Some detailed information for BCS-BEC Crossover

Transcription

1 BCS-BEC Crossover Some detailed information for BCS-BEC Crossover BCS Theory 李泽阳 April 3, 2015 School of Physics, Peking University BCS-BEC Crossover April 3, / 31

2 BCS-BEC Crossover Table of Contents 1 Background Information 2 Attraction between electrons The Mean Field Hamiltonian and Cooper pair Application 3 Reference School of Physics, Peking University BCS-BEC Crossover April 3, / 31

3 BCS-BEC Crossover Background Information Critics on phenomenological theory These theories cannot explain the phenomena microscopically School of Physics, Peking University BCS-BEC Crossover April 3, / 31

4 BCS-BEC Crossover Background Information Critics on phenomenological theory These theories cannot explain the phenomena microscopically Hence we need BCS theory, which is basically an electron-phonon correlation theory School of Physics, Peking University BCS-BEC Crossover April 3, / 31

5 Background Information What is correlation theory? Note (Correlation theory) Correlation theory of electron is widely used in Condensed Matter Physics, and mostly have three basic form: e-e correlation (Mott Insulator), e-ph correlation (BCS) and e-spin correlation (Giant Magnetoresistance), and e-e + e-spin correlation (Colossal Magnetoresistance) With the development of low dimension systems, other theory like 1D e-e correlation (Luttinger Liquid), 0D e-e correlation (Quantum Dot), 2D e-ph correlation (Charge Density Wave) were also invented School of Physics, Peking University BCS-BEC Crossover April 3, / 31

6 BCS-BEC Crossover Goals Before we start, I list the thing that we want to finally have: School of Physics, Peking University BCS-BEC Crossover April 3, / 31

7 BCS-BEC Crossover Goals Before we start, I list the thing that we want to finally have: The excited spectrum (Zero temperature) School of Physics, Peking University BCS-BEC Crossover April 3, / 31

8 BCS-BEC Crossover Goals Before we start, I list the thing that we want to finally have: The excited spectrum (Zero temperature) The ground state (Zero temperature) School of Physics, Peking University BCS-BEC Crossover April 3, / 31

9 BCS-BEC Crossover Goals Before we start, I list the thing that we want to finally have: The excited spectrum (Zero temperature) The ground state (Zero temperature) The critical temperature T c (Finite temperature) School of Physics, Peking University BCS-BEC Crossover April 3, / 31

10 BCS-BEC Crossover Attraction between electrons Attraction between electrons We firstly introduce how the well-known attraction between two electrons was invited in this case School of Physics, Peking University BCS-BEC Crossover April 3, / 31

11 BCS-BEC Crossover Attraction between electrons Attraction between electrons We firstly introduce how the well-known attraction between two electrons was invited in this case A naïve picture is: School of Physics, Peking University BCS-BEC Crossover April 3, / 31

12 Attraction between electrons Attraction between electrons We firstly introduce how the well-known attraction between two electrons was invited in this case A naïve picture is: p 2 p 4 phonon scattering p 1 p 3 Figure: Due to the so-called non-locality of phonon, the two electron gained a effective attraction by same phonon School of Physics, Peking University BCS-BEC Crossover April 3, / 31

13 BCS-BEC Crossover Attraction between electrons An interacting process between electron and phonon is described well in the Hamiltonian: H int = g ψ α(r)ψ α (r)φ(r)d 3 r 1 Detailed calculation of screening effect is not shown here School of Physics, Peking University BCS-BEC Crossover April 3, / 31

14 Attraction between electrons An interacting process between electron and phonon is described well in the Hamiltonian: H int = g ψ α(r)ψ α (r)φ(r)d 3 r This form is derived based on such a mechanism: the electron perceive the interaction by addition energy due to polarization by the vibration(phonon): e n(r)k(r r )divp(r )d 3 rd 3 r 1 Detailed calculation of screening effect is not shown here School of Physics, Peking University BCS-BEC Crossover April 3, / 31

15 Attraction between electrons An interacting process between electron and phonon is described well in the Hamiltonian: H int = g ψ α(r)ψ α (r)φ(r)d 3 r This form is derived based on such a mechanism: the electron perceive the interaction by addition energy due to polarization by the vibration(phonon): e n(r)k(r r )divp(r )d 3 rd 3 r where K represents the form of interacting, and always replaced by a e δ(r r ) due to screening effect 1, and hence the Hamiltonian is derived 1 Detailed calculation of screening effect is not shown here School of Physics, Peking University BCS-BEC Crossover April 3, / 31

