quantum limit is suppressed by an unusual spin-orbit induced Zeeman effect, and electron interactions at high magnetic fields may give rise to distinct phases such as Wigner crystals and charge density waves. Further investigation in high fields will be warranted to ascertain these behaviors and to uncover new phenomena in bismuth and other semimetals.

TOPOLOGICAL PHASES

During the last few years, a new class of materials, “topological phases,” has emerged as an exciting frontier of science. These materials have a number of remarkable properties that were not previously envisaged, and strong magnetic fields have played an important role in their exploration.

Topological insulators and topological superconductors are materials that have an energy gap for electronic excitations in their interior, but of necessity have low-energy excitations located at their boundaries. Transport properties, such as electrical conduction, may be dominated by these surface excitations.

The quantum mechanical ground states of these materials are characterized by “topological quantum numbers,” which cannot change their values as one continuously varies an external parameter such as the pressure applied to the system, unless the system undergoes a phase transition in which the energy gap closes and reopens again. Topological phases have quantum numbers that are different from those of ordinary, nontopological phases, so they represent a conceptually distinct state of matter. Mathematically, the topological quantum numbers characterize the way the ground state evolves if one changes certain parameters in the Hamiltonian, which affect its explicit form but do not affect the ground state energy. The set of such ground states forms a surface in the quantum-mechanical Hilbert space, and surfaces corresponding to different topological phases are distinguished from each other in much the same way as the surface of a sphere is topologically distinct from the surface of a donut—the surfaces cannot be deformed into each other without tearing and re-stitching.

The earliest known examples of topological insulators were in fact the quantized Hall states of 2D electron systems at low temperatures, and in a strong magnetic field. It was shown in the 1980s that the quantized Hall conductance of these systems is related to a winding number, which describes the way in which the ground-state wave function evolves when one varies the boundary conditions of the system. This winding number is an integer that, in principle, may be positive or negative and arbitrarily large.

By contrast, for the topological insulators of greatest current interest, the topological numbers have only two possible values, typically equal to 1 in the topological phase and 0 in the trivial non-topological phase. (Such numbers are often denoted Z2 numbers, in contrast to the case of arbitrary integers, which are denoted



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