Microscopic Derivation of Ginzburg Landau Theory. Mathematics and Quantum Physics

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1 Microscopic Derivation of Ginzburg Landau heory Robert Seiringer IS Austria Joint work with Rupert Frank, Christian Hainzl, and Jan Philip Solovej J. Amer. Math. Soc. 25 (2012), no. 3, Mathematics and Quantum Physics Rome, July 8 12, 2013 R. Seiringer Microscopic Derivation of Ginzburg Landau heory July 11, 2013 Nr. 1

2 Abstract of the talk I will discuss how the Ginzburg Landau (GL) model of superconductivity arises as an asymptotic limit of the microscopic Bardeen Cooper Schrieffer (BCS) model. he asymptotic limit may be seen as a semiclassical limit and one of the main difficulties is to derive a semiclassical expansion with minimal regularity assumptions. It is not rigorously understood how the BCS model approximates the underlying many-body quantum system. I will formulate the BCS model as a variational problem, but only heuristically discuss its relation to quantum mechanics. R. Seiringer Microscopic Derivation of Ginzburg Landau heory July 11, 2013 Nr. 2

3 Superconductivity and Superfluidity Superconductivity is the phenomenon that certain materials have zero electrical resistance below a critical temperature. his is a quantum phenomenon on a macroscopic scale. A brief history of superconductivity: 1911 Onnes discovers superconductivity experimentally 1950 Ginzburg and Landau provide a phenomenological macroscopic model for superconductivity 1957 Bardeen, Cooper and Schrieffer propose a microscopic theory and introduce the concept of Cooper pairs 1959 Gor kov gives a derivation of GL theory from BCS theory In addition, important contributions from Bogoliubov, de Gennes,... he related phenomenon of superfluidity concerns fluids with zero viscosity. While originally discovered in liquid helium, it is currently being explored in experiments on ultracold atomic gases. R. Seiringer Microscopic Derivation of Ginzburg Landau heory July 11, 2013 Nr. 3

4 he Ginzburg Landau model Let C R 3 be a compact set and let A and W be vector and scalar potentials on C. Set [ ED GL (ψ)= B 1 ( i + 2A(x)) ψ(x) 2 + B 2 W (x) ψ(x) 2 B 3 D ψ(x) 2 + B 4 ψ(x) 4] dx C Here, B 1, B 3, B 4 > 0, B 2 R and D R are coefficients. Ginzburg Landau energy E GL D = inf ψ ED GL(ψ) A minimizing ψ describes the macroscopic variations in the superfluid density. he normal state corresponds to ψ 0, while ψ > 0 means superfluidity (or supercond.). Question: Is the optimal ψ 0 or not? For us, C = [0, 1] 3 and ψ satisfies periodic boundary conditions (torus) One is often interested in minimizing over both ψ and A, adding an additional field energy term. For us, A is fixed (but arbitrary). R. Seiringer Microscopic Derivation of Ginzburg Landau heory July 11, 2013 Nr. 4

5 he BCS model State of the system described by a 2 2 operator-valued matrix (op. in L 2 (R 3 ) L 2 (R 3 )) ( ) γ α Γ = with 0 Γ 1 ᾱ 1 γ Here, 0 γ 1 is the 1-particle density matrix, and α the Cooper-pair wavefunction. [( ) ] F BCS (Γ) := r ( ih + ha(x)) 2 µ + h 2 W (x) γ + r Γ ln Γ + V (h 1 (x y)) α(x, y) 2 dx dy C R 3 Again C = [0, 1] 3, Γ is periodic and r stands for the trace per unit volume. BCS energy F BCS = inf Γ F BCS (Γ) he normal state corresponds to α 0, while α > 0 describes Cooper pairs. Question: Is the optimal α 0 or not? R. Seiringer Microscopic Derivation of Ginzburg Landau heory July 11, 2013 Nr. 5

