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arXiv:2309.17003 · Physical review. B./Physical review. B

Energy minimization of paired composite fermion wave functions in the spherical geometry

Greg J. Henderson, Gunnar Möller, Steven H. Simon

Published Dec 11, 2023• 5 citations
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Abstract

We perform the energy minimization of the paired composite fermion (CF) wave functions, proposed by M\"oller and Simon (MS) [Phys. Rev. B 77, 075319 (2008)] and extended by Yutushui and Mross (YM) [Phys. Rev. B 102, 195153 (2020)], where the energy is minimized by varying the CF pairing function, in the case of an approximate model of the Coulomb interaction in the second Landau level for pairing channels $\ensuremath{\ell}=\ensuremath{-}1,3,1$, which are expected to be in the Pfaffian, anti-Pfaffian, and particle-hole symmetric (PH) Pfaffian phases, respectively. It is found that the energy of the $\ensuremath{\ell}=\ensuremath{-}1$ MS wave function can be reduced substantially below that of the Moore-Read wave function at small system sizes; however, in the $\ensuremath{\ell}=3$ case the energy cannot be reduced much below that of the YM trial wave function. Nonetheless, both our optimized and unoptimized wave functions with $\ensuremath{\ell}=\ensuremath{-}1,3$ extrapolate to roughly the same energy per particle in the thermodynamic limit. For the $\ensuremath{\ell}=1$ case, the optimization makes no qualitative difference and these PH-Pfaffian wave functions are still energetically unfavorable. The effective CF pairing is analyzed in the resulting wave functions, where the effective pairing for the $\ensuremath{\ell}=\ensuremath{-}1,3$ channels is found to be well approximated by a weak-pairing BCS ansatz and the $\ensuremath{\ell}=1$ wave functions show no sign of emergent CF pairing.

