BCS PRELIMINARY NOTES (IN PROGRESS) BY L. G. MOLINARI

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1 BCS PRELIMINARY NOTES (IN PROGRESS) BY L. G. MOLINARI 1. The BCS Hamiltonian In electronic systems the low temperature properties are determined by the longlived quasi-particles in an energy shell B T near the Fermi surface. Because of the Debye cutoff, the interaction mediated by phonons v 2 s 2 U ph (,ω) = g ω 2 vs 22θ(ω D v s ) isattractiveintheenergyshell ǫ ǫ F < ω D andthis, atlowenoughtemperatures, leads to the formation of Cooper pairs with binding energy ω D exp( 2/gρ F ) characterising a superconductive phase, with critical temperature B T C. Cooper obtained this important result (1956) by solving a 2-particle problem in presence of a filled Fermi sea 3. The BCS theory (1957) 1 is a full many-electron model, characterized by the attractive interaction that arises in the static limit, gδ(x x ), which captures the essence 2: (1) ˆK = dx δ µν (ˆψ µ x ˆψν )(x) g 2 (ˆψ µ ˆψ ν ˆψ ν ˆψµ )(x) µν where x = 1 2m (p+ e c A)2 +U(x) µ and the Debye cut-off is understood for the interaction. By the exclusion principle, only two spin configurations are allowed, and are equivalent: ˆψ ˆψ ˆψ ˆψ = ˆψ ˆψ ˆψ ˆψ. The quartic interaction gets simplified by replacing pairs of operators with their mean values, ˆψ ˆψ ˆψ ˆψ ˆψ ˆψ ˆψ ˆψ + ˆψ ˆψ ˆψ ˆψ This introduces a complex field, which behaves as an order parameter that can be related to the Ginzburg Landau field: (2) (x) = g ˆψ (x)ˆψ (x) The thermal average is calculated with the effective Hamiltonian, that no longer conserves the number of electrons (3) ˆK eff = ˆK 0 + dx (x) ˆψ (x)ˆψ (x)+ ˆψ (x)ˆψ (x) (x). Date: 19 dec At the time, Leon Cooper and Robert Schrieffer were respectively post-doc and graduate student of John Bardeen. Read the nice hystorical account by Hoddeson 7 1

2 2 PRELIMINARY NOTES (IN PROGRESS) BY L. G. MOLINARI 1.1. The Hartree approximation. To gain some understanding of the approximation, let ˆK = ˆK 0 + ˆK 1, where ˆK 1 is the quartic term, and consider the (thermal) interaction picture. A thermal average of field operators is T ˆψ(x 1 )ˆψ(x 2 )... K = T U I( β,0)ˆψ(x 1 )ˆψ(x 2 )... K0 U I ( β,0) K0. where x = (x, τ). Consider the discretization of Dyson s T-product expansion: ( U I ( β,0) = T exp 1 ) g (ˆψ ˆψ )(x+ )(ˆψ ˆψ )(x) x = T exp + g (ˆψ ˆψ )(x+ )(ˆψ ˆψ )(x) x where, because with time-ordering, the quadratic operators commute. The fourfermion interaction may be splitted with the introduction of an auxiliary complex field (x). At each point x the following complex integral applies g exp AB = d 2 z πg exp 1 ( 1 g z 2 +za+bz With z = (x), we obtain a product of integrals which defines a Gaussian functional integral, where all pairs of operators commute because of T -ordering: U I ( β,0) =T d 2 (x) exp 1 πg g (x) 2 1 ( ˆψ ˆψ + ˆψ ˆψ )(x) x = 1 D T exp 1 Z S, S = dx( 1 g 2 + ˆψ ˆψ + ˆψ ˆψ ), Z = D exp 1 g dx (x) 2. The partition function is Z = Z 0 U I ( β,0) K0, with Z 0 = tr(e βk0 ) and U I ( β,0) K0 = 1 D T e 1 S, (4) Z K 0 Now comes the approximation: the main contribution to the functional integral comesfromtheauxiliaryfield (x)thatmaximisestheboltzmannweight T e S/. The extremum condition for a variation δ is an equation for (x). By retaining only linear terms: ) T exp( 1 S, +δ ) K0 = T exp( 1 S, ) 1+ dxδ (x) g 1 (x)+ ˆψ (x)ˆψ (x) +... K0 The first variation is zero for (5) (x) = g Te 1 S, ˆψ (x)ˆψ (x) K0 e 1 S, K0 This is an equation for (x), which appears on both sides(time-dependence cancels because of equal times). The integral (6) simplifies: (6) U I ( β,0) K0 N T e 1 S, K 0 = e β ˆK 0 e β ˆK eff

