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Physical review. B./Physical review. B

Application of Fermi hypernetted-chain theory to spin-polarized higher-order fractional quantum Hall states

J. W. Kim, H. Kim, N. Kim

Published Apr 6, 2017• 0 citations
Publisher

Abstract

We apply Fermi hypernetted-chain theory to study the spin polarization of higher-order fractional quantum Hall (FQH) states at filling factors in between the primary FQH sequences, $\ensuremath{\nu}=p/({q}_{e}p\ifmmode\pm\else\textpm\fi{}1)$, where ${q}_{e}$ is an even integer and $p$ is a nonzero integer. The filling factors related to the higher-order FQH states include $\ensuremath{\nu}=3/8$, $4/11$, $5/13$, $5/17$, $4/13$, $6/17$, $7/11$, and so on. We use a model of strongly interacting fermions with different spin degrees of freedom to explain the states beyond primary FQH sequences. We calculate the correlation energy, the radial distribution function, as well as the static structure function associated with the Halperin wave function adopted for the mixture states of fermions with different spins. The results are comparable with those from the residual interaction between composite fermions.

DOI
10.1103/physrevb.95.155109

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We apply Fermi hypernetted-chain theory to study the spin polarization of higher-order fractional quantum Hall (FQH) states at filling factors in between the primary FQH sequences, \ensuremathν=p/(qep\ifmmode±\else\textpm\fi1)\ensuremath{\nu}=p/({q}_{e}p\ifmmode\pm\else\textpm\fi{}1)\ensuremathν=p/(qe​p\ifmmode±\else\textpm\fi1), where qe{q}_{e}qe​ is an even integer and ppp is a nonzero integer. The filling factors related to the higher-order FQH states include \ensuremathν=3/8\ensuremath{\nu}=3/8\ensuremathν=3/8, 4/114/114/11, 5/135/135/13, 5/175/175/17, 4/134/134/13, 6/176/176/17, 7/117/117/11, and so on. We use a model of strongly interacting fermions with different spin degrees of freedom to explain the states beyond primary FQH sequences. We calculate the correlation energy, the radial distribution function, as well as the static structure function associated with the Halperin wave function adopted for the mixture states of fermions with different spins. The results are comparable with those from the residual interaction between composite fermions.
The Fermi-hypernetted-chain (FHNC) theory and the effective hypernetted-chain method are applied to study the composite-fermion (CF) states of the fractional quantum Hall effect. Using this theory we compute, in the thermodynamic limit, the correlation energy, radial distribution function, and static structure factor for all unprojected CF wave functions. The unprojected excitation gaps for $\ensuremath{\nu}=1/3,1/5$ were obtained by adopting in the FHNC a scheme previously used to compute nuclear matter excitation spectra. The results obtained so far are consistent with Monte Carlo simulations and small-number exact diagonalizations.
The Fermi hypernetted-chain theory is applied to study the half-filled state of the fractional quantum Hall effect in the thermodynamic limit. We study in detail the radial distribution function, the correlation energy, and the quasiparticle-quasihole excitation spectrum of an unprojected Fermi wave function of the form ${\ensuremath{\psi}}_{\ensuremath{\nu}=1/2}^{\mathrm{Fermi}}={\ensuremath{\Pi}}_{j<k}^{N}{(z}_{j}\ensuremath{-}{z}_{k}{)}^{2}\mathrm{Det}{{\ensuremath{\varphi}}_{\stackrel{\ensuremath{\rightarrow}}{k}}(\stackrel{\ensuremath{\rightarrow}}{r})},$ a possible candidate to describe the half-filled state. Adopting a technique originating from nuclear physics, we compute the effective mass of the fermion excitations near the Fermi surface for this wave function. We find it to be exactly the bare mass of the electron, in accordance with the mean field approximation of not imposing the lowest Landau level constraint. Similar calculations were performed on other related wave functions, which, based on the composite fermion picture, describe the half-filled state of the electrons as a limit of infinite-filled composite fermion Landau levels.
We discuss the relationship between fractional quantum Hall (FQH) states at filling factor $\ensuremath{\nu}=p/(2p+1)$ and quantum spin chains. This series corresponds to the Jain series $\ensuremath{\nu}=p/(2mp+1)$ with m=1m=1m=1 where the composite fermion picture is realized. We show that the FQH states with toroidal boundary conditions beyond the thin-torus limit can be mapped to effective quantum spin S=1S=1S=1 chains with ppp spins in each unit cell. We calculate energy gaps and the correlation functions for both the FQH systems and the corresponding effective spin chains, using exact diagonalization and the infinite time-evolving block decimation (iTEBD) algorithm. We confirm that the mass gaps of these effective spin chains are decreased as ppp is increased which is similar to S=pS=pS=p integer Heisenberg chains. These results shed new light on a link between the hierarchy of FQH states and the Haldane conjecture for quantum spin chains.
The hypernetted-chain theory is applied to study hierarchical states in the fractional quantum Hall effect. It is noted that a class of wave functions introduced by Girvin [Phys. Rev. B 29, 6012 (1984)] and MacDonald, Aers, and Dharma-wardana [Phys. Rev. B 31, 5529 (1985)], based on charge-conjugation procedures, is of the extended shadow wave-function type. The correlation energy, pair distribution function, and static structure function have been calculated in the thermodynamic limit at various filling factors. The results obtained agree with those of previous calculations performed with a finite number of electrons.
The hypernetted-chain theory is applied to study the fractional quantum Hall effect with the Laughlin wave functions. A new method is proposed to include the effect of the elementary diagrams, which improves upon the commonly used modified hypernetted-chain approximation. The correlation energy, the pair distribution function, as well as the magnetoroton excitation spectrum have been computed within this method. The results obtained are in very good agreement with the available Monte Carlo estimates. The method is generalizable to treat other wave functions, like those corresponding to the hierarchical states or those of composite fermion type.
We introduce an exactly solvable fermion chain that describes a ν=1/3 fractional quantum Hall (FQH) state beyond the thin-torus limit. The ground state of our model is shown to be unique for each center-of-mass sector, and it has a matrix product representation that enables us to exactly calculate order parameters, correlation functions, and entanglement spectra. The ground state of our model shows striking similarities with the BCS wave functions and quantum spin-1 chains. Using the variational method with matrix product ansatz, we analytically calculate excitation gaps and vanishing of the compressibility expected in the FQH state. We also show that the above results can be related to a ν=1/2 bosonic FQH state.