Mathematical Analysis of the BCS-Bogoliubov Theory

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1 Mathematical Analysis of the BCS-Bogoliubov Theory Shuji Watanabe Division of Mathematical Sciences, Graduate School of Engineering Gunma University 1 Introduction Superconductivity is one of the historical landmarks in condensed matter physics Since Onnes found out the fact that the electrical resistivity of mercury drops to zero below the temperature 4K in 1911, the zero electrical resistivity is observed in many metals and alloys Such a phenomenon is called superconductivity, and the magnetic properties of superconductors as well as their electric properties are also astonishing For example, the magnetic flux is excluded from the interior of a superconductor This phenomenon was observed first by Meissner in 1933, and is called the Meissner effect In 1957 Bardeen, Cooper and Schrieffer [1] proposed the highly successful quantum theory called the BCS theory The superconducting state and the Hamiltonian they dealt with are called the BCS state and the BCS Hamiltonian, respectively In 1958 Bogoliubov [] obtained the results similar to those in the BCS theory using the canonical transformation called the Bogoliubov transformation This theory is called the Bogoliubov theory The ground state of the BCS Hamiltonian is discussed by several authors In 1961 Mattis and Lieb [5] studied the wavefunction of the ground state of the BCS Hamiltonian under the condition that in the ground state, all the electrons in the neighborhood of the Fermi surface are paired See Richardson [7] and von Delft [3] for the ground state of the BCS Hamiltonian without the condition just above From the viewpoint of C -algebra, Gerisch and Rieckers [4] studied a class of BCS-models to show that there is a unique C -dynamical system for each BCS-model In this paper, first, we reformulate the BCS-Bogoliubov theory of superconductivity from the viewpoint of linear algebra We define the BCS Hamiltonian on C M, where M is a positive integer We discuss selfadjointness and symmetry of the BCS Hamiltonian as well as spontaneous symmetry breaking Beginning with the gap equation, we give the well-known expression for the BCS state and find the existence of an energy gap We also show that the BCS state has a lower energy than the normal state Second, we introduce a new superconducting state explicitly and show from the viewpoint of linear algebra that this new state has a lower energy than the BCS state Third, beginning with our new gap equation, we show from the viewpoint of linear algebra that we arrive at the results similar to those in the BCS-Bogoliubov theory See Watanabe [8] for more details Let L, K max > 0 be large enough and let us fix them For n 1,n,n 3 Z, set Λ= π L n 1,n,n 3 ) R 3 : π L n 1 + n + n 3 K max Here we do not let K max = for simplicity Let the number of all the elements of Λ be M and let wave vector k belong to Λ 1 }

2 The number n kσ of electrons with wave vector k and spin σ σ = spin up), spin down)) is equal to 0 or 1, and so the number of the states n k,n k,n k,n k,, k, k, Λ is equal to M Here, n k, n k =0, 1, and the elements k and k are arranged in a certain order We therefore choose, as our Hilbert space H, H = C M and denote each standard unit vector in H = C M : e i =0,, 0, i 1, 0,, 0), i =1,,, M by each state above for simplicity For example, we denote e 1 =1, 0, 0,, 0), e =0, 1, 0,, 0) by 0, 0, 0,, 1, 0, 0,, respectively Moreover, we denote e M =0, 0,, 0, 1) by 1, 1, 1, Here the symbol 0, 0, 0, corresponds to the state n k = n k = 0 for all k Λ, and 1, 0, 0, to the state n k = 1 and n k = n k σ = 0 for all k Λ \k} and for all σ =, Moreover, 1, 1, 1, corresponds to the state n k = n k = 1 for all k Λ We abbreviate 0, 0, 0, to 0 and call it the vacuum vector in H = C M We denote by, ) the inner product of H = C M Creation and annihilation operators We assume that each creation operator and each annihilation operator depend both on wave vector k Λ and on spin σ of an electron We denote the creation operator resp the annihilation operator) by Ckσ resp by C kσ) Note that, n k,n k, n k,n k =0, 1) stands for the corresponding standard unit vector in H = C M,as mentioned in the preceding section Definition 1 Ck, n k,n k, = 1) δ, n 1,nk k 1, n k,, Ck, n k, n k, = 1) δ 0,nk, n k +1,n k,, where the symbol denotes the number of electrons arranged at the left of the symbol n k above Ck, n k,n k, = 1) δ, n 1,nk k, n k 1,, Ck, n k, n k, = 1) δ 0,nk, n k,n k +1,, where the symbol denotes the number of electrons arranged at the left of the symbol n k above

