Microscopic theory of phase transitions and nonlocal corrections for free energy of a superconductor
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] S e p Microscopic theory of phase transitions and nonlocal corrections for free energy of asuperconductor.
K.V. Grigorishin ∗ and B.I. Lev † Boholyubov Institute for Theoretical Physics of the Ukrainian NationalAcademy of Sciences, 14-b Metrolohichna str. Kiev-03680, Ukraine. (Dated: November 19, 2018)The new approach to the microscopic description of the phase transitions starting from the onlyfirst principles was developed on an example of the transition normal metal-superconductor. Thismeans mathematically, that the free energy is calculated in the range of temperatures, which in-cludes a point of pase transition, without introducing any artificial parameters similar to an orderparameter, but only starting from microscopic parameters of Hamiltonian. Moreover the theoremsabout connection of a vacuum amplitude with thermodynamics potentials are realized. The func-tional of a superconductor’s free energy in a magnetic field was obtained with help the developedmethod. The obtained functional is generalization of Ginzburg-Landau functional for the case ofarbitrary value of a gap, arbitrary spatial inhomogeneities and nonlocal magnetic response. Theexplicit expressions for the extremals of this functional were obtained in the low-temperature limitand the high-temperature limit at the condition of slowness of gap’s changes.
PACS numbers: 64.60.Bd, 74.20.-z, 74.20.De, 74.20.Fg
I. FORMULATION OF THE PROBLEM
Exact microscopic description of a phase transition is opened problem of modern theoretical physics. The problemcan be solved for several simplest models only, but different phenomenological approaches exist for the rest cases. Theessence of the matter lies in the following. Basic problem of statistical mechanics is calculation of a partition function(of grand canonical ensemble in a total case) Z or calculation of a density matrix b ρ : Z = Sp (cid:16) e − β ( b H − µ b N ) (cid:17) ≡ e − β Ω , b ρ = 1 Z e − β ( b H − µ b N ) , (1)where b H = b H + b V is full Hamiltonian of a system, b N is the particle operator, µ is the chemical potential, Ω is thegrand thermodynamics potential. Replacement of Hamiltonian b H in canonical ensemble by Hamiltonian b H − µ b N ingrand canonical ensemble leads to the shift of reference of particle’s energy from zero to Fermi surface: ε ( k F ) = 0.Hence, the potential Ω plays a part of Helmholtz free energy with a reference of particle’s energy from Fermi surface.Therefore we shall call the grand thermodynamics potential by free energy for brevity. Let’s transform the partitionfunction (1) to a form: Z = Sp (cid:16) e − β ( c H − µ b N ) e β ( c H − µ b N ) e − β ( b H − µ ˆ N ) (cid:17) = Sp (cid:16) Z b ρ e U ( β ) (cid:17) = Z R ( β ) , (2)where Z is the partition function for a system of noninteracting particles, e U ( β ) = e + β ( ˆ H − µ b N ) e − β ( b H − µ b N ) = ∞ X n =0 ( − n n ! Z β dτ . . . Z β dτ n b T n b H I ( τ ) . . . b H I ( τ n ) o (3)is the evolution operator in the interaction representation (it describes evolution of the system in imaginary time it → τ , b T is the ordering operator in time), b H I ( β ) = e + β ( ˆ H − µ ˆ N ) b V e − β ( ˆ H − µ ˆ N ) is the interaction operator of particlesin interaction representation, R ( β ) = h b U ( β ) i = Sp (cid:16)b ρ e U ( β ) (cid:17) = ∞ X n =0 ( − n n ! Z β dτ . . . Z β dτ n Sp (cid:16)b ρ b T n b H ( τ I ) . . . b H I ( τ n ) o(cid:17) (4) ∗ Electronic address: [email protected] † Electronic address: [email protected] is the vacuum amplitude of the system. The averaging hi ≡ Sp ( b ρ . . . ) is done over ensemble of noninteracting particles. It is necessary to note, that the vacuum amplitude describes dynamics of the system in the multiparticlestate Φ under the influence of the internal interaction, unlike Green function describing dynamics of a particle inone-particle state φ k under the influence of its interaction with other particles.The partition function Z can be found exactly for any system. If particles interact, then the situation becomescomplicated essentially. The only several models can be solved exactly [2, 3], but in the rest cases the values (1)can not be found exactly and it is necessary to develop a perturbation theory. Solution of the basic problem of thestatistical mechanics reduces to the calculation of a transition amplitude ”vacuum-vacuum” (4). In order to formulateour problem let’s consider the case of zero temperature T = 0. In this case the time is real, and the vacuum amplitudeis determined by the following: R ( t ) = h Φ | e U ( t − t ) | Φ i t =0 = h Φ | U ( t ) | Φ i e iW t = ∞ X n =0 ( − i ) n n ! Z t dt . . . Z t dt n D Φ | b T n b H I ( t ) . . . b H I ( t n ) o | Φ E , (5)where b H I ( t ) = e + it ( ˆ H − µ ˆ N ) b V e − it ( ˆ H − µ ˆ N ) is operator of particles’ interaction in the interaction representation, W isground state energy of a system without interaction. The averaging hi ≡ h Φ | . . . | Φ i is done over the ensemble of noninteracting particles. Knowing a vacuum amplitude we can calculate the ground state energy of the system of theinteracting particles E using the theorem [1]: E = W + lim t →∞ (1 − iδ ) i ddt ln R ( t ) , (6)where δ is infinitely small value, but ∞ · δ → ∞ . The theorem is correct if the limit transition[ ddt ln R ( t )] ∞ (1 − iδ ) = iW + | h Φ | Ψ i | ( − iE ) e − iE ∞ e − E δ ·∞ | h Φ | Ψ i | e − iE ∞ e − E δ ·∞ = iW − iE (7)is possible. This transition is possible at the condition when the symmetries of ground state of the system withinteraction | Ψ i and without interaction | Φ i are identical: h Φ | Ψ i 6 = 0 . (8)The expressions (7) and (8) mean: 1)potential of interaction is being switched slowly in the system in the groundstate without interaction, 2) the ground state of the system with interaction is being obtained by continuous wayfrom the ground state without interaction while the switching of the interaction ( adiabatic hypothesis ). For nonzerotemperature the analog of the theorem (6) has a form: U = − ∂∂β ln Z − ∂∂β ln R ( β ) , (9)where U is the internal energy of a system. Moreover, we can calculate the free energy:Ω = − β ln Z − β ln R ( β ) . (10)If the wave functions are orthogonal h Φ | Ψ i = 0 - the adiabatic hypothesis is not valid, then the symmetries of thesystem with interaction and without one is different. This means, that an initial system without interaction suffersphase transition stipulated by the interaction. A nonfulfilment of the adiabatic hypothesis means nonfulfilment of thetheorem (6). For the system with the broken symmetry we can calculate a vacuum amplitude on the free propagators G ( k , t ) and use the formula (6). However we shall find a wrong ground state energy E . This means that a stateexists with more low energy than the found value. Moreover, in consequence of the breakdown of the condition (8) asystem becomes unstable: Γ-matrix, which determines vacuum amplitude R ( t ), one-particle propagator G ( k , t ) andtwo-particle propagator K ( k , k , t ) in stair approximation, has the form:Γ( t ) = ce − αt + c ′ e + αt . (11)The value of Γ is increasing infinitely at t → ∞ [1, 4], it means an instability of the system. Similar instability wasobserved experimentally as the process of formation of a charge density wave in TbTe [6].From the aforesaid we can see, that perturbation theory making possible to calculate the vacuum amplitude (ther-modynamics function U , Ω and so on) in ranges including a point of phase transition (for example, temperature T C )doesn’t exist. In other words, it is not possible to find R ( β ) in the system with condensed phase starting from the firstprinciples. However the phenomenological approach exists for calculation of Ω. It assumes, that at the temperature T < T C the condensed phase is characterized by some order parameter η exists. Near a transition point for typeII phase transition or near a point of overcooling for type I phase transition the parameter η is small, and the freeenergy can be represented in a form of Landau expansion:Ω( T, V, µ, h ) = Ω + Z d r ( aη + bη + g ( ∇ η ) − ηh ) , (12)where h is external field. The equilibrium value of order parameter η is determined by an extremal of the functional(12): δ Ω δη | η = η = 0. However, the expansion (12) is correct in the range ( T C − T ) /T C ≪ η can be foundin this range only.In order to obtain η at any temperature the concept of quasi-averages ( anomalous averages ) is introduced. Inthe paper [5] generalization of the method of quasi-averages has been represented in terms of Green function -Nambu-Gor’kov formalism [7, 8]. It is postulated , that in the condensed phase, in addition to normal propagators G ( k , σ, t ′ − t ) = − i h Ψ | T { C k σ ( t ′ ) C + k σ ( t ) }| Ψ i , anomalous propagators F exist, for example: Ferromagnetic : F fer = − i h Ψ | T { C k ↑ ( t ′ ) C + k ↑ ( t ) }| Ψ i Solid, liquid : F sol = − i h Ψ | T { C k + q σ ( t ′ ) C + k σ ( t ) }| Ψ i (13) Superconductor with singlet pairing : F sup = − i h Ψ | T { C − k ↓ ( t ′ ) C k ↑ ( t ) }| Ψ i They are proportional to the according order parameters. The anomalous propagators F can not be obtained bysummation of diagrams consisting of normal propagators G only. This fact is result of different symmetries of aperturbed state and an unperturbed state.For construction of the self-consistent perturbation theory the ”sourse term” b H S is introduced in Hamiltonianinstead of the interaction operator b V . The sourse term is induced by the order parameter (for ferromagnetic -magnetic field orientating spins, for superconductor - a sourse of Cooper pairs). The sourse term inducts specificationstructure and changes symmetry of a system: Φ → Φ ′ , such that h Φ ′ | Ψ i 6 = 0. This means that a system has aninternal long range field H , generated by the sourse b H S , and according order parameter η , which is function of H anddepends on the interaction constant λ and temperature T : η = η λT ( H ). In turn, H is function of η : H = H λT ( η ).Hence, these two equations can be combined: η = η λT ( H λT ( η )) (14)and they can be solved in η . This means, that order parameter is determined in self consistent way. In Green functionformalism this fact has the following form: the free matrix propagator G (anomalous part is zero F = 0) and thedressed propagator G (with normal G part and anomalous F parts) obey Dyson equation [4, 5]: G − = G − − Σ ( G ) , (15)and what’s more the mass operator Σ ( G ) is determined by self consistent way. This means, that elements of diagramsfor the mass operator are dressed propagators G , which contain the sought mass operator. Artificiality of theintroduction of the sourse b H S lies in the fact that the expression for a mass operator is postulated . Moreover, Σ can not be obtained by summation of the diagram consisting of free propagators G only, that is Σ ( G ) = . Forexample, Σ has the form for a superconductor: Σ ( ω, p ) = [ − Z ( ω, p )] ω + χ ( ω, p ) τ + ϕ ( ω, p ) τ + e ϕ ( ω, p ) τ , (16)where Z is some coefficient, the field χ determines a shift of chemical potential µ at the transition to the superconduc-tive state, the fields ϕ and e ϕ play a part of order parameter in superconductivity - a gap, and specter of excitationsis represented by them E ( p ) = ε ( p ) + ϕ ( p ) + e ϕ ( p ); , τ i are unit matrix and Pauli matrixes. The dependence ofall fields on frequency ω takes into account a delay and a damping of quasi-particles. In Hartree-Fock approximationwe have Z = 1 and Σ doesn’t depend on ω . We can suppose e ϕ = 0, χ = 0 and denote ϕ ( p ) ≡ ∆( p ), then∆( T = 0) = λV Z + ∞−∞ dω π Z d p (2 π ) iF ( ω, p , ∆) , ∆( T ) = λV T n =+ ∞ X n = −∞ Z d p (2 π ) iF ( ω n , p , ∆) , (17)where λ < ω n = (2 n + 1) πT . We cansee, that the order parameter is determined by the anomalous propagator F , and the equations (17) are equations ofself consistency for order parameter ∆ as specific case of the total equation (14).The described approach is phenomenological too, because the anomalous propagators and corresponding orderparameters are introduced to the theory from elsewhere. This means that we select the required states from allpossible state artificially. In this sense this approach likes Landau approach (12). The sourse of Cooper pairs forsuperconductor or the magnetic field H = J gµ B h S z i for ferromagnetic ( J is the exchange integral, g is the gyromagneticrelation, µ B is Bohr magneton, h S z i is the average projection of spin onto axis z ) can not be interpreted as real field.So, for iron the real internal magnetic field is ∼ oersted, but for the ordering of spins it is necessary the effectivemagnetic field H ∼ oersted. Thus, self magnetization has nonmagnetic nature evidently.As it has been noted in [9], in spite of all progresses reached in description of phase transitions and in calculationof main characteristics of a system in critical region, the basic problem of phase transitions is not solved: calculationexplicit expressions for thermodynamical functions of a system in ranges, which include a point of phase transitionas function of temperature, external fields and microscopic parameters of Hamiltonian . It is necessary to have adescription of phase transitions on microscopic level. This presupposes a direct calculation of free energy Ω( T, V, h )from first principles, but we must not construct it.At the present moment the two approaches can be separated for solution of the formulated problem. In the papers[9–11] a partition function of Ising model is calculated by the method of collective variables, which are oscillationmodes of spin moment. Then a partition function can be written via these variables as a functional. Investigation ofEuler-Lagrange equations shows, that among a set of collective variables the variable connected with order parameterexists. In the paper [12] a partition function is calculated by the saddle point method for classical system with shortrange attraction and repulsion between particles. It has been shown, that the free energy can be represented by theform which is analogous to Landau expansion where the saddle point plays a role the order parameter. The saddlepoint method has the sense as method of separation of states giving largest contribution in a partition function and itis equivalent to the mean field method. With the help of the saddle point method the processes of cluster formationcan be investigated. The cluster formation can be considered as a phase transition from spatially homogeneousdistribution to spatially inhomogeneous distribution. In the papers [13–16] both short range potentials and longrange potentials were considered. Critical temperature and critical concentration, when clusters form in a system,dependence of size of a cluster on temperature have been obtained.One of the important applications of the microscopic theory of phase transitions is description of thermodynamicsand electrodynamics of superconductors. It is necessary to know the functional of free energy Ω( β, ∆ , A ), where A is potential of magnetic field. Then the equations δ Ω δ ∆ = 0, δ Ω δ A = 0 will describe equilibrium states of the condensedphase and normal phase. Two basic methods for obtaining of the sought equations exist. First of them is joint solutionof Gor’kov equations and the equation of self consistency (17). At the temperature T → T C , when ∆ /T C →
0, thesolution of these equation can be represented in the form of series in degrees of ∆. Moreover, magnetic penetrationdepth is bigger then Pippard coherent length l , hence the potential A changes a little on a coherent length. As aresult we have the well-known Ginzburg-Landau equation.Another method, proposed in [17], is the direct calculation of a vacuum amplitude R ( β ). The concept lies in thefact that we consider electrons in a normal metal propagating in random ”field” of thermodynamic fluctuations oforder parameter ∆ q , where q is small wave-vector. The operator of the interaction of electrons with the fluctuationscan be written as: b H int = X p h ∆ q b C + p + b C + − p − + ∆ ∗ q b C − p − b C p + i , (18)where p ± = p ± q /
2. A correction to the thermodynamics potential from any interaction is represented via thevacuum amplitude R : ∆Ω = − T ln R ( β ) ≈ − T [ R ( β ) − . (19)Then using Wick theorem we can represent (4) via the propagators. For the correction of second order we have:∆Ω = − T Z /T dτ Z /T dτ h b T τ ( b H int ( τ ) b H int ( τ )) i = − T Z /T dτ Z /T dτ | ∆ q | X p G ( p + , τ − τ ) G ( − p − , τ − τ ) . (20)The correction ∆Ω is represented via the free propagators G of normal state only - we consider normal metal at T > T c , where the fluctuation sourse of Cooper pair (18) acts. As a result we have Landau expansion:Ω s − Ω n = X q (cid:20) α ( T ) | ∆ q | + b | ∆ q | + γq | ∆ q | (cid:21) , (21)where α ( T ) ∝ ( T − T ) , b, γ are expansion coefficients.In our opinion, this approach is not successively microscopic, because the artificial element is used - the externalsourse of Cooper pairs (18), that implies some seed order parameter. As for calculation of the correction ∆Ω the freepropagators G of normal phase is used only, the condensed phase is considered as fluctuations against the backgroundof normal phase. This means, that we can obtain the limit Ω( T → T ) only.Ginzburg-Landau equations are correct for description of thermodynamics and electrodynamics of a superconductorat the following restrictions:1. The gap is much less than critical temperature. Then the parameter ∆( r , T ) /T C ≪ T → T C or in the range H → H C (intensity ofmagnetic field is near the second critical magnetic field H C ).2. ∆( r , T ) changes slowly on the coherent length l ( T ), which is size of a Cooper pair.3. Magnetic field H ( r ) = rot A ( r ) changes slowly on the coherent length, that is the magnetic penetration depthis λ ( T ) ≫ l (0). This means, that electrodynamics of a superconductor is local.In the papers [18, 19] the equations has been proposed, where the first restriction is absent. These equations wereobtained from Gor’kov equations and they are the generalization of Ginzburg-Landau equations for the case ofarbitrary value of ∆( r , T ) /T . However spatial inhomogeneities