Magnetic quantumcriticality 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 k_{B}T.
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 spinsinglet
superconductor (a SC state) and a superconductor with microscopic coexistence 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 spinwaves, 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 quantumcritical region, which is a highly nontrivial 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 g_{c}.
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 temperatureindependent relaxation rate characteristic of the pink quantumcritical regime as first predicted in paper 2.
Supporting evidence for such a quantumcritical interpretation also appears in spinecho 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 crosssection at finite energy. Just such a resonance has been observed in a number of experiments (J. RossatMignod et al.
Physica C 185189, 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=g_{c} 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 "nonmagnetic" Zn impurities on the spin resonance mode, as noted under a separate category.
PAPERS

Universal quantum critical dynamics of twodimensional
antiferromagnets, S. Sachdev and J. Ye, Physical Review Letters 69,
2411 (1992); condmat/9204001.

Universal magnetic properties of La_{2x}Sr_{x}CuO_{4}
at intermediate temperatures, A.V. Chubukov and S. Sachdev, Physical
Review Letters 71, 169 (1993); 71, 2680 (1993) (E); condmat/9301027.

Comment by A. Millis and Chubukov and Sachdev reply on
"Universal magnetic properties of La_{2x}Sr_{x}CuO_{4}
at intermediate temperatures", Physical Review Letters 71, 3615
(1993).

Theory of twodimensional quantum antiferromagnets with
a nearlycritical ground state, A.V. Chubukov, S. Sachdev, and J. Ye,
Physical Review B 49, 11919 (1994); condmat/9304046.

Polylogarithm identities in a conformal field theory
in three dimensions, S. Sachdev, Physics Letters B 309, 285
(1993); hepth/9305131.

Quantum antiferromagnets in two dimensions,
S. Sachdev, Low dimensional quantum field theories for condensed matter
physicists, Yu Lu, S. Lundqvist, and G. Morandi eds., World Scientific,
Singapore (1995); condmat/9303014.

Universal behavior of the spinecho decay rate in La_{2}CuO_{4},
A.V. Chubukov, S. Sachdev, and A. Sokol, Physical Review B 49, 9052
(1994); condmat/9310052.

Quantumcritical behavior in a two layer antiferromagnet,
A.W. Sandvik, A.V. Chubukov, and S. Sachdev, Physical Review B 51,
16483 (1995); condmat/9502012.

Quantum phase transitions in spin systems and the
high temperature limit of continuum quantum field theories, S.
Sachdev, 19th IUPAP International Conference on Statistical Physics, Hao
Bailin ed., World Scientific, Singapore (1996); condmat/9508080.

Quantum
phase transitions, S. Sachdev, Physics World 12, No 4, 33
(April 1999).

Quantum phase transitions in antiferromagnets and
superfluids, S. Sachdev and M. Vojta, Physica B 280, 333
(2000); condmat/9908008.

Quantum criticality: competing ground states in low
dimensions, S. Sachdev, Science 288, 475 (2000); condmat/0009456.

Comment on "Critical spin dynamics of the 2D quantum Heisenberg antiferromagnets: Sr_{2}CuO_{2}Cl_{2} and Sr_{2}Cu_{3}O_{4}Cl_{2}", S. Sachdev and O.A. Starykh, condmat/0101394.

Spin ordering quantum transitions of superconductors in a magnetic field, E. Demler, S. Sachdev and Y. Zhang, Physical Review Letters 87, 067202 (2001); condmat/0103192.

Bond operator theory of doped antiferromagnets: from Mott insulators with bondcentered charge order, to superconductors with nodal fermions, K. Park and S. Sachdev, Physical Review B 64, 184510 (2001); condmat/0104519.

Spin and charge order in Mott insulators and dwave
superconductors, S. Sachdev, Journal of Physics and Chemistry of Solids 63, 2269 (2002); condmat/0108238.

Quantum phase transitions of correlated electrons in two dimensions, S. Sachdev, Lectures at the International Summer School on Fundamental Problems
in Statistical Physics X, AugustSeptember 2001, Altenberg, Germany, Physica A 313, 252 (2002); condmat/0109419.

Magnetic field tuning of charge and spin order in the cuprate superconductors, A. Polkovnikov, S. Sachdev, M. Vojta, and E. Demler, Proceedings of Physical Phenomena at High Magnetic FieldsIV, Santa Fe, October 1925, 2001, G. Boebinger, Z. Fisk, L.P. Gor'kov,
A. Lacerda, and J.R. Schieffer eds, World Scientific, Singapore (2002), International Journal of Modern Physics B 16, 3156 (2002); condmat/0110329.

`Deconfined' quantum critical points , T. Senthil, A. Vishwanath, L. Balents, S. Sachdev,
and M. P. A. Fisher, Science 303, 1490 (2004); condmat/0311326.

Quantum criticality beyond the LandauGinzburgWilson paradigm, T. Senthil, L. Balents, S. Sachdev, A. Vishwanath, and M. P. A. Fisher, Physical Review B 70, 144407 (2004); condmat/0312617.

Deconfined criticality critically defined, T. Senthil, L. Balents, S. Sachdev, A. Vishwanath, and M. P. A. Fisher, Journal of the Physics Society of Japan 74
Suppl. 1 (2005); condmat/0404718.

Phenomenological lattice model for dynamic spin and charge fluctuations in the cuprates, M. Vojta and S. Sachdev, Proceedings of Spectroscopies of Novel Superconductors, Sitges, Spain, July 1116, 2004, Journal of the Physics and Chemistry of Solids, 67, 11 (2006); condmat/0408461.
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