The idea that thermal and quantum fluctuations cause static stripes to melt into a fluctuating state with dynamic correlations has often
been discussed theoretically (6, 9, 16), but the
experimental evidence remains sparse and seldom direct (31). Our state-of-the-art numerical
calculations have shown that in the disordered
phase, stripes maintain their characteristic antiphase behavior and periodicity in a fluctuating
form, while being robust to variations in parameters, cluster size, and boundary condition. The
fluctuating stripe order observed up to such high
temperatures is a strong piece of corroborating
evidence that these phenomena are strong enough
to affect all electronic properties in the phase diagram. In particular, they may have a bearing on the
controversy in previous studies (24) over the true
ground state of microscopic models for cuprates; a
benchmark of dynamical properties determined
numerically is highly desired to go beyond comparisons of solely static properties (32).
REFERENCES AND NOTES
1. J. M. Tranquada, B. J. Sternlieb, J. D. Axe, Y. Nakamura,
S. Uchida, Nature 375, 561–563 (1995).
2. R. Comin, A. Damascelli, Annu. Rev. Condens. Matter Phys. 7,
3. J. M. Tranquada, D. J. Buttrey, V. Sachan, J. E. Lorenzo,
Phys. Rev. Lett. 73, 1003–1006 (1994).
4. I. A. Zaliznyak, J. P. Hill, J. M. Tranquada, R. Erwin, Y. Moritomo,
Phys. Rev. Lett. 85, 4353–4356 (2000).
5. A. P. Ramirez et al., Phys. Rev. Lett. 76, 3188–3191 (1996).
6. V. J. Emery, S. A. Kivelson, O. Zachar, Phys. Rev. B 56,
7. S. Kivelson, E. Fradkin, V. Emery, Nature 393, 550–553 (1998).
8. J. Zaanen, O. Y. Osman, H. V. Kruis, Z. Nussinov, J. Tworzydlo,
Philos. Mag. B 81, 1485–1531 (2001).
9. S. A. Kivelson et al., Rev. Mod. Phys. 75, 1201–1241 (2003).
10. V. Cvetkovic, Z. Nussinov, S. Mukhin, J. Zaanen, Europhys. Lett.
81, 27001 (2007).
11. J. M. Tranquada, in Handbook of High-Temperature
Superconductivity, J. R. Schrieffer, J. S. Brooks, Eds. (Springer,
2007), chap. 6.
12. M. Vojta, Adv. Phys. 58, 699–820 (2009).
13. J. M. Tranquada et al., Nature 429, 534–538 (2004).
14. S. M. Hayden, H. A. Mook, P. Dai, T. G. Perring, F. Doğan,
Nature 429, 531–534 (2004).
