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arXiv:2406.12825 · Physical review. B./Physical review. B

Theory of Coulomb driven nematicity in a multivalley two-dimensional electron gas

Vladimir Calvera, Agnes Valenti, Sebastian D. Huber, Erez Berg, Steven Allan Kivelson

Published Apr 18, 2025• 7 citations
arXivHTMLPDF

Abstract

The properties of a two-dimensional electron gas (2DEG) in a semiconductor host with two valleys related by an underlying ${C}_{4}$ rotational symmetry are studied using Hartree-Fock and various other many-body approaches. A familiar artifact of the HF approach is a degeneracy between the valley-polarized---``Ising nematic''---and spin-polarized---ferromagnetic---phases, which is inconsistent with recent variational Monte Carlo (VMC) results. Correlation effects, computed either within the random phase approximation (RPA) or the T-matrix approximation, enhance the valley susceptibility relative to the spin susceptibility. Extrapolating the results to finite interaction strength, we find a direct first-order transition from a symmetry-unbroken state to a spin-unpolarized Ising nematic fluid with full valley polarization, in qualitative agreement with VMC. The RPA results are also reminiscent of experiments on the corresponding 2DEG in AlAs heterostructures.

DOI
10.1103/physrevb.111.155135

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The properties of a two-dimensional electron gas (2DEG) in a semiconductor host with two valleys related by an underlying C4{C}_{4}C4​ rotational symmetry are studied using Hartree-Fock and various other many-body approaches. A familiar artifact of the HF approach is a degeneracy between the valley-polarized---``Ising nematic''---and spin-polarized---ferromagnetic---phases, which is inconsistent with recent variational Monte Carlo (VMC) results. Correlation effects, computed either within the random phase approximation (RPA) or the T-matrix approximation, enhance the valley susceptibility relative to the spin susceptibility. Extrapolating the results to finite interaction strength, we find a direct first-order transition from a symmetry-unbroken state to a spin-unpolarized Ising nematic fluid with full valley polarization, in qualitative agreement with VMC. The RPA results are also reminiscent of experiments on the corresponding 2DEG in AlAs heterostructures.
The two-dimensional electron gas is of fundamental importance in quantum many-body physics. We study a minimal extension of this model with C_{4} (as opposed to full rotational) symmetry and an electronic dispersion with two valleys with anisotropic effective masses. Electrons in our model interact via Coulomb repulsion, screened by distant metallic gates. Using variational Monte Carlo simulations, we find a broad intermediate range of densities with a metallic valley-polarized, spin-unpolarized ground state. Our results are of direct relevance to the recently discovered "nematic" state in AlAs quantum wells. For the effective mass anisotropy relevant to this system, m_{x}/m_{y}≈5.2, we obtain a transition from an anisotropic metal to a valley-polarized metal at r_{s}≈12 (where r_{s} is the dimensionless Wigner-Seitz radius). At still lower densities, we find a (possibly metastable) valley and spin-polarized state with a reduced electronic anisotropy.
Starting from the quantum Monte Carlo (QMC) prediction for the ground-state energy of a clean two-dimensional one-valley (2D1V) electron gas, we estimate the energy correction due to scattering sources present in actual devices such as AlAs quantum wells and GaAs heterostructures. We find that the effect of uncorrelated disorder, in the lowest (second) order in perturbation theory, is to enhance the spin susceptibility leading to its eventual divergence. In the density region where the Born approximation is able to reproduce the experimental mobility, the prediction for the spin susceptibility yielded by perturbation theory is in very good agreement with the available experimental evidence.
We have studied the nematic state of the two dimensional electron gas (2DEG) at halffilled Landau level (LL). Our motivation comes from experiments in which anisotropic transport in the 2DEG under high magnetic field and at low temperature was observed. Based on a model of the nematic state proposed by Oganesyan, Fradkin and Kivelson, we investigate this state and compare it with other competing states proposed by other groups. First, we investigate at what LL the nematic state becomes energetically favorable as compared to the isotropic state. Our studies indicate that this occurs at the second excited LL. Moreover, we compare the energy of the nematic state with that of the stripe state obtained within the Hartree-Fock approximation (HF) and we conclude