3 BCS 3 N is a normalization factor. The equation for corresponds to eq.(2) with the effective Hamiltonian (7) (see Zagosin, 10). Exercise 1.1. Prove that, if A and B commute, then dxdy exp( 1 g z 2 +za+bz) = πgexp(gab), z = x+iy 1.2. Matrix formulation. In the operator ˆK 0 an integration by parts and an anticommutation bring ˆψ ˆψ x to ˆψ xˆψ up to a constant2. Then: ˆK eff = dx ˆψ x ˆψ + ˆψ x ˆψ + ˆψ ˆψ + ˆψ ˆψ The Hamiltonian is now written in a matrix form introduced by Nambu 8: (7) ˆK eff = dxψ (x)(k x Ψ)(x) (8) K x = x (x) (x) x ˆψ (x), Ψ(x) = ˆψ (x), Ψ (x) = ˆψ (x), ˆψ (x) The components of Ψ and Ψ anticommute (note that (Ψ r ) = (Ψ ) r ): (9) {Ψ r (x),ψ s (y)} = δ rsδ 3 (x y), {Ψ r,ψ s } = {Ψ r,ψ s } = 0 As the effective Hamiltonian is quadratic in the fields, the model can be solved lie a theory of independent particles or a Hartree theory, with the self-consistency eq.(2), named gap equation. Two equivalent approaches are presented: one, by de Gennes, generalises the canonical transformation introduced by Bogoljubov and Valatin (1958) for homogeneous systems; the other one is based on Green functions, introduced by Gor ov in and here expressed with Nambu s matrix formalism The Bogoljubov - de Gennes equations. The matrix operator K x acts on the Hilbert space L 2 (R 3 ) C 2 and is self-adjoint. It has real eigenvalues, and the eigenvectors form an orthonormal basis. The eigenvalue equation (10) x (x) (x) x ua (x) v a (x) gives the pair of Bogoljubov - de Gennes equations: = E a ua (x) v a (x) (u a )(x)+ (x)v a (x) = E a u a (x) (v a )(x) (x)u a (x) = E a v a (x) If (u a, v a ) solve them with eigenvalue E a > 0, then ( v a, u a ) are a solution with eigenvalue E a. The equations (10) with eigenvalues ±E a may be written jointly: ua (x) v (11) K a (x) ua (x) v x = a (x) Ea 0 v a (x) u a (x) v a (x) u a (x) 0 E a 2 The operators x and x differ by the sign of the term linear in p, if any.

