Crystallization for Coulomb and Riesz interactions as a consequence of the Cohn-Kumar conjecture

Petrache, Mircea; Serfaty, Sylvia

doi:10.1090/proc/15003

Crystallization for Coulomb and Riesz interactions as a consequence of the Cohn-Kumar conjecture
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Proc. Amer. Math. Soc. 148 (2020), 3047-3057 Request permission

Abstract:

The Cohn-Kumar conjecture states that the triangular lattice in dimension 2, the $E_8$ lattice in dimension 8, and the Leech lattice in dimension 24 are universally minimizing in the sense that they minimize the total pair interaction energy of infinite point configurations for all completely monotone functions of the squared distance. This conjecture was recently proved by Cohn-Kumar-Miller-Radchenko-Viazovska in dimensions 8 and 24. We explain in this note how the conjecture implies the minimality of the same lattices for the Coulomb and Riesz renormalized energies as well as jellium and periodic jellium energies, hence settling the question of their minimization in dimensions 8 and 24.

References

Amandine Aftalion and Sylvia Serfaty, Lowest Landau level approach in superconductivity for the Abrikosov lattice close to $H_{c_2}$, Selecta Math. (N.S.) 13 (2007), no. 2, 183–202. MR 2361092, DOI 10.1007/s00029-007-0043-7
A. Aftalion, X. Blanc, and F. Nier, Lowest Landau level functional and Bargmann spaces for Bose-Einstein condensates, J. Funct. Anal. 241 (2006), no. 2, 661–702. MR 2271933, DOI 10.1016/j.jfa.2006.04.027
Fabrice Bethuel, Haïm Brezis, and Frédéric Hélein, Ginzburg-Landau vortices, Progress in Nonlinear Differential Equations and their Applications, vol. 13, Birkhäuser Boston, Inc., Boston, MA, 1994. MR 1269538, DOI 10.1007/978-1-4612-0287-5
L. Bétermin, M. Petrache, and L. de Luca, Crystallization to a square lattice for a 2-body potential, arXiv:1907.06105.
Laurent Bétermin and Etienne Sandier, Renormalized energy and asymptotic expansion of optimal logarithmic energy on the sphere, Constr. Approx. 47 (2018), no. 1, 39–74. MR 3742809, DOI 10.1007/s00365-016-9357-z
Xavier Blanc and Mathieu Lewin, The crystallization conjecture: a review, EMS Surv. Math. Sci. 2 (2015), no. 2, 225–306. MR 3429164, DOI 10.4171/EMSS/13
S. Borodachov, D. Hardin, and E. Saff, Discrete Energy on Rectifiable Sets, to appear in Springer Mathematical Monographs.
Alexei Borodin and Sylvia Serfaty, Renormalized energy concentration in random matrices, Comm. Math. Phys. 320 (2013), no. 1, 199–244. MR 3046995, DOI 10.1007/s00220-013-1716-z
J. M. Borwein, M. L. Glasser, R. C. McPhedran, J. G. Wan, and I. J. Zucker, Lattice sums then and now, Encyclopedia of Mathematics and its Applications, vol. 150, Cambridge University Press, Cambridge, 2013. With a foreword by Helaman Ferguson and Claire Ferguson. MR 3135109, DOI 10.1017/CBO9781139626804
J. S. Brauchart, D. P. Hardin, and E. B. Saff, The next-order term for optimal Riesz and logarithmic energy asymptotics on the sphere, Recent advances in orthogonal polynomials, special functions, and their applications, Contemp. Math., vol. 578, Amer. Math. Soc., Providence, RI, 2012, pp. 31–61. MR 2964138, DOI 10.1090/conm/578/11483
Luis Caffarelli and Luis Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations 32 (2007), no. 7-9, 1245–1260. MR 2354493, DOI 10.1080/03605300600987306