that, for the samples studied experimentally, the nematic state might be more stable. In our study we have used two different methods which both have advantages and disadvantages. The first method used is the Fermi-hyper-netted chain (FHNC) which provides results valid in the thermodynamic limit (infinite size system), however, it is accurate only for a low density system. The second method used is the Monte Carlo method (MC) which can be used on a finite-size system and, thus, the issue arises of how to extrapolate in the thermodynamic limit (finite-size effect). The results obtained from both methods are in good agreement and indicate that the nematic state might be a viable candidate to explain the experimental findings. In order to compare our results for the nematic state to those obtained for the stripe state, we needed to include the kinetic energy contribution beyond the familiar hw_c/2 term in the case of the nematic state which comes from the deformed geometry of the Fermi sea. For the stripe state we have carried out a HF calculation for a more realistic potential for a 2DEG which includes the effects of the finite confinement of the electron wave function along the z-direction.
Nematicity is a well-known property of liquid crystals and has been recently discussed in the context of strongly interacting electrons. An electronic nematic phase has been seen in many experiments in certain strongly correlated materials, in particular, in the pseudogap phase generic to many hole-doped cuprate superconductors. Recent measurements in high Tc superconductors have shown that even if the lattice is perfectly rotationally symmetric, the ground state can still have strongly nematic local properties. Our study of the two-dimensional one-band Hubbard model provides strong support for the recent experimental results on local rotational C4 symmetry breaking. The variational cluster approach is used here to show the possibility of an electronic nematic state and the proximity of the underlying symmetry-breaking ground state within the Hubbard model. We identify this nematic phase in the overdoped region and show that the local nematicity decreases with increasing electron filling. Our results also indicate that strong Coulomb interaction may drive the nematic phase into a phase similar to the stripe structure. The calculated spin (magnetic) correlation function in momentum space shows the effects resulting from real-space nematicity.
The problem of the many-body enhancement of the static spin susceptibility at long wavelengths and its relation to the quasiparticle effective mass is investigated for a normal electron gas in two-dimensional space as a function of the electronic density. We start from a discussion of the results of the simple Hartree-Fock approximation for various interaction potentials and proceed to develop a complete theory. We find that the effects of the electron-electron interaction are significantly larger than in the familiar three-dimensional case. Our approach is based on a new self-consistent scheme which goes beyond the simple random-phase approximation by explicitly allowing for charge- and spin-fluctuation-induced vertex corrections of the Hubbard type. We show that when the latter are neglected, the many-body enhancement of the spin susceptibility can be cast in a remarkably simple and elegant analytic form.
We present a study of spin-unpolarized and spin-polarized two-dimensional uniform electron liquids using variational and diffusion quantum Monte Carlo (VMC and DMC) methods with Slater-Jastrow-backflow trial wave functions. Ground-state VMC and DMC energies are obtained in the density range 1 ≤ r s ≤ 40 . Single-particle and many-body finite-size errors are corrected using canonical-ensemble twist-averaged boundary conditions and extrapolation of twist-averaged energies to the thermodynamic limit of infinite system size. System-size-dependent errors in Slater-Jastrow-backflow DMC energies caused by partially converged VMC energy minimization calculations are discussed. We find that, for 1 ≤ r s ≤ 5 , optimizing the backflow function at each twist lowers the twist-averaged DMC energy at finite system size. However, nonsystematic system-size-dependent effects remain in the DMC energies, which can be partially removed by extrapolation from multiple finite system sizes to infinite system size. The DMC energies in the thermodynamic limit are used to parametrize a local spin density approximation correlation functional for inhomogeneous electron systems. Our zero-temperature phase diagram shows a single transition from a paramagnetic fluid to a hexagonal Wigner crystal at r s = 35 ( 1 ) , with no region of stability for a ferromagnetic fluid. Published by the American Physical Society 2024