4 4 PRELIMINARY NOTES (IN PROGRESS) BY L. G. MOLINARI The ortho-normalization and completeness of the doublets in Hilbert space may be expressed in matrix form: ub (x) v dx b (x) ua (x) v a (x) (12) = δ v b (x) u b (x) v a (x) u a (x) ab I 2 ua (x) v a (x) ua (y) v a (y) (13) = δ(x y) I v a (x) u a (x) v a (y) u a (y) 2 a 1.4. Diagonalization of the many-body Hamiltonian. The matrix relation (11) suggests that the many body Hamiltonian is diagonalized by the following transformation to new operators: ˆψ (x) ˆψ (x) (14) = a This and the adjoint are, in detail: (15) ˆψ (x) = a ua (x) v a (x) v a (x) u a (x) u a (x)ˆα a v a (x)ˆβ a, ˆψ (x) = a ˆαa ˆβ a u a (x)ˆα a v a (x)ˆβ a (16) ˆψ (x) = a v a (x)ˆα a +u a (x)ˆβ a, ˆψ (x) = a v a (x)ˆα a +u a (x)ˆβ a Inversion is done with the aid of (12): ˆαa ua (x) v (17) ˆβ a = dx a (x) v a (x) u a (x) ˆψ (x) ˆψ (x) The adjoint operators are also obtained. The transformation is canonical i.e. the new operators have canonical anticommutation relations: (18) {ˆα a, ˆα b } = δ ab, {ˆβ a, ˆβ b } = δ ab (all other anticommutators vanish). By eq.(11) (K x Ψ)(x) = ua (x) v a (x) Ea 0 ˆαa v a (x) u a (x) 0 E a ˆβ a a Evaluationof ˆK eff = dxψ KΨand(12)giveadiagonaloperatorforquasiparticles (bogolons): (19) ˆKeff = U 0 + a E a ˆα aˆα a + ˆβ aˆβ a whereu 0 = a E a. Thegroundstateisdefinedby ˆα a BCS = 0and ˆβ a BCS = 0 for all a The gap equation. The change of basis (15) simplifies the gap equation: (x) = g ab u a(x)v b (x) ˆα aˆα b v a (x)u b (x) ˆβ aˆβ b = g a u a(x)v a (x)1 2n(E a ) wheren(e a ) is the Fermi-Diracoccupationnumberofthe statewith energy E a. Then: (x) = g ( ) β (20) u a (x)v a (x)tanh 2 E a a The equation must be solved self-consistently with the Bogoljubov - de Gennes equations for u a and v a.

5 BCS 5 Remar 1.2. As the gap function depends on temperature, the amplitudes u a, v a as well as the energies E a and BCS depend on T. Exercise 1.3. Show that Ω = 2 β a log( 2cosh 1 2 βe a). Exercise 1.4. Show that the average density of electrons is: (21) n(x) = a u a (x) 2 n a + v a (x) 2 (1 n a ), n a = 1 e βea Nambu - Gorov theory. There are advantages in studying the BCS model with the Green function formalism. The imaginary time evolution of operators is O(τ) = e τk/ Oe τk/, where K is the effective hamiltonian (7). The equation of motion of Ψ(x,τ) is: τ Ψ r(x,τ) =e 1 τk Ψ r (x),ke 1 τk =e 1 τk dx Ψ r (x),ψ s (x )(K s sψ s )(x )e 1 τk =e 1 τk dx {Ψ r (x),ψ s (x )}(K s sψ s )(x )e 1 τk =(K rs Ψ s )(x,τ) Let us introduce the thermal Nambu propagator (22) G(x,x ) = T Ψ(x)Ψ (x ) It is a matrix with components G(x,x T ψ (x)ψ ) = (x ) T ψ (x)ψ (x ) T ψ (x)ψ (x ) T ψ (x)ψ (x = ) G(x,x ) F(x,x ) F (x,x ) G(x,x) Note the sign and the exchange of x and x in one component. The correlators F and F are named anomalous and vanish in the normal phase. In particular: (23) (x) = gf(x,x + ) The equation of motion of the Nambu propagator, (24) τ +K x G(x,x ) = δ 4 (x x ) I 2 simplifies in Matsubara (odd) frequency space: i ωn + x (x) (25) G(x,x ;iω (x) i ω n n ) = δ 3 (x x )I 2 x G(x,x G(x,x,iω n ) =,iω n ) F(x,x,iω n ) (26) F (x,x,iω n ) G(x,x, iω n ) Exercise 1.5. The propagators can be represented as expansions in the Bogoljubov - de Gennes eigenstates. Show that: (27) (28) G(x,x,iω n ) = a F(x,x,iω n ) = a u a (x)u a (x ) iω n E a / + v a(x)v a (x ) iω n +E a / u a(x)v a (x ) iω n E a / + v a(x)u a (x ) iω n +E a /