J. W. S. Cassels, On a problem of Rankin about the Epstein zeta-function, Proc. Glasgow Math. Assoc. 4 (1959), 73–80 (1959). MR 117193, DOI 10.1017/S2040618500033906
Patrick Chiu, Height of flat tori, Proc. Amer. Math. Soc. 125 (1997), no. 3, 723–730. MR 1396970, DOI 10.1090/S0002-9939-97-03872-0
Henry Cohn, A conceptual breakthrough in sphere packing, Notices Amer. Math. Soc. 64 (2017), no. 2, 102–115. MR 3587715, DOI 10.1090/noti1474
Henry Cohn and Abhinav Kumar, Universally optimal distribution of points on spheres, J. Amer. Math. Soc. 20 (2007), no. 1, 99–148. MR 2257398, DOI 10.1090/S0894-0347-06-00546-7
Henry Cohn, Abhinav Kumar, Stephen D. Miller, Danylo Radchenko, and Maryna Viazovska, The sphere packing problem in dimension 24, Ann. of Math. (2) 185 (2017), no. 3, 1017–1033. MR 3664817, DOI 10.4007/annals.2017.185.3.8
H. Cohn, A. Kumar, S. D. Miller, D. Radchenko, and M. Viazovska, Universal optimality of the E8 and Leech lattices and interpolation formulas, arXiv:1902.05438.
C. Cotar and M. Petrache, Equality of the Jellium and Uniform Electron Gas next-order asymptotic terms for Coulomb and Riesz potentials, arXiv:1707.07664.
Renaud Coulangeon and Achill Schürmann, Energy minimization, periodic sets and spherical designs, Int. Math. Res. Not. IMRN 4 (2012), 829–848. MR 2889159, DOI 10.1093/imrn/rnr048
P. H. Diananda, Notes on two lemmas concerning the Epstein zeta-function, Proc. Glasgow Math. Assoc. 6 (1964), 202–204 (1964). MR 168537, DOI 10.1090/s0025-5718-52-99389-7
Veikko Ennola, A lemma about the Epstein zeta-function, Proc. Glasgow Math. Assoc. 6 (1964), 198–201 (1964). MR 168536, DOI 10.1017/S2040618500035024
Veikko Ennola, On a problem about the Epstein zeta-function, Proc. Cambridge Philos. Soc. 60 (1964), 855–875. MR 168535, DOI 10.1017/S0305004100038330
Y. Ge, E. Sandier, On lattices with finite Coulombian interaction energy in the plane, arXiv:1307.2621.
Dorian Goldman, Cyrill B. Muratov, and Sylvia Serfaty, The $\Gamma$-limit of the two-dimensional Ohta-Kawasaki energy. Droplet arrangement via the renormalized energy, Arch. Ration. Mech. Anal. 212 (2014), no. 2, 445–501. MR 3176350, DOI 10.1007/s00205-013-0711-z
D. P. Hardin, E. B. Saff, and B. Simanek, Periodic discrete energy for long-range potentials, J. Math. Phys. 55 (2014), no. 12, 123509, 27. MR 3390564, DOI 10.1063/1.4903975
Douglas P. Hardin, Edward B. Saff, Brian Z. Simanek, and Yujian Su, Next order energy asymptotics for Riesz potentials on flat tori, Int. Math. Res. Not. IMRN 12 (2017), 3529–3556. MR 3693658, DOI 10.1093/imrn/rnw049
David de Laat and Frank Vallentin, A breakthrough in sphere packing: the search for magic functions, Nieuw Arch. Wiskd. (5) 17 (2016), no. 3, 184–192. Includes an interview with Henry Cohn, Abhinav Kumar, Stephen D. Miller and Maryna Viazovska. MR 3643686
Thomas Leblé, Logarithmic, Coulomb and Riesz energy of point processes, J. Stat. Phys. 162 (2016), no. 4, 887–923. MR 3456982, DOI 10.1007/s10955-015-1425-4
Thomas Leblé and Sylvia Serfaty, Large deviation principle for empirical fields of log and Riesz gases, Invent. Math. 210 (2017), no. 3, 645–757. MR 3735628, DOI 10.1007/s00222-017-0738-0
M. Lewin, E. Lieb, and R. Seiringer, A floating Wigner crystal with no boundary charge fluctuations, to appear in Phys. Rev. B.
