# One-dimensional conductors

Denis Jerome, Claude Bourbonnais

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Denis Jerome, Claude Bourbonnais. One-dimensional conductors. Physics World, 1998. hal-05011712

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Electrons behave very strangely when they are confined in one dimension.

As standard theory breaks down, their spin and charge separate, with
the electrons exhibiting a wide range of different behaviours

**One-dimensional conductors**

Claude Bourbonnais and Denis Jéréome

OVER the second half of this century,
scientists have made remarkable progress
in understanding the properties of elec-
trons in condensed matter. This success
has primarily been due to describing
the electrons in terms of a collection
of states, with each state labelled by its
momentum and spin. Electrons are
fermions and therefore obey the Pauli ex-
clusion principle, which means that two
fermions cannot have the same momen-
tum and spin quantum numbers. The
exclusion principle forces electrons to
occupy a range of momentum states up
to a so-called Fermi surface. In three
dimensions the Fermi surface is a sphere.

This simple picture of valence elec-
trons – developed in the late 1920s by
Pauli and Sommerfeld, and further ex-
tended by Bloch – has until recently pro-
vided a more-than-adequate description
for the low-lying energy excitations of
most conductors and semiconductors.
In particular, Fermi theory predicts that
the electronic specific heat of metals is
proportional to temperature, and that
the magnetic susceptibility is independ-
ent of temperature – just as happens in
common metals such as aluminium and
copper. In metals the Fermi energy – the
energy of the electrons on the Fermi sur-
face – is much greater than the thermal
energy, kT, where k is the Boltzmann
constant and T is temperature.

When electrons interact

This Fermi picture was improved by
Landau in the 1950s, when he showed
that the theory could be extended to take
into account the interactions between
electrons and “holes”. (A hole is the
result of the gap left in momentum space
when an electron is excited above the
Fermi surface.) In Landau’s Fermi-liquid
theory, the interaction among electrons,
which can be strong, creates and scatters

1 Bechgaard salts

<figure><img src="image_2.png" /></figure>

JEAN-CHALLES ROCHER, INSTITUT DES MATERIAUX DE NANTES

<figure><img src="image_3.png" /></figure>

<figure><img src="image_4.png" /><figcaption>(a) A schematic representation of a TMTSF (or TMTTF) Bechgaard salt with the selenium (sulphur) atoms shown in yellow, the carbon atoms in brown and the hexafluorophosphate ions in green and blue. (The hydrogen atoms are not shown.) The 1-D character of the salt is obvious. The a direction is into the page (along the stack); the b direction is approximately vertical; and the c direction is approximately horizontal (along the long axis of the individual molecules). (b) A side view of a Bechgaard salt with the electron orbitals shown in green and yellow. (c) Detail of a single molecule from the stack.</figcaption></figure>

particles above the Fermi surface, and
holes below it. The phase space available
for these events is restricted by the exclu-
sion principle, so the excitations become
extremely stable near the Fermi surface
and merge into new entities, known as
quasi-particles or quasi-holes.

According to the Landau picture,
these elementary excitations behave as if
there is no interaction between the elec-
trons at all. The number of quasi-parti-
cles with a certain spin and momentum
— known as the probability distribution
function — exhibits a step-like disconti-
nuity at the Fermi energy, although the
step is smaller than that found for non-
interacting particles in a Fermi liquid.
The other difference between the inter-
acting and non-interacting cases is that
the mass of the particles is different.

However, electron interactions also
affect the surroundings of each quasi-
particle, which actually moves in the
average field produced by the other
quasi-particles. This field acts as a re-
storing force for long-wavelength dis-
turbances or fluctuations of the Fermi
liquid, giving rise to collective oscillations
in both the spin and charge densities. In
a three-dimensional Fermi liquid, these
oscillation modes are coupled to the
quasi-particle excitations and acquire a
finite lifetime. In one dimension, how-
ever, the physics is very different. For
example, the oscillation modes cannot
decay at low energy and their lifetime
becomes infinite.