6 6 PRELIMINARY NOTES (IN PROGRESS) BY L. G. MOLINARI and recover the gap equation (20) by evaluating the Matsubara sum (x) = g 1 F(x,x,iω n )e iωnη β n 1.7. Perturbative expansion. When = 0, eq.(25) is solved by the normal Nambu propagator G n (x,x ), which can be used to transform (25) into a Dyson equation (integration and summation of repeated variables is implicit): G(x,y) = G n (x,y)+ 1 0 (29) G n(x,x )D(x )G(x,y), D(x) = 0 In BCS model the self-energy D is local and time-independent. When this description is inadeguate, one has to consider a microscopic model with the actual phonon-electron interaction. The Dyson s equation becomes (30) G(x,y) = G n (x,y)+g n (x,x )S(x,x )G(x,y). where S is a non-local self-energy matrix. In a 1-phonon exchange approximation, S(x,y) = 1 G(x,y)U0 ph (x y), the coupled equations for G and F are: G(x,y) = G n (x,y)+g n (x,x )S 11 (x,x )G(x,y)+G n (x,x )S 12 (x,x )F(x,y) F(x,y) = G n (x,x )S 12 (x,x )G(y,x )+G n (x,x )S 11 (x,x )F(x,y). with the addition of the gap equation. 2. Homogeneous systems In homogeneous problems there is no external field and is constant. An analytic solution is found in momentum space The Bogoljubov - Valatin canonical transformation. We see for a solution of the Bogoljubov - de Gennes equations of the form u (x) ei x u (31) = v (x) V v Then ξ ξ u v = E u v where ξ = ǫ µ are the single-particle energies (normal phase) measured with respect to the chemical potential. The homogeneous system admits a nontrivial solution if (32) E = ξ (the positive root is selected for stability). The energy gap separating the Fermi surface ξ = 0 from the lowest excitation, profoundly modifies the properties of the electron gas at low temperatures. The amplitudes solve the normalization condition u 2 + v 2 = 1 and the condition ξ u + v = E u. The latter gives v = (E ξ ) u, with solutions u 2 = 1 ( 1+ ξ ), v 2 = 1 ( 1 ξ ) (33) 2 E 2 E

7 BCS Figure 1. The parameters u 2 and v 2 as functions of ξ for = 0.1. In the normal phase E = ξ ; then: u = θ(ǫ µ) and v = θ(µ ǫ ). The equation ξ u + v = E u gives v 2 = (E ξ )u v i.e. the useful relation: (34) u v = 2E The expansion of the field operators in the two canonical basis, ˆψ (x) ˆψ (x) = e i x â, V â = e i x u v ˆα, V v u ˆβ implies the Bogoljubov - Valatin transformation: â, u v â = ˆα (35), v u ˆβ and the Hermitian conjugate. Inversion gives: (36) (37) ˆα = ū â, + v â,, ˆβ = v â, +ū â,, ˆα = u â, +v â, ˆβ = v â, +u â, The operators ˆα and ˆβ annihilate, for all vectors, the state (38) BCS = (ū + v a a ) 0 which reads as a sea of Cooper pairs 3. In the normal phase ( = 0) it coincides with the filled Fermi sphere. ˆα BCS = q (ū q + v q â q â q )ˆα (ū + v â â ) 0 = q (ū q + v q â q â q )(ū v â, â â + v ū â, ) 0 = 0 3 In 2 Bardeen, Cooper and Schrieffer (1957) introduced the state with variational parameters u and v with u 2 + v 2 = 1 for normalization. Minimization of BCS ˆK eff BCS with respect to the parameters yields the same results presented here. Bogoljubov and Valatin independently simplified the theory by their canonical transformation 1, 9.