are slow and electrodynamics is local.Our aim is the description of phase transitions on microscopic level starting from the first principles only. Mathe-matically this means to develop a method of calculation of the partition function (1) (the free energy Ω) in ranges oftemperatures and parameters of interaction, which include a point of pase transition, without introducing any artificialparameters of type of order parameter η and sourses of ordering b H S , but starting from microscopic parameters ofHamiltonian ˆ H = ˆ H + ˆ V only. The theorems (6,9,10) about connection of a vacuum amplitude with thermodynamicspotentials must be realized. Thus, in microscopic theory of phase transitions the equation of self-consistency (14), theanomalous propagators F ( p , ω ) (13) and Landau functional (12) must be deduced, but they must be not postulated.To solve the problem means to develop the perturbation theory for the vacuum amplitudes R ( t ) and R ( β ), which iscorrect both normal phase and condensed phase. Phase transition normal metal - superconductor has been consideredas an example. Since in Nambu-Gor’kov formalism (the method of anomalous propagators) any phase transition canbe described [5], then our method can be generalized to the rest transitions (ferromagnetism, waves of charge andspin density, crystallization and so on). Further we can apply the developed method in order to calculate free energyof a superconductor at arbitrary temperatures, spatial inhomogeneities, magnetic fields and currents, moreover withnonlocal magnetic response. The obtained expression will be the generalization of Ginzburg-Landau functional (12)in above mentioned sense.In the section II we consider the instability of normal Fermy system at the switching of attraction between particles.Mathematically this is expressed in the fact that a two-particle propagator K , calculated on free one-particle propaga-tors in stair approximation, has a pole α which doesn’t belong to a free propagator K and situated on complex axis.This means a presence of bound states of particles in the system (with the binding energy | α | ) and evolution of thesystem in time as ∼ e | α | t . On the other hand, a two-particle propagator has a pole at the energy of the bound state ifthe bound state of two isolated particles exists. The residue in this pole is product of Bethe-Solpiter amplitudes ηη + - the amplitudes of pairing.In the section III we generalizes the result of two-particles problem to a many-particle system. It follows from theidentity principle, that amplitudes of pairing are determined by dynamics of all particles of the system, and observedvalues of the amplitudes of pairing are the result of the averaging over a system. Thus, the collective (condensate)of pairs exists. In order to find an one-particle propagator G S and to generalize the two-particle problem to amany-particle case we proposed the method of an uncoupling of correlations. The method considers interaction of aadditional fermion with fluctuations of pairing (formation and decay of the pairs). As a result of such interaction,the law of quasi-particles’ dispersion changes as ε ( k ) → p ε ( k ) + ∆∆ + , where ∆ and ∆ + are amplitudes of pairingplaying a role of a gap and they are analog of Bethe-Solpiter amplitudes in two-particle problem. Gor’kov equationsand the existence of anomalous propagators F and F + follow from Dyson equation for the described above process.This fact means breakdown of a global gauge symmetry, namely number of particles is not conserved in the course of apresence of a pairs’ condensate. After calculation of particles’ interaction with fluctuations of pairing all characteristicsof a system must be calculated over the new vacuum with broken symmetry.For calculation of observed values of the amplitude of pairing ∆ and ∆ + it is necessary to know a vacuum amplitude R ( t ) of a system. The vacuum amplitude must be calculated over the new vacuum with broken symmetry. This givespossibility to use the theorem about connection of a vacuum amplitude with ground state energy of a system. In thesection IV the method of uncoupling of correlations is proposed. The method allows to represent a vacuum amplitudevia anomalous propagators F and F + . Thus we obtain a functional of ground state energy over the fields ∆ and ∆ + .An extremal of the obtained functional is the equation of self consistency for the parameter ∆ in Nambu-Gor’kovformalism. This means, that order parameter is averaged Bethe-Solpiter amplitude over all system. In the section Vwe generalize the results of two previous sections for the case of nonzero temperature. Using the method of uncouplingof correlations we calculate a vacuum amplitude R ( β ) and a functional of free energy Ω(∆ , T ) over the fields ∆ and∆ + . In a high-temperature limit T → T C the expansion of free energy in powers of | ∆ | has a form of Landauexpansion. This fact proves, that the averaged over a system the amplitudes of pairing ∆ and ∆ + have properties,which are analogous to the properties of an order parameter in a phenomenological theory. In the section VI weconsider the case of a pairing with nonzero momentum of a pair’s center of mass.In the sections VII and VIII using the developed method of microscopic description of a phase transition we obtainedthe functional of free energy of a spatial inhomogeneous superconductor in magnetic field. The functional generalizesGinzburg-Landau functional in cases of arbitrary temperatures, arbitrary spatial inhomogeneities and nonlocality ofmagnetic response. The free energy has a form of Ginzburg-Landau functional in the high-temperature limit at thecondition of slowness of gap’s change in space. Corresponding equations of superconductor’s state demonstrate thenonlinear connection between the current and the magnetic field. In the low-temperature limit in the case of weakfield H ≪ H C at the condition of slowness of gap’s change the nonlocal connection between the current and the fieldappears, which is the long-wave limit of Pippard law. The last fact proves the nonlocality of the obtained functionalof free energy. II. THE INSTABILITY OF NORMAL STATE AND TWO-PARTICLE DYNAMICS
Let we have a system from N noninteracting fermions in volume V . In ideal Fermy gas propagation of a particle withmomentum k , energy ε ≈ v F ( | k | − k F ) counted off Fermy surface (we are using system of units, where ¯ h = k B = 1)and spin σ is described by the free propagator: G ( k , t ) = ( − i h Φ | C k ,σ ( t ) C + k ,σ (0) | Φ i , t > i h Φ | C + k ,σ (0) C k ,σ ( t ) | Φ i , t ≤ ) = − iθ t A e − i | ε | t + iθ − t B e i | ε | t ,G ( k , t ) = Z dω π G ( k , ω ) e − iωt = ⇒ G ( k , ω ) = 1 ω − ε ( k ) = ω + εω − ε = A ω − | ε | + B ω + | ε | , (22)where A = 12 (cid:18) ε | ε | (cid:19) , B = 12 (cid:18) − ε | ε | (cid:19) , θ t = (cid:26) , t > , t < (cid:27) , (23) C k ,σ ( t ) and C + k ,σ ( t ) are creation and annihilation operators in Heisenberg representation. Now let attractive forceacts between particles. The force is described by matrix element of interaction potential: h l , − l | b V | k , − k i = λw l w k < , w k = (cid:26) , ε ( k ) < ω D , ε ( k ) > ω D (cid:27) , (24)moreover interacting particles have opposite spins. This model potential is correct if a range of interaction r is muchsmaller than average distance between particles: r p N/V ∼ r k F ≪
1, that means ”slowness” of collisions. In turn,account of Fermy statistic, in the limit of ”slow” collisions particles can scatter with opposite spins only.The mass operator Σ( G ) in stair approximation is determined by so called Γ-matrix (amplitude of scattering),which is solution of the equation represented by the diagram in Fig.1. For the case q = 0 Γ-matrix has a form:Γ = λw k ′ w k − iλ R d k (2 π ) dω π w k G ( k , α − ω ) , G ( k , ω ) = λw k ′ w k − | λ | mk F π ln (cid:12)(cid:12)(cid:12) ω D α − (cid:12)(cid:12)(cid:12) . (25)This expression has a pole in the point α . In its neighborhood the expression has a view:Γ( α → α ) = − π mk F ± i | α | α ± i | α | , α ( λ →
