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Physical Review B

Numerical evidence for non-Abelian quantum liquids in the lowest Landau level

Arkadiusz Wójs

Published Jan 27, 2009• 5 citations
Publisher

Abstract

Signatures of non-Abelian statistics are sought in correlated liquids of composite fermions (CFs) responsible for a ``second generation'' of fractional quantum Hall effect. The hierarchy stems from the zero-energy state of a model CF interaction by means of a flux attachment procedure converting this uniquely correlated state of ``first-generation'' CFs to a filled shell of second-generation ${\text{CF}}^{\ensuremath{\ast}}$'s. Quasiholes of this state do not obey Abelian statistics. The hierarchy is confirmed numerically, including known states at filling factors ${\ensuremath{\nu}}_{e}=4/11$ and 3/8 and a hypothetical state at ${\ensuremath{\nu}}_{e}=9/25$.

DOI
10.1103/physrevb.79.041304

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Signatures of non-Abelian statistics are sought in correlated liquids of composite fermions (CFs) responsible for a second generation'' of fractional quantum Hall effect. The hierarchy stems from the zero-energy state of a model CF interaction by means of a flux attachment procedure converting this uniquely correlated state of first-generation'' CFs to a filled shell of second-generation CF\ensuremath∗{\text{CF}}^{\ensuremath{\ast}}CF\ensuremath∗'s. Quasiholes of this state do not obey Abelian statistics. The hierarchy is confirmed numerically, including known states at filling factors \ensuremathνe=4/11{\ensuremath{\nu}}_{e}=4/11\ensuremathνe​=4/11 and 3/8 and a hypothetical state at \ensuremathνe=9/25{\ensuremath{\nu}}_{e}=9/25\ensuremathνe​=9/25.
We use exact numerical diagonalization in the search of fractional quantum Hall states with non-Abelian quasiparticle statistics. For the (most promising) states in a partially filled second Landau level, the search is narrowed to the range of filling factors 7/3 < v e < 8/3. In this range, the analysis of energy spectra and correlation functions, calculated including finite width and Landau level mixing, supports the prominent non-Abelian candidates at v e = 5/2 (paired Moore–Read "pfafian" state) and 12/5 (clustered Read–Rezayi "parafermion" state). Outside of this range, the noninteracting composite fermion model with four attached flux quanta is validated, yielding the family of quantum liquids with fractional, but Abelian statistics. The borderline v e = 7/3 state is shown to be adiabatically connected to the Laughlin liquid, but its short-range correlations are significantly different.
We study 'second generation' of fractional quantum Hall states which contain a partially filled composite fermion (CF) Landau level (LL). We explain the role of CF-CF interaction in the incompressibility of the underlying quantum electron liquid. In particular, using exact diagonalization on a Haldane sphere, we determine two- and three-body CF correlation functions for these liquids and show that the CFs form a paired state (instead of a Laughlin liquid) when filling v = 1/3 of their second LL. At v = 1/2 we show that the CFs group into even larger clusters rather than form a Moore—Read paired state. We also address the problem of spin polarization of interacting CFs and predict transition to the partially unpolarized ground state in sufficiently narrow quantum wells and/or weak magnetic fields.
Two- and three-body correlations of incompressible quantum liquids are studied numerically. Pairing of composite fermions (CFs) in the 1/3-filled second CF Landau level is found at ${\ensuremath{\nu}}_{e}=4∕11$. It is explained by reduced short-range repulsion due to ring-like single-particle charge distribution. Although Moore-Read state of CFs is unstable in the 1/2-filled second CF level, condensation of its quasiholes is a possible origin of incompressibility at ${\ensuremath{\nu}}_{e}=4∕11$. Electron pairing occurs at ${\ensuremath{\nu}}_{e}=7∕3$ and 13∕313∕313∕3, but with different pair-pair correlations. Signatures of triplets are found at higher fillings.
Two-and three-body correlation functions (number of pairs or triplets vs. relative angular momentum) of electrons or Laughlin quasielectrons (i.e., composite fermions in their first excited Landau level) are studied numerically in several fractional quantum Hall liquids.It is shown directly that the νe = 4/11 liquid (corresponding to a ν = 1/3 filling of composite fermions in their first excited Landau level) is a paired state of quasielectrons, hence interpreted as a condensate of "second-generation" quasiholes of Moore-Read ν = 1/2 state of composite fermions.
Using exact numerical diagonalization we have studied correlated many-electron ground states in a partially filled second Landau level.We consider filling fractions ν = 1/2 and 2/5, for which incompressible quantum liquids with non-Abelian anion statistics have been proposed.Our calculations include finite layer width, Landau level mixing and arbitrary deformation of the interaction pseudopotential.Computed energies, gaps, and correlation functions support the non-Abelian ground states at both ν = 1/2 ("Pfaffian") and ν = 2/5 ("parafermion" state).
The composite fermion (CF) theory gives both a phenomenological description for many fractional quantum Hall (FQH) states, as well as a microscopic construction for large scale numerical calculation of these topological phases. The fundamental postulate of mapping FQH states of electrons to integer quantum Hall (IQH) states of CFs, however, was not formally established. The Landau level (LL) projection needed for the microscopic calculations is in some sense uncontrolled and unpredictable. We rigorously derive the unitary relationship between electrons and the CFs, showing the latter naturally emerge from special interactions within a single LL, without resorting to any projection by hand. In this framework, all FQH states topologically equivalent to those described by the conventional CF theory (e.g., the Jain series) have exact model Hamiltonians that can be explicitly derived, and we can easily generalize to FQH states from interacting CFs. Our derivations reveal fundamental connections between the CF theory and the pseudopotential/Jack polynomial constructions, and argue that all Abelian CF states are physically equivalent to the IQH states, while a plethora of non-Abelian CF states can be systematically constructed and classified. We also discuss about implications to experiments and effective field theory descriptions based on the descriptions with CFs as elementary particles.