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arXiv:2010.05367

On the Colloidal Phase of the Homogeneous Electron Fluid

Tom Banks, Bingnan Zhang

Published Oct 11, 2020• 0 citations

Abstract

We provide semi-rigorous arguments that the Homogeneous Electron Fluid (HEF) has a colloidal phase separating the Wigner Crystal from the high density fluid phase. Near the crossover between crystal and fluid ground state energies, the argument is quite general and valid for practically any quantum transition between a crystal and a more amorphous phase. In this regime, the colloid is a gel and its "Goldstone" modes are flows of irregular fluid droplets separated by crystalline walls. A metal insulator transition occurs when a single bubble of fluid spans the entire system. Beyond this transition the colloid is a sol and its properties depend on the existence of meta-stable finite crystallites with negative surface tension. If these exist, the sol phase has lower energy than the homogeneous fluid. In the two dimensional HEF, Kivelson and Spivak argued that such negative surface tension objects always exist, at least in the form of stripes. We argue that the existence of stripes also implies the existence of finite negative tension elliptical crystallites and the detailed competition between the finite crystallite phase and striped phases is difficult to calculate. We also provide weaker arguments that finite crystallites exist in three dimensions. This provides evidence for our claim that the gapless excitations at non-zero wavenumber, observed in the numerical calculations of\cite{haule}\cite{gapless} are quantum remnants of these crystallites (Bosonic quasi-particles) in the limit of vanishing surface tension. Finally, we suggest a Landau mean field theory for the second order quantum phase transition between the fluid and sol phases.

DOI: 10.48550/arxiv.2010.05367

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We provide semi-rigorous arguments that the Homogeneous Electron Fluid (HEF) has a colloidal phase separating the Wigner Crystal from the high density fluid phase. Near the crossover between crystal and fluid ground state energies, the argument is quite general and valid for practically any quantum transition between a crystal and a more amorphous phase. In this regime, the colloid is a gel and its "Goldstone" modes are flows of irregular fluid droplets separated by crystalline walls. A metal insulator transition occurs when a single bubble of fluid spans the entire system. Beyond this transition the colloid is a sol and its properties depend on the existence of meta-stable finite crystallites with negative surface tension. If these exist, the sol phase has lower energy than the homogeneous fluid. In the two dimensional HEF, Kivelson and Spivak argued that such negative surface tension objects always exist, at least in the form of stripes. We argue that the existence of stripes also implies the existence of finite negative tension elliptical crystallites and the detailed competition between the finite crystallite phase and striped phases is difficult to calculate. We also provide weaker arguments that finite crystallites exist in three dimensions. This provides evidence for our claim that the gapless excitations at non-zero wavenumber, observed in the numerical calculations of\cite{haule}\cite{gapless} are quantum remnants of these crystallites (Bosonic quasi-particles) in the limit of vanishing surface tension. Finally, we suggest a Landau mean field theory for the second order quantum phase transition between the fluid and sol phases.

We show that there can be no direct first order transition between a Fermi liquid and an insulating electronic (Wigner) crystalline phase in a clean two-dimensional electron gas in a metal-oxide-semiconductor field-effect transistor (MOSFET); rather, there must always exist intermediate ``micro-emulsion'' phases, and an accompanying sequence of continuous phase transitions. Among the intermediate phases which we find are a variety of electronic liquid crystalline phases, including stripe-related analogues of classical smectics and nematics. The existence of these phases can be established in the neighborhood of the phase boundaries on the basis of an {\it asymptotically exact} analysis, and reasonable estimates can be made concerning the ranges of electron densities and device geometries in which they exist. They can occur in clean Si MOSFETs in the range of densities in which an ``apparent metal to insulator transition'' has been observed in existing experiments. We also point out that, in analogy with the Pomaranchuk effect in

^3

He, the Wigner crystalline phase has higher spin entropy than the liquid phase, leading to an increasing tendency to crystallization with increasing temperature.