Fermi-liquid theory has been ex-
tremely successful. It can, for example,
explain the properties of heavy fermion
metals – intermetallic compounds such
as the uranium–platinum alloy UPt₃, or
the cerium–copper–silicon compound
CeCu₂Si₂. In heavy fermion metals the
interaction between the rare-earth local
magnetic moments and the free elec-
## The many kinds of one-dimensional electron behaviour

The energy spectrum, $\varepsilon(k)$, of the electrons in a conductor depends on their momentum, $k$, which is usually a 3-D vector. At zero temperature, the electron states are filled up to the Fermi energy, $E_F = \varepsilon(k_F)$, where $k_F$ is the Fermi momentum, and the Fermi surface – the constant-energy surface that encloses all occupied electron states – is a sphere in momentum space. In a 1-D conductor, however, the Fermi surface reduces to two points at $\pm k_F$ (figure a). The Fermi surface for an array of parallel, non-interacting 1-D conductors is two planes that intersect the $k$-axis at $\pm k_F$.

One unusual feature of such systems is that it is possible to connect all of the electron states on one plane to all of the hole states on the other plane with a single “nesting” wavevector, $Q$, with magnitude $2k_F$. As predicted 40 years ago by Rudolf Peierls and Herbert Fröhlich, a charge-density wave is set up when the electron and hole states are coupled by a phonon. In a charge-density wave the electronic density is modulated at twice the Fermi wavevector (the wavelength associated with $k_F$) and, when coupled to the lattice, drives a lattice deformation at the same wavevector.

The setting up of a charge-density wave is one example of an “instability” in an electron system. Another instability is superconductivity. The time-reversal symmetry property, $\varepsilon(k, \uparrow) = \varepsilon(-k, \downarrow)$, is preserved in 1-D and, taken alone, this would lead to superconductivity as soon as the slightest attractive interaction (e.g. a phonon-mediated coupling) came into play. However, if the electrons and holes experience Coulomb interactions in 1-D, the charge-density wave and superconducting instabilities compete, and the end result is a Luttinger liquid.

For a quasi-1-D conductor, the Fermi surface is warped (figure b) but the salient features of 1-D physics persist down to a temperature of $T \sim t_D/k$, where $t_D$ is a measure of the interaction between the 1-D conductors and $k$ is the Boltzmann constant. Below this temperature the electrons are no longer confined in 1-D and spread out because the coherent electron motion transverse to the chains is no longer washed out by thermal fluctuations. Density-wave instabilities cannot develop in isotropic 2-D or 3-D systems because a given vector can connect no more than two $k$-states on the Fermi surface. This means that only a superconducting instability can develop (figure c).

a

<figure><img src="image_5.png" /></figure>

b

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c

<figure><img src="image_7.png" /></figure>

(a) In a strictly 1-D metal, the Fermi surface consists of two Fermi points located at $\pm k_F$. Electron (red dots) and hole excitations (white dots) separated by the nesting vector, $Q = 2k_F$, favour a charge-density wave instability (top). However, this also interferes with pairs of electron or hole states involved in superconductivity (bottom) and leads to the formation of a Luttinger liquid. (b) For weakly coupled metallic organic chains, the Fermi surface is made up of two slightly bent curves in 2-D (planes in 3-D). The nesting vector, $Q$, can pick up a transverse component and the interference becomes gradually suppressed at low temperatures. The bending of the Fermi surface is related to $t_D$, the coupling between neighbouring 1-D chains. (c) The Fermi surface for an isotropic metal is a sphere. There is no nesting, so superconductivity – in which the electrons and holes form pairs – is the most likely instability.

<figure><img src="image_8.png" /></figure>

trons makes their mass considerably heavier than the free electron mass. The low-temperature behaviour of these materials can be explained in terms of an assembly of almost independent quasi-particles states with much increased masses.

However, Fermi-liquid theory also has its limitations, and needs various additions to explain certain physical phenomena. The most spectacular of these is superconductivity – the vanishing of electrical resistance and the expulsion of magnetic flux below a certain transition temperature. In the Bardeen–Cooper–Schrieffer (BCS) model of superconductivity, it is the attractive two-body interaction between quasi-particles with opposite spin and momentum that favours the formation of the bound electron pairs that leads to superconducting behaviour. This “instability” of the Fermi liquid to the formation of these Cooper pairs can explain superconductivity in a large number of metallic or intermetallic compounds. The attraction between the electrons in the Cooper pair is mediated by vibrations of the lattice or “phonons”.

## Physics in reduced dimensions

One remarkable feature of Fermi-liquid theory is that it is

highly sensitive to the spatial dimensionality of the electron system. Indeed, in the early 1950s Sin-Ito Tomonaga, who shared the Nobel prize with Feynman and Schwinger for his work on quantum electrodynamics, showed that Fermi-liquid theory can break down in one-dimensional (1-D) systems. Later, in the 1960s, Joaquin Luttinger of Columbia University in New York proposed a model of a 1-D metal that proved to be solvable and in which the only stable low-lying excitations were quantized spin and charge waves that obeyed boson statistics.