15. G. Xu et al., Nat. Phys. 5, 642–646 (2009).
16. J. Zaanen, M. L. Horbach, W. van Saarloos, Phys. Rev. B 53,
17. M. Eschrig, Adv. Phys. 55, 47–183 (2006).
18. J. Zaanen, O. Gunnarsson, Phys. Rev. B 40, 7391–7394
19. K. Machida, Physica C 158, 192–196 (1989).
20. S. R. White, D. J. Scalapino, Phys. Rev. Lett. 80, 1272–1275
21. G. Hager, G. Wellein, E. Jeckelmann, H. Fehske, Phys. Rev. B 71,
22. C. C. Chang, S. Zhang, Phys. Rev. Lett. 104, 116402 (2010).
23. S. R. White, D. J. Scalapino, Phys. Rev. B 92, 205112 (2015).
24. P. Corboz, T. M. Rice, M. Troyer, Phys. Rev. Lett. 113, 046402
25. Y. F. Kung et al., Phys. Rev. B 93, 155166 (2016).
26. V. I. Iglovikov, E. Khatami, R. T. Scalettar, Phys. Rev. B 92,
27. See supplementary materials.
28. C.-C. Chen et al., Phys. Rev. Lett. 105, 177401 (2010).
29. M. Jarrell, J. E. Gubernatis, Phys. Rep. 269, 133–195 (1996).
30. K. Yamada et al., Phys. Rev. B 57, 6165–6172 (1998).
31. S. Anissimova et al., Nat. Commun. 5, 3467 (2014).
32. J. P. F. LeBlanc et al., Phys. Rev. X 5, 041041 (2015).
33. E. Müller-Hartmann, A. Reischl, Eur. Phys. J. B 28, 173–183
34. C. Stock et al., Phys. Rev. B 71, 024522 (2005).
35. P. Dai, H. A. Mook, R. D. Hunt, F. Doğan, Phys. Rev. B 63,
36. H. A. Mook, P. Dai, F. Doğan, Phys. Rev. Lett. 88, 097004 (2002).
37. V. Hinkov et al., Science 319, 597–600 (2008).
38. B. O. Wells et al., Science 277, 1067–1071 (1997).
39. Y. S. Lee et al., Phys. Rev. B 60, 3643–3654 (1999).
40. S. Wakimoto et al., Phys. Rev. B 60, R769–R772 (1999).
41. S. Wakimoto et al., Phys. Rev. B 61, 3699–3706 (2000).
42. M. Matsuda et al., Phys. Rev. B 61, 4326–4333 (2000).
43. M. Matsuda et al., Phys. Rev. B 62, 9148–9154 (2000).
44. M. Matsuda et al., Phys. Rev. B 65, 134515 (2002).
45. H. Kimura et al., Phys. Rev. B 59, 6517–6523 (1999).
46. J. M. Tranquada, N. Ichikawa, K. Kakurai, S. Uchida,
J. Phys. Chem. Solids 60, 1019–1023 (1999).
47. J. M. Tranquada et al., Phys. Rev. B 54, 7489–7499 (1996).
48. N. Ichikawa et al., Phys. Rev. Lett. 85, 1738–1741 (2000).
We thank A. Kampf, S. Kivelson, W.-S. Lee, Y. Lee, D. Scalapino,
R. Scalettar, J. Tranquada, and J. Zaanen for helpful discussions.
Supported by the U.S. Department of Energy (DOE), Office of Basic
Energy Sciences, Division of Materials Sciences and Engineering,
under contract DE-AC02-76SF00515; the Alexander von Humboldt
Foundation via a Feodor Lynen fellowship (C.B.M.); and the
University of Tennessee’s Science Alliance Joint Directed Research
and Development program, a collaboration with Oak Ridge National
Laboratory (S.J.). Computational work was performed on the
Sherlock cluster at Stanford University and on resources of the
National Energy Research Scientific Computing Center, supported
by DOE under contract DE-AC02-05CH11231. Data supporting
this manuscript are stored on the Sherlock cluster at Stanford
University and are available from the corresponding author upon
request. Source code for the simulations can be found at
https://github.com/cmendl/hubbard-dqmc. Sample input files are
included for the three-band Hubbard model as studied in this work.
Materials and Methods
Figs. S1 to S10
19 September 2016; resubmitted 30 January 2017
Accepted 2 October 2017
Observations and modeling of the
elastogravity signals preceding direct
Martin Vallée,1 Jean Paul Ampuero,2 Kévin Juhel,1 Pascal Bernard,1
Jean-Paul Montagner,1 Matteo Barsuglia3
After an earthquake, the earliest deformation signals are not expected to be carried by the
fastest (P) elastic waves but by the speed-of-light changes of the gravitational field.
However, these perturbations are weak and, so far, their detection has not been accurate
enough to fully understand their origins and to use them for a highly valuable rapid
estimate of the earthquake magnitude. We show that gravity perturbations are particularly
well observed with broadband seismometers at distances between 1000 and 2000 kilometers
from the source of the 2011, moment magnitude 9.1, Tohoku earthquake. We can accurately
model them by a new formalism, taking into account both the gravity changes and the
gravity-induced motion. These prompt elastogravity signals open the window for minute
time-scale magnitude determination for great earthquakes.
Earthquakes involve the displacement of large amounts of mass, which modifies the gravity field. This effect is not restricted to a permanent gravity change due to the fi- nal mass redistribution [e.g., (1–3)] but is
also induced by the transient density perturba-
tions carried by seismic waves. During the wave
propagation, an observer feels attracted by the
compressed parts of the medium and repelled
by its dilated parts, with a global net effect de-
pending on the earthquake mechanism. The grav-
ity perturbations are transmitted at the speed of
light (3.105 km/s), far faster than the first-arriving
(P) elastic waves that travel at 6 to 10 km/s in the
crust and upper mantle. Additionally, at distances
close to a large earthquake, it is difficult to esti-
mate the event magnitude from the information
provided by elastic waves, even when the area is
densely instrumented by seismometers. In the
case of the 2011, moment magnitude (Mw) 9.1,
Tohoku earthquake, the near-real-time mag-
nitude provided by the authoritative Japan Mete-
orological Agency (4) was 7.9 and was corrected
only 3 hours later to 8.8 (5). This underestimation
is due to the fact that real-time local magnitudes
are generally derived from instrumental peak
amplitudes, which are poorly correlated with
moment magnitude when the earthquake is
large. Detection of the gravity perturbations
would provide a much faster method for esti-
mating the size of fault ruptures.
The theoretical relations between the elastic
and gravitational fields are well known [e.g., (6)],
and analytical computations predicted the expected gravity change DgP before the arrival of
the P waves in full-space (7) and half-space (8)
1164 1 DECEMBER 2017 • VOL 358 ISSUE 6367 sciencemag.org SCIENCE
1Institut de Physique du Globe de Paris, Sorbonne Paris Cité,
Université Paris Diderot, CNRS, France. 2Seismological
Laboratory, California Institute of Technology, Pasadena, CA,
USA. 3AstroParticule et Cosmologie, Université Paris Diderot,
CNRS/IN2P3, Commissariat à l’Energie Atomique et aux
Energies Alternatives, Irfu, Observatoire de Paris, Sorbonne
Paris Cité, France.
*Corresponding author. Email: email@example.com