Discussion session on New Challenges for Coulomb Gases Mathieu LEWIN mathieu.lewin@math.cnrs.fr (CNRS & Universit´ e de Paris-Dauphine) Conference on “Mathematical challenges in classical & quantum statistical mechanics” Venice, August 2017 Mathieu LEWIN (CNRS / Paris-Dauphine) Coulomb gases 1 / 23
Homogeneous Electron Gas (HEG) Coulomb gas Renormalized Energy Jellium Sine- β process, Brownian carousel One-Component Plasma (OCP) Uniform Electron Gas (UEG) Mathieu LEWIN (CNRS / Paris-Dauphine) Coulomb gases 2 / 23
Instability of Coulomb Gas ◮ Long range of Coulomb force = ⇒ no simple thermodynamics Ω N = N 1 / 3 Ω where Ω fixed domain with | Ω | = 1 /ρ N 5 / 3 1 d µ ( x ) d µ ( y ) ˆ ˆ � min ∼ min | x j − x k | 2 | x − y | x i ∈ Ω N N →∞ µ proba Ω Ω 1 ≤ j < k ≤ N � �� � Cap(Ω) Particles accumulate close to the boundary of Ω N Mathieu LEWIN (CNRS / Paris-Dauphine) Coulomb gases 3 / 23
One- & Two-Component Plasma ◮ Jellium = OCP : negatively charged particles in uniform positive background good approx. to interiors of white dwarfs (fully ionized atoms) electrons in a solid Local Density Approximation of Density Functional Theory classical OCP appears in many areas of Mathematics and Physics ◮ TCP : mixture of positive and negative charges classical collapse: need short range regularization (or T > T c > 0 in 2D) quantum: stable only when one kind are fermions (Dyson ’67, Conlon-Lieb-Yau ’88, Lieb-Solovej ’04, Dyson-Lenard ’67, Lieb-Thirring ’75) functional integrals, Euclidean Field Theory, Sine-Gordon transformation (Siegert ’60, Edwards-Lenard ’62, Albeverio-Høegh Krohn ’73, Fr¨ ohlich ’76, Park ’77, Fr¨ ohlich-Park ’78,...) 2D-TCP: Berezinski-Kosterlitz-Thouless (BKT) phase transition (Kosterlitz-Thouless ’73, Fr¨ ohlich-Spencer ’81) here, we focus mainly on Jellium Mathieu LEWIN (CNRS / Paris-Dauphine) Coulomb gases 4 / 23
Jellium N | x j − y | dy + ρ 2 1 ˆ 1 ˆ ˆ dx dy � � E ρ (Ω , x 1 , ..., x N ) = | x j − x k | − ρ 2 | x − y | Ω Ω Ω 1 ≤ j < k ≤ N j =1 1 x 1 ,..., x N ∈ Ω N E ρ (Ω N , x 1 , ..., x N ) = ρ 4 / 3 e cl e cl Jell ( ρ ) = lim min Jell (1) | Ω N | N →∞ N | Ω N | → ρ T ˆ E ρ (Ω N , x 1 ,..., xN ) f cl (Ω N ) N e − Jell ( T , ρ ) = − lim | Ω N | log dx 1 · · · dx N T N →∞ N | Ω N | → ρ = ρ 4 / 3 f cl Jell ( T ρ − 1 / 3 , 1) ρ → 0 f cl Similar definition f Jell ( T , ρ ) in quantum case, with f Jell ( T , ρ ) ∼ Jell ( T , ρ ) (Lieb-Narnhofer ’73) Mathieu LEWIN (CNRS / Paris-Dauphine) Coulomb gases 5 / 23
Phase diagram of Jellium Jones-Ceperley, Phys. Rev. Lett. ’96 Mathieu LEWIN (CNRS / Paris-Dauphine) Coulomb gases 6 / 23
Phase diagram of classical Jellium � 4 π � 1 / 3 e 2 ρ 1 / 3 Γ = www.lanl.gov/projects/dense-plasma-theory/ 3 k B T Brush-Salin-Teller ’66, ... Mathieu LEWIN (CNRS / Paris-Dauphine) Coulomb gases 7 / 23
With spin (electrons): Para/Ferromagnet transition Zong-Lin-Ceperley, Phys. Rev. E. ’02 Drummond et al , Phys. Rev. B. ’04 Mathieu LEWIN (CNRS / Paris-Dauphine) Coulomb gases 8 / 23
