A fundamental concept in physics is the Fermi surface, the constant-energy surface in momentum space encompassing all the occupied quantum states at absolute zero temperature. In 1960, Luttinger postulated that the area enclosed by the Fermi surface
should remain unaffected even when electron-electron interaction is turned on, so long as the interaction does not cause a phase transition. Understanding what determines the Fermi surface size is a crucial and yet unsolved problem in strongly interacting systems such as high-$T_{c}$ superconductors. Here we present a precise test of the Luttinger theorem for a two-dimensional Fermi liquid system where the exotic quasi-particles themselves emerge from the strong interaction, namely for the Fermi sea of composite fermions (CFs). Via direct, geometric resonance measurements of the CFs Fermi wavevector down to very low electron densities, we show that the Luttinger theorem is obeyed over a significant range of interaction strengths, in the sense that the Fermi sea area is determined by the density of the textit{minority carriers} in the lowest Landau level. Our data also address the ongoing debates on whether or not CFs obey particle-hole symmetry, and if they are Dirac particles. We find that particle-hole symmetry is obeyed, but the measured Fermi sea area differs quantitatively from that predicted by the Dirac model for CFs.
Two-dimensional interacting electrons exposed to strong perpendicular magnetic fields generate emergent, exotic quasiparticles phenomenologically distinct from electrons. Specifically, electrons bind with an even number of flux quanta, and transform
into composite fermions (CFs). Besides providing an intuitive explanation for the fractional quantum Hall states, CFs also possess Fermi-liquid-like properties, including a well-defined Fermi sea, at and near even-denominator Landau level filling factors such as $ u=1/2$ or $1/4$. Here, we directly probe the Fermi sea of the rarely studied four-flux CFs near $ u=1/4$ via geometric resonance experiments. The data reveal some unique characteristics. Unlike in the case of two-flux CFs, the magnetic field positions of the geometric resonance resistance minima for $ u<1/4$ and $ u>1/4$ are symmetric with respect to the position of $ u=1/4$. However, when an in-plane magnetic field is applied, the minima positions become asymmetric, implying a mysterious asymmetry in the CF Fermi sea anisotropy for $ u<1/4$ and $ u>1/4$. This asymmetry, which is in stark contrast to the two-flux CFs, suggests that the four-flux CFs on the two sides of $ u=1/4$ have very different effective masses, possibly because of the proximity of the Wigner crystal formation at small $ u$.
The interplay between different orders is of fundamental importance in physics. The spontaneous, symmetry-breaking charge order, responsible for the stripe or the nematic phase, has been of great interest in many contexts where strong correlations ar
e present, such as high-temperature superconductivity and quantum Hall effect. In this article we show the unexpected result that in an interacting two-dimensional electron system, the robustness of the nematic phase, which represents an order in the charge degree of freedom, not only depends on the orbital index of the topmost, half-filled Landau level, but it is also strongly correlated with the magnetic order of the system. Intriguingly, when the system is fully magnetized, the nematic phase is particularly robust and persists to much higher temperatures compared to the nematic phases observed previously in quantum Hall systems. Our results give fundamental new insight into the role of magnetization in stabilizing the nematic phase, while also providing a new knob with which it can be effectively tuned.
In a number of widely-studied materials, such as Si, AlAs, Bi, graphene, MoS2, and many transition metal dichalcogenide monolayers, electrons acquire an additional, spin-like degree of freedom at the degenerate conduction band minima, also known as v
alleys. External symmetry breaking fields such as mechanical strain, or electric or magnetic fields, can tune the valley-polarization of these materials making them suitable candidates for valleytronics. Here we study a quantum well of AlAs, where the two-dimensional electrons reside in two energetically degenerate valleys. By fabricating a strain-inducing grating on the sample surface, we engineer a spatial modulation of the electron population in different valleys, i.e., a valley superlattice in the quantum well plane. Our results establish a novel manipulation technique of the valley degree of freedom, paving the way to realizing a valley-selective layered structure in multi-valley materials, with potential application in valleytronics.
The enigmatic even-denominator fractional quantum Hall state at Landau level filling factor $ u=5/2$ is arguably the most promising candidate for harboring Majorana quasi-particles with non-Abelian statistics and thus of potential use for topological
quantum computing. The theoretical description of the $ u=5/2$ state is generally believed to involve a topological p-wave pairing of fully spin-polarized composite fermions through their condensation into a non-Abelian Moore-Read Pfaffian state. There is, however, no direct and conclusive experimental evidence for the existence of composite fermions near $ u=5/2$ or for an underlying fully spin-polarized Fermi sea. Here, we report the observation of composite fermions very near $ u=5/2$ through geometric resonance measurements, and find that the measured Fermi wavevector provides direct demonstration of a Fermi sea with full spin polarization. This lends crucial credence to the model of $5/2$ fractional quantum Hall effect as a topological p-wave paired state of composite fermions.
