*Herb Fertig. Photo by Ric Cradick at IU Photographics.*

The real action of “high” tech devices occurs in “low” dimensional systems — those with one or more dimensions fewer than the three-dimensional space of our customary world. The branch of physics that contends with those low dimensional worlds (and which shades into engineering) is the largest in terms of number of practitioners and their published results — some 50,000 papers in the last 25 years. Accordingly, for so vast a field, last year’s KITP program on “Low Dimensional Electron Systems” was the longest condensed matter program ever, spanning almost a half-year.

As the program name indicates, the relevant particles are electrons. To be an electron system in two dimensions means that particle movement is confined by various means to a single plane (as in transistors and in the multitudinous aggregates of transistors that make up computer chips). Electron systems in one dimension confine particle movement to a line, and such a one-dimensional system is called a “quantum wire,” and one type of quantum wire is called a “carbon nanotube.” Electron systems within a very small region of space, sometimes referred to as a “zero-dimensional system,” are called “quantum dots.”

Matter behaves differently in different dimensions. “Condensed matter in low dimensions often unlocks physics that is inaccessible or non-existent in three dimensions,” according to the program description. Those differences are what the participants in the “Low Dim” program sought to find and explore and explain.

In addition to satisfying purely scientific curiosity about the often startling and profound nature of low-dimensional system realities, research into these realms has led to virtually all of what we think of as “high technology.” And if the program’s subjects and directions are any indication, much more is surely to come: “…some of the most important practical advances in materials physics in the last decade involve semiconductors of low-dimensionality and/or small structure at the nanometer scale,” according to the program organizers.

### Quantum Hall Only 2-D

Principal organizer Herb Fertig (of Indiana University) singles out two major topical emphases that dominated the program: (1) quantum Hall effects (integer and fractional, whose separate discoveries have already garnered separate Nobel prizes) and (2) graphene (See article on adjacent page).

Klaus von Klitzing discovered the quantum Hall effect in 1980 in a two-dimensional electron gas in a silicon-MOS-based system (“MOS,” a much-used acronym in condensed matter physics, stands for “metal- oxide-semiconductor,” characterizing the three successive layers of an important class of electronic devices). Two years later, Horst Störmer and Dan Tsui discovered the fractional quantum Hall effect in a high-quality gallium arsenide sample prepared by UCSB’s Art Gossard.

The quantum Hall effects are purely two-dimensional phenomena that show that resistance can be quantized. Until von Klitzing’s discovery, most everybody who thought about resistance thought it took on a continuous range of values for a given system. Nobody, in other words, thought possible values of resistance would be determined directly by the rules of quantum mechanics; but they are, but only (so far) in two-dimensional systems.

The discoveries of the quantum Hall effects have been so momentous because they showed conclusively that there are phenomena in lower dimensions that exist only in lower dimensions, and thereby opened up whole new “lower” vistas for discovery in two, one, or zero dimensional systems.

One possibility much discussed in the program pertains to the relevance of the fractional quantum Hall effect to the dream of quantum computing — a paradigm for computing completely different from and vastly more powerful than our current binary mode. (See “Topological Quantum Computing: The Devil Is Not in the Details,” 2006 KITP Newsletter).

The fractional quantum Hall effect, explained by theorist Robert Laughlin (who shared the 1998 Nobel Prize with the experimentalist discoverers Störmer and Tsui), occurs when electrons are cooled to low temperature and subjected to high magnetic fields. The electrons organize themselves in a highly correlated state in which the ground state and low-energy excitations of the system are insensitive to local perturbations. The discovery of the quantized values of the Hall resistance and its independence from device characteristics means that the effect is robust – exactly the kind of physical system one is looking for in quantum computing.

In the system that is most exciting experimentally, the proposed “qubit” (the hypothetical fundamental unit for quantum computing analogous to a binary bit) is the presence or absence of a neutral fermionic excitation associated with a pair of electrical charges. When these two charges are far apart, the information is delocalized over the whole system.