0) = − ω D exp (cid:26) − π mk F | λ | (cid:27) . (26) G ( k , ) G ( k ’’, ) k , q - kk , ’ q - k’ ’ V(k’-k’’) k , ’ q - k’ ’ == +; q - kk , V(k-k’) q - k’q - k k , ’ k , G ( q - k ’’, ) Figure 1: The mass operator Σ expressed via Γ( q , α )-matrix. Bethe-Solpiter equation for Γ( q , α )-matrix. If with the help of Fourier transformation to pass from ω -representation to t -representation, then we shall have theexpression (11), which increases infinitely at t → ∞ . Hence, according to the first diagram in Fig.1, we have the samebad behavior of the mass operator Σ. The presence of a pole in imaginary axis means instability of a system. In ouropinion, this result can be interpreted as follows: in a system because of the interaction (24) the strong fluctuationsexists, but the free propagator G doesn’t consider these perturbations. In turn, it leads to the instable solution ofDyson equation. This means, that besides the interaction between particles V ( k ) the interaction of particles withaforesaid fluctuations exists too. For stability of the solution, dressed propagators G S , which considers scattering onthe fluctuations, must be in the equation for the mass operator Σ and scattering amplitude Γ in Fig.(1) instead thefree propagators G .Let’s determine a two-particle propagator by the expression: K ( x , x ; x , x ) = h Ψ | T b C ( x ) b C ( x ) b C + ( x ) b C + ( x ) | Ψ i K ( x , x ; x , x ) = h Φ | T b C ( x ) b C ( x ) b C + ( x ) b C + ( x ) | Φ i = G ( x , x ) G ( x , x ) , (27)where K is a propagator in a system with interaction, K is a free two-particle propagator and it is equal to aproduct of one-particle propagators G , x ≡ ( ξ, t ). The free and dressed two-particle propagators are connected byBethe-Solpiter equation (in an operator view): K = K + K iυK = K + K i Γ K ⇒ Γ = υ + iυK Γ , (28)where υ ( x , x ; x , x ) = V ( r − r ) δ ( x − x ) δ ( x − x ) δ ( t − t ). The last equation in (28) for the scattering amplitudeΓ is represented graphically in Fig.1. We can see from the equation (28), that a presence of the pole in Γ means apresence the same pole in the two-particle propagator K . As it is well known [20, 22–25], a two-particle propagatorhas a pole structure at the values of energy corresponding to a bound state . The pole | α | (26) doesn’t belong to thefree propagator K , but it appears as a result of the attraction (24) λ <
0. Hence, the pole means a presence of boundstates of two particles in a system with the binding energy E s ≈ | α | .For investigation of the bound state and calculation of of particles’ interaction with above described fluctuationslet’s consider the problem of two particles at first. Previously we considered dynamics of two selected particles beingin a field of the rest particles of a system. Therefore the propagator K determined dynamics of all particles of asystem in such approximation. Now we shall consider the system consisting from two particles only being in the stateΦ s with the energy E s . Interaction between them is described by the potential V ( k ) ≡ V :( H + H + V )Φ s = E s Φ s . (29)Let’s determine a propagator for two particles as kernel of the integral operator finding Φ( t ) known Φ ( t ′ ):Φ( ξ , ξ , t, t ′ ) = − Z K ( ξ , ξ , t ; ξ ′ , ξ ′ , t ′ )Φ ( ξ ′ , ξ ′ , t ′ ) dξ ′ dξ ′ . (30)Then the two-particle propagator can be written in Fourier representation as [20]: − K ( ξ , ξ ; ξ ′ , ξ ′ ; E ) = i X s Φ s ( ξ , ξ )Φ ∗ s ( ξ ′ , ξ ′ ) E − E s + iγ . (31)If the bound state is among states s , then the pole of the function K ( ξ , ξ , E ) corresponds to the bound state s at areal E , where E is equal to energy of the bound state E s .The residue in a pole E = E s is Φ s ( ξ , ξ )Φ ∗ s ( ξ ′ , ξ ′ ). As K hasn’t a pole in E = E s , where E s is the energy ofthe bound state, then a pole of K means a presence of a pole at Γ according to the equation (28). For the function K ( x , x ; x ′ , x ) ′ with different times the expression can be written: iK ( ξ , ξ , τ ; ξ ′ , ξ ′ , τ ′ ; E ) = X s Π s ( ξ , ξ , τ ; ξ ′ , ξ ′ , τ ′ ) E − E s + iγ , (32)where we denoted that t − t = τ ′ , t − t ′ = τ ′ , t + t = 2 t, t ′ + t ′ = 2 t ′ . This expression is the formula (31) at τ = τ ′ = 0.The residue Π s ( ξ , ξ , τ ; ξ ′ , ξ ′ , τ ′ ) for the bound state can be written in multiplicative form as in the case τ = τ ′ = 0: Π s = η s ( ξ , ξ , τ ) η + s ( ξ ′ , ξ ′ , τ ′ ) . (33)The values η and η + are Bethe-Solpiter amplitudes [22–25]. They are connected with the wave functions Φ s as (inmomentum representation ξ ≡ k ): Φ s ( k , k ) = R η s ( k , k , ǫ ) dǫ π . In the equation for K (28), K can be neglectednear a pole corresponding to a bound state. Resulting homogeneous equation has the solution (33), moreover thefunction η s ( ξ , ξ , τ ) satisfies the equation η = iK υη ⇐⇒ η s ( k , k , ǫ ) = iG ( k , E s / − ǫ ) G ( k , E s / ǫ ) Z V ( k ) η s ( k + q , k − q , ǫ ′ ) dǫ ′ π d k (2 π ) . (34)The term in the sum (32), corresponding to the bound state E s , can be represented by the diagram in Fig.2, where + k + q , k - q S E E i k -qk +q k S E E iK = k , k k Figure 2: The two-particle propagator for isolated pair of particles in neighborhood of a pole corresponding to the bound state E s . the dotted line means the multiplier E − E s + iγ - propagation of two particles in bound state, and the blocks are η s ( k , k , ǫ ) and η + s ( k + q , k − q , ǫ ) - transition amplitudes in bound state and back. The diagram in Fig.2 can beinterpreted by the following way. Two particles with momentums k and k form a bound state with energy E s withamplitude η s ( k , k ). Further, the two particles propagate together. Then the bound state can decay with amplitude η + s ( k + q , k − q ). As a result the two free particles appear with momentums k + q and k − q . III. THE UNCOUPLING OF CORRELATIONS AND A MULTIPARTICLE DYNAMICS.
In the previous section we considered dynamics of two isolated particles. Now we have to generalize the obtainedresults to the multi-particle case - propagation of two interacting particles in a system of identical fermions. Thissituation differs from the previous case by the following conditions:1. Each pair of fermions is in field of all the rest particles.2. All particles of a system are identical. Moreover, the average size of a pair l ∼ / p | α | m ≫ p V /N is morebig than average distant between particles, that means the wave packages of pairs overlap strongly.Mathematically this means, that the amplitudes η and η + are not solution of the equation (34), which is correct forisolated pair only. Now the amplitudes are determined by dynamics of all particles of the system, and their observedvalue is result of an averaging over a system . Pairs in such system are effective, namely two fermions having formeda bound state with an amplitude η s ( k , k , ǫ ) (Fig.2) are not fixed pair: one from partners in a pair can leave thebound state with a fermion from another pair with amplitude η + s ( k + q , k − q , ǫ ). Thus, the collective of pairs(condensate) exists.Let’s consider the two-particle propagator K E → E s represented in Fig.2. As it has been noted earlier, fermions withopposite momentums and and opposite spins form a pair. Let’s suppose, that corresponding amplitudes of pairingdon’t depend on time. In order to obtain an one-particle propagator G S we shall use the method of uncouplingof correlations considered in the Appendix A, and we shall be acting analogously to Fig.11. The procedure of anuncoupling is represented in Fig.3. We connect the entering line and the outgoing line corresponding to particles withmomentum − k and energy parameter − ω each. As a result we have the intermediate propagator G . Since partnersin each pair are not fixed in consequence of the identity principle and strong intersection of wave packages of pairs,that is a condensate of pairs exists, then it is necessary to cut the dotted line - the propagator of a pair E − E s + iγ . Thenthe points of a joining of the dotted lines correspond to interaction with the effective field (in accordance with therules of diagram technics in the Appendix A). The fluctuation of pairing play a role of the above mentioned effectivefield. The amplitudes of such interaction we denote as − i ∆( k , − k ) and i ∆ + ( k , − k ). These values correspond to theamplitudes of the two-particle problem η s ( k , − k ) and η + s ( k , − k ) to the extent that their observed values is result ofaveraging over a system h η s ( k , − k ) i ∼ ∆ h η + s ( k , − k ) i ∼ ∆ + in consequence of statistical correlations between pairsand they are determined by dynamics of all system’s particles. k ,- kk ,- k G k ,G k ,-G k ,G k ,G S k , ...++= S E E i - kk - kk Figure 3: The procedure of uncoupling of correlations for a two-particle propagator of a pair being in a field of all rest fermionof a system. The result of the uncoupling is the dressed one-particle propagator G S ( k , ω ), as a consequence of interaction of afree fermion with fluctuations of pairing. The result of the procedure of uncoupling of correlations means the follows. Let an additional particle withmomentum k , ω propagates through a system of identical fermions. In the process of propagation a particle can formbound states with other fermions according to the following mechanism. Some pair of fermions decays in componentswith momentums − k , − ω and k , ω with amplitude i ∆ + . Second particle of the decayed pair is in state of the additionalparticle ( k , ω ) and it is identical to the additional particle. The second particle propagates through a system further.First particle of decayed pair forms bound state with the initial additional particle with amplitude − i ∆. Anew formedpair replenishes the condensate of pairs in a system. Thus, the dressed propagator G S takes into account interactionof a particle, initially described by free propagator G , with fluctuations of pairing. Intensity of the interaction is theamplitudes − i ∆ and i ∆ + .Starting from the aforesaid, we can write the mass operator for such process (Fig.4) as − i Σ = − i ∆ iG ( − k, − ω ) i ∆ + ⇒ Σ = ∆∆ + ω + ε ( k ) . (35)0 i + i = iG (- k , ) -i Figure 4: The diagram for the mass operator Σ describing interaction of a fermion with fluctuations of pairing.