DOI
10.1103/physrevb.108.245128

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We perform the energy minimization of the paired composite fermion (CF) wave functions, proposed by M"oller and Simon (MS) [Phys. Rev. B 77, 075319 (2008)] and extended by Yutushui and Mross (YM) [Phys. Rev. B 102, 195153 (2020)], where the energy is minimized by varying the CF pairing function, in the case of an approximate model of the Coulomb interaction in the second Landau level for pairing channels \ensuremathℓ=\ensuremath−1,3,1\ensuremath{\ell}=\ensuremath{-}1,3,1\ensuremathℓ=\ensuremath−1,3,1, which are expected to be in the Pfaffian, anti-Pfaffian, and particle-hole symmetric (PH) Pfaffian phases, respectively. It is found that the energy of the \ensuremathℓ=\ensuremath−1\ensuremath{\ell}=\ensuremath{-}1\ensuremathℓ=\ensuremath−1 MS wave function can be reduced substantially below that of the Moore-Read wave function at small system sizes; however, in the \ensuremathℓ=3\ensuremath{\ell}=3\ensuremathℓ=3 case the energy cannot be reduced much below that of the YM trial wave function. Nonetheless, both our optimized and unoptimized wave functions with \ensuremathℓ=\ensuremath−1,3\ensuremath{\ell}=\ensuremath{-}1,3\ensuremathℓ=\ensuremath−1,3 extrapolate to roughly the same energy per particle in the thermodynamic limit. For the \ensuremathℓ=1\ensuremath{\ell}=1\ensuremathℓ=1 case, the optimization makes no qualitative difference and these PH-Pfaffian wave functions are still energetically unfavorable. The effective CF pairing is analyzed in the resulting wave functions, where the effective pairing for the \ensuremathℓ=\ensuremath−1,3\ensuremath{\ell}=\ensuremath{-}1,3\ensuremathℓ=\ensuremath−1,3 channels is found to be well approximated by a weak-pairing BCS ansatz and the \ensuremathℓ=1\ensuremath{\ell}=1\ensuremathℓ=1 wave functions show no sign of emergent CF pairing.
In a recent proposal of the half-filled Landau level, the composite fermions are taken to be Dirac particles and particle-hole symmetric. Cooper pairing of these composite fermions in different angular momentum channels, $\ensuremath{\ell}$, can give rise to different kinds of Pfaffian states. In addition to the well-known Moore-Read Pfaffian and anti-Pfaffian states, a new putative particle-hole symmetric Pfaffian state, corresponding to the sss-wave pairing channel, was also proposed. However, the possible underlying pairing mechanism is not clear at all. In this work we provide a specific pairing mechanism for realizing some of these Pfaffian states. We show that there can be nonzero pairing in angular momentum channels $|\ensuremath{\ell}|\ensuremath{\ge}1$ depending on the magnitude of a coupling constant. There is a quantum phase transition from the Dirac composite Fermi-liquid state to Cooper pairing states in angular momentum channels $|\ensuremath{\ell}|\ensuremath{\ge}1$ as the coupling constant is tuned across its critical point value. Surprisingly the particle-hole symmetric $\ensuremath{\ell}=0$ channel pairing turns out to be impossible irrespective of the size of the coupling constant.
Recently, a Dirac (particle-hole symmetric) description of composite fermions in the half-filled Landau level (LL) was proposed [D. T. Son, Phys. Rev. X 5, 031027 (2015)], and we study its possible consequences on BCS (Cooper) pairing of composite fermions (CFs). One of the main consequences is the existence of anisotropic states in single-layer and bilayer systems, which was previously suggested in Jeong and Park [J. S. Jeong and K. Park, Phys. Rev. B 91, 195119 (2015)]. We argue that in the half-filled LL in the single-layer case the gapped states may sustain anisotropy, because isotropic pairings may coexist with anisotropic ones. Furthermore, anisotropic pairings with the addition of a particle-hole symmetry-breaking mass term may evolve into rotationally symmetric states, i.e., Pfaffian states of Halperin-Lee-Read (HLR) ordinary CFs. On the basis of the Dirac formalism, we argue that in the quantum Hall bilayer at total filling factor 1, with decreasing distance between the layers, weak pairing of ppp-wave paired CFs is gradually transformed from Dirac to ordinary, HLR-like, with a concomitant decrease in the CF number. Global characterization of low-energy spectra based on the Dirac CFs agrees well with previous calculations performed by exact diagonalization on a torus. Finally, we discuss features of the Dirac formalism when applied in this context.
We numerically assess model wave functions for the recently proposed particle-hole-symmetric Pfaffian (``PH-Pfaffian'') topological order, a phase consistent with the recently reported thermal Hall conductance [M. Banerjee et al., Nature 559, 205 (2018)] at the ever enigmatic $\ensuremath{\nu}=5/2$ quantum Hall plateau. We find that the most natural Moore-Read-inspired trial state for the PH-Pfaffian, when projected into the lowest Landau level, exhibits a remarkable numerical similarity on accessible system sizes with the corresponding (compressible) composite Fermi liquid. Consequently, this PH-Pfaffian trial state performs reasonably well energetically in the half-filled lowest Landau level, but is likely not a good starting point for understanding the $\ensuremath{\nu}=5/2$ ground state. Our results suggest that the PH-Pfaffian model wave function either encodes anomalously weak ppp-wave pairing of composite fermions or fails to represent a gapped, incompressible phase altogether.
Topological pairing of composite fermions has led to remarkable ideas, such as excitations obeying non-Abelian braid statistics and topological quantum computation. We construct a ppp-wave paired Bardeen-Cooper-Schrieffer (BCS) wave function for composite fermions in the torus geometry, which is a convenient geometry for formulating momentum space pairing as well as for revealing the underlying composite-fermion Fermi sea. Following the standard BCS approach, we minimize the Coulomb interaction energy at half filling in the lowest and the second Landau levels, which correspond to filling factors $\ensuremath{\nu}=1/2$ and $\ensuremath{\nu}=5/2$ in GaAs quantum wells, by optimizing two variational parameters that are analogous to the gap and the Debye cutoff energy of the BCS theory. Our results show no evidence for pairing at $\ensuremath{\nu}=1/2$ but a clear evidence for pairing at $\ensuremath{\nu}=5/2$. To a good approximation, the highest overlap between the exact Coulomb ground state at $\ensuremath{\nu}=5/2$ and the BCS state is obtained for parameters that minimize the energy of the latter, thereby providing support for the physics of composite-fermion pairing as the mechanism for the 5/25/25/2 fractional quantum Hall effect. We discuss the issue of modular covariance of the composite-fermion BCS wave function, and calculate its Hall viscosity and pair correlation function. By similar methods, we look for but do not find an instability to sss-wave pairing for a spin-singlet composite-fermion Fermi sea at half-filled lowest Landau level in a system where the Zeeman splitting has been set to zero.
We present two two-body Hamiltonians that approximate the exact particle-hole Pfaffian wave function with their ground states for all the system sizes where this wave function has been numerically constructed to date. The approximate wave functions have high overlap with the original and reproduce well the low-lying entanglement spectrum and structure factor. The approximate generating Hamiltonians are obtained by an optimization procedure where three to four pseudopotentials are varied in the neighbourhood of second Landau level Coulomb interaction or of a noninteracting model. They belong to a finite region in the variational space of Hamiltonians where each point approximately generates the particle-hole Pfaffian. We diagonalize the identified Hamiltonians for up to 20 electrons and find that for them the particle-hole Pfaffian shift appears energetically more favorable. The possibility to interpret the data in terms of composite fermions is discussed.