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arXiv:cond-mat/0509436 · Physical Review B

On the quantum melting of the two-dimensional Wigner crystal

Xavier Waintal

Published Feb 13, 2006• 35 citations

Abstract

The Fermi liquid-Wigner crystal transition in a two-dimensional electronic system is revisited with a focus on the nature of the fixed node approximation done in quantum Monte Carlo calculations. Recently, we proposed [Phys. Rev. Lett. 94, 046801 (2005)] that for intermediate densities, a hybrid phase (with the symmetry of the crystal but otherwise liquidlike properties) is more stable than both the liquid and the crystal phase. Here we confirm this result both in the thermodynamic and continuum limit. The liquid-hybrid transition takes place at ${r}_{s}^{*}=31.5\ifmmode\pm\else\textpm\fi{}0.5$. We find that the stability of the hybrid phase with respect to the crystal one is tightly linked to its delocalized nature. We discuss the implications of our results for various transition scenarios (quantum hexatic phase, supersolid, multiple exchange, and microemulsions) as proposed in the literature.

DOI: 10.1103/physrevb.73.075417

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Physical Review B•Oct 27, 2004•220 citations

The Fermi liquid-Wigner crystal transition in a two-dimensional electronic system is revisited with a focus on the nature of the fixed node approximation done in quantum Monte Carlo calculations. Recently, we proposed [Phys. Rev. Lett. 94, 046801 (2005)] that for intermediate densities, a hybrid phase (with the symmetry of the crystal but otherwise liquidlike properties) is more stable than both the liquid and the crystal phase. Here we confirm this result both in the thermodynamic and continuum limit. The liquid-hybrid transition takes place at

{r}_{s}^{*}=31.5\ifmmode\pm\else\textpm\fi{}0.5

. We find that the stability of the hybrid phase with respect to the crystal one is tightly linked to its delocalized nature. We discuss the implications of our results for various transition scenarios (quantum hexatic phase, supersolid, multiple exchange, and microemulsions) as proposed in the literature.

We generalize a path integral Monte Carlo formula to investigate the phase boundary between the fractional quantum Hall liquid and the Wigner crystal for a two-dimensional electron (fermion) system in the presence of strong magnetic fields and at low temperatures. At very low temperature, the state of the system is found to be a solid at filling factors $\ensuremath{\nu}=1∕7$,

1∕9

, and

1∕11

, while the state remains liquidlike at $\ensuremath{\nu}=1∕3$ and

1∕5

. The temperature-density phase diagrams are obtained at low filling factors. At $\ensuremath{\nu}=1∕7$ and a fixed electron density, the calculated melting temperature is $\ensuremath{\sim}220\phantom{\rule{0.3em}{0ex}}\mathrm{mK}$, in good agreement with some experimental results. The nature of the phase transition is suggested to be continuous.

For intermediate values of the Coulomb energy to Fermi energy ratio

r_s

, the ground state of a few spinless fermions confined on a two dimensional torus is the quantum superposition of a floppy Wigner molecule with delocalized vacancies and of a Fermi liquid of intersticial particles. This raises the question of the existence of an unnoticed liquid-solid phase between the Fermi liquid and the Wigner crystal for fermionic systems in two dimensions.

Accurate diffusion Monte Carlo technique within the fixed-node approximation has been used to determine the equation of state of the two-dimensional electron gas in three phases: the normal paramagnetic liquid, the ferromagnetic (spin-polarized) liquid, and the Wigner crystal. Our backflow fixed-node calculation, which is a significant improvement over two previous conflicting Slater-Jastrow calculations, shows that there is only one clear phase transition from the paramagnetic liquid to the Wigner crystal as the density decreases. Unlike the three-dimensional case, we do not see the stability of the polarized liquid phase at intermediate densities before the crystallization.

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.