3 On the basis of the definition we regard each of the creation and annihilation operators as a linear operator on H = C M The definition immediately gives the following lemma Lemma a) The annihilation operator C kσ is a bounded linear operator on H = C M, and its adjoint operator coincides with the creation operator Ckσ b) The operators C kσ and Ckσ satisfy the canonical anticommutation relations on H = C M : where A, B} = AB + BA Remark 3 C kσ,ck σ } = δ kk δ σσ, C kσ, C k σ } = C kσ, Ck σ } =0, n k,n k,n k,n k, = C k ) nk C k ) nk C k ) nk C k ) nk 0 3 The BCS Hamiltonian Let m and μ stand for the electron mass and the chemical potential, respectively Here, m, μ > 0 Set ξ k = ħ k /m) μ The BCS Hamiltonian [1] is given by H = ξ k CkσC kσ + U k, k Ck C k C k C k,σ=, Here, U k, k is a function of k and k, and satisfies U k, k 0, U k,k = U k, k, U k, k = U k, k and U k, k =0 Proposition 31 The BCS Hamiltonian H is a bounded, selfadjoint operator on H = C M The bounded, selfadjoint operator G =,σ=, C kσ C kσ generates a strongly continuous unitary group e iαg} α R on H = CM As is shown just below, the transformation e iαg gives rise to a phase transformation of the creation the annihilation) operator Proposition 3 Let G and H be as above Then, for α R, e iαg C kσ e iαg = e iα C kσ, e iαg C kσ eiαg = e iα C kσ Consequently, e iαg He iαg = H Remark 33 The transformation e iαg leaves the BCS Hamiltonian H invariant In this case the BCS Hamiltonian H is said to have global U1) symmetry 3

4 4 Spontaneous symmetry breaking Definition 41 Nambu and Jona-Lasinio) Let G be as above Suppose that there is the ground state Ψ 0 H= C M of the BCS Hamiltonian H The global U1) symmetry is said to be spontaneously broken if there is a bounded linear operator A on H = C M satisfying Ψ 0, [ G, A]Ψ 0 ) 0 Lemma 4 Set A = C k C k in the definition above Then Ψ 0, [ G, C k C k ]Ψ 0 )= Ψ 0,C k C k Ψ 0 ) Remark 43 If Ψ 0,C k C k Ψ 0 ) 0, then the global U1) symmetry is spontaneously broken Remark 44 The concept of spontaneous symmetry breaking was introduced first by Nambu and Jona-Lasinio [6] in 1961 This plays an important role in quantum mechanics such as the BCS-Bogoliubov theory and quantum gauge field theory 5 An energy gap for excitation from the BCS state Let Δ k be a function of k Λ We assume the existence of the following Δ k : Δ k 0 and Δ k =Δ k, and is a solution to the gap equation [1], []) Δ k = 1 k Λ U k, k Δ k ξ k +Δ k Let θ k be a function of k Λ and let it satisfy [1], []) Δ k satisfies sin θ k = Δ k ξ k +Δ k, cos θ k = ξ k ξ k +Δ k with 0 θ k π/ Note that θ k = θ k We denote by G B the following bounded, selfadjoint operator on H = C M : G B = i θ k C k C k C k C k ) We set Ψ BCS = e ig B 0 H= C M and call it the BCS state [1], []) Lemma 51 BCS) ) } Ψ BCS = cos θk + sin θ k Ck C k 0 4