Spin and charge waves are long-wavelength modulations of the spin and charge densities that are analogous to acoustic waves in solids. And just like purely harmonic acoustic waves, these spin and charge waves do not interact. This leads to a complete separation of the spin and charge degrees of freedom. This separation is one of the key features of what is called a Luttinger liquid.

There is no phase space available for these modes to decay into, so their lifetime becomes infinite. Moreover, all of the static and dynamical properties of the system at large distances are governed by the velocities of these excitations, and by stiffness constants that depend on the original interactions
## 2 Many phases in few dimensions

<figure><img src="image_9.png" /></figure>

A phase diagram for the Bechgaard salts as a function of temperature (y-axis) and pressure (x-axis). On the left-hand side, the sulphur series (TMTTF) has a Luttinger-liquid conducting state ($LL_{p+\sigma}$) that becomes an insulating Luttinger liquid ($LL_p$) below the localization temperature, $T_p$. Further cooling results in either spin-Peierls (SP) or localized antiferromagnetic (AF) order. Spin-Peierls order is non-magnetic and results from the dimerization of the lattice and the spins. On the right-hand side, a selenide-based salt (TMTSF) slowly changes (starting around $T_c$) from a Luttinger liquid to a Fermi liquid (FL) as it is cooled, and eventually becomes antiferromagnetic or superconducting.

among electrons. Finally, various properties of 1-D metals – such as their response to applied electric or magnetic fields – are characterized by various power-law singularities of the form $T^{-\alpha}$, where $T$ is temperature and the exponent $\alpha$ depends on the stiffness constants. This form of behaviour is generic for a variety of 1-D models, and systems that exhibit it are called Luttinger liquids.

A real challenge for the validity of the Landau-Fermi theory was the discovery of high-temperature superconductivity in 2-D copper oxides in 1986/7. It was not so much the large value of the transition temperature, $T_c$, that created problems, but rather the numerous puzzling properties of the “normal” phase above $T_c$. For instance, the resistivity varies linearly with temperature over several orders of magnitude, whereas Fermi theory suggests that it should vary quadratically. Even more dramatic is the behaviour of the resistivity perpendicular to the copper-oxide planes (the c-axis), which exhibits an insulating character with a negative temperature coefficient. The magnetic properties of the cuprates also differ from the predictions of Fermi-liquid theory.

Philip Anderson of Princeton University in the US has taken advantage of these experimental features to propose an extension of the Luttinger-liquid picture in which the circular Fermi surface is replaced by a collection of 1-D Luttinger liquids, one for each point on the Fermi surface (see Physics World 1995 December pp37–40). Anderson's theory has triggered a passionate controversy that is still unresolved.

Before this debate started, several research groups had decided to study materials in which Fermi-liquid theory was expected to break down. These were the 1-D organic conductors in which the strong directionality of the intermolecular bonds force the electrons to travel along 1-D trajectories.

This quest started in 1964 when Bill Little of Stanford University in California suggested that the attraction between two electrons in a molecular wire with strongly polarizable side groups would be much stronger than the interaction

between the electrons in the Cooper pairs of the BCS model for conventional superconductors. In Little's model the interaction between electrons remains attractive right up to the Fermi energy (i.e. about 10 000 K), rather than the vibrational or phonon energy (about 100 K) in the BCS model. Consequently this would lead to superconductivity at much higher temperatures. Although Little's theory was unable to explain the transition temperatures measured in real low-dimensional organic conductors – about 1 K in 1-D and around 12 K in 2-D – his work triggered extensive studies of low-dimensional conductors.

Organic conductors are made of packed and flat charged molecules in an organic salt. Since these flat molecules interact weakly in the solid state, they retain their flat shape, which results in a good intermolecular overlap of the electron clouds along the packing direction, making electron delocalization possible along that direction.

The first organic conductor to show metal-like conductivity was the 1-D compound tetrathiafulvalene-tetracyanoquinoxalimethane (TTF-TCNQ). In 1973 Alan Heeger and Anthony Garito of the University of Pennsylvania, and Dwaine Cowan and co-workers of Johns Hopkins University in Baltimore, showed that this compound exhibits a metal-insulator transition at 54 K, which they attributed to the Peierls instability of a 1-D electron system (see box).