6 Remarks about the BCS model [( ) ] F BCS (Γ) = r ( ih + ha(x)) 2 µ + h 2 W (x) γ + r Γ ln Γ + V (h 1 (x y)) α(x, y) 2 dx dy C R 3 Can be heuristically derived from a many-body Hamiltonian for spin 1 2 fermions with two-body interaction V via two simplifications. First, one restricts to quasi-free states, and second one drops the direct and exchange term in the interaction energy. Microscopic data: chemical potential µ, temperature, interaction potential V Macroscopic data: vector magnetic potential A, scalar electric potential W What is h? It is the ratio of the microscopic and macroscopic scale. echnical assumptions: V real-valued, V (x) = V ( x) and V L 3/2 (R 3 ) W and A periodic and Ŵ (p), Â(p) (1 + p ) l1 R. Seiringer Microscopic Derivation of Ginzburg Landau heory July 11, 2013 Nr. 6

7 he normal state Let us first discuss the non-superfluid case, i.e., (( )) inf F BCS γ 0 = inf {r Hγ + r (γ ln γ + (1 γ) ln(1 γ))} 0 γ γ 0 γ 1 = r ln (1 ) + e H/ with H = ( ih + ha(x)) 2 + h 2 W (x) µ. his infimum is attained iff Γ is the normal state ) Γ normal = ( γ normal γ normal, γ normal = ( 1 + e H/ ) 1. Order of magnitude of free energy: By Weyl s law, F BCS ( ) Γ normal = r ln (1 ) + e H/ (2πh) 3 ln(1 + e p 2 / )dp as h 0. R 3 R. Seiringer Microscopic Derivation of Ginzburg Landau heory July 11, 2013 Nr. 7

8 he critical temperature Define c (h) := sup { 0 : F BCS c (h) := inf { 0 : F BCS < F BCS ( )} Γ normal = F BCS ( )} Γ normal Lemma 1. c :=lim h 0 c (h) =lim h 0 c (h) exists in [0, ) and is characterized by inf spec (K + V ) < 0 if 0 < c, inf spec (K + V ) 0 if c, where K = ( µ) coth(( µ)/2 ) in L 2 (R 3 ). Note that c does not depend on the macroscopic A or W. In the following, we shall assume that V and µ are such that c > 0, and that the eigenvalue 0 of K c + V is simple. his is satisfied, e.g., if V 0 (and 0). Let α 0 denote the normalized eigenfunction of K c + V corresponding to its eigenvalue 0. R. Seiringer Microscopic Derivation of Ginzburg Landau heory July 11, 2013 Nr. 8

9 Main results: asymptotics of energy and minimizers HEOREM 1. Fix D R and let = c (1 h 2 D). For appropriate B 1,..., B 4, with E GL D F BCS = F BCS (Γ normal ) + h ( ED GL + o(1) ) = inf ψ E GL D (ψ) and const. h2 o(1) const. h 1/5 for small h. HEOREM 2. If Γ is an approximate minimizer of F BCS in the sense that F BCS (Γ) F BCS (Γ normal ) + h ( ED GL + ϵ ) for some small ϵ > 0, then the corresponding α can be decomposed as α = h 2 ( ψ(x) α0 ( ih ) + α 0 ( ih )ψ(x) ) + σ at = c (1 h 2 D), with C R 3 σ(x, y) 2 dx dy const. h 3/5 and ED GL (ψ) ED GL + ϵ + const. h 1/5 R. Seiringer Microscopic Derivation of Ginzburg Landau heory July 11, 2013 Nr. 9

10 Remarks on energy asymptotics: F BCS = F BCS (Γ normal ) + h ( ED GL + o(1) ) F BCS (Γ normal ) Ch 3, hence GL theory gives an O(h 4 ) correction to the main term. For smooth enough A and W, one could also expand F BCS (Γ normal ) to order h. We bound directly the energy difference, however!... asymptotics of almost minimizers: hat is, α = h 2 ( ψ(x) α0 ( ih ) + α 0 ( ih )ψ(x) ) + σ α(x, y) = 1 2h 2 (ψ(x) + ψ(y)) α 0 ( x y h ) 1 + σ(x, y) h 2 ψ ( ) ( x+y x y ) 2 α0 h o appreciate σ(x, y) 2 dx dy const. h 3/5, note that for the main term h 2 ψ((x + y)/2)α 0 ((x y)/h) 2 dx dy = const. h 1. R. Seiringer Microscopic Derivation of Ginzburg Landau heory July 11, 2013 Nr. 10