The delocalization of the qubit is enabled by a subtle topological structure of the fractional quantum Hall state. In turn, the delocalization makes the qubit robust against errors that plague other architectures for quantum computing. Hence, the approach is known as “topological” quantum computing.

The qubit components — the delocalized neutral fermion and the charged excitations – are examples of “quasiparticles” in the fractional quantum Hall effect. As a result of the neutral fermions, the charged quasiparticles are non-Abelian “anyons.” All particles are typically classified according to their statistics as either “fermions” or “bosons,” except for quasiparticles first discovered in conjunction with the fractional quantum Hall two-dimensional systems (of which there are many) whose quantum states range continuously between fermionic and bosonic and are therefore called “anyonic.”

The words themselves, “quasiparticles” and “anyons,” convey how exotic and how potentially technologically transformative are the phenomena that exist only in the lower dimensional worlds.

In addition to the quantum Hall effects and the new and exciting study of graphene, another program emphasis – perhaps as much a matter of technique as of topic – concerns the use of atomic systems as analogues for investigating complex matter systems — a research mode pioneered, in part, through the concatenation of two KITP programs run simultaneously in 2004. One program looked at cold atom systems or Bose-Einstein condensates, and the other at strongly correlated electron systems.

Bose-Einstein condensation (a big development in atomic physics acknowledged with more than one Nobel Prize) produced a coherent quantum state of cold atoms that Einstein had predicted. The big realization, given impetus via the conjunction of the two 2004 programs, is that these coherent quantum states can be used to construct models of condensed matter systems that are more controllable than had been imagined and attained heretofore.

Via the analogues between atomic and condensed matter states, said Fertig, “We are motivated to look at things that might not have occurred to us.”

One big area for investigation is the Hubbard model, an idealized model for the simplest correlated electron systems. Despite the fact that the model is simple and has long been known, said Fertig, “We still don’t know what the phase diagram is, don’t know what kind of phases the system can have. We have some ideas, but there aren’t rigorous proofs."

“A lot of people,” said Fertig, “believe the Hubbard model in some form applies to the high temperature superconductors [whose mechanism is the biggest unsolved problem in condensed matter physics for the last 20 years]. We would like to know if we can use the Hubbard model or some variant to describe the high temperature superconductors.”

### Simulating Hubbard

Atomic gases, said Fertig, afford a possible way of simulating the Hubbard model in experiment, so that a set of outstanding theoretical questions might be addressed experimentally.

Fertig said that the program featured an “experimentalist of the week” to offer the theorists insights on progress being made. For most condensed matter theorists, said Fertig, “It is very important to be in touch with our experimental colleagues because we theorists are often wrong if we just guess what is going to happen in these systems.”

Essentially what atomic physicists have done is to make a lattice-like structure for positioning atoms via interference patterns created with lasers. That structure is like a crystal made out of light.

“The complications of this atomic model,” said Fertig, “capture what we think are the most important complications of the Mott insulators in a low dimensional electron system, which may be key to understanding high temperature superconductivity.” Mott insulators are essentially materials that should theoretically function as conductors of electricity, but which act (contrary to expectation) as insulators especially at low temperature, as a consequence of the correlations induced by the strong interactions between the electrons.

The “Low Dim” program was designed deliberately to discourage talks on the mainstream research approaches to and developments in high temperature superconductivity because the subject now commands the attention of so many researchers that including it more centrally would have entailed doubling the length of the already longest condensed matter theory program in KITP history.

It is hard to imagine what technology will look like 15 or 25 or 50 years from now except to assume it will be different, and a significant driver of the difference is all but certain to come out of the Low Dim world contemplated in this program. But Fertig, as exemplar of the theorist devotee of that Low Dim world, seems almost indifferent to the dazzling lure of transformative technologies. He is in it, seemingly, for the sheer pleasure of knowing, rather than picking material fruits.