Quantum Statistical Derivation of a Ginzburg-Landau Equation

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1 Journal of Modern Physics, 4, 5, Published Online October 4 in SciRes. Quantum Statistical Derivation of a Ginzburg-Landau Euation Shigeji Fujita, Akira Suzuki Department of Physics, University at Buffalo, State University of New York, Buffalo, USA Department of Physics, Faculty of Science, Tokyo University of Science, Tokyo, Japan asuzuki@rs.kagu.tus.ac.jp Received 4 August 4; revised September 4; accepted 6 September 4 Copyright 4 by authors and Scientific Research Publishing Inc. This work is licensed under the Creative Commons Attribution International License (CC BY). Abstract The pairon field operator Ψ ( rt, ) evolves, following Heisenberg s euation of motion. If the Hamiltonian H contains a condensation energy α ( < ) and a repulsive point-like interparticle interaction βδ ( r r), β ( > ), the evolution euation for Ψ is non-linear, from which we derive the Ginzburg-Landau (GL) euation: α Ψ ( r) + β Ψ ( r) Ψ ( r) = for the GL wave function ( r) Ψ ru, n uu where denotes the state of the condensed Cooper pairs (pairons), and n the pairon density operator (u and u are kind of suare root density operators). The GL euation with α = ε g ( T ) holds for all temperatures ( T ) below the critical temperature Tc, ε T is the T-dependent pairon energy gap. Its solution yields the condensed pairon β ε. The T-dependence of the expansion parameters near T c ob- α = c, β = constant is confirmed. where g ( ) density ( ) ( ) n T = Ψ r = g ( T) tained by GL: ct ( T) Keywords Ginzburg-Landau Euation, Complex-Order Parameter, Coherent Length, Cooper Pair (Pairon), Pairon Density Operator, T-Dependent Pairon Energy Gap. Introduction In 95 Ginzburg and Landau (GL) [] proposed a revolutionary idea that below the critical temperature T c a How to cite this paper: Fujita, S. and Suzuki, A. (4) Quantum Statistical Derivation of a Ginzburg-Landau Euation. Journal of Modern Physics, 5,

2 superconductor has a complex order parameter (also known as a GL wave function) Ψ just as a ferromagnet possesses a real order parameter (spontaneous magnetization). Based on Landau s theory of second-order phase transition [], GL expanded the free energy density f ( r ) of a superconductor in powers of small Ψ ( r ) and Ψ r : ( ) f 4 r r r r, () m ( ) = f + α Ψ ( ) + β Ψ ( ) + Ψ( ) where f, α and β are constants, and m is the superelectron mass. To include the effect of a magnetic field B, they used a uantum replacement: ( i ) A, charge =, () where A is a vector potential generating B = A, and added a magnetic energy term B µ. The integral of the free energy density f ( r ) over the sample volume Ω gives the Helmholtz free energy F. After minimizing F with variation in Ψ and A j, GL obtained two euations: with the density condition: m ( i A) ( r) α ( r) β ( r) ( r) Ψ + Ψ + Ψ Ψ =, () i j = ( Ψ Ψ Ψ Ψ ) Ψ ΨA, (4) m m ( ) ( ) ( ) Ψ r Ψ r = n s r = superelecton density, (5) where j is the current density. The superelectron model has difficulties. Since electrons are fermions, no two electrons can occupy the same particle state by Pauli s exclusion principle. Cooper [] introduced Cooper pairs in 956. Bardeen, Cooper and Schrieffer (BCS) published a classic paper [4] on the superconductivity in 957 based on the phonon exchange attraction between two electrons. The center-of-mass (CM) of Cooper pairs (pairons) move as bosons, which is shown in Section. We view that the Bose-condensed pairons can generate the superconducting state. We modify the density condition (5) to ( ) Ψ r = condensed pairon density at the state. (6) Euation () is the celebrated Ginzburg-Landau euation, which is uantum mechanical and nonlinear. Since the smallness of Ψ is assumed, the GL euation was thought to hold only near T c, Tc T Tc. Below T c there is a supercondensate whose motion generates a supercurrent and whose presence generates gaps in the elementary excitation energy spectra. The GL wave function Ψ represents the uantum state of this supercondensate. In the present work, we derive the GL wave euation microscopically and show that the GL euation is valid for all temperature below T c.. Moving Cooper Pairs (Pairons) Fujita, Ito and Godoy in their book [5] discussed the moving pairons. We briefly summarize their theory and main results here. The energy w of a moving pairon can be obtained from v wa ( k, ) = { ε( k+ ) + ε ( k+ )} a( k, ) d ka ( k, ), (7) π ( ) which is Cooper s euation, Euation () of his 956 Physical Review []. The prime on the k'-integral means the restriction on the integration domain arising from the phonon-exchange attraction, see below. We note that the net momentum is a constant of motion, which arises from the fact that the phonon exchange is an internal process, and hence cannot change the net momentum. The pair wave functions a ( k, ) are coupled with respect to the other variable k, meaning that the exact (or energy-eigenstate) pairon wavefunctions are superposi- 56