11 ED GL (ψ)= he coefficients in the GL functional C [ B 1 1/2 ( i + 2A) ψ 2 + B 2 W ψ 2 B 3 D ψ 2 + B 4 ψ 4 ]dx Let t be the Fourier transform of 2K c α 0 = 2V α 0, where α 0 2 = 1. Let g 1 (z) = e2z 2ze z 1 z 2 (1 + e z ) 2, g 2 (z) = g 1(z) + 2g 1 (z)/z. hen the matrix B 1 and the numbers B 2, B 3 and B 4 are given by (β c = 1 (B 1 ) ij = β2 c 16 c ) t(p) 2 ( δ ij g 1 (β c (p 2 µ)) + 2β c p i p j g 2 (β c (p 2 µ)) ) dp R (2π) 3, B 1 > 0 3 B 2 = β2 c t(p) 2 g 1 (β c (p 2 dp µ)) 4 R (2π) 3, 3 B 3 = β ( ) c t(p) 2 cosh 2 βc dp 8 2 (p2 µ) R (2π) 3, B 3 > 0, 3 B 4 = β2 c t(p) 4 g 1(β c (p 2 µ)) dp 16 R p 2 µ (2π) 3, B 4 > 0. 3 R. Seiringer Microscopic Derivation of Ginzburg Landau heory July 11, 2013 Nr. 11

12 Main results: Asymptotics of the critical temperature For every ψ Hper(C), 1 [ ED GL (ψ)= B 1/2 1 ( i + 2A) ψ 2 + B 2 W ψ 2 B 3 D ψ 2 + B 4 ψ ]dx 4 C is an affine-linear, non-increasing function of D with ED GL (0) = 0. hus, E GL D is a non-positive, non-increasing and concave function of D. Let D c = sup{d R : E GL D = 0} = inf{d R : E GL D < 0} = B 1 3 inf spec (( i + 2A)B 1 ( i + 2A) + B 2 W ) Corollary 1. lim h 0 c (h) c c h 2 = lim h 0 c (h) c c h 2 = D c R. Seiringer Microscopic Derivation of Ginzburg Landau heory July 11, 2013 Nr. 12

13 Key Semiclassical Estimates For ψ H 2 loc (Rd ) and t sufficiently nice, let denote the operator = h 2 (ψ(x)t( ih ) + t( ih )ψ(x)) he effective Hamiltonian on L 2 (R d ) C 2 is ( ( ih + ha(x)) 2 µ + h H = 2 W (x) (ih + ha(x)) 2 + µ h 2 W (x) ) HEOREM 3. Let f(z) = ln (1 + e z ) and φ(p) = 1 2 p 2 µ tanh( β 2 (p2 µ)). hen h d ( ) β r [f(βh ) f(βh 0 )] = h 2 E 1 + h 4 E 2 + O(h 6 ) ψ 6 H 1 (C) + ψ 2 H 2 (C), for explicit E 1 and E 2. Moreover, the off-diagonal entry α of [1 + e βh ] 1 satisfies ( ) α h 2 (ψ(x)φ( ih ) + φ( ih )ψ(x)) H const. h3 d/2 ψ H2 (C) + ψ 3 1 H 1 (C) t(p) R. Seiringer Microscopic Derivation of Ginzburg Landau heory July 11, 2013 Nr. 13

14 Conclusions Rigorous derivation of Ginzburg-Landau theory, starting from the BCS model. For weak external fields and close to the critical temperature, GL arises as an effective theory on the macroscopic scale. he relevant scaling limit is semi-classical in nature. Some open problems: reat physical boundary conditions reat self-consistent magnetic fields Derive BCS theory from many-body quantum mechanics R. Seiringer Microscopic Derivation of Ginzburg Landau heory July 11, 2013 Nr. 14