16 Attraction between electrons Because the system is electron & phonon system, Feynman Diagram in momentum space has such rule: For solid line (electron fermi line), G (0) (p) = and for wavy line (phonon bose line), D (0) (k) = 1 ω ϵ(p) + i(0 + )sgnε(p) ω 2 0 (k) ω 2 ω 2 0 (k) + i(0+ ) School of Physics, Peking University BCS-BEC Crossover April 3, / 31

17 Attraction between electrons After several calculation, we can have the D for phonon in the diagram p 2 p 4 phonon scattering p 1 p 3 g 2 D(ε 3 ε 1 ; p 3 p 1 ) = g 2 u 2 (p 3 p 1 ) 2 (ε 3 ε 1 ) 2 u 2 (p 3 p 1 ) 2 2 reason can be seen later School of Physics, Peking University BCS-BEC Crossover April 3, / 31

18 Attraction between electrons After several calculation, we can have the D for phonon in the diagram p 2 p 4 phonon scattering p 1 p 3 g 2 D(ε 3 ε 1 ; p 3 p 1 ) = g 2 u 2 (p 3 p 1 ) 2 (ε 3 ε 1 ) 2 u 2 (p 3 p 1 ) 2 Mostly, the electron is around Fermi Surface 2, up ph ω D, and also ε 3 ε 1 ω D, (continued by next slide) 2 reason can be seen later School of Physics, Peking University BCS-BEC Crossover April 3, / 31

19 BCS-BEC Crossover Attraction between electrons Continue In such case, we can have g 2 D = g 2 which means the interaction can be constantly considered as an attraction School of Physics, Peking University BCS-BEC Crossover April 3, / 31

20 BCS-BEC Crossover Attraction between electrons Continue In such case, we can have g 2 D = g 2 which means the interaction can be constantly considered as an attraction Before we go any further, we should claim something School of Physics, Peking University BCS-BEC Crossover April 3, / 31

21 Attraction between electrons Continue In such case, we can have g 2 D = g 2 which means the interaction can be constantly considered as an attraction Before we go any further, we should claim something 1 The attraction is an additional term The Coulomb interaction still exists School of Physics, Peking University BCS-BEC Crossover April 3, / 31

22 Attraction between electrons Continue In such case, we can have g 2 D = g 2 which means the interaction can be constantly considered as an attraction Before we go any further, we should claim something 1 The attraction is an additional term The Coulomb interaction still exists 2 Which force dominate causes serious different in our project: BCS-BEC Crossover School of Physics, Peking University BCS-BEC Crossover April 3, / 31

23 Attraction between electrons Continue In such case, we can have g 2 D = g 2 which means the interaction can be constantly considered as an attraction Before we go any further, we should claim something 1 The attraction is an additional term The Coulomb interaction still exists 2 Which force dominate causes serious different in our project: BCS-BEC Crossover 3 BCS theory next consider the overall interaction as a constant local attraction, but it actually isn t School of Physics, Peking University BCS-BEC Crossover April 3, / 31

24 Attraction between electrons Continue In such case, we can have g 2 D = g 2 which means the interaction can be constantly considered as an attraction Before we go any further, we should claim something 1 The attraction is an additional term The Coulomb interaction still exists 2 Which force dominate causes serious different in our project: BCS-BEC Crossover 3 BCS theory next consider the overall interaction as a constant local attraction, but it actually isn t 4 Electrons are almost-free, and have square dispersion relation ω k 2 School of Physics, Peking University BCS-BEC Crossover April 3, / 31

25 The Mean Field Hamiltonian and Cooper pair Now we get into BCS theory The Hamiltonian can be considered as H = H 0 + V = λ 2 ψ α(x) [ ( iħ + e A(x)/c) 2 2m ψ α(x)ψ β (x)ψ β(x)ψ α (x)d 3 x where the second term is a constant local attraction µ ] ψ α (x)d 3 x School of Physics, Peking University BCS-BEC Crossover April 3, / 31

26 The Mean Field Hamiltonian and Cooper pair Now we get into BCS theory The Hamiltonian can be considered as H = H 0 + V = λ 2 ψ α(x) [ ( iħ + e A(x)/c) 2 2m ψ α(x)ψ β (x)ψ β(x)ψ α (x)d 3 x µ ] ψ α (x)d 3 x where the second term is a constant local attraction From another point, we consider the difference between this Hamiltonian and the H 0 Obviously, the ground state of H 0 can be considered as a vacuum state, say vac, or N stands for a N-(real) particle and no out-of-fermi-sea electrons state School of Physics, Peking University BCS-BEC Crossover April 3, / 31

