thermally driven convection. In this context,
the governing parameter is the Rossby number
(Ro), which measures the influence of rotation on
the system (a small Rossby number corresponds
to a fast-rotating state). The cycle period in
our set of simulations is shown to scale as R;1 o
(Fig. 3), in contrast to dimensional inferences
from a kinematic, linear mean-field dynamo (3),
which instead predicts cycle periods proportional to Ro.
Our numerical simulations operate in a nonlinear regime in which the magnetic force alters
the force balance sustaining the large-scale flows
(10). In Fig. 3B, we show the systematic acceleration of the differential rotation that modifies
the electromotive force to trigger the polarity inversion of the mean azimuthal magnetic field.
The amplitude of these fluctuations in the differential rotation is small (~1%), similar to the ones
observed on the Sun. A detailed analysis of our
simulations (fig. S8) reveals that the torque applied by the large-scale magnetic field controls
these modulations. The magnetic cycle period
decreases when the amplitude of the differential
rotation modulation increases, indicating that
nonlinear feedback of the Lorentz force on the
large-scale differential rotation is driving polarity reversals and setting the cycle period.
Although restricted in the stellar parameter
range they span, our simulation results suggest
a single trend of cycle period with rotational influence, quantified by the Rossby number, which
can accommodate both the Sun and existing
stellar data within a single dynamo branch, rather
than multiple branches. The scatter about the
mean relationship observed between cycle period
and rotation rate (Fig. 2A) can be partly attributed to the sensitive dependence of the cycle
period on luminosity. The remaining scatter remains to be explained and could originate from
structural factors such as the exact depth of the
convection zone or the exact shape of the differential rotation, which have not been explored
yet. These considerations reinstate the Sun to
the status of an ordinary solar-type star and a
robust calibration point for stellar astrophysics.
REFERENCES AND NOTES
1. S. L. Baliunas et al., Astrophys. J. 438, 269–287
2. J. C. Hall, G. W. Lockwood, B. A. Skiff, Astron. J. 133, 862–881
3. R. W. Noyes, N. O. Weiss, A. H. Vaughan, Astrophys. J. 287,
4. S. H. Saar, A. Brandenburg, Astrophys. J. 524, 295–310
5. E. Böhm-Vitense, Astrophys. J. 657, 486–493 (2007).
6. A. S. Brun, R. A. Garcia, G. Houdek, D. Nandy, M. Pinsonneault,
Space Sci. Rev. 196, 303–356 (2015).
7. H. Hotta, M. Rempel, T. Yokoyama, Science 351, 1427–1430
8. M. Ghizaru, P. Charbonneau, P. K. Smolarkiewicz, Astrophys. J.
715, L133–L137 (2010).
9. P. J. Käpylä, M. J. Mantere, A. Brandenburg, Astrophys. J.
755, L22 (2012).
10. K. Augustson, A. S. Brun, M. Miesch, J. Toomre, Astrophys. J.
809, 149 (2015).
11. L. D. V. Duarte, J. Wicht, M. K. Browning, T. Gastine, Mon. Not.
R. Astron. Soc. 456, 1708–1722 (2016).
12. P. K. Smolarkiewicz, P. Charbonneau, J. Comput. Phys. 236,
13. A. Brandenburg, F. Krause, R. Meinel, D. Moss, I. Tuominen,
Astron. Astrophys. 213, 411–422 (1989).
14. D. Gubbins, K. Zhang, Phys. Earth Planet. Inter. 75, 225–241
15. S. M. Tobias, Astron. Astrophys. 322, 1007–1017
16. M. L. DeRosa, A. S. Brun, J. T. Hoeksema, Astrophys. J. 757,
17. R. W. Noyes, L. W. Hartmann, S. L. Baliunas, D. K. Duncan,
A. H. Vaughan, Astrophys. J. 279, 763–777 (1984).
18. J. C. Hall, G. W. Henry, G. W. Lockwood, Astron. J. 133,
19. M. Mayor et al., The Messenger 114, 20–24 (2003).
20. R. F. Díaz et al., Astron. Astrophys. 585, A134 (2016).
21. Gaia Collaboration et al., Astron. Astrophys. 595, A1
22. Gaia Collaboration et al., Astron. Astrophys. 595, A2
23. G. Torres, Astron. J. 140, 1158–1162 (2010).
24. T. S. Metcalfe, R. Egeland, J. van Saders, Astrophys. J. 826,
25. V. See et al., Mon. Not. R. Astron. Soc. 462, 4442–4450
26. F. Ochsenbein, P. Bauer, J. Marcout, Astron. Astrophys. Suppl.
Ser. 143, 23–32 (2000).
