Magnetic quantum-criticality and the collective spin resonance in the high temperature superconductors

Early neutron scattering experiments on the high temperature superconductors (S. M. Hayden et al., Physical Review Letters 66, 821 (1991); B. Keimer et al., Physical Review B 46, 14034 (1992)) showed that the characteristic energy scale of spin fluctuations in the lightly doped regime appeared to be kBT. Papers 1 and 2 were the first to argue that this could be naturally understood by the proximity of a magnetic quantum critical point below its upper critical dimension. Numerous examples of such transitions are known in insulating antiferromagnets, and an introductory discussion appears under a separate category. However, papers 1,2,3,4 argued that transitions in doped antiferromagnets could also be in the same universality class, and presented evidence from numerous neutron scattering, NMR, and magnetization measurements to support this conclusion. These experiments are in systems which are superconducting at low temperatures, and so the quantum critical point is between a spin-singlet superconductor (a SC state) and a superconductor with microscopic co-existence of spin density wave order (a SC+SDW state).

These papers represent the first discussion of the anomalous normal state properties of the cuprates in terms of quantum critical point. Such a point of view has since become increasingly popular. In particular, the experiments of Greg Boebinger and collaborators (e.g. Physical Review Letters 77, 5417 (1996)) on transport in very strong magnetic fields provide specific experimental support.

The experimental detection of a quantum critical point must necessarily rely upon nonzero temperature measurements. It is therefore crucial to understand the nature of the finite temperature crossovers in the vicinity of a zero temperature critical point. The crossovers for static properties were described by S. Chakravarty, B. I. Halperin, and D. R. Nelson, Physical Review B 39, 2344 (1989), while papers 1, 2, 4 described the crossovers in the dynamics, including the first discussion of the dynamics in the novel quantum critical region. These issues are discussed in some detail in my book on quantum phase transitions, and an elementary sketch is in the figure below. See also the discussion under the category of the Ising chain in a transverse field.

Finite temperature crossovers near a zero temperature magnetic quantum critical point in insulating and doped antiferromagnets. At low temperatures above the magnetically ordered phase (blue region on the left), the excitations are a doublet of gapless spin-waves, while at low temperatures above the paramagnetic phase (blue region on the right), the excitations are a triplet of gapped quasiparticles. Neither picture is valid in pink shaded quantum-critical region, which is a highly non-trivial spin liquid with no quasiparticle excitations. The coupling constant g can be varied either by a frustrated exchange constant in an insulating antiferromagnet, or by changing the carrier concentration in a doped antiferromagnet. In the latter case, the ground state is also superconducting on both sides of gc.

Very soon after papers 2 and 3 appeared, NMR experiments at the University of Illinois (T. Imai et al., Physical Review Letters 70, 1002 (1993)) showed crossovers that could be neatly characterized by the crossovers in the theoretical model above.

Measurements of the nuclear relaxation rate in a high temperature superconductor by T. Imai et al., Physical Review Letters 70, 1002 (1993). The small x values are naturally explained by the blue "spin wave" regime above, the large x values by the blue "triplet quasiparticle" regime, while the intermediate x values show a temperature-independent relaxation rate characteristic of the pink quantum-critical regime as first predicted in paper 2.

Supporting evidence for such a quantum-critical interpretation also appears in spin-echo measurements of Fujiyama et al., Physical Review B 60, 9801 (1999). The critical point appeared to be at around a doping concentration of 0.12, which is well within the region where the ground state is superconducting, as we claimed above. Evidence for a magnetic quantum critical point at such a doping concentration also appeared in neutron scattering measurements of Aeppli et al., Science 278, 1432 (1998).

A further consequence of the above picture of the lightly doped cuprates is that the higher doping compounds should show characteristics of the blue "triplet quasiparticle" regime in the top figure. As was discussed early on in paper 4, this triplet quasiparticle would manifest itself as a sharp resonance in the neutron scattering cross-section at finite energy. Just such a resonance has been observed in a number of experiments (J. Rossat-Mignod et al. Physica C 185-189, 86 (1991); H. A. Mook, M. Yehiraj, G. Aeppli, T. E. Mason, and T. Armstrong, Physical Review Letters 70, 3490 (1993); H. F. Fong, B. Keimer, F. Dogan, I. A. Aksay, Physical Review Letters 78, 713 (1997)). Of course, these observations are in systems which are also superconducting, but, as we have stated above, the crossovers in the first figure above still apply: arguments supporting this are reviewed in paper 11, where it is shown that the main role of the onset of superconductivity is to increase the value of the effective parameter g in the action for the spin fluctuations. In particular, the nature of the g=gc quantum critical point in the superconductor remains the same as that in the insulator. Such a model for the resonance mode also leads to a satisfactory theoretical picture of the effect of "non-magnetic" Zn impurities on the spin resonance mode, as noted under a separate category.


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