This mass operator has been proposed in [20, 21], however an existence of the amplitudes ∆ , ∆ + and the equation ofself-consistency were postulated (as the anomalous averages). From Dyson equation (A6) we can obtain the dressedone-particle propagator: G S ( k , ω ) = 1 ω − ε − Σ = ω + εω − E = A S ω − E + B S ω + EG S ( k , t ) = − iθ t A S e − iEt + iθ − t B S e iEt (36) A S = 12 (cid:16) εE (cid:17) , B S = 12 (cid:16) − εE (cid:17) , where E ( k ) = p ε ( k ) + ∆ (37)is dispersion law of dressed particles (quasi-particles). The amplitude ∆ is named by gap, because minimal work forcreation of one-particle excitations is 2∆. In accordance with the definition of one-particle propagator we can write: G S ( k , t ) = ( − i h Ψ | C k ,σ ( t ) C + k ,σ (0) | Ψ i , t > i h Ψ | C + k ,σ (0) C k ,σ ( t ) | Ψ i , t ≤ ) , (38)where the system is placed in another ground state Ψ . In the state Ψ interaction of particles with fluctuation ofpairing (existence of condensate of pairs) is taken into account, moreover Ψ (∆ = 0) = Φ , G S (∆ = 0) = G . Apropagator defines occupations number of quasi-particles n k by the following way: n k = h Ψ | C + k C k | Ψ i = − i lim t → − G S ( k , t ) = B S , − n k = h Ψ | C k C + k | Ψ i = i lim t → + G S ( k , t ) = A S . (39)Hence, we can suppose, that C + k | Ψ i = p A S | Ψ , p k i , C k | Ψ i = p B S | Ψ , h k i , (40)where | Ψ , p k i and | Ψ , h k i are states with one added particle and one removed particle with momentum k accordingly.Hamiltonian of a system of free quasi-particle has a view: b H = X k E ( k ) C + k C k , (41)Chemical potential of a quasi-particles’ system equals to zero. Therefore the grand potential Ω coincides withHelmholtz free energy in a superconductive state. The states | Ψ i , | Ψ , p k i and | Ψ , h k i are eigenvectors of theHamiltonian (41): b H | Ψ i = Ω | Ψ i , b H | Ψ , p k i = (Ω + E ( k )) | Ψ , p k i , b H | Ψ , h k i = (Ω + E ( k )) | Ψ , h k i . (42)Then, using the definition (38), we can find: G S ( k , t >
0) = − i h Ψ | C k ( t ) C + k (0) | Ψ i = − i h Ψ | e i b H t C k e − i b H t C + k | Ψ i = − i h Ψ , p k | e i Ω t p A S e − i (Ω + E ( k )) t p A S | Ψ , p k i = − iA S e − iEt h Ψ , p k | Ψ , p k i = − iA S e − iEt , (43)that coincides with (36). For t ≤ ω − ε ) G = 1. On the other hand wehave G = G S / (1 + G S Σ). Moreover, let’s introduce the notations − G S Σ ≡ ∆ F + , − G S Σ ≡ ∆ + F. (44)Then, we can obtain the set of equations:( ω − ε ( k )) G S ( k , ω ) + ∆ F + ( k , ω ) = 1 (45)( ω + ε ( k )) F + ( k , ω ) + ∆ + G S ( k , ω ) = 0 . (46)These equations are Gor’kov equations in momentum representation. However, unlike phenomenological approach(where existence of the anomalous propagator F and the equation for order parameter are postulated) these equationsare obtained by microscopic way with help of the procedure of uncoupling of correlations. From the equations (45,46)we can find, that F + = − ∆ + ω − E , F = − ∆ ω − E . (47)The anomalous propagators describe creation of two fermions from the condensate of pairs - F + , formation a pair bytwo particles with leaving to the condensate - F . Moreover, F and F + are the infinity sum of the serial processes ofcreation and annihilation of pairs described by amplitudes ∆ and ∆ + . Mathematically this is expressed in the factthat F + αβ ( k , t ) = ∆ + √ ∆ + ∆ ( i h Ψ | C + − k ,β ( t ) C + k ,α (0) | Ψ i , t > i h Ψ | C + k ,α (0) C + − k ,β ( t ) | Ψ i , t ≤ ) = g αβ (cid:18) iθ t ∆ + √ ∆ + ∆ p A S B S e − iEt + iθ − t ∆ + √ ∆ + ∆ p A S B S e iEt (cid:19) , (48) F αβ ( k , t ) = ∆ √ ∆ + ∆ (cid:26) i h Ψ | C k ,α ( t ) C − k ,β (0) | Ψ i , t > i h Ψ | C − k ,β (0) C k ,α ( t ) | Ψ i , t ≤ (cid:27) = g αβ (cid:18) iθ t ∆ √ ∆ + ∆ p A S B S e − iEt + iθ − t ∆ √ ∆ + ∆ p A S B S e iEt (cid:19) , g αβ = (cid:18) (cid:19) . (49)Let’s prove the formulas (48,49). For this let’s consider the expression: h Ψ | C + − k ,β ( t ) C + k ,α (0) | Ψ i = h Ψ | e i b H t C + − k ,β e − i b H t C + k ,α | Ψ i = h Ψ , h − k ,α | e i Ω t p B S e − i (Ω + E ( k )) t p A S | Ψ , p k ,β i = p A S B S e − iE ( k ) t h Ψ , h − k ,α | Ψ , p k ,β i = p A S B S e − iE ( k ) t , (50)that corresponds to (48). In the last equality the fact has been used, that the states, created by addition of a particleto a state k , α or removing of a particle from a state − k , β at α = β , are identical in the course of existence of thepair condensate. The rest cases is proved analogously. It is not difficult to see, that F (∆ = 0) = F + (∆ = 0) = 0,because A ( k ) B ( k ) = 0. From (48) we can see, that the existence of nonzero anomalous propagators F and F + means breakdown of global gauge symmetry in a system, that is number of particles is not conserved in the course ofexistence of a pair condensate . Hence the states Φ and Ψ have different symmetries: h Φ | Ψ i = 0 . (51)However distribution function over N has a maximum at the average number of particles h N i determined by theexpression: h N i = 2 X k n k = − i Z d k (2 π ) dω π lim t → − G S ( k , t ) = 2 Z d k (2 π ) B S ( k ) ≈ N = 2 Z d k (2 π ) B ( k ) . (52)Now let’s return to the left part of the expression (25) for Γ-matrix. After considering of particles’ interaction withfluctuations of pairing we have to substitute dressed propagators G S instead free propagators G in the formula (25).It is not difficult to verify, that Γ hasn’t poles at any α and ∆ . This means, that the problem of instability of a system is removed, and we can use dressed propagator for further calculations confidently. Moreover, the absence of polesmeans the absence of bound states, because we have taken into account them in the specter of quasi-particles E ( k )(37).It is necessary to note, that the mass operator (35) and Gor’kov equation (45,46) haven’t parameter of interactionbetween particles. This means, that the amplitudes of pairing ∆ and ∆ + exists regardless of interaction betweenparticles and its type. However, as we shall see below, the interaction determines the average value of the amplitudes,that is observed in experiment. This average value is not zero in the case of attraction between particle only. IV. GROUND STATE ENERGY.A. Summary kinetic energy of particles.