5 Remark 5 In 1957 Bardeen, Cooper and Schrieffer [1] introduced the well-known expression in this lemma Corollary 53 a) Ψ BCS,C k C k Ψ BCS )= Ψ BCS,Ck C k Ψ ) 1 BCS = sin θ k b) Δ k = U k, k Ψ BCS,C k C k Ψ BCS ) k Λ We replace the ground state Ψ 0 of the BCS Hamiltonian by Ψ BCS k Λ) and set for all C k C k =Ψ BCS,C k C k Ψ BCS )+b k, C k C k = Ψ BCS,C k C k Ψ BCS) + b k Lemma 54 Set H M = ξ k CkσC kσ ) Δ k C k C k + Ck C k,σ=, + Δ k Ψ BCS,C k C k Ψ BCS ) Then the BCS Hamiltonian is rewritten as H = H M + U k, k b k b k Remark 55 The Hamiltonian H M is called the mean field approximation for the BCS Hamiltonian H We now introduce the Bogoliubov transformation of C kσ []: γ kσ = e ig B C kσ e ig B Note that the operator γ kσ and its adjoint operator γkσ are both bounded linear operators on H = C M Proposition 56 Bogoliubov) H M = ξk +Δ k γ kσ γ kσ,σ=, + ξ k } ξk +Δ k +Δ k Ψ BCS,C k C k Ψ BCS ) Corollary 57 a) The BCS state Ψ BCS is the ground state of H M, and the ground state energy E BCS is given by E BCS = } ξ k ξk +Δ k +Δ k Ψ BCS,C k C k Ψ BCS ) 5

6 b) Let E BCS be as in a) Then the spectrum of H M is given by } σ H M )= ξk +Δ k N k + N k )+E BCS N k,n k =0, 1 Remark 58 The corollary above implies that it takes a finite energy ξk +Δ k > Δ k) to excite a particle from the BCS state to an upper energy state So the function Δ k of k Λ corresponds exactly to the energy gap, and hence Δ k is called the gap function see Bardeen, Cooper and Schreiffer [1], and Bogoliubov []) We now study some properties of the operators γ kσ see Bogoliubov []) Corollary 59 The operators γ kσ and γkσ satisfy the following a) γ kσ,γk σ } = δ kk δ σσ, γ kσ, γ k σ } = γ kσ,γ k σ } =0 b) γ kσ Ψ BCS = 0 for each k Λ and for each σ =, γk = cos θ k C k sin θ k C k, c) γ k = sin θ k Ck + cos θ k C k Ck = cos θ k γ k + sin θ k γ k, d) C k = sin θ k γk + cos θ k γ k 6 The BCS and normal states Let Δ k = 0 for all k Λ Then the BCS state Ψ BCS coincides with the Fermi vacuum Ψ F H= C M Here the Fermi vacuum Ψ F corresponds to the normal state and is defined by Ψ F = k ξ k 0) C k C k 0, where the symbol k ξ k 0) stands for k Λ satisfying ξ k 0 Proposition 61 The BCS state Ψ BCS has a lower energy than the Fermi vacuum Ψ F the normal state), ie, Ψ BCS,HΨ BCS ) Ψ F,HΨ F ) = 1 < 0 ξ k +Δ k ξ k ξ k +Δ k ) 6