We demonstrate tuning of the Fermi contour anisotropy of two-dimensional (2D) holes in a symmetric GaAs (001) quantum well via the application of in-plane strain. The ballistic transport of high-mobility hole carriers allows us to measure the Fermi w
avevector of 2D holes via commensurability oscillations as a function of strain. Our results show that a small amount of in-plane strain, on the order of $10^{-4}$, can induce significant Fermi wavevector anisotropy as large as 3.3, equivalent to a mass anisotropy of 11 in a parabolic band. Our method to tune the anisotropy textit{in situ} provides a platform to study the role of anisotropy on phenomena such as the fractional quantum Hall effect and composite fermions in interacting 2D systems.
The pairing of composite fermions (CFs), electron-flux quasi-particles, is commonly proposed to explain the even-denominator fractional quantum Hall state observed at $ u=5/2$ in the first excited ($N=1$) Landau level (LL) of a two-dimensional electr
on system (2DES). While well-established to exist in the lowest ($N=0$) LL, much is unknown about CFs in the $N=1$ LL. Here we carry out geometric resonance measurements to detect CFs at $ u=5/2$ by subjecting the 2DES to a one-dimensional density modulation. Our data, taken at a temperature of 0.3 K, reveal no geometric resonances for CFs in the $N=1$ LL. In stark contrast, we observe clear signatures of such resonances when $ u=5/2$ is placed in the $N=0$ LL of the anti-symmetric subband by varying the 2DES width. This finding implies that the CFs mean-free-path is significantly smaller in the $N=1$ LL compared to the $N=0$ LL. Our additional data as a function of in-plane magnetic field highlight the role of subband index and establish that CFs at $ u=5/2$ in the $N=0$ LL are more anisotropic in the symmetric subband than in the anti-symmetric subband.
There has been a surge of recent interest in the role of anisotropy in interaction-induced phenomena in two-dimensional (2D) charged carrier systems. A fundamental question is how an anisotropy in the energy-band structure of the carriers at zero mag
netic field affects the properties of the interacting particles at high fields, in particular of the composite fermions (CFs) and the fractional quantum Hall states (FQHSs). We demonstrate here tunable anisotropy for holes and hole-flux CFs confined to GaAs quantum wells, via applying textit{in situ} in-plane strain and measuring their Fermi wavevector anisotropy through commensurability oscillations. For strains on the order of $10^{-4}$ we observe significant deformations of the shapes of the Fermi contours for both holes and CFs. The measured Fermi contour anisotropy for CFs at high magnetic field ($alpha_mathrm{CF}$) is less than the anisotropy of their low-field hole (fermion) counterparts ($alpha_mathrm{F}$), and closely follows the relation: $alpha_mathrm{CF} = sqrt{alpha_mathrm{F}}$. The energy gap measured for the $ u = 2/3$ FQHS, on the other hand, is nearly unaffected by the Fermi contour anisotropy up to $alpha_mathrm{F} sim 3.3$, the highest anisotropy achieved in our experiments.
Via the application of parallel magnetic field, we induce a single-layer to bilayer transition in two-dimensional electron systems confined to wide GaAs quantum wells, and study the geometric resonance of composite fermions (CFs) with a periodic dens
ity modulation in our samples. The measurements reveal that CFs exist close to bilayer quantum Hall states, formed at Landau level filling factors $ u=1$ and 1/2. Near $ u=1$, the geometric resonance features are consistent with half the total electron density in the bilayer system, implying that CFs prefer to stay in separate layers and exhibit a two-component behavior. In contrast, close to $ u=1/2$, CFs appear single-layer-like (single-component) as their resonance features correspond to the total density.
Interacting two-dimensional electrons confined in a GaAs quantum well exhibit isotropic transport when the Fermi level resides in the first excited ($N=1$) Landau level. Adding an in-plane magnetic field ($B_{||}$) typically leads to an anisotropic,
stripe-like (nematic) phase of electrons with the stripes oriented perpendicular to the $B_{||}$ direction. Our experimental data reveal how a periodic density modulation, induced by a surface strain grating from strips of negative electron-beam resist, competes against the $B_{||}$-induced orientational order of the stripe phase. Even a minute ($<0.25%$) density modulation is sufficient to reorient the stripes along the direction of the surface grating.