In order to calculate ground state energy it is necessary to know kinetic energy of particles and energy of theirinteraction. The operator of kinetic energy of all particles of a system has a form: c W = X k ,α v F ( k − k F ) C + k ,α C k ,α = 2 X k ε ( k ) C + k ,α C k ,α . (53)Then the corresponding average value is h W i = − i X k G ( k , t → − ) ε ( k ) = − i lim t → − X k Z dω π G ( k , ω ) e − iωt ε ( k )= 2 X k B ( k ) ε ( k ) = V ν F Z ∞− v F k F B ( ε ) εdε. (54)Here ν F = k F π v F is density of states on Fermy surface. Since the interaction V l − lk − k (24) exists in the layer − ω D <ε ( k ) < ω D only, we can suppose that G = (cid:20) G ; | ε ( k ) | > ω D G S ; | ε ( k ) | < ω D (cid:21) , A ( k ) = (cid:20) A ( k ); | ε ( k ) | > ω D A S ( k ); | ε ( k ) | < ω D (cid:21) , B ( k ) = (cid:20) B ( k ); | ε ( k ) | > ω D B S ( k ); | ε ( k ) | < ω D (cid:21) . (55)Then we can separate a normal part and a superconductive part of the kinetic energy: h W i = V ν F Z − ω D − v F k F B εdε + V ν F Z ω D − ω D B S εdε + V ν F Z ∞ ω D B εdε = W n + V ν F Z ω D − ω D B S εdε − V ν F Z ω D − ω D B εdε = W n − V ν F Z ω D − ω D (cid:18) ε E − ε | ε | (cid:19) = W n + V ν F (cid:18) ω D − ω D q ω D + ∆ + ∆ arcsinh ω D ∆ (cid:19) . (56)We can see, that the pairing leads to a loss in the kinetic energy. B. Vacuum amplitude.
Since we took account of interaction of particles with fluctuations of pairing and we discovered that the groundstate of a system | Ψ i has other symmetry as compared with the initial state | Φ i , hence the vacuum amplitude of asystem can be written as: R ( t ) = h Ψ | e U ( t − t ) | Ψ i t =0 = h Ψ | U ( t ) | Ψ i e iW t = ∞ X n =0 ( − i ) n n ! Z t dt . . . Z t dt n D Ψ | T n b H I ( t ) . . . b H I ( t n ) o | Ψ E , (57)3where b H I ( t ) = e + it ˆ H b V e − it ˆ H is operator of particles’ interaction in an interaction representation, W - ground stateenergy of a system without interaction. The averaging hi ≡ h Ψ | . . . | Ψ i is realized by ensemble of noninteractingquasi particles . The Hamiltonian of such system has the following form: b H + b V = X α X k E ( k ) C + k ,α C k ,α + 12 V X α,β,γ,δ X k , l , m , n V klmn C + l ,β C + k ,α C m ,γ C n ,δ , (58)where momentum is conserved k + l = m + n and spin is conserved α + β = γ + δ , V is volume of a system. Thesequence order of indexes of matrix elements and of creation and annihilation operators is important.The expressions for several first orders ( n = 0 , , ... ) in the expansion of vacuum amplitude are (let us suppose t > t for definiteness): R ( t ) = h Ψ | Ψ i = 1 R ( t ) = 11! 1 V Z t dt X α,β,γ,δ X k , l , m , n (cid:18) − i V klmn (cid:19) h Ψ | C + l ,β ( t ) C + k ,α ( t ) C m ,γ ( t ) C n ,δ ( t ) | Ψ i R ( t ) = 12! 1 V Z t dt Z t dt X α,β,γ,δ X k , l , m , n (cid:18) − i V klmn (cid:19) X α ′ ,β ′ ,γ ′ ,δ ′ X k ′ , l ′ , m ′ , n ′ (cid:18) − i V k ′ l ′ m ′ n ′ (cid:19) ×h Ψ | C + l ′ ,β ′ ( t ) C + k ′ ,α ′ ( t ) C m ′ ,γ ′ ( t ) C n ′ ,δ ′ ( t ) C + l ,β ( t ) C + k ,α ( t ) C m ,γ ( t ) C n ,δ ( t ) | Ψ i , (59)where C + ( t ) = e + it ˆ H C + e − it ˆ H , C ( t ) = e + it ˆ H Ce − it ˆ H are operators of creation and annihilation in Heisenbergrepresentation, that coincides with interaction representation for an ensemble of noninteracting particles.In Appendix B we propose the method of uncoupling of correlations for approximate calculation of a vacuumamplitude R ( t ). As an example Hartree-Fock normal processes have been considered there. We shall generalize thismethod for anomalous processes here. In our case the particles interact by the potential (24) V l , − l , k , − k . Hence, thevacuum amplitude has a form: R ( t ) = 1 + 11! 1 V Z t dt X α,β X k , l (cid:18) − i V l , − l , k , − k (cid:19) h Ψ | C + − l ,β ( t ) C + l ,α ( t ) C k ,α ( t ) C − k ,β ( t ) | Ψ i + 12! 1 V Z t dt Z t dt X α,β X k , l (cid:18) − i V l , − l , k , − k (cid:19) X α ′ ,β ′ X k ′ , l ′ (cid:18) − i V l ′ , − l ′ , k ′ , − k ′ (cid:19) ×h Ψ | C + − l ′ ,β ′ ( t ) C + l ′ ,α ′ ( t ) C k ′ ,α ′ ( t ) C − k ′ ,β ′ ( t ) C + − l ,β ( t ) C + l ,α ( t ) C k ,α ( t ) C − k ,β ( t ) | Ψ i + ... (60)We can uncouple correlations by the following way taking into account anticommutation of the operators C and C + : R ( t ) ≈ − V Z t dt X α,β X k , l (cid:18) − i V l , − l , k , − k (cid:19) h Ψ | C + l ,α ( t ) C + − l ,β ( t ) | Ψ ih Ψ | C − k ,β ( t ) C k ,α ( t ) | Ψ i + 12! ( − V Z t dt Z t dt X α,β X k , l (cid:18) − i V l , − l , k , − k (cid:19) X α ′ ,β ′ X k ′ , l ′ (cid:18) − i V l ′ , − l ′ , k ′ , − k ′ (cid:19) ×h Ψ | C + l ′ ,α ′ ( t ) C + − l ′ ,β ′ ( t ) | Ψ ih Ψ | C − k ′ ,β ′ ( t ) C k ′ ,α ′ ( t ) | Ψ ih Ψ | C + l ,α ( t ) C + − l ,β ( t ) | Ψ ih Ψ | C − k ,β ( t ) C k ,α ( t ) | Ψ i + . . . = 1 + R + 12! R + . . . = exp( R ) (61)As a result of the uncoupling of correlations we can see, that anomalous processes CC and C + C + give contributionto the vacuum amplitude of a system with the interaction (24) only. Another combinations of the uncoupling with anobtaining of normal propagator C + C don’t conserve a momentum. Such representation of the vacuum amplitude byuncoupled correlations is analogous to Fock approximation for normal processes, and it means a neglect of dynamiccorrelation between pairs. Then R ( t ) can be written asln R ( t ) = R ( t ) = 1 V Z t dt X α,β X k , l (cid:18) − i V l , − l , k , − k (cid:19) √ ∆ + ∆∆ + ( − i ) F + αβ ( l , t − t ) √ ∆ + ∆∆ ( − i ) F αβ ( k , t − t )4= 2 V X k , l (cid:18) − i V l , − l , k , − k (cid:19) ( − i ) F + ( l , t − t )( − i ) F ( k , t − t ) t = iλV X l Z dω π w l F + ( l , ω ) X k Z dω π w k F ( k , ω ) t, (62)where a summation over spin variables gave the multiplier 2. By analogy with Fig.12 in Appendix B, the processof uncoupling of correlations we can represent by graphically in Fig.5, where in the diagram of a scattering of twofermions from the states k and − k into the state l and − l as a result of the interaction V l , − l , k , − k , we connect thelines corresponding to oppositely directed momentums. As a result we have two anomalous propagators: F ( k , ω )and F + ( l , ω ). The obtained diagram means the following. Two fermions with opposite momentums and oppositespins appear from a condensate of pairs, interact with each other by the potential V l , − l , k , − k and go back into thecondensate of pairs. , C l , C l , C k, C k V l ,- l , k ,- k F + l F k Figure 5: The uncoupling of correlations in a vacuum amplitude for the process of scattering of two fermions by the potential ofinteraction V l , − l , k , − k from the initial state k , α and − k , β to the final states l , α and − l , β . As a result we have the anomaloustransition amplitude ”vacuum-vacuum”. Using the formulas (48) and (49) we can rewrite R ( t ) in the following form:ln R ( t ) = − iλV (cid:16) V ν F (cid:17) t Z ω D − ω D p A ( ε ) B ( ε ) dε Z ω D − ω D p A ( ε ) B ( ε ) dε = − iλV (cid:16) ν F (cid:17) ∆ arcsinh (cid:16) ω D ∆ (cid:17) t. (63)This formula was obtained from such arguments. Since the area of action of the potential is limited by the layer 2 ω D in a neighborhood of Fermy surface, then the amplitudes ∆ and ∆ + is not equal to zero in this area only. As it waspointed before, presented manner of uncoupling of correlations is analogous to Fock exchange interaction for normalprocesses. In Hartree-Fock approximation a decay of quasi-particles is absent [1]. This means, that the amplitude ofpairing is real: ∆ = ∆ + in a momentum space.For calculation of the contribution of interaction to internal energy it is necessary to use the theorem (6), whichconnect a vacuum amplitude with a ground state energy:Ω λ = i ddt ln R ( t ) = − λV X l Z dω π w l F + ( l , ω ) X k Z dω π w k F ( k , ω )= λV (cid:16) ν F (cid:17) Z ω D − ω D ∆2 E dε Z ω D − ω D ∆2 E dε = λV (cid:16) ν F (cid:17) ∆ arcsinh (cid:16) ω D ∆ (cid:17) . (64)Since λ <
0, then the interaction tries to reconstruct a system so, that the gap ∆ is as much as possible. We canuse the theorem because interaction of particle with pairing fluctuation, changing a symmetry of a system: Φ → Ψ , G → G S , was considered before switching of the interaction V l , − l , k , − k . Switching of the interaction V l , − l , k , − k ,transferring the state Ψ in some other Ψ ′ , doesn’t change symmetry of a system: h Ψ | Ψ ′ i 6 = 0. This means that theadiabatic hypothesis is correct in the case of phase transition even.5 C. Internal energy.
