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arXiv:1304.7553 · Physical Review B

Quantum Hall effect of two-component bosons at fractional and integral fillings

Ying-Hai Wu, Jainendra K. Jain

Published Jun 24, 2013• 79 citations
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Abstract

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 $\ensuremath{\nu}=2/3$, $4/5$, and $4/3$, and partially spin-polarized states at filling factors $3/4$ and $3/2$, 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 $\ensuremath{\nu}=2/3$, $4/5$, and $3/4$, 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.

DOI
10.1103/physrevb.87.245123

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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 \ensuremathν=2/3\ensuremath{\nu}=2/3\ensuremathν=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 $\ensuremath{\nu}=2/3$, $4/5$, and $3/4$, 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 \ensuremathν=4/3\ensuremath{\nu}=4/3\ensuremathν=4/3 state is more likely to be a spin-singlet state of reverse-flux-attached composite fermions at filling \ensuremathν∗=4{\ensuremath{\nu}}^{*}=4\ensuremathν∗=4. We also consider incompressible states at integral filling factors \ensuremathν=1\ensuremath{\nu}=1\ensuremathν=1 and 2. The incompressible state at \ensuremathν=1\ensuremath{\nu}=1\ensuremathν=1 is shown to be well described by the parton-based Jain spin-singlet wave function, and the incompressible state at \ensuremathν=2\ensuremath{\nu}=2\ensuremathν=2 as the spin-singlet state of reverse-flux-attached composite fermions at \ensuremathν∗=2{\ensuremath{\nu}}^{*}=2\ensuremathν∗=2, which provides an example of the bosonic integer topological phase.
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.
The recent production of synthetic magnetic fields acting on electroneutral particles, such as atoms or photons, has boosted interest in the quantum Hall physics of bosons. Adding pseudospin 1/2 to the bosons greatly enriches the scenario, as it allows them to form an interacting integer quantum Hall (IQH) phase with no fermionic counterpart. Here we show that, for a small two-component Bose gas on a disk, the complete strongly correlated regime, extending from the integer phase at filling factor $\ensuremath{\nu}=2$ to the Halperin phase at filling factor $\ensuremath{\nu}=2/3$, is well described by composite fermionization of the bosons. Moreover we study the edge excitations of the IQH state, which, in agreement with expectations from topological field theory, are found to consist of forward-moving charge excitations and backward-moving spin excitations. Finally, we demonstrate how pair-correlation functions allow one to experimentally distinguish the IQH state from competing states, such as non-Abelian spin singlet (NASS) states.
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A system at filling factor 2/3 could be a candidate for a quantum Hall ferromagnet at integer filling factor of composite fermions. Using exact diagonalization with electrons on a torus we study the transition from the singlet ground state to the polarized ground state at this filling and look for signatures of quantum Hall ferromagnetism. Differences between the fractional and corresponding integer systems are emphasised. Most interestingly, we find around the transition a low excited half-polarized state which might become the ground state in the thermodynamical limit. We study its structure and compare it to the singlet and polarized ground states. A new interpretation of the singlet state is suggested and comparison of the filling factors 2/3 and 2/5 is presented. Adding magnetic inhomogeneities into the system we investigate the stability of all the three involved states and the tendency to build up domains like in conventional ferromagnets.
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Quantum Hall systems offer the most familiar setting where strong interparticle interactions combine with the topology of single-particle states to yield novel phenomena. Despite our mature understanding of these systems, an open challenge has been to develop a microscopic theory capturing both their universal and nonuniversal properties, when the Hamiltonian is restricted to the noncommutative space of the lowest Landau level. Here we develop such a theory for the Jain sequence of bosonic fractional quantum Hall states at fillings $\ensuremath{\nu}=\frac{p}{p+1}$. Building on a lowest Landau level description of a parent composite Fermi liquid at $\ensuremath{\nu}=1$, we describe how to dope the system to reach the Jain states. Upon doping, the composite fermions fill noncommutative generalizations of Landau levels, and the Jain states correspond to integer composite fermion filling. Using this approach, we obtain an approximate expression for the bosonic Jain sequence gaps with no reference to any long-wavelength approximation. Furthermore, we show that the universal properties, such as Hall conductivity, are encoded in an effective noncommutative Chern-Simons theory, which is obtained on integrating out the composite fermions. This theory has the same topological content as the familiar Abelian Chern-Simons theory on commutative space.