[Paper Review] Particle-Hole Symmetry and the Fractional Quantum Hall States at 5/2 Filling Factor

Jian Yang|arXiv (Cornell University)|Jan 13, 2017

Quantum and electron transport phenomena3 citations

TL;DR

This paper proposes a derivative operator $ D_m $ that generates new fractional quantum Hall wave functions by acting on Laughlin states, yielding states at $ \nu = 1/(m_L - m) $. It identifies $ \Psi_{3,5} $ and $ \Psi_{1,3} $ as candidates for the $ 5/2 $ filling factor, with $ \Psi_{1,3} $ exhibiting exact particle-hole symmetry and $ \Psi_{3,5} $ closely overlapping with the particle-hole conjugate of the Moore-Read Pfaffian state.

ABSTRACT

We propose a derivative operator formed as a function of derivatives of the electron coordinates. When the derivative operator is applied to the Laughlin wave function, two new wave functions in the lowest Landau level at filling factor 1/2 are generated. For systems of 4, 6, and 8 electrons in spherical geometry, it is shown that the first wave function has nearly unity overlap with the particle-hole conjugate of the Moore-Read Pfaffian wave function, therefore together with the Moore-Read Pfaffian state forms a particle-hole conjugate pair. The second wave function has essentially perfect particle-hole symmetry itself, with a positive parity when the number of electron pairs N/2 is an even integer and and a negative parity when N/2 is an odd integer. An equivalent form suggests the first wave function forms a f-wave pairing of composite fermions, and the second wave function forms a p-wave pairing. The corresponding Non-Abelian statistics quasiparticle wave functions are also proposed.

Motivation & Objective

• To resolve the particle-hole symmetry paradox in the $ 5/2 $ fractional quantum Hall effect, where the Moore-Read Pfaffian state lacks particle-hole symmetry but the Coulomb Hamiltonian is invariant under it.
• To construct a wave function that is intrinsically particle-hole symmetric and has high overlap with the exact incompressible ground state at $ 5/2 $ filling.
• To provide a microscopic description of non-Abelian quasiparticles and their statistics in the $ 5/2 $ state via a derivative-based operator formalism.
• To extend the applicability of the quasielectron space of Laughlin states beyond $ 2/5 $ filling to $ 1/2 $ filling, including $ 5/2 $.

Proposed method

• Define a derivative operator $ D_m = \mathrm{Pf}\left( \frac{1}{(\partial_{z_i} - \partial_{z_j})^m} \right) \prod_{i<j} (\partial_{z_i} - \partial_{z_j})^m $, acting on Laughlin wave functions $ \Phi_{m_L} $ to generate new states $ \Psi_{m,m_L} $.
• Apply the operator in spherical geometry, where the magnetic flux $ N_\phi $ relates to electron number $ N $ via $ N_\phi = (m_L - m)N + (2m - m_L) $, yielding $ N_\phi = 2N+1 $ for $ \Psi_{3,5} $ and $ N_\phi = 2N-1 $ for $ \Psi_{1,3} $.
• Use exact diagonalization in spherical geometry to compute overlaps between $ \Psi_{1,3} $ and its particle-hole conjugate $ \Psi_{1,3}^{PH} $, and to assess parity and overlap with the exact ground state.
• Derive equivalent forms of $ \Psi_{3,5} $ and $ \Psi_{1,3} $ via lowest Landau level projection, revealing $ f $-wave and $ p $-wave pairing of composite fermions, respectively.
• Construct non-Abelian quasihole and quasiparticle wave functions by modifying the Pfaffian structure with insertion of coordinates $ \xi_a $, generalizing the Moore-Read Pfaffian construction.
• Demonstrate that $ \Psi_{m,m_L} $ lies within the quasielectron space of the Laughlin state, extending its validity to $ 1/2 $ filling factor.

Experimental results

Research questions

• RQ1 Can a wave function be constructed that is intrinsically particle-hole symmetric and has high overlap with the exact $ 5/2 $ ground state?
• RQ2 Does the $ \Psi_{1,3} $ state exhibit exact particle-hole symmetry, and what is its parity behavior as a function of $ N/2 $?
• RQ3 How does the $ \Psi_{3,5} $ state relate to the Moore-Read Pfaffian and its particle-hole conjugate?
• RQ4 Can the derivative operator $ D_m $ generate states that describe $ f $-wave and $ p $-wave pairing of composite fermions?
• RQ5 What is the role of the quasielectron space of the Laughlin state in describing the $ 5/2 $ state, and does it extend beyond $ 2/5 $ filling?

Key findings

• The wave function $ \Psi_{1,3} $ has a parity of $ -1 $ when $ N/2 $ is odd and $ +1 $ when $ N/2 $ is even, and satisfies $ \Psi_{1,3} = (-1)^{N/2} \Psi_{1,3}^{PH} $, indicating exact particle-hole symmetry.
• For $ N = 6 $, the overlap between $ \Psi_{1,3} $ and its particle-hole conjugate $ \Psi_{1,3}^{PH} $ is $ 0.9991 $, and for $ N = 8 $, it is $ 0.9798 $, confirming high symmetry and consistency.
• The wave function $ \Psi_{3,5} $ has nearly unity overlap ($ \sim 0.999 $) with the particle-hole conjugate of the Moore-Read Pfaffian state in spherical geometry for $ N = 4, 6, 8 $.
• The equivalent form $ \Psi_{3,5} = P_{LLL} \mathrm{Pf}\left( \frac{1}{(u_i^*v_j^* - u_j^*v_i^*)^3} \right) \prod_{i<j} |u_i v_j - u_j v_i|^6 \Phi_2 $ reveals $ f $-wave pairing of composite fermions.
• The equivalent form $ \Psi_{1,3} = P_{LLL} \mathrm{Pf}\left( \frac{1}{u_i^*v_j^* - u_j^*v_i^*} \right) \prod_{i<j} |u_i v_j - u_j v_i|^2 \Phi_2 $ reveals $ p $-wave pairing of composite fermions.
• The non-Abelian quasihole wave function is constructed via $ \mathrm{Pf}\left( \frac{\prod_a (z_i - \xi_a) \prod_b (z_j - \xi_b) + (i \leftrightarrow j)}{(\partial_{z_i} - \partial_{z_j})^m} \right) $, with quasihole charge $ -1/(2(m_L - m)) $ and statistics $ 1/(2(m_L - m)) $.