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arXiv:1510.03555 · Physical Review B

Incompressible states of the interacting composite fermions in negative effective magnetic fields at ν=4/13,5/17 , and 3/10

Sutirtha Mukherjee, Sudhansu S. Mandal

Published Dec 7, 2015• 14 citations
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Abstract

By developing an algorithm for evaluating the basis states for the composite fermions with negative effective magnetic field, we perform the composite-fermion-diagonalization study for the fully spin-polarized fractional quantum Hall states at the filling factors $\ensuremath{\nu}=3/10$, 4/13, and 5/17 in the range $2/7<\ensuremath{\nu}<1/3$. These observed states correspond to a partially filled second effective Landau level, for the composite fermions carrying four vortices, with filling factor $\overline{\ensuremath{\nu}}=1/2$, 1/3, and 2/3, respectively, analogous to the previously studied states of composite fermions with two attached vortices in the range $1/3<\ensuremath{\nu}<2/5$. We show that the character of these states in the range $2/7<\ensuremath{\nu}<1/3$ replicates the same for the states in the range $1/3<\ensuremath{\nu}<2/5$ having identical $\overline{\ensuremath{\nu}}$: chiral $p$-wave pairing with an anti-Pfaffian correlation of composite fermions carrying six quantized vortices produces an incompressible state at $\ensuremath{\nu}=3/10$; an unconventional interaction between composite fermions, resulting from the suppression of fermion pairs with relative angular momentum three and producing the fractional quantum Hall effect of composite fermions in the second effective Landau level with $\overline{\ensuremath{\nu}}=1/3$ and its particle-hole conjugate filling factor 2/3, reproduces incompressible states at $4/13$ and $5/17$ filling factors. We further estimate the thermodynamic limit of the ground-state energies and calculate the lowest energy gap for the neutral collective excitations of these states.