8 8 PRELIMINARY NOTES (IN PROGRESS) BY L. G. MOLINARI ˆβ BCS = q (ū q + v q â q â q )ˆβ (ū + v â â ) 0 = q (ū q + v q â q â q )( v ū â, +ū v â â â, ) 0 = 0 Creation operators create excited states (bogolons) consisting of Cooper pairs and unpaired electrons. For example: ˆα BCS = q (ū q + v q â q â q )â 0 ˆβ BCS = q (ū q + v q â q â q )â 0 ˆα β BCS = ū (ū q + v q â q â q )â â 0 q 2.2. The gap equation. By means of (34) the gap equation (20) becomes: = g 1 ( ) β tanh V 2E 2 E θ( ω D ξ ) By introducing the density of states per unit volume and single spin component of the normal phase, ρ n (ξ) = 1 δ(ξ ξ ) V the sum in -space is changed into an integral in energy, 1 = g dξρ n (ξ) tanh(1 2 β ξ ) θ( ω D ξ ) 2 ξ2 + 2 With the assumption that the density is almost constant in the energy shell ξ < ω D it simplifies to: (39) 1 ωd gρ n (0) = dξ tanh(1 2 β ξ ) 0 ξ2 + 2 gap equation where ρ(0) is the density of states at the Fermi energy, g is the squared coupling constant of the phonon to the electron. According to the microscopic theory: z c ni g = π 2 e 2 a 0 v s M i F z c is the number of conducting electrons per ion, n i is the number of ions per unit volume, M i is the ionic mass, v s is the speed of sound. Exercise 2.1. Show that the density of states per unit volume and spin component ofthe freeelectrongasatthefermienergyisρ n (0) = 3 4 n/e F, wherenisthe density of electrons and E F is the Fermi energy. Then show that gρ n (0) = z c 6 m M i ( ) 2 vf where m is the electron s mass and v F is the Fermi velocity. v s

9 BCS The Green functions. In -space the equation of motion(25) for the Nambu propagator is algebraic i ωn ξ G(;iω i ω n +ξ n ) = I 2 i ωn +ξ G(;iω n ) = 2 ωn 2 +ξ2 + 2 i ω n ξ The normal and anomalous propagators are obtained, with E = ξ : G(,iω n ) = i ω n +ξ 2 ωn 2 = +E2 F(,iω n ) = 2 ωn 2 = +E2 The Matsubara sum in the gap equation = g β yields the expression (39). n u 2 iω n (E / ) + v 2 u v iω n (E / ) d 3 (2π) 3 F(,iω n)e iωnη iω n +(E / ) u v iω n +(E / ) Example 2.2. Show that the average number of electrons in a state (,σ) is n = 1 G(,iω n ) = 1 β 2 1 ξ tanh βe (40) 2E 2 n Exercise 2.3 (spectral density). Evaluate the spectral density of the superconducting phase ρ s (E) = u 2 δ(e E )+ v 2 δ(e +E ) (use the approximation µ). Note the presence of an energy gap of width 2 centred at E = 0 (chemical potential). (41) (42) ρ s (E) ρ n (0) = ρ n (E) = 1 4π 2 E+ E 2 2 E2 2 E E 2 2 E E µ + E E 2 2 E2 2 1 E µ µ < E < 0 E < 1 E µ + E+ E 2 2 E 1+ E 2 2 µ < E < µ 2 1+ E µ µ < E ( )3 2m 2 E +µ 2 Near the gap ρ s (E) 2ρ n (0) E / E Discussion of the gap equation. T = T c. At the critical temperature the order parameter is zero, and the gap equation is an equation for T c : 1 gρ(0) = ωd 0 logx c dξ ξ tanh(1 2 β cξ) = 0 xc 0 dx xc x tanhx = tanh(x c)log(x c ) dx logx 0 cosh 2 x dx logx cosh 2 x = logx c +log(4e C /π), x c = ω D 2 B T c = T D 2T c