Elliott H. Lieb and Heide Narnhofer, The thermodynamic limit for jellium, J. Statist. Phys. 12 (1975), 291–310. MR 401029, DOI 10.1007/BF01012066
Hugh L. Montgomery, Minimal theta functions, Glasgow Math. J. 30 (1988), no. 1, 75–85. MR 925561, DOI 10.1017/S0017089500007047
Francis Nier, À propos des fonctions thêta et des réseaux d’Abrikosov, Séminaire: Équations aux Dérivées Partielles. 2006–2007, Sémin. Équ. Dériv. Partielles, École Polytech., Palaiseau, 2007, pp. Exp. No. XII, 27 (French). MR 2385199
B. Osgood, R. Phillips, and P. Sarnak, Extremals of determinants of Laplacians, J. Funct. Anal. 80 (1988), no. 1, 148–211. MR 960228, DOI 10.1016/0022-1236(88)90070-5
Mircea Petrache and Simona Rota Nodari, Equidistribution of jellium energy for Coulomb and Riesz interactions, Constr. Approx. 47 (2018), no. 1, 163–210. MR 3742814, DOI 10.1007/s00365-017-9395-1
Mircea Petrache and Sylvia Serfaty, Next order asymptotics and renormalized energy for Riesz interactions, J. Inst. Math. Jussieu 16 (2017), no. 3, 501–569. MR 3646281, DOI 10.1017/S1474748015000201
Charles Radin, The ground state for soft disks, J. Statist. Phys. 26 (1981), no. 2, 365–373. MR 643714, DOI 10.1007/BF01013177
R. A. Rankin, A minimum problem for the Epstein zeta-function, Proc. Glasgow Math. Assoc. 1 (1953), 149–158. MR 59300, DOI 10.1017/S2040618500035668
Luz Roncal and Pablo Raúl Stinga, Fractional Laplacian on the torus, Commun. Contemp. Math. 18 (2016), no. 3, 1550033, 26. MR 3477397, DOI 10.1142/S0219199715500339
Simona Rota Nodari and Sylvia Serfaty, Renormalized energy equidistribution and local charge balance in 2D Coulomb systems, Int. Math. Res. Not. IMRN 11 (2015), 3035–3093. MR 3373044, DOI 10.1093/imrn/rnu031
Nicolas Rougerie and Sylvia Serfaty, Higher-dimensional Coulomb gases and renormalized energy functionals, Comm. Pure Appl. Math. 69 (2016), no. 3, 519–605. MR 3455593, DOI 10.1002/cpa.21570
Etienne Sandier and Sylvia Serfaty, From the Ginzburg-Landau model to vortex lattice problems, Comm. Math. Phys. 313 (2012), no. 3, 635–743. MR 2945619, DOI 10.1007/s00220-012-1508-x
Etienne Sandier and Sylvia Serfaty, 2D Coulomb gases and the renormalized energy, Ann. Probab. 43 (2015), no. 4, 2026–2083. MR 3353821, DOI 10.1214/14-AOP927
Peter Sarnak and Andreas Strömbergsson, Minima of Epstein’s zeta function and heights of flat tori, Invent. Math. 165 (2006), no. 1, 115–151. MR 2221138, DOI 10.1007/s00222-005-0488-2
Sylvia Serfaty, Coulomb gases and Ginzburg-Landau vortices, Zurich Lectures in Advanced Mathematics, European Mathematical Society (EMS), Zürich, 2015. MR 3309890, DOI 10.4171/152
Sylvia Serfaty, Systems of points with Coulomb interactions, Proceedings of the International Congress of Mathematicians—Rio de Janeiro 2018. Vol. I. Plenary lectures, World Sci. Publ., Hackensack, NJ, 2018, pp. 935–977. MR 3966749
Florian Theil, A proof of crystallization in two dimensions, Comm. Math. Phys. 262 (2006), no. 1, 209–236. MR 2200888, DOI 10.1007/s00220-005-1458-7
Maryna S. Viazovska, The sphere packing problem in dimension 8, Ann. of Math. (2) 185 (2017), no. 3, 991–1015. MR 3664816, DOI 10.4007/annals.2017.185.3.7
E. P. Wigner, On the interaction of electrons in metals, Phys. Rev. 46, (1934), 1002–1011.