Particle-hole symmetry for composite fermions: An emergent symmetry in the fractional quantum Hall effect

The particle-hole (PH) symmetry of {\em electrons} is an exact symmetry of the electronic Hamiltonian confined to a specific Landau level, and its interplay with the formation of composite fermions has attracted much attention of late. This article investigates an emergent symmetry in the fractional quantum Hall effect, namely the PH symmetry of {\em composite fermions}, which relates states at composite fermion filling factors $\nu^*=n+\bar{\nu}$ and $\nu^*=n+1-\bar{\nu}$, where the integer $n$ is the $\Lambda$ level index and $0\leq \bar{\nu}\leq 1$. Detailed calculations using the microscopic theory of composite fermions demonstrate that for low lying $\Lambda$ levels (small $n$): (i) the 2-body interaction between composite-fermion particles is very similar, apart from a constant additive term and an overall scale factor, to that between composite-fermion holes in the same $\Lambda$ level; and (ii) the 3-body interaction for composite fermions is an order of magnitude smaller than the 2-body interaction. Taken together, these results imply an approximate PH symmetry for composite fermions in low $\Lambda$ levels, which is also supported by exact diagonalization studies and available experiments. This symmetry, which relates states at electron filling factors $\nu={n+\bar{\nu}\over 2(n+\bar{\nu})\pm 1}$ and $\nu={n+1-\bar{\nu}\over 2(n+1-\bar{\nu})\pm 1}$, is not present in the original Hamiltonian and owes its existence entirely to the formation of composite fermions. With increasing $\Lambda$ level index, the 2-body and 3-body pseudopotentials become comparable, but at the same time they both diminish in magnitude, indicating that the interaction between composite fermions becomes weak as we approach $\nu=1/2$.Comment: 9 pages, 3 figures, 2 table

Current-induced gap opening in interacting topological insulator surfaces

Two-dimensional topological insulators (TIs) host gapless helical edge states that are predicted to support a quantized two-terminal conductance. Quantization is protected by time-reversal symmetry, which forbids elastic backscattering. Paradoxically, the current-carrying state itself breaks the time-reversal symmetry that protects it. Here we show that the combination of electron-electron interactions and momentum-dependent spin polarization in helical edge states gives rise to feedback through which an applied current opens a gap in the edge state dispersion, thereby breaking the protection against elastic backscattering. Current-induced gap opening is manifested via a nonlinear contribution to the system's $I-V$ characteristic, which persists down to zero temperature. We discuss prospects for realizations in recently discovered large bulk band gap TIs, and an analogous current-induced gap opening mechanism for the surface states of three-dimensional TIs.Comment: 6 pages, 2 figures, published versio

Luttinger theorem for the strongly correlated Fermi liquid of composite fermions

While an ordinary Fermi sea is perturbatively robust to interactions, the paradigmatic composite-fermion (CF) Fermi sea arises as a non-perturbative consequence of emergent gauge fields in a system where there was no Fermi sea to begin with. A mean-field picture suggests two Fermi seas, of composite fermions made from electrons or holes in the lowest Landau level, which occupy different areas away from half filling and thus appear to represent distinct states. We show that in the microscopic theory of composite fermions, which satisfies particle-hole symmetry in the lowest Landau level to an excellent degree, the Fermi wave vectors at filling factors $\nu$ and $1-\nu$ are the same, and are generally consistent with the experimental findings of Kamburov {\em et al.} [Phys. Rev. Lett. {\bf 113}, 196801 (2014)]. Our calculations suggest that the area of the CF Fermi sea may slightly violate the Luttinger area rule.Comment: 21 pages, 17 figures including supplemental material, published versio

State Counting for Excited Bands of the Fractional Quantum Hall Effect: Exclusion Rules for Bound Excitons

Exact diagonalization studies have revealed that the energy spectrum of interacting electrons in the lowest Landau level splits, non-perturbatively, into bands, which is responsible for the fascinating phenomenology of this system. The theory of nearly free composite fermions has been shown to be valid for the lowest band, and thus to capture the low temperature physics, but it over-predicts the number of states for the excited bands. We explain the state counting of higher bands in terms of composite fermions with an infinitely strong short range interaction between an excited composite-fermion particle and the hole it leaves behind. This interaction, the form of which we derive from the microscopic composite fermion theory, eliminates configurations containing certain tightly bound composite-fermion excitons. With this modification, the composite-fermion theory reproduces,for all well-defined excited bands seen in exact diagonalization studies, an exact counting for $\nu>1/3$, and an almost exact counting for $\nu\leq 1/3$. The resulting insight clarifies that the corrections to the nearly free composite fermion theory are not thermodynamically significant at sufficiently low temperatures, thus providing a microscopic explanation for why it has proved successful for the analysis of the various properties of the composite-fermion Fermi sea.Comment: 10 pages, 6 figure

Moving boundary and photoelastic coupling in GaAs optomechanical resonators

Chip-based cavity optomechanical systems are being considered for applications in sensing, metrology, and quantum information science. Critical to their development is an understanding of how the optical and mechanical modes interact, quantified by the coupling rate $g_{0}$. Here, we develop GaAs optomechanical resonators and investigate the moving dielectric boundary and photoelastic contributions to $g_{0}$. First, we consider coupling between the fundamental radial breathing mechanical mode and a 1550 nm band optical whispering gallery mode in microdisks. For decreasing disk radius from $R=5$ $\mu$m to $R=1$ $\mu$m, simulations and measurements show that $g_{0}$ changes from being dominated by the moving boundary contribution to having an equal photoelastic contribution. Next, we design and demonstrate nanobeam optomechanical crystals in which a $2.5$ GHz mechanical breathing mode couples to a 1550 nm optical mode predominantly through the photoelastic effect. We show a significant (30 $\%$) dependence of $g_{0}$ on the device's in-plane orientation, resulting from the difference in GaAs photoelastic coefficients along different crystalline axes, with fabricated devices exhibiting $g_{\text{0}}/2\pi$ as high as 1.1 MHz for orientation along the [110] axis. GaAs nanobeam optomechanical crystals are a promising system which can combine the demonstrated large optomechanical coupling strength with additional functionality, such as piezoelectric actuation and incorporation of optical gain media