DOI
10.1103/physrevb.92.235302

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By developing an algorithm for evaluating the basis states for the composite fermions with negative effective magnetic field, we perform the composite-fermion-diagonalization study for the fully spin-polarized fractional quantum Hall states at the filling factors \ensuremathν=3/10\ensuremath{\nu}=3/10\ensuremathν=3/10, 4/13, and 5/17 in the range 2/7<\ensuremathν<1/32/7<\ensuremath{\nu}<1/32/7<\ensuremathν<1/3. These observed states correspond to a partially filled second effective Landau level, for the composite fermions carrying four vortices, with filling factor \ensuremathν‾=1/2\overline{\ensuremath{\nu}}=1/2\ensuremathν​=1/2, 1/3, and 2/3, respectively, analogous to the previously studied states of composite fermions with two attached vortices in the range 1/3<\ensuremathν<2/51/3<\ensuremath{\nu}<2/51/3<\ensuremathν<2/5. We show that the character of these states in the range 2/7<\ensuremathν<1/32/7<\ensuremath{\nu}<1/32/7<\ensuremathν<1/3 replicates the same for the states in the range 1/3<\ensuremathν<2/51/3<\ensuremath{\nu}<2/51/3<\ensuremathν<2/5 having identical \ensuremathν‾\overline{\ensuremath{\nu}}\ensuremathν​: chiral ppp-wave pairing with an anti-Pfaffian correlation of composite fermions carrying six quantized vortices produces an incompressible state at \ensuremathν=3/10\ensuremath{\nu}=3/10\ensuremathν=3/10; an unconventional interaction between composite fermions, resulting from the suppression of fermion pairs with relative angular momentum three and producing the fractional quantum Hall effect of composite fermions in the second effective Landau level with \ensuremathν‾=1/3\overline{\ensuremath{\nu}}=1/3\ensuremathν​=1/3 and its particle-hole conjugate filling factor 2/3, reproduces incompressible states at 4/134/134/13 and 5/175/175/17 filling factors. We further estimate the thermodynamic limit of the ground-state energies and calculate the lowest energy gap for the neutral collective excitations of these states.
We compare quantum Hall systems at filling factors $\ensuremath{\nu}=2$ to $\ensuremath{\nu}=\frac{2}{3}$ and 25\frac{2}{5}52​, corresponding to the exact filling of two lowest electron or composite fermion (CF) Landau levels. The two fractional states are examples of CF liquids with spin dynamics. There is a close analogy between the ferromagnetic (spin polarization P=1P=1P=1) and paramagnetic (P=0)(P=0)(P=0) incompressible ground states that occur in all three systems in the limits of large and small Zeeman spin splitting. However, the excitation spectra are different. At $\ensuremath{\nu}=2$, we find spin domains at half-polarization (P=12)(P=\frac{1}{2})(P=21​), while antiferromagnetic order seems most favorable in the CF systems. The transition between P=0P=0P=0 and 1, as seen when, e.g., the magnetic field is tilted, is also studied by exact diagonalization in toroidal and spherical geometries. The essential role of an effective CF-CF interaction is discussed, and the experimentally observed incompressible half-polarized state is found in some models.
Exact diagonalization of a two-dimensional electron gas in a strong magnetic field in the disk geometry shows that there exists a filling-factor range in the second Landau level where the states significantly differ from those in the lowest Landau level. We show that the difference arises because the interaction between composite fermions is not negligible in higher Landau levels, as indicated by a substantial mixing between composite-fermion Landau-like levels, or $\ensuremath{\Lambda}$ levels. We find that the exact ground state is well reproduced by composite-fermion theory with $\ensuremath{\Lambda}$-level mixing incorporated at the lowest level of approximation. Using the same variational approach in the spherical geometry we estimate the excitation gap at filling 1∕31∕31∕3 in the second Landau level (which corresponds to 7∕37∕37∕3 of experiment).
There is increasing experimental evidence for fractional quantum Hall effect at filling factor $\ensuremath{\nu}=2+3/8$. Modeling it as a system of composite fermions, we study the problem of interacting composite fermions by a number of methods. In our variational study, we consider the Fermi sea, the Pfaffian paired state, and bubble and stripe phases of composite fermions, and find that the Fermi sea state is favored for a wide range of transverse thickness. However, when we incorporate interactions between composite fermions through composite-fermion diagonalization on systems with up to 25 composite fermions, we find that a gap opens at the Fermi level, suggesting that inter-composite fermion interaction can induce fractional quantum Hall effect at $\ensuremath{\nu}=2+3/8$. The resulting state is seen to be distinct from the Pfaffian wave function.
We investigate the feasibility of many candidate quantum Hall states for two-component bosons in the lowest Landau level. We identify interactions for which spin-singlet incompressible states occur at filling factors $\ensuremath{\nu}=2/3$, 4/54/54/5, and 4/34/34/3, and partially spin-polarized states at filling factors 3/43/43/4 and 3/23/23/2, where ``spin'' serves as a generic label for the two components. We study ground states, excitations, edge states, and entanglement spectrum for systems with up to 16 bosons and construct explicit trial wave functions to clarify the underlying physics. The composite fermion theory very accurately describes the ground states as well as excitations at $\ensuremath{\nu}=2/3$, 4/54/54/5, and 3/43/43/4, although it is less satisfactory for the $\ensuremath{\nu}=3/2$ state. For $\ensuremath{\nu}=4/3$ a ``non-Abelian spin-singlet'' state, which is the exact ground state of a three-body contact interaction, has been proposed to occur even for a two-body contact interaction; our trial wave functions are very accurate for the excitations of the three-body interaction, but they do not describe the excitations of the two-body interaction very well. Instead, we find that the $\ensuremath{\nu}=4/3$ state is more likely to be a spin-singlet state of reverse-flux-attached composite fermions at filling ${\ensuremath{\nu}}^{*}=4$. We also consider incompressible states at integral filling factors $\ensuremath{\nu}=1$ and 2. The incompressible state at $\ensuremath{\nu}=1$ is shown to be well described by the parton-based Jain spin-singlet wave function, and the incompressible state at $\ensuremath{\nu}=2$ as the spin-singlet state of reverse-flux-attached composite fermions at ${\ensuremath{\nu}}^{*}=2$, which provides an example of the bosonic integer topological phase.
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.
We propose a particle-hole symmetric theory of the Fermi-liquid ground state of a half-filled Landau level. This theory should be applicable for a Dirac fermion in the magnetic field at charge neutrality, as well as for the v = 1/2 quantum Hall ground state of nonrelativistic fermions in the limit of negligible inter-Landau-level mixing. We argue that when particle-hole symmetry is exact, the composite fermion is a massless Dirac fermion, characterized by a Berry phase of π around the Fermi circle. We write down a tentative effective field theory of such a fermion and discuss the discrete symmetries, in particular, C P. The Dirac composite fermions interact through a gauge, but non-Chern-Simons, interaction. The particle-hole conjugate pair of Jain-sequence states at filling factors n/(2n + 1) and (n + 1)/(2n + 1), which in the conventional composite fermion picture corresponds to integer quantum Hall states with different filling factors, n and n + 1, is now mapped to the same half-integer filling factor n + 1/2 of the Dirac composite fermion. The Pfaffian and anti-Pfaffian states are interpreted as d -wave Bardeen-Cooper-Schrieffer paired states of the Dirac fermion with orbital angular momentum of opposite signs, while s -wave pairing would give rise to a particle-hole symmetric non-Abelian gapped phase. When particle-hole symmetry is not exact, the Dirac fermion has a C P -breaking mass. The conventional fermionic Chern-Simons theory is shown to emerge in the nonrelativistic limit of the massive theory.