## The many phases of Bechgaard salts

So what is unusual about physics in one dimension? Firstly, there exists one feature in 1-D that is usually absent in higher dimensions. All particle states with a wavevector, $k$, close to the Fermi surface have an energy equal but opposite in sign to the hole states at $k+Q$, where $Q$ is the “nesting” vector. The interactions between the particle and holes states can then cause the charge (or magnetization) to be spatially modulated with a wavevector, $Q$. (This feature is still preserved to some extent if the Fermi surface becomes slightly warped.)

What makes 1-D conductors so different from the usual multi-dimensional metals is the reduction in the phase space available for scattering events. This means that both superconducting and density wave instabilities cannot be considered separately. Both interfere on an equal footing and conspire to form a Luttinger liquid.

All this theory has received brilliant experimental support from a series of measurements on the Bechgaard salts. This range of organic conductors – discovered in 1980 by Klaus Bechgaard and colleagues at the Oersted Institute in Copenhagen – has proved to be a remarkable playground for exploring the relevance of Fermi-liquid theory to 1-D electrons. Indeed, within months of their discovery, one of us (DJ) and co-workers in Orsay, working with Bechgaard, observed superconductivity in these remarkable compounds.

The generic formula for Bechgaard salts is $(TM)_2X$ where $TM$ is a flat molecule that donates electrons and $X$ is a negative ion that supplies a hole. $TM$ is usually tetra-methyl-tetraselenafulvalene (TMTSF), a molecule that contains four selenium atoms, ten carbon atoms and 12 hydrogen atoms, or a similar molecule known as TMTTF that contains sulphur instead of selenium. $X$ is typically hexafluorophosphate $(PF_6^-)$, perchlorate $(ClO_4^-)$ or something similar.

These molecules naturally stack in 1-D structures with two $TM$ molecules and one $X$ ion per unit cell (figure 1). The two organic molecules contribute one electron to each unit cell.
The overlap between the electron clouds of neighbouring molecules in the stacking direction is about ten times larger than that between the stacks in the transverse *b* direction, and about 500 times larger than that along the *c* direction.

This anisotropy gives the Fermi surface a very unusual shape. It is quasi planar and intersects the $k_a$ axis at $\pm k_F$, where the Fermi momentum, $k_F$, is given by $\rho\pi/4a$. Here $\rho$ is the density of carriers per unit cell and $a$ is the length of the unit cell. It is this particular shape of the Fermi surface, together with interactions between the electrons and holes, that leads to the formation of a Luttinger liquid.

At low temperatures (~20 K), true long-range 3-D order is able to develop because there is always some non-zero coupling between the stacks. The type of order depends on the nature of the dominant interstack coupling: magnetic coupling (i.e. interactions between the spins) leads to antiferromagnetism (i.e. alternate spins point in opposite directions), whereas electron hopping from stack to stack leads to superconductivity (figure 2).

However, at high temperatures – where the coupling between molecular stacks is washed out by thermal fluctuations – the Bechgaard salts become truly 1-D. Theory predicts that the system minimizes the Coulomb repulsion among carriers by promoting localization of the charge carriers, reminiscent of what was first proposed in the context of isotropic systems by Mott in 1944, and was later formalized in a model proposed by Hubbard in the early 1960s. This limiting form – in which the Luttinger liquid is an insulator with a non-zero spin susceptibility – is indeed observed with $(T\text{MTTF})_2X$ compounds below the localization temperature, $T_p$ (figure 2). The electrical resistivity increases by several orders of magnitude as the system is cooled through $T_p$, while the spin susceptibility remains unaffected. Above $T_p$ the Luttinger liquid is a conductor.

Nuclear magnetic resonance (NMR) experiments in Orsay confirmed the separation of spin and charge in these molecular systems. The nuclear spin-lattice relaxation rate, $T_1$, varies as $T^{-1} \sim T^{k_p}$, where $T$ is temperature and $k_p$ is the stiffness constant of charge waves (which depends on the details of the electron interaction). The relaxation rate probes the electron spin correlations at low temperature. In the non-interacting Fermi-liquid limit, $k_p = 1$ and the relaxation rate is proportional to temperature. For Luttinger liquids, however, the stiffness constant tends to zero below the localization temperature, and the relaxation rate should become independent of temperature. This is exactly what is observed for the $(T\text{MTTF})_2X$ series. The $T_1$ data also suggest that strong antiferromagnetic spin fluctuations persist below 50 K in the selenide-based Bechgaard salts, although they remain conducting.

