Precise Experimental Test of the Luttinger Theorem and Particle-Hole Symmetry for a Strongly Correlated Fermionic System

Editors' Suggestion

Precise Experimental Test of the Luttinger Theorem and Particle-Hole Symmetry for a Strongly Correlated Fermionic System

Md. Shafayat Hossain, M. A. Mueed, M. K. Ma, K. A. Villegas Rosales, Y. J. Chung, L. N. Pfeiffer, K. W. West, K. W. Baldwin, and M. Shayegan

Phys. Rev. Lett. 125, 046601 – Published 20 July 2020

Abstract

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 quasiparticles themselves emerge from the strong interaction, namely, for the Fermi sea of composite fermions (CFs). Via direct, geometric resonance measurements of the CFs’ Fermi wave vector 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 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.

Received 30 March 2020
Accepted 24 June 2020

DOI:https://doi.org/10.1103/PhysRevLett.125.046601

Physics Subject Headings (PhySH)

Ballistic transport Composite fermions Fermi surface Quantum Hall effect

Quantum wells Two-dimensional electron gas

Hall bar

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Md. Shafayat Hossain , M. A. Mueed, M. K. Ma, K. A. Villegas Rosales, Y. J. Chung, L. N. Pfeiffer, K. W. West, K. W. Baldwin, and M. Shayegan

Department of Electrical Engineering, Princeton University, Princeton, New Jersey 08544, USA

Article Text (Subscription Required)

Click to Expand

Supplemental Material (Subscription Required)

Click to Expand

References (Subscription Required)

Click to Expand

Issue

Vol. 125, Iss. 4 — 24 July 2020

Reuse & Permissions

Access Options

Author publication services for translation and copyediting assistance advertisement

Images

Figure 1
(a) Top panels: Electrons (at $B = 0$ ) and CFs (at $ν = 1 / 2$ ) in real space. Bottom panels: Fermi seas of electrons and CFs at $n = 3.20 \times 10^{10} {cm}^{- 2}$ , and their respective Fermi wave vectors ( $k_{F}$ and $k_{F}^{*}$ ), in reciprocal space. (b) The lowest LL at $ν = 1 / 2$ and its evolution away from $ν = 1 / 2$ at a fixed density ( $n = 3.20 \times 10^{10} {cm}^{- 2}$ ) and varying magnetic field. The shaded regions denote the occupation of the lowest LL by electrons (blue) and holes (yellow). Our experimental data show that, out of the cases (i) to (iii) as described in the text, the CF Fermi sea area is determined by the density of minority carriers [case (ii)], namely, by electrons ( $n_{\min} = n$ ) for $ν < 1 / 2$ and by holes [ $n_{\min} = n (1 - ν) / ν$ ] for $ν > 1 / 2$ , regardless of the interaction.
Reuse & Permissions
Figure 2
Overview of our GR technique and magnetotransport data. (a) Our experimental technique consists of patterning a one-dimensional superlattice (shown in blue) on the sample surface to induce a small, periodic density perturbation of period $a$ in the 2DES. A representative scanning electron micrograph shown on the right attests to the uniformity of the stripes. When the cyclotron orbit of the CFs becomes commensurate with $a$ , the $i = 1$ GR occurs. (b) Magnetoresistance traces over a wide range of 2DES densities $n$ , taken at $T = 0.30 K$ , plotted against $1 / ν$ , showing pronounced GR resistance minima on the flanks of $ν = 1 / 2$ (vertical arrows), even at very low $n$ . The values of $n$ (in units of $10^{10} {cm}^{- 2}$ ) are given for each trace. (c)–(d) Expanded view of CF GR features, plotted against $B^{*}$ and $ν$ . The observed GR minima positions exhibit clear asymmetry with respect to $ν = 1 / 2$ ( $B^{*} = 0$ ). Vertical (dash-dotted) blue, (solid) red, and (dashed) green lines mark the expected positions for the $i = 1$ GR for fully spin-polarized CFs according to the fixed density model, minority-carrier model, and Dirac theory, respectively; see text for a description of the models. The blue lines in (c) are exactly symmetric in their positions with respect to $B^{*} = 0$ . Also, the blue and red lines coincide for $ν < 1 / 2$ ( $B^{*} > 0$ ). The experimental data best match the predictions of the minority-carrier model (red vertical lines). The differences between the observed minima positions and the predictions of the fixed density model and Dirac theory are also clearly visible.
Reuse & Permissions
Figure 3
CF Fermi wave vectors determined from the measured GR minima plotted against density $n$ . The symbols represent experimental data from samples with modulation periods $a = 190 nm$ (solid squares), $a = 200 nm$ (open squares), and 225 nm (open triangles), respectively. Blue, red, and green curves represent the calculated $k_{F}^{*}$ based on $k_{F}^{*} = (4 π n)^{1 / 2}$ , $k_{F}^{*} = (4 π n_{\min})^{1 / 2}$ , and Dirac theory, respectively. For each model, the results of calculations are shown in different ranges of $a$ where the experimental data were taken. For a description of the multiple curves for the Dirac theory and minority-carrier model, see Ref. [38]. The experimental data match the minority-carrier expression (red curves) very well. The top axes give the LL mixing parameter $κ$ .
Reuse & Permissions
Figure 4
The filling-factor positions of the observed CF GR minima $ν_{GR}$ at different densities, plotted against $a / l_{B}$ . The red and green curves are the predictions of the minority-carrier model and Dirac theory, respectively (expressions are given in the inset).
Reuse & Permissions

Physical Review Letters