10 10 PRELIMINARY NOTES (IN PROGRESS) BY L. G. MOLINARI Figure 2. The spectral density (µ = 20, = 0.6). The thin line is the square root E of the normal phase. The gap is centred on the chemical potential. The approximations are justified by T D /T c 1. With C , the result is ( (43) B T c = ω D exp 1 ) gρ(0) T = 0. The gap equation becomes: with solution (44) 0 = ω D sinh 1 gρ(0) 1 ωd gρ(0) = 1 dξ 0 ξ The following universal ratio is then obtained: ( 2 ω D exp 1 ) gρ(0) (45) 0 B T c = πe C The Ginzburg - Landau limit of BCS The Ginzburg-Landau theory can be derived from the microscopic BCS model. NearthetransitionlineH = H c (T),thefunction issmall,andthedysonequation (29) for G(x,y;iω n ) can be solved by iteration: G = G n + 1 G ndg n + 1 2G ndg n DG n + 1 3G ndg n DG n DG n The truncation to third order in evaluates the anomalous correlator F(x,y,iω n ) and the Green function G(x,y,iω n ) in terms of the normal Green function and the

11 BCS 11 gap function: (46) (47) F(1,2,iω n ) = 1 G n(1,3,iω n ) (3)G n (2,3, iω n ) G n(1,3,iω n ) (3)G n (4,3, iω n ) (4)G n (4,5,iω n ) (5)G n (2,5, iω n ); G(1,2,iω n ) = G n (1,2,iω n ) 1 2 G n(1,3,iω n ) (3)G n (4,3, iω n ) (4)G n (4,2,iω n ) The space variables 3,4,5 are integrated. G n are the normal Green functions for independent particles in a static magnetic field. The equations are the starting point for Gor ov s derivation 6 of the two Ginzburg-Landau equations (1959). Eq.(46) with 2 = 1 + and summation of Matsubara frequencies, is a cubic expansion of the gap equation for, and provides the first G.L. equation with order parameter ψ : (48) 1 (1) = Q(1,2) (2)+R(1,2,3,4) (2) (3) (4) g with weight functions Q(1,2) = 1 2 G n (1,2,iω n )G n (1,2, iω n ) β n R(1,2,3,4) = 1 4 β G n (1,2,iω n )G n (3,2, iω n )G n (3,4,iω n )G n (1,4, iω n ) n Eq.(47) is an expansion for the Green function, that is used to to evaluate the super-current, and yields the second G.L. equation. The derivation of G.L. equations relies crucially on the large difference among the length scales involved. We need some preliminaries.... References 1 N. N. Bogoljubov, On a new method in the theory of superconductivity, Nuovo Cim. 7 n.6 (1958) J. Bardeen, L. N. Cooper and J. R. Schrieffer, Theory of superconductivity, Phys, Rev. 108 (1957) L. N. Cooper, Bound electron pairs in a degenerate Fermi gas, Phys. Rev. 104 (1956) P. G. de Gennes, Superconductivity of metals and alloys, Benjamin, L. P. Gor ov, On the energy spectrum of superconductors, Sov. Phys. JETP 34(7) n.3 (1958) L. P. Gor ov, Microscopic derivation of the Ginzburg-Landau equations in the theory of superconductivity, Sov. Phys. JETP 36(9) n.6 (1959) L. Hoddeson, John Bardeen and the theory of superconductivity: a study of insight, confidence, perseverance, and collaboration, J. Supercond. Nov. Magn. 21 (2008) Y. Nambu, Quasi-particles and gauge invariance in the theory of superconductivity, Phys. Rev. 117 n.3 (1960) J. G. Valatin, Comments on the theory of superconductivity, Nuovo Cim. 7 n.6 (1958) Alexander Zagosin, Quantum theory of many-body systems, Graduate Texts in contemporary physics, Springer Jian-Xin Zhu, Bogoliubov-de Gennes method and its applications, Lecture Notes in Physics 924, Springer 2016.