## Inside the Luttinger liquid

All these results confirm that the charge carriers are confined along the molecular stacks. Crudely speaking one can view this confinement as a result of a reduction in the transverse

<figure><img src="image_10.png" /><figcaption>3 Two liquids at once</figcaption></figure>

The longitudinal resistivity, $\rho_a$, exhibits metallic behaviour by falling with temperature (green line). At first the transverse resistivity, $\rho_c$, increases as $T$ gets smaller (red line). This is Luttinger-liquid behaviour. Below 100 K, however, $\rho_c$ falls with temperature, signalling the onset of instability in the Luttinger liquid and the progressive recovery of a Fermi liquid. The steep rise at low temperatures is due to the transition from a metal to an insulator.

hopping rate due to the correlations of Luttinger liquid. A carrier is more likely to stay on a stack bound to another carrier (with opposite charge) than it is to jump onto a neighbouring stack. In other words, the intrastack Coulombic attraction between electrons and holes makes the system more 1-D than band-structure calculations would lead us to expect. 1-D confinement is quite pronounced on the left-hand side of the phase diagram (figure 2). However, as one moves to the right, the localization temperature, $T_p$, becomes lower, causing the intrastack interactions to become weaker and 1-D behaviour less pronounced.

The big question is: what is left of the Luttinger liquid when metallic behaviour is stabilized in the selenide-based Bechgaard salts at low temperatures? One could try to answer this question by watching the temperature dependence of the longitudinal resistivity, $\rho_a$. However, $\rho_a$ exhibits approximately Fermi-liquid behaviour (i.e. $\rho_a \sim T^2$) in a broad temperature regime between 20 K and 150 K. To find out if non-Fermi-liquid features survive the stabilization of a conducting phase, we need to seek out more subtle effects.

Studying electric transport along the transverse direction with the weakest coupling, the *c* direction, has helped because it probes the physics in the two other directions we are interested in. The coupling along *c* is so weak that no coherent carrier motion can be established at any temperature, and transport can only proceed via the hopping of individual particles. For this to happen a particle (or quasi-particle) has to emerge from the Luttinger liquid – in which its spin and charge are separated – by recombining its charge and spin components. The particle can then hop onto a neighbouring stack and decay into the Luttinger liquid there.

This process leads to a transverse resistivity: $\rho_c(T) \approx (t_c)^{-2}T^{-2\alpha}$, where $t_c$ is the coupling in the *c* direction, and $\alpha = \nu_4(K_p + K_p^{-1} - 2)$. The density of states near the Fermi energy, $E_F$, is proportional to $(E - E_F)^\alpha$, where $E$ is energy. The exponent $\alpha$ is infinite for Luttinger liquids ($K_p = 0$), and zero for Fermi liquids ($K_p = 1$). This implies that there will be a dip in the density of states at the Fermi level in a Luttinger liquid. Clear evidence for such a pronounced dip in $(T\text{MTSF})_2X$ was recently obtained from photoemission spectra by Marco Grioni and colleagues at the Ecole Polytechnique Fédérale in Lausanne, Switzerland.

Another sign of non-Fermi-liquid behaviour would be the co-existence of insulating behaviour in the *c* direction and metallic behaviour along the stacks (the *a* direction) – and such behaviour was observed in experiments at Orsay last year (figure 3). The maximum in $\rho_c(T)$, observed around 100 K, signals the beginning of the gradual crossover from the Luttinger liquid to more conventional Fermi-liquid behaviour, and this crossover is complete at around 10 K. The resistance along the intermediate coupling transverse direction, $\rho_b$, also differs from the longitudinal resistance, $\rho_a$, in this temperature regime.

Optical conductivity experiments – in which a laser is shone on the sample and its conductivity, $\sigma$, is measured as a
## 4 Evidence for Luttinger liquids

<figure><img src="image_11.png" /></figure>

(a) The optical conductivity, $\sigma$, along the chain in the normal state of $(\text{TMTSF})_2\text{PF}_6$ as a function of frequency, $\omega$, at 25 K. The spectrum shows the remnant of a 1-D gap at $\omega = 2\Delta$ and a power-law decrease $(\sigma - \omega^{-1.3})$ at high frequency that is a hallmark of the Luttinger liquid. (b) The anomalous magnetoresistance of $(\text{TMTSF})_2\text{PF}_6$ at 9 Kbar. Magnetoresistance is a change of electrical resistance in the presence of an applied magnetic field (H). In zero field the system is a superconductor below 1.2 K (red line). When a magnetic field is applied along the c direction, the system develops a huge magnetoresistance with a resistivity minima that moves deep into the normal state. This is reminiscent of the Luttinger-liquid state found in the sulphur series at low pressure.

