Microscopic Derivation of GinzburgLandau Theory Robert Seiringer - PowerPoint PPT Presentation

Microscopic Derivation of Ginzburg–Landau Theory Robert Seiringer IST Austria Joint work with Rupert Frank, Christian Hainzl, and Jan Philip Solovej J. Amer. Math. Soc. 25 (2012), no. 3, 667–713 Mathematics and Quantum Physics Rome, July 8–12, 2013 R. Seiringer – Microscopic Derivation of Ginzburg–Landau Theory – July 11, 2013 Nr. 1

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. • The asymptotic limit may be seen as a semiclassical limit and one of the main difficulties is to derive a semiclassical expansion with minimal regularity as- sumptions . • 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 Theory – July 11, 2013 Nr. 2

Superconductivity and Superfluidity Superconductivity is the phenomenon that certain materials have zero electrical resis- tance below a critical temperature . This 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 , . . . The 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 Theory – July 11, 2013 Nr. 3

The Ginzburg–Landau model Let C ⊂ R 3 be a compact set and let A and W be vector and scalar potentials on C . Set ∫ [ B 1 | ( − i ∇ + 2 A ( x )) ψ ( x ) | 2 + B 2 W ( x ) | ψ ( x ) | 2 − B 3 D | ψ ( x ) | 2 + B 4 | ψ ( x ) | 4 ] E GL D ( ψ )= dx C Here, B 1 , B 3 , B 4 > 0 , B 2 ∈ R and D ∈ R are coefficients. E GL = inf ψ E GL D ( ψ ) Ginzburg–Landau energy D A minimizing ψ describes the macroscopic variations in the superfluid density . The 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 Theory – July 11, 2013 Nr. 4

The 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 α ¯ 1 − ¯ γ Here, 0 ≤ γ ≤ 1 is the 1-particle density matrix, and α the Cooper-pair wavefunction . [( ) ] ( − ih ∇ + hA ( x )) 2 − µ + h 2 W ( x ) F BCS (Γ) := Tr γ + T Tr Γ ln Γ T ∫∫ C× R 3 V ( h − 1 ( x − y )) | α ( x, y ) | 2 dx dy + Again C = [0 , 1] 3 , Γ is periodic and Tr stands for the trace per unit volume . F BCS = inf Γ F BCS (Γ) BCS energy T T The 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 Theory – July 11, 2013 Nr. 5

Remarks about the BCS model [( ) ] ( − ih ∇ + hA ( x )) 2 − µ + h 2 W ( x ) F BCS (Γ) = Tr γ + T Tr Γ ln Γ T ∫∫ C× R 3 V ( h − 1 ( x − y )) | α ( x, y ) | 2 dx dy + • 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 T , 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 . • Technical assumptions : V real-valued, V ( x ) = V ( − x ) and V ∈ L 3 / 2 ( R 3 ) W and A periodic and � W ( p ) , | � A ( p ) | (1 + | p | ) ∈ ℓ 1 R. Seiringer – Microscopic Derivation of Ginzburg–Landau Theory – July 11, 2013 Nr. 6

The normal state Let us first discuss the non -superfluid case, i.e., (( γ )) 0 0 ≤ γ ≤ 1 F BCS inf = 0 ≤ γ ≤ 1 { Tr Hγ + T Tr ( γ ln γ + (1 − γ ) ln(1 − γ )) } inf T 0 1 − γ ( 1 + e − H/T ) = − T Tr ln with H = ( − ih ∇ + hA ( x )) 2 + h 2 W ( x ) − µ . This infimum is attained iff Γ is the normal state ( γ normal ) ( 1 + e H/T ) − 1 0 Γ normal T γ normal = , = . T T 1 − γ normal 0 T Order of magnitude of free energy: By Weyl’s law , ∫ ( 1 + e − H/T ) ( ) T R 3 ln(1 + e − p 2 /T ) dp F BCS Γ normal = − T Tr ln ∼ − as h → 0 . T T (2 πh ) 3 R. Seiringer – Microscopic Derivation of Ginzburg–Landau Theory – July 11, 2013 Nr. 7

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

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

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

The coefficients in the GL functional [� ] ∫ � 2 � � + B 2 W | ψ | 2 − B 3 D | ψ | 2 + B 4 | ψ | 4 E GL 1 / 2 ( − i ∇ + 2 A ) ψ D ( ψ )= � B 1 � dx C Let t be the Fourier transform of 2 K T c α 0 = − 2 V α 0 , where ∥ α 0 ∥ 2 = 1 . Let g 1 ( z ) = e 2 z − 2 ze z − 1 g 2 ( z ) = g ′ , 1 ( z ) + 2 g 1 ( z ) /z . z 2 (1 + e z ) 2 Then the matrix B 1 and the numbers B 2 , B 3 and B 4 are given by ( β c = T − 1 ) c ∫ R 3 t ( p ) 2 ( ) ( B 1 ) ij = β 2 dp δ ij g 1 ( β c ( p 2 − µ )) + 2 β c p i p j g 2 ( β c ( p 2 − µ )) c (2 π ) 3 , B 1 > 0 16 ∫ B 2 = β 2 dp R 3 t ( p ) 2 g 1 ( β c ( p 2 − µ )) c (2 π ) 3 , 4 ∫ R 3 t ( p ) 2 cosh − 2 ( ) B 3 = β c dp 2 ( p 2 − µ ) β c (2 π ) 3 , B 3 > 0 , 8 ∫ R 3 t ( p ) 4 g 1 ( β c ( p 2 − µ )) B 4 = β 2 dp c (2 π ) 3 , B 4 > 0 . p 2 − µ 16 R. Seiringer – Microscopic Derivation of Ginzburg–Landau Theory – July 11, 2013 Nr. 11

Main results: Asymptotics of the critical temperature For every ψ ∈ H 1 per ( C ) , [� ] ∫ � 2 � � + B 2 W | ψ | 2 − B 3 D | ψ | 2 + B 4 | ψ | 4 � B 1 / 2 E GL D ( ψ )= ( − i ∇ + 2 A ) ψ dx � 1 C is an affine-linear, non-increasing function of D with E GL D (0) = 0 . Thus, E GL is a non-positive, non-increasing and concave function of D . Let D D c = sup { D ∈ R : E GL = 0 } D = inf { D ∈ R : E GL < 0 } D = B − 1 inf spec (( − i ∇ + 2 A ) B 1 ( − i ∇ + 2 A ) + B 2 W ) 3 Corollary 1. T c ( h ) − T c T c ( h ) − T c lim = lim = − D c T c h 2 T c h 2 h → 0 h → 0 R. Seiringer – Microscopic Derivation of Ginzburg–Landau Theory – July 11, 2013 Nr. 12