3 tions of the pair wavefunctions (, ) a k. Euation (7) can be solved as follows. We assume that the energy Then, ε( ) ε( ) ( + ) + ε( + ) w w is negative: w <. (8) k+ + k+ >. Rearranging the terms in Euation (7) and dividing it by w ε k k, we obtain Here (, ) = { ε( + ) + ε( + ) } ( ) a k k k w C. (9) C ( ) v ( π ) ( k ) d ka, is k-independent. Introducing Euation (9) in Euation (7), and dropping the common factor C ( ), we obtain ( π ) { ε( ) ε( ) w } v = k+ + k+ + d k (). () We now assume a free electron model. The Fermi surface is a sphere of the radius (momentum): k ( ) F F = mε F where m represents the effective mass of an electron. The energy ε ( k ) is given by k, () kf The prime on the k-integral in Euation () means the restriction: k ε( k ) εk =. () m ( ) ( ) < ε k+, ε k+ < ωd. (4) We may choose the polar axis along as shown in Figure. The integration with respect to the azimuthal angle simply yields the factor π. The k-integral can then be expressed by where k D is given by ( π ) π kf + kd cosθ k dk = 4π dθsinθ k cos F + θ + ε k + ( 4 ) v w m D D F After performing the integration and taking the small- and small- ( ) (5) k m ω k (6) k k limits, we obtain D F Figure. The range of the interaction variables (k, θ) is restricted to the circular shell of thickness k D. 56

4 where the pairon ground-state energy w is given by vf w = w +, (7) ωd w =, v V Ω. (8) exp { vd ( )} As expected, the zero-momentum pairon has the lowest energy. The excitation energy is continuous with no energy gap. Euation (7) was first obtained by Cooper (unpublished) and it is recorded in Schrieffer s book (Ref. [6], Euations ()-(5)). The energy w increases linearly with momentum ( = ) for small. This behavior arises since the pairon density of states is strongly reduced with increasing momentum and dominates the -increase of the kinetic energy. This linear dispersion relation means that a pairon moves like a massless particle with a common speed v F. The center-of-mass (CM) of pairons move as bosons. We can show this as follows. Second-uantized operators for a pair of electrons (i.e., electron pairons) are defined by where c s and B B c c, B = c c (9) k k 4 4 c s satisfy the Fermi anti-commutation rules: [ ] c c c c + c c = δ c c = k, k k k k k k, k, k, k + + Using Euation () we compute commutation relations for pair operators and obtain where [ ] () B, B B B B B = () n n if k = k and k = k 4 cc 4 if k= k and k= / k4, B 4 = cc if k k and k k4 B = otherwise ( B ) BB () =, (), (4) k k k k n c c n c c are number operators for electrons. Using Euations (9)-(4), we obtain Hence ( ) ( ) ( ) ( ) n B B = B n n + B B B = B B n. (5) n n n = n n n =, indicating that the eigenvalues n of n are n = or. Thus, the number operator in the k - representation nk BkBk with both k and specified, has the eigenvalues or. We now introduce B B k k k k k k, (6) and calculate B, n and obtain B, n = ( n + n + ) B = B, n, B = B, (7) from which it follows straightforwardly that the eigenvalues n of n are: [7] with the eigenstates n =,,, (8), = B, = BB,. (9) 56