Some rigorous results on 3D Jellium ◮ Thermodynamics Existence (Lieb-Narnhofer ’73) ◮ States and screening BBGKY / KMS / KS / DLR (Gruber-Lugrin-Martin ’79-80) Cluster expansions & Debye screening (Brydges ’78, Brydges-Federbush ’80, Imbrie ’83) Clustering = ⇒ Euclidean invariance (Gruber-Martin ’80, Gruber-Martin-Oguey ’82) Sum rules, charge fluctuations (Gruber-Lebowitz-Martin ’81, Gruber-Lugrin-Martin ’80, Lebowitz-Martin ’84) ◮ Behavior at large density e Jell ( ρ ) = c TF ρ 5 / 3 − c D ρ 4 / 3 + o ( ρ 4 / 3 ) ρ →∞ (Graf-Solovej ’94) Jell ( ρ ) = c Foldy ρ 5 / 4 + o ( ρ 5 / 4 ) ρ →∞ (Lieb-Solovej ’01–06) e bos | γ 0 ( x ) | 2 f bos/fer ( T , ρ ) = f bos/fer ( T , ρ ) ± ρ ´ dx + o ( ρ 4 / 3 ) ρ →∞ (Seiringer ’06) Jell free R 3 2 | x | β → 0 Mathieu LEWIN (CNRS / Paris-Dauphine) Coulomb gases 9 / 23
Riesz gases Adding background ≡ second-order Taylor expansion ≡ screening Background works for any interaction potential w ( x ) ∼ ∞ | x | − s for d − 2 ≤ s < d Riesz gases 1 for s � = 0, s | x | s w s ( x ) = − log | x | for s = 0. s > d (short range): well defined thermodynamics without background d − 2 ≤ s < d (long range): background necessary s < d − 2: background does not screen enough, unstable = ⇒ natural family in statistical mechanics, even includes hard spheres ( s → ∞ ) � s d / T s , Γ = ρ (classical) parameters: s , T , ρ (quantum) Mathieu LEWIN (CNRS / Paris-Dauphine) Coulomb gases 10 / 23
Crystallization conjecture & analytic number theory Crystallization conjecture T = 0 in classical case (Blanc-Lewin, EMS Rev. ’15) 2D: Riesz gas crystallized on hexagonal lattice ∀ s ≥ 0 3D: on BCC lattice for 1 ≤ s ≤ 3 / 2 and FCC lattice for s ≥ 3 / 2 If particles on lattice L and s > d , then s × energy = 1 1 � | ℓ | s = ζ L ( s ) = Epstein Zeta Function 2 ℓ ∈L\{ 0 } which admits analytic extension on C \ { d } , with pole at d independent of L 2D: Epstein is minimal for hexagonal lattice ∀ s ≥ 0 (Rankin ’53, Cassels ’59, Ennola ’64, Diandana ’64, Montgomery ’88) 3D: BCC-FCC conjectured, FCC known for s ≫ 1 Theorem (Analytic extension) If crystallization on lattice L , then e Jel ( s ) = ζ L ( s ) / s. At T = 0 , classical Jellium = analytic extension in s of long range case! (Borwein-Borwein-Shail ’89, Borwein-Borwein-Straub ’14, Lewin-Lieb ’15) Mathieu LEWIN (CNRS / Paris-Dauphine) Coulomb gases 11 / 23
1D Riesz gases ◮ s = − 1 : Crystallization at all T ≥ 0 and all ρ > 0, classical and quantum (Kunz ’74, Aizenman-Martin ’80, Brascamp-Lieb ’75, Jansen-Jung ’14) ◮ T = � = 0 : Crystallization for all 1 � = s ≥ 0 (Nijboer-Ventevogel ’79, Borodin-Serfaty ’13, Sandier-Serfaty ’14, Brauchart-Hardin-Saff ’12, Lebl´ e ’16) ◮ � = 0 : No breaking of translations for T > 0 and s ≥ 0 (Fr¨ ohlich-Pfister ’81, Baus ’80, Alastuey-Jancovici ’81, Chakravarty-Dasgupta ’81, Martinelli-Merlini ’84, Requardt-Wagner ’90) T long range short range B 1 K 2 T s − 1 0 1 crystal log Coulomb Sine- β Mathieu LEWIN (CNRS / Paris-Dauphine) Coulomb gases 12 / 23