On the basis of above obtained results we can write expression for internal energy of a system (at the temperature T = 0 internal energy coincides with free energy):Ω = h W i + Ω λ = − i lim t → − X k dω π G ( k , ω ) e − iωt ε ( k ) − λV X k Z dω π w k F + ( k , ω ) X k Z dω π w k F ( k , ω ) . (65)As it was pointed before, the gap is real ∆ = ∆ + , then F ( k, ω ) = F + ( k , ω ). We can see, that the energy depends onthe unknown amplitude of pairing ∆, which corresponds to Bethe-Salpeter amplitude η in the two-particle problem.The amplitude ∆ is determined by dynamics of all particles of a system, and its observed value is a result of averagingover a system. The procedure of averaging means mathematically, that the observer value of ∆ minimizes the internalenergy: d Ω d ∆ = 0 = ⇒ ( − i )∆ = λV X k Z dω π w k F ( k , ω ) , (66)that coincides with (17). It means that the self-consistency equation for order parameter in Nambu-Gor’kov formalismis an extremal of the obtained free energy functional (65), and the order parameter is the averaged Bethe-Salpeteramplitude over a system .The functional (65) can be written in an explicit form in quadratures:Ω = Ω n − V ν F Z ω D − ω D (cid:18) ε E − ε | ε | (cid:19) + V ν F g Z ω D − ω D ∆2 E dε Z ω D − ω D ∆2 E dε, (67)where Ω n is the energy of a normal phase, g = λ ν F is the effective interaction constant. Value of the energy on theextremal (66) is Ω min = Ω n + V ν F ω D − ω D q ω D + ∆ ) . (68)Thus, we solved the basic problem of statistical mechanics (at zero temperature): the calculation of a partitionfunction (free energy) and, in particular, of a vacuum amplitude in a system of interacting particles for case, when theinteraction causes a phase transition, that is symmetry of a system changes at a switching of the interaction. Unlikeother methods, this result was obtained from first principles without introducing any artificial parameters of type oforder parameter, but starting from parameters of the Hamiltonian only . V. NONZERO TEMPERATURES.A. Normal and anomalous propagators.
In the sections III and IV we have described microscopic mechanism of formation of long-range order in a systemat zero temperature. In this section we shall formulate the approach for case of nonzero temperatures. Let wehave a system from N noninteracting fermions being in volume V at temperature T . Then we must use Matsubarapropagators, where time t is complex: t → − iτ , τ ∈ [0 , β ]. In ideal Fermy gas propagation of a particle withmomentum k , energy ε ≈ v F ( | k | − k F ) and spin σ is described by the free propagator: G ( k , τ = τ − τ ) = − i Sp nb ρ C k ,σ ( τ ) C + k ,σ ( τ ) o , τ > i Sp nb ρ C + k ,σ ( τ ) C k ,σ ( τ ) o , τ ≤ = − iθ τ ( g +0 A e −| ε | τ + g − B e | ε | τ ) + iθ − τ ( g − A e −| ε | τ + g +0 B e | ε | τ ) ,G ( k , τ ) = 1 β n =+ ∞ X n = −∞ G ( k , ω n ) e − iω n τ , G ( k , ω n ) = 12 Z β − β G ( k , τ ) e iω n τ dτG ( k , ω n ) = iiω n − ε ( k ) = i iω n + ε ( iω n ) − ε = i A iω n − | ε | + i B iω n + | ε | , (69)6where g +0 = 1 e −| ε | β + 1 , g − = 1 e | ε | β + 1 , ω n = (2 n + 1) πβ , (70) C k ,σ ( τ ) and C + k ,σ ( τ ) are operators of creation and annihilation in Heisenberg representation. b ρ is density matrix ofnoninteracting particles: b ρ = exp ( Ω − b H + µ b NT ) = exp Ω − P k ,σ k m C k ,σ C + k ,σ + µ b NT = exp ( Ω − P k ,σ ε ( k ) C k ,σ C + k ,σ T ) , (71)where ε ( k ) = k m − µ ≈ v F ( k − k F ) is kinetic energy of particles counted off from Fermy surface.Now let an attracting force acts between particles. The force is described by the matrix element of interaction (24).In this case the instability of a system appears again with regard to a pairing (as in the section II) and Γ-matrix hasa pole structure: Γ(0 ,
0) = λ λ mk F π ln γω D πT ≈ − π mk F T C T − T C , T C = 2 γπ ω D (cid:18) − | λ | ν F (cid:19) , (72)where ν F = mk F π is density of states on Fermy surface. We can see that bound states exist in a system whiletemperature is not higher than critical temperature T C - at higher temperatures particles have large kinetic energy,so that an attraction between them leads to a scattering only.Generalization of the two particle problem on the multiparticle case is done analogously to the section III. In thecase of nonzero temperature the mass operator has a form: − Σ( k , ω n ) = ( − ∆) iG +0 ( − k , ω n )( − ∆ + ) = − ∆∆ + iω n + ε ( k ) . (73)Then it follows from Dyson equation, that a dressed propagator has a view:1 G = 1 G S − i Σ ⇒ G S ( k , ω n ) = iiω n − ε ( k ) − Σ( k , ω n )= i iω n + ε ( iω n ) − E ( k ) = i A S iω n − | ε | + i B S iω n + | ε | . (74)It can be written with help of a total definition of Green function in ( k , t )-space: G S ( k , τ ) = τ − τ ) = − i Sp nb ̺ C k ,σ ( τ ) C + k ,σ ( τ ) o , τ > i Sp nb ̺ C + k ,σ ( τ ) C k ,σ ( τ ) o , τ ≤ = − iθ τ ( g + S A S e − Eτ + g − S B S e Eτ ) + iθ − τ ( g − S A S e − Eτ + g + S B S e Eτ ) , (75)where g + S = 1 e − Eβ + 1 , g − S = 1 e Eβ + 1 . (76) b ̺ is the density matrix of noninteracting quasi-particles : b ̺ = exp ( Ω − P k ,σ E ( k ) C k ,σ C + k ,σ T ) . (77)It should be noted, that the state described by the density matrix ̺ has another symmetry in comparison with theinitial state ρ . Occupation numbers n ( k ) is determined by the following manner: n ( k ) = − i lim τ → − G ( k , τ ) = g − A + g + B, lim T → n ( k ) = B ( k ) . (78)Let’s introduce the designations: − G Σ ≡ ∆ F + , − G + Σ + ≡ ∆ + F.. (79)7Then Dyson equation can be rewritten in a form of Gor’kov equations:( iω n − ε ) G + ∆ F + = i (80)( iω n + ε ) F + + G ∆ = 0 . (81)The expressions for anomalous propagators follow from Gor’kov equations: F + ( k , ω n ) = − i ∆ + ( iω n ) − E ( k ) , F ( k , ω n ) = ( F + ( k , ω n )) + = i ∆( iω n ) − E ( k ) , (82)These expressions are analogous to the expressions (47). We can write the anomalous propagators in ( k ,t)-space andin a form of a vacuum average of creation and annihilation operators: F + αβ ( k , τ ) = ∆ + √ ∆ + ∆ ( i Sp { b ̺ C + − k ,β ( τ ) C + k ,α ( τ ) } , τ > i Sp { b ̺ C + k ,α ( τ ) C + − k ,β ( τ ) } , τ ≤ ) = ig αβ ∆ + √ ∆ + ∆ p A S B S (cid:2)(cid:0) g + S e − Eτ − g − S e Eτ (cid:1) θ τ − (cid:0) g + S e Eτ − g − S e − Eτ (cid:1) θ − τ (cid:3) , (83) F αβ ( k , τ ) = ∆ √ ∆ + ∆ (cid:26) − i Sp { b ̺ C k ,α ( τ ) C − k ,β ( τ ) } , τ > − i Sp { b ̺ C − k ,β ( τ ) C k ,α ( τ ) } , τ ≤ (cid:27) = ig αβ ∆ √ ∆ + ∆ p A S B S (cid:2) − (cid:0) g + S e − Eτ − g − S e Eτ (cid:1) θ τ + (cid:0) g + S e Eτ − g − S e − Eτ (cid:1) θ − τ (cid:3) , (84)that is analogous to the expressions (48) and (49). B. Kinetic energy and entropy.