5 . Ginzburg-Landau Euation at K Let us take a three dimensional (D) superconductor such as tin (Sn) and lead (Pb). Both metals form face-centered cubic (fcc) crystals. They are in superconducting states at K. The system ground-state wave function Ψ ( r ) is a constant in the normalization volume Ω and vanishes at the boundary: Ψ We may assume a periodic rectangular-box with side-lengths ( x, y, z) constant within the volume Ω, r = () at the boundary. ( ) L L L along the cubic lattice, L = Na, N, j= xyz,,, () j j j where a is the lattice constant. We introduce a one-body density operator n and the density matrix n ab for the treatment of a many-particle system. The density operator n can be expanded in the form: n = µ Pµ µ, Pµ () where { } P µ denote the relative probabilities that particle states { } following normalization condition: µ µ are occupied. It is customary to adopt the n = Pµ = tr{ n} = N, () µ where the symbol tr means a one-body trace and N is the particle number. The density operator n is Hermitean: Let us introduce kind of a suare root density operator u such that = n is. We will show that the revised G-L wave function Ψ ( r ) is con- This u is not Hermitean but nected with uu n n n. (4) = uu. (5) ( ) u Ψ r = r, (6) where denotes the condensed pairon state. For a running ring super current [8] we may choose π = pm = m, m =, ±, ±,, (7) L where L is the ring circumference. The m are very small numbers ( m N) states m are practically the same as the ground state ( m = ) since m N. The energies of the excited and L a. The excited states are semi-stable because N particles must be redistributed when going from an excited state m to the ground state. The natural decay times are measured in days.,t ψ r,t which satisfy the Bose commutation rules: Let us introduce boson field operators ψ ( r ) and ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ψ r, ψ r ψ r ψ r ψ r ψ r = δ r r, ( r) ( r ) = ( r) ( r ) = ( t ) ψ, ψ ψ, ψ, the time omitted. ( ) ( r r ) ( ) ( ) ( ) and ( ) where δ δ x x δ y y δ z z δ x x is Dirac s delta-function. We take a system characterized by many-boson Hamiltonian H: H = r h i + r r d ψ ( r) ( r, r) ψ ( r) d d ψ ( r) ψ ( r ) φ( r r ) ψ ( r ) ψ ( r) where h( r, p ) is a single-boson Hamiltonian and φ ( ) obtained from the Heisenberg euation of motion is (8) (9) r r is a pair potential energy. The field euation 564

6 ( r, t) ψ ( ) ( ) ( ) i = ψ, t, H = h, i ψ, t + d r φ( ) ψ (, t) ψ (, t) ψ (, t) t r r r r r r r r r. (4) We note that the field euation is nonlinear in the presence of a pair potential. We can exprress r and r by where ψ ( r ) and ( ) ( ), ψ ( ) r = Φ ψ r r = r Φ, (4) ψ r are boson operators and Φ and Φ represent vacuum state vectors satisfying ( ) ψ ( ) Φ ψ r = r Φ =, Φ Φ =. (4) In the Heisenberg picture (HP) the boson states are time-independent and boson operators ψ and ψ move following the field Euation (4). The single particle Hamiltonian h contains the kinetic energy h ( p ), which depends on the momentum p only, and the boson-condensation Hamiltonian h e which arises from the phonon exchange attraction. The ground state wave function Ψ ( r ) is flat as seen from Euation (). There are no gradients and no material currents. Hence, we obtain ( ) ( ) ( ) ( ) ( ) ( ) p Ψ r = r Ψ r = h h i, (4) where the superscript () means the ground state average. The ground-state energy of the pairon is negative and is given by w in Euation (8). Hence we may choose ( ) ( ) ( ) ( ) w α w α Ψ r = Ψ r, = <. (44) The pairon has charge magnitude e and a size. For Pb the pairon linear size is about Å. Because of the Colomb repulsion and Pauli s exclusion principle two pairons repel each other at short distances. We may represent this repulsion by a point-like pair potential: ( ( ) v ) δ ( ) v( ) φ r r = r r, constant >. (45) Using this and the random phase (factorization) approximation we obtain ( ) v r δ ( r r ) Ψ ( r ) Ψ ( r ) Ψ ( r) = v Ψ ( r) Ψ ( r) ( t-dependence and superscript omitted) d. Gathering the results (4), (44) and (46), we obtain ( r, t) Ψ i = wψ ( r, t) + v Ψ( r, t) Ψ ( r, t). t ( superscripts ommited) For the steady state the time derivative vanishes, yielding w ( ) v ( ) ( ) Ψ + Ψ Ψ = (46) (47) r r r. (48) This is precicely, the GL euation, Euation () with α = w and β = v. In our derivation we assumed that pairons move as bosons, which is essential. Bosonic pairons can multiply occupy the condensed momentum state while fermionic superelectrons cannot. The correct density condition (6) instead of (5) must therefore be used. 4. Discussion We derived the GL euation from first principles. In the derivation we found that the particles that are described by the GL wave function Ψ ( r ) must be bosons. We take the view that Ψ ( r ) represents the bosonically condensed pairons. This explains the uantum nature of the GL wave function. The nonlinearity of the GL euation arises from the point-like repulsive inter-pairon interaction. In 95 when Ginzburg and Landau published their work, the Cooper pair (pairon) was not known. They simply assumed the 565