7 7 A new superconducting state Set E k = ξ k +Δ k, k Λ and set B k = C k C k We abbreviate sin θ k resp cos θ k )to S k resp to C k ) We consider the following vector in H = C M : Ψ= Ψ BCS +Φ 1+Φ, Φ), where Φ = 1 p, p Λ U p, p We prepare some lemmas C p Sp + C p p) S E p + E p γ p γ p γ p γ p Ψ BCS Lemma 71 a) Ψ BCS, Φ) = 0 b) H M Φ=E BCS Φ+ E p U p, p C p Sp + C p p) S E p + E p p, p Λ γ p γ p γ p γ p Ψ BCS c) Ψ, H M Ψ) = E BCS Φ, Φ) p, p Λ Up, p C p Sp + ) C p S p E p + E p Set H = H H M Then H = U k, k Bk B k C k S k Bk + B k)+c k S k C k S k } Lemma 7 Let H be as above a) H Ψ BCS = U k, k Sk C k γ k γ k γ k γ k Ψ BCS b) Ψ BCS,H Ψ BCS )=0 c) Φ, H Ψ BCS )= 1 p, p Λ Up, p C p Sp + ) C p S p E p + E p Lemma 73 a) B k Φ = ) U C k S k Sk γk γ k Φ+C k, p C k Sp + ) C p S k k γp γ E k + E p Ψ BCS p p Λ U k, p C k Sp C k S + ) C p S k k γk γ E k + E k γ p γ p Ψ BCS p p Λ b) Φ, B k Φ) = C k S k Φ, Φ) p Λ Uk, p C k Sp + ) } C p S k E k + E p ) 7

8 c) Φ, Bk B kφ) [ = C k S k C k S k Φ, Φ) p Λ U k, p C k Sp + ) C p S k E k + E p ) + U ) k,p C k Sp + Cp Sk }] E k + E p ) Uk, k +4C k S k C k S C k S k + C k S k ) k E k + E k ) + ) ) Ck C k + ) U k, p U k,p C k S S k S p + Cp Sk C k Sp + Cp Sk k E k + E p )E k + E p ) p Λ Let E = Ck U C k + S k S k k, k 1+Φ, Φ) ) ) U k, p U k,p C k Sp + Cp Sk C k Sp + Cp Sk E k + E p )E k + E p ) p Λ +4 C k S k C k S k Uk, k U C k S k + C k S k ) k, k 1+Φ, Φ) E k + E k ) Note that E <0 Lemma 74 Let H and E be as above Then Φ, H Φ) = 1+Φ, Φ) } E We now show that the state Ψ above has a lower energy than the BCS state Ψ BCS Theorem 75 The state Ψ has a lower energy than the BCS state Ψ BCS, and hence than the Fermi vacuum Ψ F,ie, 8 A new gap equation Ψ, HΨ) Ψ BCS,HΨ BCS )= E <0 We use the BCS state Ψ BCS to deal with the expectation values of the operators C k C k and Ck C k But we originally need to use the ground state of the BCS Hamiltonian to deal with the expectation values of such operators The ground state of the BCS Hamiltonian is studied by several authors See Mattis and Lieb [5], Richardson [7] and von Delft [3] for example They assumed that U k, k is a negative constant if k and k both belong to the 8

9 neighborhood of the Fermi surface, and 0 otherwise So little is known about the ground state when U k, k does not satisfy the assumption just above We therefore try to use our superconducting state Ψ in the preceding section instead This is because our state Ψ has a lower energy than the BCS state Ψ BCS To this end we begin with a new gap equation Let Δ k be a function of k Λ We assume the existence of the following Δ k : Δk satisfies Δ k 0 and Δ k = Δ k, and is also a solution to the new gap equation Δ k = 1 k Λ U k, k Δ k ξk + Δ k 1 4D ) k, D + where D k = 1 4 p Λ Uk,p ξ k + Δ k + 1 ξ p + Δ p ξ k ξ p ξ k + Δ k ξ p + Δ p ), D = k Λ D k Remark 81 A numerical calculation gives 4D k /D +) O10 17 ) in the case of aluminum So it is expected that Δ k is nearly equal to Δ k and that Δ k 0 Let θ k be a function of k Λ and let it satisfy sin θ k = Δ k ξk + Δ k, cos θ k = ξ k ξ k + Δ k with 0 θ k π/ Note that θ k = θ k We denote by G B the following bounded, selfadjoint operator on H = C M : G B = i θ k C k C k C k C k ) We set Ψ BCS = e i G e B 0 H= C M Lemma 8 ΨBCS = cos θ k + sin θ k Ck C k ) } 0 Corollary 83 ) ΨBCS,C k C k ΨBCS = ΨBCS,C k C k Ψ BCS ) = 1 sin θ k 9

