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High-temperature superconductivity

The secret of the hourglass

Nature volume 471, pages 314316 (17 March 2011) | Download Citation

The finding that a cobalt oxide insulator's magnetism is similar to that of cuprate superconductors lends support to the popular but contentious idea that stripe-like electronic order is present in the latter materials. See Letter p.341

One hundred years after its discovery, superconductivity is still an active field of research. On page 341 of this issue, Boothroyd et al.1 describe experimental results on an insulating material that offer insight into the physics of one of the most intriguing families of superconductors — the copper oxides, or cuprates.

Conventional superconductivity — that which occurs in simple metals such as lead and aluminium — was explained back in 1957 by Bardeen, Cooper and Schrieffer, in what is known as the BCS theory2. But in 1986, a different, high-temperature form of superconductivity was discovered in complex cuprates3. This discovery rumbled like an earthquake through the physics community, because the superconducting transition temperatures (Tc), below which these materials conduct electricity without resistance, were much too high to be explained by BCS theory. What causes superconductivity in the cuprates is still much of a mystery, but intensive research has shown that the ground rules of the quantum physics governing the electron systems in the cuprates are very different from those of conventional systems: in contrast to the featureless quantum gas formed by the electrons in conventional superconductors, the high-Tc electrons seem to form highly organized types of 'quantum matter', exhibiting a richness of electronic behaviour that challenges even the diversity found in classical complex fluids4.

With their unique ability to measure how the magnetic properties of this 'electron stuff' fluctuate in time, experiments based on the scattering of neutrons have played a key part in discerning some of the signatures of the cuprates' electron quantum matter. Such measurements revealed that these magnetic fluctuations behave in a peculiarly organized manner, nicknamed the hourglass spectrum (Fig. 1). Boothroyd et al.1 now demonstrate that the magnetic fluctuations in an insulating material, for which it is established that the electrons freeze out in a complex 'stripe' pattern, look very similar to those of the cuprates, thereby supporting the popular but controversial idea that such stripes are also present in superconducting cuprates — although in a quantum 'dynamical' form5.

Figure 1: The hourglass magnetic spectrum.
Figure 1

a, Boothroyd et al.1 find that the collective magnetic 'vibrations' (spin waves) in a cobalt oxide insulator have an hourglass shape when plotted as a function of their energy and inverse wavelengths (kx, ky); the colours indicate the intensity of the spin waves as measured from neutron scattering. b, In the superconducting copper oxides, the neutron-scattering results display a very similar shape, except that at low energies a gap appears that is indicative of a collective quantum melting of stripe-like electronic order5. (Graphic based on a sketch by A. Boothroyd.)

In conventional superconductors, electronic quantum zero-point motions dominate to such a degree that the electrons form a nearly featureless quantum gas, just leaving room for the formation of the electron pairs that, according to BCS theory, are responsible for superconductivity2. Such pairs are also formed in the cuprates, albeit in a mysteriously sturdy form. But there is much more going on in these materials. A large body of experimental work has shown that the cuprates' high-Tc superconducting state seems to coexist4, to a greater or lesser degree, with a wealth of exotic quantum organizational phenomena, including static and dynamical stripes, spontaneous 'diamagnetic' currents, and quantum nematics (quantum versions of the liquid crystals used in liquid-crystal displays).

Among these exotic forms of electronic order, the stripes have a special status because they were the first to be identified — in fact as a surprising output6 of computer simulations that I generated as a young postdoc in 1987. This story starts with the particulars of Cu ions4, having the effect that the electrons in the copper oxide lattice repel each other much more strongly than in conventional superconductors. As a result, the cuprate electron world is more like traffic on a congested highway than like the 'quantum fog' of conventional metals. In the stoichiometric cuprates, the electron-traffic density is so high that the traffic jams completely, forming what is known as a Mott insulator. To get the electrons moving, some of them are removed by chemical doping, and one enters an 'underdoped' regime of quantum mechanical stop-and-go traffic — and here the exotic orders emerge.

By dismissing all collective electronic quantum fluctuations, the physics of such dense quantum traffic can be computed, and my calculations6 showed that the electron traffic spontaneously forms complex patterns, which we nicknamed stripes. These consist of 'rivers', in which the electron motions are relatively free, separated by domains in which the traffic continues to be completely jammed. The electron spins in this stripe phase interact through short-range quantum fluctuations, which cause the spins to freeze out in a special 'incommensurate' antiferromagnetic order: within the insulating domains, the orientation of the spin of each electron is opposite to those of its neighbours, and the rivers act as 'domain walls' in this antiferromagnet. Disappointingly, in the cuprates, my computations insisted that this static stripe phase should insulate instead of superconduct. But this turned out to have been a blessing in disguise when it became gradually clear in the 1990s that the static stripes explain why doped Mott insulating oxides that do not contain copper as a rule form insulators: invariably, their electrons freeze in the sturdy stripes of my computer code.

Starting from the mid-1990s, evidence accumulated that stripes might also have a role in the cuprates, albeit in a 'dynamical' way that is not that well understood: in underdoped superconducting cuprates, stripes would be present in the form of strong correlations at short timescales, but at longer timescales they would fall prey to a quantum melting driven by collective quantum fluctuations4,5. Consistent with this hypothesis, it was found that stripes become static only in some special cases7, and that static stripes have a detrimental effect on superconductivity. Neutron-scattering experiments were pivotal in collecting that evidence. These experiments measure spin fluctuations in the materials as a function of their energy and of their wavenumber (inverse wavelength), and in this energy–wavenumber space the spin-fluctuation spectrum of the underdoped cuprates is shaped like an hourglass (Fig. 1). It was argued5 that this could be explained in terms of collective 'vibrations' (spin waves) of the stripe-phase incommensurate antiferromagnetic order, except that at low energies a gap appears in this spectrum that was interpreted as the signature of the collective quantum melting of the stripe phase as a whole.

However, an equally credible case was made8 that the hourglass spectrum could instead be explained in terms of spin excitations in a rather weakly interacting gas of itinerant electrons, and a debate regarding the interpretation of the hourglass spectrum evolved that rages up to the present day. All along, the problem for the dynamical-stripe interpretation was that the modelling of the spin waves involved a lot of assumptions. In this regard, Boothroyd and colleagues' study1 makes a big difference. The authors perform a neutron-scattering experiment on a material that falls outside the family of cuprate superconductors — a cobalt oxide insulator — and that is known to display stripes9 in a simple static form6. They show that the material exhibits an hourglass spin-fluctuation spectrum (Fig. 1a) strikingly similar to that of the cuprates (Fig. 1b); the only difference is seen at low energies, where the cuprate 'quantum gap' is absent in the cobalt oxide. This similarity lends support to the hypothesis that the hourglass spin-fluctuation spectrum in the cuprate superconductors arises from dynamical stripes4,5.

Boothroyd and colleagues' results arrive at a time when the reality of complex quantum matter in underdoped cuprates is becoming mainstream wisdom. Perhaps we already know so much about these materials that research should be refocused on the greatest mystery of all4: that increased levels of doping make the complex quantum stuff gradually fade away, and that the best superconductors are found at the point where the electron traffic starts to resemble the quantum fog of the simple metals.


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  1. Jan Zaanen is at the Instituut-Lorentz for Theoretical Physics, Leiden University, 2300 RA Leiden, the Netherlands.

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Correspondence to Jan Zaanen.

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