27. Astroquery package; http://astroquery.readthedocs.io/en/
28. K. H. Schatten, J. M. Wilcox, N. F. Ness, Sol. Phys. 6, 442–455
29. S. T. Fletcher et al., Astrophys. J. 718, L19–L22
30. R. Simoniello et al., Astrophys. J. 765, 100 (2013).
31. R. K. Ulrich, T. Tran, Astrophys. J. 768, 189 (2013).
We thank P. Smolarkiewicz for help and advice on using the
EULAG code, J. F. Cossette for discussions about the modeling
of convection inside stars, and R. Garcia for discussions on
stellar rotation. We acknowledge support from Canada’s Natural
Sciences and Engineering Research Council. This work was
also partially supported by the Institut National des Sciences
de l’Univers/Programme National Soleil-Terre, the Agence
Nationale de la Recherche 2011 Blanc Toupies, European
Research Council grant STARS2 207430, and Centre National
d’Etudes Spatiales Solar Orbiter and PLA TO grants. This work made
use of the VizieR database (26), through the Astroquery package
(27). This work also made use of data from the European Space
Agency mission Gaia ( www.cosmos.esa.int/gaia), processed by the
Gaia Data Processing and Analysis Consortium ( www.cosmos.esa.
int/web/gaia/dpac/consortium). The EULAG-MHD code can be
accessed at www.astro.umontreal.ca/~paulchar/grps/eulag-mhd.
html (subject to U.S. export restrictions). The simulation
outputs are downloadable from www.astro.umontreal.ca/sun/,
and the data for the stellar sample are tabulated in the
Materials and Methods
Figs. S1 to S9
Tables S1 and S2
14 November 2016; accepted 5 June 2017
Tuning quantum nonlocal effects in
Mark B. Lundeberg,1 Yuanda Gao,2 Reza Asgari,3,4 Cheng Tan,2 Ben Van Duppen,5
Marta Autore,6 Pablo Alonso-González,6,7 Achim Woessner,1 Kenji Watanabe,8
Takashi Taniguchi,8 Rainer Hillenbrand,9,10 James Hone,2
Marco Polini,11 Frank H. L. Koppens1,12*
The response of electron systems to electrodynamic fields that change rapidly in space
is endowed by unique features, including an exquisite spatial nonlocality. This can
reveal much about the materials’ electronic structure that is invisible in standard probes
that use gradually varying fields. Here, we use graphene plasmons, propagating at
extremely slow velocities close to the electron Fermi velocity, to probe the nonlocal
response of the graphene electron liquid. The near-field imaging experiments reveal a
parameter-free match with the full quantum description of the massless Dirac electron
gas, which involves three types of nonlocal quantum effects: single-particle velocity
matching, interaction-enhanced Fermi velocity, and interaction-reduced compressibility.
Our experimental approach can determine the full spatiotemporal response of an
The quantum physics of electron systems in- volves complex short-distance interactions and motions that depend sensitively on elec- tron correlations and Fermi-surface defor- mations (1, 2). Although optical techniques
have been applied to study correlated electron
materials (3), far-field optical probes can wash
out purely short-range effects in disorder-free
systems as they probe the response to electrical
fields with long length scales (4). In contrast,
when free electron systems are driven by elec-
tric fields varying rapidly in both time and space,
This aspect of electron response—known as
nonlocality or spatial dispersion in conductivity—
arises due to the internal spreading of energy
via the moving electrons. Even in ambient con-
ditions, the spatial dispersion in an electron
system retains a detailed connection to Fermi-
surface and electron-electron correlation effects,
and hence it provides a unique window into
quantum theories of electron systems without
requiring extremes of low temperature or high