1D classical Riesz gas at s = 0 N -particle classical probability density on S 1 at β = 1 / T ≡ exact quantum ground state 2 of N bosons with ∼ g π 2 / r 2 interactions, β = 1 + √ 1 + 2 g (Sutherland ’72, Forrester ’84) Haldane’s formula 2 ( r ) = − T a m � r 4 Tm 2 cos(2 π mr ) + o ( r − 2 ) r →∞ ρ T π 2 r 2 + m ≥ 1 (Haldane, Phys. Rev. Lett. ’81) BKT-type transition − T T > 1 / 2 π 2 r 2 ρ T 2 ( r ) ∼ a 1 r →∞ r 4 T cos(2 π r ) T < 1 / 2 (Forrester ’84) Mathieu LEWIN (CNRS / Paris-Dauphine) Coulomb gases 13 / 23
2D Riesz gas ( s = 0 , 1 ) ◮ Mermin-Wagner does not apply, but seems valid (Baus ’80, Alastuey-Jancovici ’81, Martinelli-Merlini ’84, Requardt-Wagner ’90) . Common belief: “solid” phase with (algebraic) quasi-long-range positional order and long range orientational order intermediate hexatic phase (Kosterlitz-Thouless-Halperin-Nelson-Young) (Muto-Aoki ’99, He-Cui-Ma-Liu-Zou ’03) ◮ For many years, computer simulations indicated a 1st order solid-fluid transition (Gann-Chakravarti-Chester ’78, de Leeuw-Perram ’82, Caillol-Levesque-Weis-Hansen ’82) 2D boltzons with 1 / r interaction ( s = 1) Clark-Casula-Ceperley, Phys. Rev. Lett ’09 Mathieu LEWIN (CNRS / Paris-Dauphine) Coulomb gases 14 / 23
Mean-field limit for confined Riesz gases Forget background and put external confining potential N V ext ( x j ) + 1 � � E V ext ( x 1 , ..., x N ) = w ( x j − x k ) N j =1 1 ≤ j < k ≤ N Bonitz et al, Phys. Plasma ’08 Theorem (Mean-field limit) � ˆ � min x j E V ext ( x 1 , ..., x N ) R d V ext d µ + 1 ¨ N →∞ min − → R 2 d w ( x − y ) d µ ( x ) d µ ( y ) N 2 µ proba � � − E Vext ´ log R dN exp � ˆ T − T N →∞ min − → R d V ext d µ N µ proba � + 1 ¨ ˆ R 2 d w ( x − y ) d µ ( x ) d µ ( y ) + T R d µ log µ 2 (Messer-Spohn ’82, Kiessling ’89, ..., Pecot lectures by Rougerie ’14) Rmk. with other convention E V ext � N E V ext , effective temperature T / N → 0 Mathieu LEWIN (CNRS / Paris-Dauphine) Coulomb gases 15 / 23
X The next order × N 1 /d µ MF ( X ) X N � δ N 1 /d ( x j − X ) ⇀ µ Jellium N 1 � δ x j ⇀ µ MF j =1 N j =1 Theorem (2nd order) Assume w =Riesz with max(0 , d − 2) ≤ s < d . If N 1 − s d T → T 0 ∈ [0 , ∞ ) then � − E V ext � = N e MF − δ 0 ( s ) ˆ − T log R dN exp log N T d � ˆ � � ˆ � T 0 − δ 0 ( s ) � � s s R d f cl d ) + N s , T 0 , µ MF ( x ) dx + R d µ MF log µ MF + o ( N d Jell d [Weak CV of states holds as well] Sandier-Serfaty ’14, Borodin-Serfaty ’13, Petrache-Serfaty ’15, Rougerie-Serfaty ’14, Lebl´ e-Serfaty ’17, Bauerschmidt-Bourgade-Nikula-Yau ’17,... Rmk. Also true for s > d ! Should not matter that V confining and w =Riesz Mathieu LEWIN (CNRS / Paris-Dauphine) Coulomb gases 16 / 23
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