We show that there can be no direct first-order transition between a Fermi liquid and an insulating electronic (Wigner) crystalline phase in a clean two-dimensional electron gas in a metal-oxide-semiconductor field-effect transistor (MOSFET); rather, there must always exist intermediate ``microemulsion'' phases, and an accompanying sequence of continuous phase transitions. Among the intermediate phases which we find are a variety of electronic liquid crystalline phases, including stripe-related analogues of classical smectics and nematics. The existence of these phases can be established in the neighborhood of the phase boundaries on the basis of an asymptotically exact analysis, and reasonable estimates can be made concerning the ranges of electron densities and device geometries in which they exist. They likely occur in clean Si MOSFETs in the range of densities in which an ``apparent metal to insulator transition'' has been observed in existing experiments. We also point out that, in analogy with the Pomaranchuk effect in

^{3}\mathrm{He}

, the Wigner crystalline phase has higher spin entropy than the liquid phase, leading to an increasing tendency to crystallization with increasing temperature.

Abstract In recent experiments, external anisotropy has been a useful tool to tune different phases and study their competitions. In this paper, we look at the quantum Hall charge density wave states in the N = 2 Landau level. Without anisotropy, there are two first-order phase transitions between the Wigner crystal, the 2-electron bubble phase, and the stripe phase. By adding mass anisotropy, our analytical and numerical studies show that the 2-electron bubble phase disappears and the stripe phase significantly enlarges its domain in the phase diagram. Meanwhile, a regime of stripe crystals that may be observed experimentally is unveiled after the bubble phase gets out. Upon increase of the anisotropy, the energy of the phases at the transitions becomes progressively smooth as a function of the filling. We conclude that all first-order phase transitions are replaced by continuous phase transitions, providing a possible realisation of continuous quantum crystalline phase transitions.

We use quantum Monte Carlo methods to calculate the zero-temperature phase diagram of the two-dimensional homogeneous electron gas. We find a transition from a paramagnetic fluid to an antiferromagnetic triangular Wigner crystal at density parameter r(s)=31(1) a.u. and a transition to a ferromagnetic crystal at r(s)=38(5) a.u. The fully spin-polarized fluid is never stable. We search for, but do not find, the ferromagnetic "hybrid" phase proposed by H. Falakshahi and X. Waintal [Phys. Rev. Lett. 94, 046801 (2005)10.1103/PhysRevLett.94.046801].

Variational and diffusion quantum Monte Carlo methods are employed to investigate the zero-temperature phase diagram of the three-dimensional homogeneous electron gas at very low density. Fermi fluid and body-centered cubic Wigner crystal ground-state energies are determined using Slater-Jastrow-backflow and Slater-Jastrow many-body wave functions at different densities and spin polarizations in finite simulation cells. Finite-size errors are removed using twist-averaged boundary conditions and extrapolation of the energy per particle to the thermodynamic limit of infinite system size. Unlike previous studies, our results show that the electron gas undergoes a first-order quantum phase transition directly from a paramagnetic fluid to a body-centered cubic crystal at density parameter

{r}_{\text{s}}=86.6(7)

, with no region of stability for an itinerant ferromagnetic fluid.

We develop a microscopic theory of the electronic nematic phase proximate to an isotropic Fermi liquid in both two and three dimensions. Explicit expressions are obtained for the small amplitude collective excitations in the ordered state; remarkably, the nematic Goldstone mode (the director wave) is overdamped except along special directions dictated by symmetry. At the quantum critical point we find a dynamical exponent of

z=3,

implying stability of the Gaussian fixed point. The leading perturbative effect of the overdamped Goldstone modes leads to a breakdown of Fermi-liquid theory in the nematic phase and to strongly angle-dependent electronic self energies around the Fermi surface. Other metallic liquid-crystal phases, e.g., a quantum hexatic, behave analogously.