function of the laser frequency – can also provide evidence for non-Fermi-liquid behaviour in the form of an infrared gap in the recorded spectrum. This approach was pioneered by John Eldridge at the University of British Columbia in Canada and by Tom Timusk at McMaster University, also in Canada, in the 1980s.

The infrared gap is a remnant at low temperature of the high-temperature Luttinger liquid. The conduction is very weak below the gap frequency, which corresponds to the minimum energy needed to promote single-particle excitations. For higher frequencies, the Luttinger-liquid picture predicts that $\sigma$ will follow a power law, $\sigma \sim \omega^{16K_p}$. Behaviour of this type with $\sigma \sim \omega^{-1.3}$ over more than a decade in frequency (and corresponding to $K_p \approx 0.23$) has been identified in a series of experiments on different compounds by George Gruner of the University of California at Los Angeles and Leo Degiorgi

of ETH Zurich in Switzerland (figure 4a).

## Magnets and ladders

The unusual behaviour of the Bechgaard salt $(\text{TMTSF})_2\text{X}$ is also revealed through its response to an applied magnetic field. Classically the motion of electrons sitting on a nearly planar Fermi surface should not be affected by a transverse field. (If anything, the field will increase the 1-D character of any quasi-1-D electrons.) However, a huge transverse magnetoresistance in the low-temperature region of the normal phase $(\text{TMTSF})_2\text{X}$ has recently been observed at Orsay (figure 4b). This is evidence of a restoration of the Luttinger-liquid character, and is accompanied by an increase in anti-ferromagnetic fluctuations.

Until now, organic conductors were the only materials to present such a 1-D character over a wide temperature regime. However, new cuprate-based conductors called spin ladders – pairs of $\text{CuO}_2$ chains linked by oxygen atoms between the coppers – have recently demonstrated strong 1-D behaviour. In the actual material, $\text{Sr}_{14-x}\text{Ca}_x\text{Cu}_2\text{O}_{4+\delta}$, the copper ladders are intrinsically hole-doped and very similar to the organics – they are metallic along the ladder axis and insulating in the perpendicular direction.

The charge localization, which takes place below 50 K, can be suppressed at high pressure (above 3 GPa) and superconductivity becomes observable around 10 K. Unlike organic conductors, however, a spin gap opens up when the charge and spin separate in a spin ladder. This allows metallic conduction along ladders to co-exist with very low spin susceptibility, a sure sign of non-Fermi-liquid behaviour. The behaviour of this spin gap has been the subject of much theoretical speculation. Recent experiments at Orsay have shown that the spin gap vanishes when the superconducting state is stabilized at high pressure.

## Outlook

As we have seen, in addition to superconductivity, one-dimensional organic conductors display a wealth of phenomena in their normal phase that the well established Fermi-liquid theory of metals is unable to explain. The recent discovery of 1-D copper-oxide spin ladders appears to represent a bridge between 1-D organic conductors and 2-D cuprate superconductors. Our dreams would come true if the quasi-1-D physics in the organic conductors and spin ladders could contribute to our understanding of electrons in metals and, ultimately, to an explanation of high-temperature superconductivity in the cuprates.

## Further reading

C Bourbonnais 1995 in *Strongly-interacting Fermions and High-temperature Superconductivity* (ed) B Doucot and J Zinn-Justin pp307–369 (North Holland, Amsterdam)

D Jérôme 1991 *Science* **252** 1509–1514

A Mayaffre et al. 1998 *Science* **279** 345–348

J Moser et al. 1998 *Euro. Physical J.* **B1** 39–46

M Uehara et al. 1996 *J. Physical Soc. Japan* **65** 2764–2767

J Voit 1995 *Rep. Progress in Phys.* **58** 977–1116

**Claude Bourbonnais** is in the Centre de Recherche en Physique du Solide et Département de Physique, Université de Sherbrooke, Sherbrooke, Québec, Canada J1K 2R1. **Denis Jérôme** is in the Laboratoire de Physique des Solides, Université Paris-Sud, 91405 Orsay, France