10 We introduce another Bogoliubov transformation of C kσ : γ kσ = e i e G B C kσ e i e G B Note that the operator γ kσ and its adjoint operator γ kσ are both bounded linear operators on H = C M Corollary 84 The operators γ kσ and γ kσ satisfy the following a) γ kσ, γ k σ } = δ kk δ σσ, γ kσ, γ k σ } = γ kσ, γ k σ } =0 b) γ kσ ΨBCS = 0 for each k Λ and for each σ =, γ k = cos θ k C k sin θ k C k c), γ k = sin θ k Ck + cos θ k C k C k = cos θ k γ k + sin θ k γ k d), C k = sin θ k γ k + cos θ k γ k Set Ẽk = ξk + Δ k, k Λ and set B k = C k C k We abbreviate sin θ k resp cos θ k ) to S k resp to C k ) We now consider the following vector in H = C M : Ψ = Ψ BCS + Φ ), 1+ Φ, Φ where Φ = 1 For all k Λ, set p, p Λ Corollary 85 Δk = k Λ U k, k U p, p C p S p + C p S p ) Ẽ p + Ẽp γ p γ p γ p γ p Ψ BCS ) C k C k = Ψ, C k C k Ψ ) Ck C k = Ψ, C k C k Ψ Ψ, C k C k Ψ) + b k, + b k Lemma 86 Set H M = ξ k Ckσ C kσ ) Δ k C k C k + Ck C k,σ=, + ) Δ k Ψ, C k C k Ψ Then the BCS Hamiltonian is rewritten as H = H M + U k, k b k b k 10

11 Remark 87 The Hamiltonian H M as well as H M is also the mean field approximation for the BCS Hamiltonian H Proposition 88 H M =,σ=, + ξ k + Δ k γ kσ γ kσ ξ k ξ k + Δ k + Δ k Ψ, C k C k Ψ) } Corollary 89 a) The state Ψ BCS is the ground state of HM, and the ground state energy ẼBCS is given by Ẽ BCS = ξ k ξk + Δ k + Δ k Ψ, C k C k Ψ) } b) Let ẼBCS be as in a) Then the spectrum of H M is given by ) } σ HM = ξk + Δ k N k + N k )+ẼBCS N k,n k =0, 1 Remark 810 We see from the corollary above that it takes a finite energy ξk + Δ k > Δ k ) to excite a particle from the state Ψ BCS to an upper energy state So Δ k as well as Δ k corresponds exactly to the energy gap, and hence Δ k as well as Δ k is the gap function Remark 811 Beginning with our new gap equation we arrive at the results similar to those in the BCS-Bogoliubov theory References [1] J Bardeen, L N Cooper and J R Schrieffer, Theory of superconductivity, Phys Rev ), [] N N Bogoliubov, A new method in the theory of superconductivity I, Soviet Phys JETP ), [3] J von Delft, Superconductivity in ultrasmall metallic grains, Ann Phys ), 1 60 [4] T Gerisch and A Rieckers, Limiting dynamics, KMS-states, and macroscopic phase angle for weakly inhomogeneous BCS-models, Helv Phys Acta ), [5] D Mattis and E Lieb, Exact wave functions in superconductivity, J Math Phys 1961),

12 [6] Y Nambu and G Jona-Lasinio, Dynamical model of elementary particles based on an analogy with superconductivity, Phys Rev ), [7] R W Richardson, A restricted class of exact eigenstates of the pairing-force Hamiltonian, Phys Lett ), [8] S Watanabe, Superconductivity and the BCS-Bogoliubov theory, preprint 007) 1