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arXiv:2201.03156 · SciPost Physics

$m^*$ of two-dimensional electron gas: A neural canonical transformation study

Hao Xie, Linfeng Zhang, Lei Wang

Published Jun 14, 2023• 15 citations
arXivHTMLPDF

Abstract

The quasiparticle effective mass m^\ast m * of interacting electrons is a fundamental quantity in the Fermi liquid theory. However, the precise value of the effective mass of uniform electron gas is still elusive after decades of research. The newly developed neural canonical transformation approach [Xie et al., J. Mach. Learn. 1, (2022)] offers a principled way to extract the effective mass of electron gas by directly calculating the thermal entropy at low temperature. The approach models a variational many-electron density matrix using two generative neural networks: an autoregressive model for momentum occupation and a normalizing flow for electron coordinates. Our calculation reveals a suppression of effective mass in the two-dimensional spin-polarized electron gas, which is more pronounced than previous reports in the low-density strong-coupling region. This prediction calls for verification in two-dimensional electron gas experiments.

DOI
10.21468/scipostphys.14.6.154

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m∗m^*m∗ of two-dimensional electron gas: A neural canonical transformation study
The quasiparticle effective mass m^\ast m * of interacting electrons is a fundamental quantity in the Fermi liquid theory. However, the precise value of the effective mass of uniform electron gas is still elusive after decades of research. The newly developed neural canonical transformation approach [Xie et al., J. Mach. Learn. 1, (2022)] offers a principled way to extract the effective mass of electron gas by directly calculating the thermal entropy at low temperature. The approach models a variational many-electron density matrix using two generative neural networks: an autoregressive model for momentum occupation and a normalizing flow for electron coordinates. Our calculation reveals a suppression of effective mass in the two-dimensional spin-polarized electron gas, which is more pronounced than previous reports in the low-density strong-coupling region. This prediction calls for verification in two-dimensional electron gas experiments.
Within the infinite series of ring (or bubble) diagram approximation for the electronic self-energy as appropriate for the long-range Coulomb interaction, we calculate the density-dependent T=0T=0T=0 Fermi-liquid quasiparticle effective-mass renormalization in two-dimensional (2D) electron systems in the random-phase approximation. We find that the quasiparticle effective mass increases very strongly with increasing rs{r}_{s}rs​, exhibiting an unexpected divergence at a critical rs{r}_{s}rs​ value ($\ensuremath{\sim}16$ in the ideal 2D system). We discuss the possibility for and the consequences of such an interaction-induced effective-mass divergence as the electron density decreases and critically comment on the likely theoretical scenarios which could lead to such a low-density effective-mass divergence in electron liquids.
We discuss the potential advantages of calculating the effective mass of quasiparticles in the interacting electron liquid from the low-temperature free energy vis-a-vis the conventional approach, in which the effective mass is obtained from approximate calculations of the self-energy, or from a quantum Monte Carlo evaluation of the energy of a variational ``quasiparticle wave function''. While raw quantum Monte Carlo data are presently too sparse to allow for an accurate determination of the effective mass, the values estimated by this method are numerically close to the ones obtained in previous calculations using diagrammatic many-body theory. In contrast to this, a recently published parametrization of quantum Monte Carlo data for the free energy of the homogeneous electron liquid yields effective masses that considerably deviate from previous calculations and even change sign for low densities, reflecting an unphysical negative entropy. We suggest that this anomaly is related to the treatment of the exchange energy at finite temperature.
We present a variational density matrix approach to the thermal properties of interacting fermions in the continuum. The variational density matrix is parametrized by a permutation equivariant many-body unitary transformation together with a discrete probabilistic model. The unitary transformation is implemented as a quantum counterpart of neural canonical transformation, which incorporates correlation effects via a flow of fermion coordinates. As the first application, we study electrons in a two-dimensional quantum dot with an interaction-induced crossover from Fermi liquid to Wigner molecule. The present approach provides accurate results in the low-temperature regime, where conventional quantum Monte Carlo methods face severe difficulties due to the fermion sign problem. The approach is general and flexible for further extensions, thus holds the promise to deliver new physical results on strongly correlated fermions in the context of ultracold quantum gases, condensed matter, and warm dense matter physics.
The real-space variation quantum Monte Carlo (VMC) and diffusion quantum Monte Carlo (DMC) are used to calculate the energy bands and the effective mass of the paramagnetic and ferromagnetic two-dimensional uniform electron liquid (2D-UEL). The many-body finite-size errors are minimized by performing simulations for three system sizes with the number of electrons, N = 146 , 218, and 302, for paramagnetic and N = 91 , 139, and 151 for ferromagnetic systems. We consider 2D-UEL to be within the density range 1 ≤ r s ≤ 10 . The VMC and DMC results predict that the effective mass m * of the paramagnetic 2D-UEL at high density r s = 1 is very close to 1, suggesting that the effective mass renormalization due to electron-electron interaction is negligible. We find that m * of the paramagnetic 2D-UEL obtained by the VMC and DMC methods increases by r s but with different slopes. Our VMC and DMC results for ferromagnetic 2D-UEL indicate that m * decreases rapidly by reducing the density due to the strong suppression of the electron-electron interaction.
We have carried out a comprehensive investigation of the quasiparticle properties of a two-dimensional electron gas, interacting via the long-range Coulomb interaction in the presence of bare mass anisotropy (i.e., with an elliptic noninteracting Fermi surface) by calculating the self-energy, the spectral function, the scattering rate, and the effective mass within the leading-order dynamical self-energy approximation. Our theory is exact in the high-density limit. We find anisotropic features of quasiparticle properties that are not captured by the commonly used isotropic approximation where the anisotropic effective mass is replaced by the isotropic averaged density-of-states mass. Some of these interesting results are as follows: (1) The many-body renormalization of the quasiparticle spectrum becomes highly anisotropic as the quasiparticle energy increases away from the Fermi energy; (2) the interaction-induced inelastic-scattering rate features a strong anisotropy, exhibiting an abrupt jump at different injected energies depending on the momentum direction of the injected electron; (3) the effective-mass enhancement is larger (smaller) for the light (heavy) mass, showing that the anisotropy is reduced by interactions. Our results and analysis show that the unjustified neglect of the mass anisotropy can lead to an incorrect description of quasiparticle properties of the anisotropic system and inaccurate estimates of physical quantities of interest although the use of an equivalent isotropic approximation using the density-of-states effective mass as is commonly and uncritically performed in the literature, works as a reasonable approximation in many situations. In addition to the complete random phase approximation theory for the anisotropic quasiparticles, we also provide a theory using the simpler plasmon-pole approximation, commenting on its validity for anisotropic self-energy calculations. We comment also on the interaction effect on the Fermi-surface topology, finding that the elliptic shape of the bare Fermi surface is preserved with suppressed ellipticity in the interacting system to a high degree of accuracy except in the very strongly interacting limit (and for very high bare mass anisotropy). Our theory provides a complete generalization of the existing isotropic many-body theory of interacting electrons to the corresponding anisotropic systems.