A Two-Dimensional Electron Liquid Solidifies in a Magnetic Field

A Two-Dimensional Electron Liquid Solidifies in a Magnetic Field

Physicists from the Georgia Institute of
Technology have developed a theory that describes, in a unified manner, the
coexistence of liquid and pinned solid phases of electrons in two dimensions
under the influence of a magnetic field. The theory also describes the
transition between these phases as the field is varied.

Physicists
from the Georgia Institute of Technology have developed a theory that describes,
in a unified manner, the coexistence of liquid and pinned solid phases of
electrons in two dimensions under the influence of a magnetic field. The theory
also describes the transition between these phases as the field is varied. The
theoretical predictions by Constantine Yannouleas and Uzi Landman, from Georgia
Tech’s School of Physics, aim to explain and provide insights into the origins
of experimental findings published last year by a team of researchers from
Princeton, Florida State and Purdue universities. The research appears in the
October 27 edition of the journal Physical
Review B
.

The
experimental discovery in 1982 of a new Hall conductance step at a fraction
ν=1/m with m=3, that is at  (1/3)e2/h
(with more conductance steps, at other m, found later) – where h is the Planck
constant and e is the electron charge – was made for  two-dimensional electrons at low temperatures
and strong magnetic fields and was greeted with great surprise.  The theoretical explanation of this finding a
year later by Robert Laughlin in terms of a new form of a quantum fluid, earned
him and the experimentalists Horst Störmer and Daniel Tsui the 1998 Nobel Prize
with the citation “for the discovery of a new form of quantum fluid with fractionally
charged excitations.” These discoveries represent conceptual breakthroughs in
the understanding of matter, and the fractional quantum Hall effect (FQHE) liquid
states, originating from the highly correlated nature of the electrons in these
systems, have been termed as new states of matter.

“The
quantum fluid state at the 1/3 primary fraction is the hallmark of the FQHE,
whose theoretical understanding has been formulated around the antithesis
between a new form of quantum fluid and the pinned Wigner crystal,” said Landman,
Regents’ and Institute Professor in the School of
Physics, F.E. Callaway Chair and director of the Center for Computational
Materials Science (CCMS) at Georgia Tech. “Therefore, the
discovery of pinned crystalline signatures in the neighborhood of the 1/3 FQHE fraction,
measured as resonances in the microwave spectrum of the two-dimensional
electron gas and reported in the Physical Review Letters in September 2010 by a
group of researchers headed by Daniel Tsui, was rather surprising,” he added.

Indeed,
formation of a hexagonally ordered two-dimensional electron solid phase, a so
called Wigner crystal (WC) named after the Nobel laureate physicist Eugene
Wigner who predicted its existence in 1934, has been anticipated for smaller
quantum Hall fractional fillings, ν, of the lowest Landau level populated by
the electrons at high magnetic fields, for example ν = 1/9, 1/7 and even 1/5.
However, the electrons in the ν=1/3 fraction were believed to resist
crystallization and remain liquid.

The
Georgia Tech physicists developed a theoretical formalism that, in conjunction
with exact numerical solutions, provides a unified microscopic approach to the
interplay between FQHE liquid and Wigner solid states in the neighborhood of
the 1/3 fractional filling. A major advantage of their approach is the use of a
single class of variational wave functions for description of both the quantum
liquid and solid phases. 

“Liquid
characteristics of the fractional quantum Hall effect states are associated
with symmetry-conserving vibrations and rotations of the strongly interacting
electrons and they coexist with intrinsic correlations that are crystalline in
nature,” Senior Research Scientist Yannouleas and Landman wrote in the opening section of their paper.
“While the electron densities of the fractional quantum Hall effect liquid
state do not exhibit crystalline patterns, the intrinsic crystalline
correlations which they possess are reflected in the emergence of a sequence of
liquid states of enhanced stability, called cusp states, that correspond in the
thermodynamic limit to the fractional quantum Hall effect filling fractions
observed in Hall conductance measurements,” they added.

The
key to their explanation of the recent experimental observations pertaining to
the appearance of solid characteristics for magnetic fields in the neighborhood
of the 1/3 filling fraction is their finding that “away from the exact
fractional fillings, for example near ν=1/3, weak pinning perturbations, due to
weak disorder, may overcome the energy gaps between adjacent good angular
momentum symmetry-conserving states. The coupling between these states
generates broken-symmetry ground states whose densities exhibit spatial
crystalline patterns. At the same time, however, the energy gap between the
ground state at ν=1/3 and adjacent states is found to be sufficiently large to
prevent disorder-induced mixing, thus preserving its quantum fluid nature.” 

Furthermore,
the work shows that the emergence of the crystalline features, via the pinning
perturbations, is a consequence of the aforementioned presence of crystalline
correlations in the symmetry-conserving states. Consequently, mixing rules that
govern the nature of the disorder-pinned crystalline states have been
formulated and tested.  Extrapolation of
the calculated results to the thermodynamic limit shows development of a
hexagonal Wigner crystal with enhanced stability due to quantum correlations.

“In
closing, the nature of electrons in the fractional quantum Hall regime continues
now for close to three decades to be a subject of great fascination, a research
field that raises questions whose investigations can lead to deeper conceptual
understanding of matter and many-body phenomena, and a rich  source of surprise and discovery,” said
Landman.

This work was
supported by the Office of Basic Energy
Sciences of the US Department of Energy.

 

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