological order in the ν=2/5 fractional quantum Hall effect

Abstract

Topological phases of matter are distinguished by topological invariants, such as Chern numbers and topological spins, that quantize their response to electromagnetic currents and changes of ambient geometry. Intriguingly, in the ν = 2/5 fractional quantum Hall effect, prominent theoretical approaches -the composite fermion theory and conformal field theory -have constructed two distinct states, the Jain composite fermion (CF) state and the Gaffnian state, for which many of the topological indices coincide and even the microscopic ground state wave functions have high overlap with each other in system sizes accessible to numerics. At the same time, some aspects of these states are expected to be very different, e.g., their elementary excitations should have either Abelian (CF) or non-Abelian (Gaffnian) statistics. In this paper we investigate the close relationship between these two states by considering not only their ground states, but also the low-energy charged excitations. We show that the low-energy physics of both phases is spanned by the same type of quasielectrons of the neighbouring Laughlin phase. The main difference between the two states arises due to an implicit assumption of short-range interaction in the CF approach, which causes a large splitting of the variational energies of the Gaffnian excitations. We thus propose that the Jain phase emerges as an effective Abelian low-energy description of the Gaffnian phase when the Hamiltonian is dominated by two-body interactions of sufficiently short range. arXiv:1907.12572v1 [cond-mat.str-el]

Stability, phase transitions, and numerical breakdown of fractional Chern insulators in higher Chern bands of the Hofstadter model

“…Future work in this area may involve, for example, an analysis of the role of interaction range. There is currently a wealth of numerical evidence to suggest that Abelian Jain states favor short-range interactions [41,50,51,57], whereas exotic fractional quantum Hall states may be stabilized exclusively via long-range interactions [99,100]. It is important to establish where |C| > 1FCIs fall on this spectrum to facilitate reliable device engineering.…”

Gaffnian and Haffnian: Physical relevance of nonunitary conformal field theory for the incompressible fractional quantum Hall effect

“…One important question to ask is if the Gaffnian state and the Jain ground state at ν = 2 5 are topologically equivalent: that any topological indices computed from these two microscopic wave functions are identical [14]. If ĤG is gapped in the thermodynamic limit in the L = 0 sector, then the statement has to be true even if the gap closes in some other L sector.…”

“…One should note that even with the assumption of conformal invariance, the value of c = 2 is not yet supported by the microscopic CF theory. This is because the quasihole excitations from each CF level with the CF construction are not orthonormal with each other, and there are missing states after the projection into the LLL [14,42]. These missing states will effectively reduce the density of states at each momentum sector.…”

“…Indeed, we would expect gapless boundary states to develop at the interface of the two Hamiltonians (or at certain values of λ when level crossing occurs). This could the be underlying difference between some of the known distinct FQH phases [14], although in this case all topological properties of the highest-density state (i.e., the ground state) are equivalent for the two phases.…”

Microscopic theory for nematic fractional quantum Hall effect

“…We can now see that the FQHN phase at ν = 1/3 is related to the Haffnian phase, which also occurs at ν = 1/3 but with a different topological shift as compared to the Laughlin phase. WhileĤ haff is conjectured to be gapless from the conformal field theory perspective, a finite gap may open as λ decreases from 1 (in analogy to the gap opening away from the Gaffnian model Hamiltonian [41]), leading to an incompressible ground state with different topological properties (though the quasihole excitations may not be non-Abelian [42]). This interesting connection will be explored in future works.…”