27 BCS-BEC Crossover The Mean Field Hamiltonian and Cooper pair The stability of metal is ensured by the energy cost of excitation of two electrons out of Fermi surface: E = ħ2 k 2 1 2m + ħ2 k 2 2 2m 2µ 0 School of Physics, Peking University BCS-BEC Crossover April 3, / 31

28 The Mean Field Hamiltonian and Cooper pair The stability of metal is ensured by the energy cost of excitation of two electrons out of Fermi surface: E = ħ2 k 2 1 2m + ħ2 k 2 2 2m 2µ 0 However, if we introduce the attraction, things are different because there can be some case (near Fermi surface) that E = ħ2 k 2 1 2m + ħ2 k 2 2 2m 2µ 0 School of Physics, Peking University BCS-BEC Crossover April 3, / 31

29 BCS-BEC Crossover The Mean Field Hamiltonian and Cooper pair Due to this, the N is not the lowest energy state, and hence the ground state should be something like Ψ 0 = N + N N N N School of Physics, Peking University BCS-BEC Crossover April 3, / 31

30 The Mean Field Hamiltonian and Cooper pair Due to this, the N is not the lowest energy state, and hence the ground state should be something like Ψ 0 = N + N N N N This causes a big difference, for the following term may not vanished: Ψ 0 ψ ψ Ψ0 0 School of Physics, Peking University BCS-BEC Crossover April 3, / 31

31 The Mean Field Hamiltonian and Cooper pair Due to this, the N is not the lowest energy state, and hence the ground state should be something like Ψ 0 = N + N N N N This causes a big difference, for the following term may not vanished: Ψ 0 ψ ψ Ψ0 0 Now and after, the notation A def Ψ 0 A Ψ 0 is used School of Physics, Peking University BCS-BEC Crossover April 3, / 31

32 BCS-BEC Crossover The Mean Field Hamiltonian and Cooper pair Consider the electron has addition freedom - spin, and the spin is not involved in the calculation, we have to carefully take them into the contraction of indices Definition ( (x) λ ψ ψ = e iθ ) This quantity describe actually the non-trivial part in the situation, and is actually independent of (x) Hereinafter we just use School of Physics, Peking University BCS-BEC Crossover April 3, / 31

33 The Mean Field Hamiltonian and Cooper pair Consider the electron has addition freedom - spin, and the spin is not involved in the calculation, we have to carefully take them into the contraction of indices Definition ( (x) λ ψ ψ = e iθ ) This quantity describe actually the non-trivial part in the situation, and is actually independent of (x) Hereinafter we just use The benefit to introduce such an definition is based on such an statement: occasionally, the ψ (x)ψ (x) doesn t vary too much and can be expressed by ψ ψ = ψ ψ (ψ ψ ψ ψ ) while the second term is relatively small School of Physics, Peking University BCS-BEC Crossover April 3, / 31

34 The Mean Field Hamiltonian and Cooper pair Hence, approximately, we can express the Hamiltonian by 3 H = [ ( iħ ) ψ α(x) 2 { λ ψ + λ ] 2m µ ψ α (x)d 3 x } (x)ψ (x) ψ (x)ψ (x) + ψ (x)ψ (x) ψ (x)ψ (x) d 3 x ψ (x)ψ (x) ψ (x)ψ (x) d 3 x 3 Here we omit the potential Am which causes nothing in this case School of Physics, Peking University BCS-BEC Crossover April 3, / 31

35 BCS-BEC Crossover The Mean Field Hamiltonian and Cooper pair It s straight forward to use Fourier Transformation of x, for the can be eventually written by a number rather than operator Notice that the transform is discrete 4 4 so that avoid some critical issues happened when there are uncountable infinite dimension School of Physics, Peking University BCS-BEC Crossover April 3, / 31

36 The Mean Field Hamiltonian and Cooper pair It s straight forward to use Fourier Transformation of x, for the can be eventually written by a number rather than operator Notice that the transform is discrete 4 Hence, we have H eff = k,α ξ k c kα c kα + k c k c k + k c k c k k c k c k where ξ k = ħ 2 k 2 /2m µ indicates the energy cost to excite an electron, the c and c for creation and annihilation operator, the eff for effective mean-field Hamiltonian 4 so that avoid some critical issues happened when there are uncountable infinite dimension School of Physics, Peking University BCS-BEC Crossover April 3, / 31

37 The Mean Field Hamiltonian and Cooper pair Next we want to diagonalize the Hamiltonian It s easy to rewrite it in: H eff = k { ( ) ( c ξ k c k e iθ ) ( ) } c k k e iθ + ξ k ξ k c k k c k c k School of Physics, Peking University BCS-BEC Crossover April 3, / 31