7 superelectron model. Our microscopic derivation allows us to interpret the expansion parameters as follows. The α represents the pairon condensation energy, and β the repulsive interaction strength: β = constant >. (49) BCS showed [4] that the ground-state energy W for the BCS system is where ( ) ωd W = ωdd( ) w, w, (5) exp { vd ( )} D is the density of states per spin at the Fermi energy and w the pairon ground-state energy. Hence we obtain Euation (44): α = w < at T =. In the original work [] GL considered a superconductor in the vicinity of the critical temperature T c, where Ψ is small. Gorkov [9]-[] used Green s functions and interrelated the GL and the BCS theory near T c. We derived the original GL euation by examining the superconductor at K from the condensed pairons point of view. The transport property of a superconductor below T c is dominated by the Bose-condensed pairons. Since there is no distribution, the ualitative property of the condensed pairons cannot change with temperature. The pairon size (the minimum of the coherence length derivable directly from the GL euation) naturally exists. There is only one supercondensate whose behavior is similar for all temperatures below T c ; only the density of condensed pairons can change. Thus, there is a uantum nonlinear euation (48) for Ψ ( ) r valid for all temperateures below T c. The pairon energy spectrum below T c has a discrete ground-state energy, which is separated from the energy continuum of moving pairons [5]. Inspection of the pairon energy spectrum with a gap suggests that Solving Euation (48) with Euation (6), we obtain ( ) α = ε T <, T > T. (5) n( T) β εg ( T) indicating that the condensed density n ( T ) is proportional to the pairon energy gap ( T ) g c = Ψ =, (5) ε g. We now consider an ellipsoidal macroscopic sample of a type I superconductor below T c subject to a weak magnetic field H applied along its major axis. Because of the Meissner effect, the magnetic fluxes are expelled from the body, and the magnetic energy is higher by µ H Ω in the super state than in the normal state. If the field is sufficiently raised, the sample reverts to the normal state at a critical field H c, which can be computed in terms of the free-energy expression () with the magnetic field included. We obtain after using Euations (6) and (5) Hc ( µβ ) εg ( T) n( T) indicating that the measurements of H c give the T-dependent n ( ), (5) T approximately. The field-induced transition corresponds to the evaporation of condensed paions, and not to their break-up into electrons. Moving pairons by construction have negative energies while uasi electrons have positive energies. Thus, the moving pairons are more numerous at the lowest temperatures, and they are dominant elementary excitations. Since the contribution of the moving pairons was neglected in the above calculation, Euation (5) contains approximation, see below. We stress that the pairon energy gap ε g is distinct from the BCS energy gap Δ, which is the solution of ( ε + ) ωd = vd ( ) dε tanh. (54) ( ε + ) kt B In the presence of a supercondensate the energy-momentum relation for an unpaired (uasi) electron changes: ( ) ( ) ε k m ε E ε +. (55) k F k k Since the density of condensed pairons changes with the temperature T, the gap Δ is T-dependent and is 566

