ScienceWeek

CONDENSED MATTER: ON BOSE METALS

The following points are made by P. Phillips and D. Dalidovich (Science 2003 302:243):

1) Before the advent of quantum mechanics, the microscopic origin of electrical conduction in metals remained shrouded in a central mystery: How do electrons avoid the atoms in a dense material? Realizing that electrons move as waves, Felix Bloch (1905-1983) (1) proposed that electrons surf the atoms in a crystal by adjusting their wavelength to fit the periodicity of the lattice. Bloch's view was radical in that electron-electron interactions were assumed absent and defects were assumed to provide only a minor correction. We know now that both of these assumptions are wrong. For example, electron interactions can generate new many-body states such as superconductivity and magnetism, and defects can destroy perfect conduction leading to electron localization (2). Even in the extreme case in which single electrons (3,4) are localized by defect scattering, superconductivity still obtains. How is this possible?

2) At low temperatures, lattice-mediated attractive interactions between electrons produce a resistance-less state in which the charge carriers are electron pairs called "Cooper pairs". Pair formation, however, is not a sufficient condition for superconductivity. Superconductivity obtains when all of the Cooper pairs phase lock into a single quantum state. Such macroscopic occupation of a single quantum state is not possible for fermions as a result of the Pauli exclusion principle. However, Cooper pairs with radii of gyration that are smaller than the interpair spacing are bosons, and there is no exclusion principle for bosons. Hence, macroscopic occupation of a single quantum state is permissible, and it is the resultant phase coherence that thwarts localization.

3) Superconductivity in the localized electron regime can be thought of as Bose-Einstein condensation, a phenomenon which has received much attention recently with the myriad of experiments reporting Bose superfluidity in optical lattices of alkali atoms (5). In the other extreme, the interpair spacing is smaller than the Cooper-pair radius, and the Cooper pair --> boson mapping breaks down. Nonetheless, phase coherence obtains.

4) In summary: The conventional theory of metals is in crisis. In the past 15 years, there has been an unexpected sprouting of metallic states in low-dimensional systems, directly contradicting conventional wisdom. For example, bosons are thought to exist in one of two ground states: condensed in a superconductor or localized in an insulator. However, several experiments on thin metal-alloy films have observed that a metallic phase disrupts the direct transition between the superconductor and the insulator. The authors analyze the experiments on the insulator-superconductor transition and argue that the intervening metallic phase is bosonic. All relevant theoretical proposals for the Bose metal are discussed, particularly the recent idea that the metallic phase is glassy. The implications for the putative vortex-glass state in the copper oxide superconductors are examined.

References (abridged):

1. F. Bloch, Z. Physik. 52, 555 (1928)

2. E. Abrahams, P. W. Anderson, D. C. Licciardello, T. V. Ramakrishnan, Phys. Rev. Lett. 42, 673 (1979)

3. M. Ma, P. Lee, Phys. Rev. B 32, 5658(1985)

4. G. Kotliar, A. Kapitulnik, Phys. Rev. B 53, 3146 (1986)

5. M. H. Anderson et al., Science 269, 198(1995)

Science http://www.sciencemag.org

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Notes:

Because the particles of systems whose behavior can be described only by the rules of quantum mechanics occupy a discontinuous spectrum of energy states, only special (i.e., non-Boltzmann) statistics can be applied to energy distributions in such systems. From the standpoint of the mathematics of statistical physics, the essential general constraint of quantum statistics is that in partition functions of quantum systems it is sums over energy levels that must be used rather than integrals over phase space.

In general, quantum statistics is concerned with the equilibrium distribution of elementary particles of a particular type among the various possible quantized energy states, with an assumption that these particles are indistinguishable. Quantum statistics, in turn, takes one of two forms, depending on distribution constraints: Fermi-Dirac statistics or Bose-Einstein statistics.

In "Fermi-Dirac statistics", the Pauli exclusion principle is obeyed, so that no identical particles (called "fermions" if they obey this condition) can be in the same quantum state (as specified by the set of quantum numbers that define such a state). In a Fermi-Dirac system, the exchange of two identical fermions (e.g., two electrons) does not affect the probability of distribution, but it does involve a change in the sign of the wave function (the exchange is "antisymmetric").

In Bose-Einstein statistics, the Pauli exclusion principle is not obeyed, so that any number of identical particles (called "bosons" if they obey this condition) can be in the same quantum state. In a Bose-Einstein system, the exchange of two bosons of the same type affects neither the probability of distribution nor the sign of the wave function (the exchange is "symmetric").

At high temperatures and low concentrations, both forms of quantum statistics reduce to classical Boltzmann statistics.

In quantum mechanics, electrons, protons, and neutrons have an intrinsic angular momentum known as "spin", and a magnetic moment parallel or antiparallel to that angular momentum. When electrons are combined together to form an atom or ion, there is a resultant angular momentum which is a combination of the intrinsic spin of the electrons and the angular momentum due to their motion about the nucleus, and this is the "spin" of the atom or ion. Atoms or ions with non-zero spin are magnetic atoms or ions. The idea of electron spin was first proposed by Goudsmit and Uhlenbeck in 1925 to explain the splitting of atomic spectroscopic emission lines in the presence of a magnetic field. Elementary particle spin involves a virtual rotation about the axis of the particle, which means only two spin states are possible, one clockwise and one counterclockwise.

All particles in nature are either fermions or bosons, with fermions (always elementary particles) having half-integer spin (spin-states characterized by half-integer multiples of Planck's constant divided by 2 pi), and bosons (all other particles) having integer spin (spin-states characterized by integer multiples of Planck's constant divided by 2 pi).

What is important in this context is that particular real systems can be manipulated in the laboratory into a condition in which quantum behavior becomes both apparent and controlling. An example is the Bose-Einstein condensate, a system long ago predicted but first experimentally realized in 1995, a system in which a gas of atoms at extremely low temperature becomes a gas of bosons obeying Bose-Einstein statistics.

In general, "Bose-Einstein condensation" is a phenomenon occurring in a macroscopic system consisting of a relatively large number of bosons at a sufficiently low temperature (microkelvins down to nanokelvins) in which a significant fraction of the particles occupy a single quantum state of lowest energy (the ground state). In an atomic Bose-Einstein condensate, several thousand atoms essentially become a single atom, a "superatom", and this effect has been observed experimentally with atoms of rubidium and lithium, the atoms trapped and cooled by special methods. The excitement in contemporary physics concerning Bose-Einstein condensates derives from the expectation that these manipulable real systems can illuminate the fundamentals of quantum mechanics, superfluidity, superconductivity, the properties and interactions of atoms, laser physics, and nonlinear optics, i.e., some of the most important research areas in modern physics.

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