38 The Mean Field Hamiltonian and Cooper pair Next we want to diagonalize the Hamiltonian It s easy to rewrite it in: H eff = k { ( ) ( c ξ k c k e iθ ) ( ) } c k k e iθ + ξ k ξ k c k k c k c k This unitary matrix can be diagonalized simply School of Physics, Peking University BCS-BEC Crossover April 3, / 31

39 The Mean Field Hamiltonian and Cooper pair Next we want to diagonalize the Hamiltonian It s easy to rewrite it in: H eff = k { ( ) ( c ξ k c k e iθ ) ( ) } c k k e iθ + ξ k ξ k c k k c k c k This unitary matrix can be diagonalized simply We introduce a transformation: ( ) ( ck uk e = iθ ) ( ) v k αk e iθ v k u k c k Continued by next slide β k School of Physics, Peking University BCS-BEC Crossover April 3, / 31

40 The Mean Field Hamiltonian and Cooper pair Continue with 2u k v k = ξk u 2 k v2 k = ξ k ξ 2 k + 2 School of Physics, Peking University BCS-BEC Crossover April 3, / 31

41 The Mean Field Hamiltonian and Cooper pair Continue with 2u k v k = ξk u 2 k v2 k = ξ k ξ 2 k + 2 And hence can derive a clear diagonalized Hamiltonian: H eff = k { ξ 2k + 2 (a k α k + β k β k 1) + ξ k } k c k c k School of Physics, Peking University BCS-BEC Crossover April 3, / 31

42 Application Excited Spectrum Hence, we have a finite-gap spectrum, for exciting a α or β (also called Bogoliubov particle) all cost at least ξk 2 + 2, as illustrated below: E V + V k 0 k Figure: Excited spectrum; red line for ground BCS state, and solid line for single excited state; k 0 stands for ξ k0 = 0 School of Physics, Peking University BCS-BEC Crossover April 3, / 31

43 BCS-BEC Crossover Application Next we want to get the relationship between Ψ 0 and vac School of Physics, Peking University BCS-BEC Crossover April 3, / 31

44 BCS-BEC Crossover Application Next we want to get the relationship between Ψ 0 and vac The Ψ 0 satisfies α k Ψ 0 = 0 β k Ψ 0 = 0 School of Physics, Peking University BCS-BEC Crossover April 3, / 31

45 Application Next we want to get the relationship between Ψ 0 and vac The Ψ 0 satisfies α k Ψ 0 = 0 β k Ψ 0 = 0 The next derivation is to obtain Ψ 0 in terms of vac and c, c ; however, it is not too simple Firstly, we construct the α, β in terms of c α k = u k c k e iθ v k c k β k = e iθ v k c k + u kc k School of Physics, Peking University BCS-BEC Crossover April 3, / 31

46 Application Here I just give the solution Ψ 0 = k (u k + e iθ v k c k c k ) vac To verify, we can see that α k Ψ 0 = 0 β k Ψ 0 = 0 School of Physics, Peking University BCS-BEC Crossover April 3, / 31

47 BCS-BEC Crossover Application Summary So far, we already have the excited spectrum (though unknown what is), and obtain the vac and Ψ 0 If we want to know the critical temperature (ie, = 0) or know the precise form of the spectrum, we have to know λ ψ ψ 5 5 Going on or not depends on whether the time is enough School of Physics, Peking University BCS-BEC Crossover April 3, / 31

48 Application After several calculation, we have self-consistent equation = λ 2 which is called gap equation d 3 k (2π) 3 ξk School of Physics, Peking University BCS-BEC Crossover April 3, / 31

49 Application After several calculation, we have self-consistent equation = λ 2 d 3 k (2π) 3 ξk which is called gap equation Note that we have already explained why the attractive interaction in BCS Hamiltonian applies only for those two electrons which are lying within an energy sell of the thickness ħω D from the Fermi surface That means, the sum over only ξ k ħω D, and leads to (u k, v k ) = (1, 0) or (0, 1) School of Physics, Peking University BCS-BEC Crossover April 3, / 31

50 BCS-BEC Crossover Application Illustration ħω D ħω D Figure: Interaction within a shell, the solid line represents the Fermi Surface, and the two dashed line represents the boundary of possible interaction-involved region School of Physics, Peking University BCS-BEC Crossover April 3, / 31