8 determined from Euation (54) (originated in the BCS energy gap euation). Two unpaired electrons can be bound by the phonon-exchange attraction to form a moving pairon whose energy w is given by w = w + vf <, (56) { } ωd ( ) ε w ( ε ) = vd d + +. (57) Note that w is T-dependent since Δ is. At T c, Δ = and the lower band edge w is eual to the pairon ground-state energy w. We may then write We call ( T ) w ( ) = w + εg T + vf, εg ( T) w w. (58) ε g the pairon energy gap. The two gaps (, ε g ) have similar T-behavior; they are zero at T c and they both grow monotonically as temperature is lowered. The rhs of Euation (54) is a function of ( T, ) ; T c is a regular point such that a small variation δt T T generates a small variation in Δ. Hence we obtain c ( T) b( T T), T T T, b constant =. (59) c c c Using similar arguments we get from Euations (57)-(59) ( T) c( T T), T T T, c constant ε =. (6) g c c c As noted earlier, moving pairons have finite (zero) energy gaps in the super (normal) states, which makes Euation (5) approximate. But the gaps disappear at T c, and hence the linear-in- ( Tc T) behavior should hold for the critical field H c : H = d( T T), T T T, d = constant, (6) c c c c which is supported by experimental data. Tunneling and photo absorption data []-[6] appear to support the linear law in Euation (6). In the original GL theory [], the following signs and T-dependence of the expansion parameters ( α, β ) near T c were assumed and tested: ( T) α ct <, β = constant >, (6) c all of which are reestablished by our microscopic calculations. 5. Conclusions In summary we reached a significant conclusion that the GL euation is valid for all temperatures below T c. The most important results in the GL theory include GL s introduction of a coherent length [] and Abrikosov s prediction of a vortex structure [7], both concepts holding not only near T c but for all temperatures below T c. In the present work the time evolution of the system is described through the field euation for the boson op-,t ψ r,t. The resulting euation erators ψ ( r ) and ( ) ( r, t) Ψ i = h( r, i r) Ψ (, t) v ( ) r + Ψ r, t Ψ ( r, t) (6) t may be useful in deriving the Josephson-Feynman euation and describing dynamical Josephson effect [8] [9]. Operators u and u are non-hermitean, see Euation (5). Hence, a off-diagonal long range order simply arises from the definition of Ψ ( r ) r u. GL treated the effect of the magnetic field applied based on the super electron model. We shall treat this effect based on the moving pairons model separately. References [] Ginzburg, V.L. and Landau, L.D. (95) Journal of Experimental and Theoretical Physics (USSR),, 64. [] Landau, L.D. and Lifshitz, E.M. (98) Statistical Physics, Part I. rd Edition, Pergamon Press, Oxford,

9 [] Cooper, L.N. (956) Physical Review, 4, [4] Bardeen, J., Cooper, L.N. and Schrieffer, J.R. (957) Physical Review, 8, [5] Fujita, S., Ito, K. and Godoy, S. (9) Quantum Theory of Conducting Matter: Superconductivity. Springer, New York, 77-79, [6] Schrieffer, J.R. (964) Theory of Superconductivity. Benjamin, New York. [7] Dirac, P.A.M. (958) Principle of Quantum Mechanics. 4th Edition, Oxford University Press, London, 6-8. [8] File, J. and Mills, R.G. (96) Physical Review Letters,, 9. [9] Gorkov, I. (958) Soviet Physics JETP, 7, 55. [] Gorkov, I. (959) Soviet Physics JETP, 9, 64. [] Gorkov, I. (96) Soviet Physics JETP,, 998. [] Glaever, I. (96) Physical Review Letters, 5, [] Glaever, I. (96) Physical Review Letters, 5, [4] Glaever, I. and Megerle, K. (96) Physical Review,,. [5] Glover III, R.E. and Tinkham, M. (957) Physical Review, 8, 4. [6] Biondi, M.A. and Garfunkel, M. (959) Physical Review, 6, [7] Abrikosov, A.A. (957) Soviet Physics JETP, 5, 74. [8] Josephson, B.D. (96) Physics Letters,, [9] Josephson, B.D. (964) Reviews of Modern Physics, 6,

On the Heisenberg and Schrödinger Pictures

Journal of Modern Physics, 04, 5, 7-76 Published Online March 04 in SciRes. http://www.scirp.org/ournal/mp http://dx.doi.org/0.436/mp.04.5507 On the Heisenberg and Schrödinger Pictures Shigei Fuita, James