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 University, Florida

State University, and Purdue

University. The research

appears in *Physical Review B*.

The experimental discovery in 1982 of a new Hall conductance step at a fraction

?=1/m with m=3, that is at (1/3)e^{2}/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 2D 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,” says

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 2D 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 adds.

Indeed, formation of a hexagonally ordered 2D 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, ?, of the lowest Landau level populated by

the electrons at high magnetic fields, for example ? = 1/9, 1/7, and even 1/5.

However, the electrons in the ?=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 add.

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 ?=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 ?=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,” says Landman.