51 BCS-BEC Crossover Application Thus, = λ 2 ξ k ħω D d 3 k (2π) 3 ξ 2 k + 2 School of Physics, Peking University BCS-BEC Crossover April 3, / 31

52 Application Thus, = λ 2 ξ k ħω D d 3 k (2π) 3 ξk Several replacement makes ħωd 1 1 = λ dωn(ω) ħω D ω School of Physics, Peking University BCS-BEC Crossover April 3, / 31

53 Application Thus, = λ 2 ξ k ħω D d 3 k (2π) 3 ξk Several replacement makes ħωd 1 1 = λ dωn(ω) ħω D ω Occasionally, ω D is relatively small, and hence the fluctuation of N(ω) is not rapid; a N(0) = mk F /2π 2 ħ 2 is enough here School of Physics, Peking University BCS-BEC Crossover April 3, / 31

54 Application Thus, = λ 2 ξ k ħω D d 3 k (2π) 3 ξk Several replacement makes ħωd 1 1 = λ dωn(ω) ħω D ω Occasionally, ω D is relatively small, and hence the fluctuation of N(ω) is not rapid; a N(0) = mk F /2π 2 ħ 2 is enough here So, ħωd 1 = gn(0) 0 [ 1 dω ω 2 + = gn(0) log ħω D + ] ω School of Physics, Peking University BCS-BEC Crossover April 3, / 31

55 BCS-BEC Crossover Application Approximation that ħω D gives = 2ħω D e 1/gN(0) 001ħω D School of Physics, Peking University BCS-BEC Crossover April 3, / 31

56 BCS-BEC Crossover Application Approximation that ħω D gives = 2ħω D e 1/gN(0) 001ħω D The last estimation is based information about some typical metal School of Physics, Peking University BCS-BEC Crossover April 3, / 31

57 Application Generalization to finite temperature As we already mentioned in the Goals, the estimation of T c is based on the finite temperature field theory Fortunately, we don t have to change a lot, but the definition of A should be taken Definition ( A = Tr[e βh eff A] Tr[e βh eff ] ) An important application is ψ (x)ψ (x) = Tr[e βh effψ (x)ψ (x)] Tr[e βh eff] School of Physics, Peking University BCS-BEC Crossover April 3, / 31

58 Application With some detailed calculation 6, we derived similar gap equation: d 3 k 1 1 = λ ξ k ħω D (2π) 3 tanh (βħe k /2) 2E k = λ ħωd 1 ( dωn(ω) ħω D 2 ω 2 + tanh 2 βħ ) ω /2 It s easy to see that decreases when temperature increases When = 0, there comes the critical temperature We then solve it 6 It s too boring, not easy to understand for a neophyte, and may cost too long for a presentation; we simply gives the result here School of Physics, Peking University BCS-BEC Crossover April 3, / 31

59 BCS-BEC Crossover Application Critical Temperature At T c, naturally, we have = 0 School of Physics, Peking University BCS-BEC Crossover April 3, / 31

60 Application Critical Temperature At T c, naturally, we have = 0 Based on this we have ħωd 1 = gn(0) 0 dω ω tanh(ħω/2k BT c ) ħω D 2k B Tc dx = gn(0) 0 x tanh x ( xc ) 1 = gn(0) ln x c ln x 0 cosh 2 x dx ) x c + = gn(0) (ln x c + ln 4eγ π + f(x c) gn(0) ln x c where γ given by some NB math tool School of Physics, Peking University BCS-BEC Crossover April 3, / 31

61 BCS-BEC Crossover Application Hence, ln ( ħωd 2k B T c 4eγ π ) = 1 gn(0) School of Physics, Peking University BCS-BEC Crossover April 3, / 31

62 Application Hence, ( ) ħωd ln 4eγ = 1 2k B T c π gn(0) We eventually have k B T c = 2eγ π ħω De 1/gN(0) 113ħω D e 1/gN(0) This derivation is confirmed by some simple metals, like Al School of Physics, Peking University BCS-BEC Crossover April 3, / 31

63 Application Hence, ( ) ħωd ln 4eγ = 1 2k B T c π gn(0) We eventually have k B T c = 2eγ π ħω De 1/gN(0) 113ħω D e 1/gN(0) This derivation is confirmed by some simple metals, like Al BCS theory can also explain critical property of the superconducting and superfluity; however, we don t have enough time So it s better to stop here School of Physics, Peking University BCS-BEC Crossover April 3, / 31

64 BCS-BEC Crossover Reference Ryuichi Shindou Quantum Statistical Physics PKU Press, v3 edition School of Physics, Peking University BCS-BEC Crossover April 3, / 31