In order to calculate a free energy it is necessary to know kinetic energy of particles of a system, energy of theirinteraction and entropy of collective excitations. Average kinetic energy of all particles of a system is h W i = − i X k G ( k , τ → − ) ε ( k ) = − iβ lim τ → − X k n =+ ∞ X n = −∞ G ( k , ω n ) e − iω n t ε ( k )= 2 X k ( g − A + g + B ) ε ( k ) = V ν F Z ∞− v F k F ( g − A + g + B ) εdε. (85)Since the interaction V l − lk − k (24) exists only in the layer − ω D < ε ( k ) < ω D , then we can suppose that g − = (cid:20) g − ; | ε ( k ) | > ω D g − S ; | ε ( k ) | < ω D (cid:21) , g + = (cid:20) g +0 ; | ε ( k ) | > ω D g + S ; | ε ( k ) | < ω D (cid:21) . (86)Hence, one may write expression for kinetic energy separating normal and superconductive parts by analogy (56): h W i = W n − V ν F Z ω D − ω D (cid:0) g − − g +0 (cid:1) ε | ε | dε + V ν F Z ω D − ω D (cid:0) g − S − g + S (cid:1) ε E dε = W n + V ν F Z ω D − ω D tanh (cid:18) β | ε | (cid:19) ε | ε | dε − V ν F Z ω D − ω D tanh (cid:18) βE (cid:19) ε E dε. (87)In the limit of low temperatures β → ∞ this expression reduces to the expression (56). If we suppose that ∆ = 0,then we shall have W = W n .At temperature T = 0 a gas of collective excitation exists - boholons with the spectrum E = p ε ( k ) + ∆ . Sinceboholons are product of decay of Cooper pairs on fermions, hence occupation numbers of states by boholons are f S ( k ) = 1 e βE + 1 = 12 (cid:18) − tanh (cid:18) βE (cid:19)(cid:19) . (88)8Then entropy of a system is S = − X k [ f ( k ) ln f ( k ) + (1 − f ( k )) ln(1 − f ( k ))]= S − V ν F Z ω D − ω D [ f S ln f S + (1 − f S ) ln(1 − f S )] dε + 2 V ν F Z ω D − ω D [ f ln f + (1 − f ) ln(1 − f )] dε. (89)Here we separated the normal part again, where f = ( e β | ε | + 1) − , so that S = S n at ∆ = 0. The multiplier ”2”appeared as result of summation over spin states. C. Vacuum amplitude.
In the previous subsections we considered the interaction of particles with fluctuations of pairing, and we found,that the state of a system described by the density matrix b ̺ has another symmetry in comparison with the initialstate b ρ . Hence vacuum amplitude of a system can be written in a form: R ( β ) = h b U ( β ) i = Sp (cid:16)b ̺ e U ( β ) (cid:17) = ∞ X n =0 ( − n n ! Z β dτ . . . Z β dτ n Sp (cid:16)b ̺ T n b H ( τ I ) . . . b H I ( τ n ) o(cid:17) , (90)where b H I ( τ ) = e + τ ˆ H b V e − τ ˆ H is the interaction operator of particles in interaction representation. The averaging Sp (cid:16)b ̺ e U ( β ) (cid:17) is made over ensemble of noninteracting quasi-particles . The potential V l , − l , k , − k acts between particles(24). Hence we can write (by analogy with (60)) expended expression for vacuum amplitude: R ( β ) = 1 + 11! 1 V Z β dτ X α,γ X k , l (cid:18) − V l , − l , k , − k (cid:19) Sp nb ̺ C + − l ,γ ( τ ) C + l ,α ( τ ) C k ,α ( τ ) C − k ,γ ( τ ) o + 12! 1 V Z β dτ Z β dτ X α,γ X k , l (cid:18) − V l , − l , k , − k (cid:19) X α ′ ,γ ′ X k ′ , l ′ (cid:18) − V l ′ , − l ′ , k ′ , − k ′ (cid:19) × Sp nb ̺ C + − l ′ ,γ ′ ( τ ) C + l ′ ,α ′ ( τ ) C k ′ ,α ′ ( τ ) C − k ′ ,γ ′ ( τ ) C + − l ,γ ( τ ) C + l ,α ( τ ) C k ,α ( τ ) C − k ,γ ( τ ) o + ..., (91)where we took into account that Sp { b ̺ } = 1. Then, by analogy with the equation (61), we can uncouple correlationsby the following way taking into account anticommutation of operators C and C + : R ( β ) ≈ − V Z β dτ X α,γ X k , l (cid:18) − V l , − l , k , − k (cid:19) Sp nb ̺ C + l ,α ( τ ) C + − l ,γ ( τ ) o Sp nb ̺ C − k ,γ ( τ ) C k ,α ( τ ) o + 12! ( − V Z β dτ Z β dτ X α,γ X k , l (cid:18) − V l , − l , k , − k (cid:19) X α ′ ,γ ′ X k ′ , l ′ (cid:18) − V l ′ , − l ′ , k ′ , − k ′ (cid:19) × Sp nb ̺ C + l ′ ,α ′ ( τ ) C + − l ′ ,γ ′ ( τ ) o Sp nb ̺ C − k ′ ,γ ′ ( τ ) C k ′ ,α ′ ( τ ) o Sp nb ̺ C + l ,α ( τ ) C + − l ,γ ( τ ) o Sp nb ̺ C − k ,γ ( τ ) C k ,α ( τ ) o + . . . = 1 + R + 12! R + . . . = exp( R ) (92)Let’s take into account that our approximation is analogous to Fock approximation for normal processes. Then wecan suppose ∆ = ∆ + , hence F = − F + . Then R ( t ) can be written asln R ( β ) = R ( β ) = 1 V Z β dτ X α,γ X k , l (cid:18) − V l , − l , k , − k (cid:19) iF αγ ( l , τ − τ ) iF αβ ( k , τ − τ )= 2 V X k , l (cid:18) V l , − l , k , − k (cid:19) F ( l , τ → − ) F ( k , τ → − ) β βλV X k w k β n =+ ∞ X n = −∞ F ( k , ω n ) X k w k β n =+ ∞ X n = −∞ F ( k , ω n )= − βλV (cid:16) ν F (cid:17) Z ω D − ω D tanh (cid:18) βE (cid:19) ∆2 E dε Z ω D − ω D tanh (cid:18) βE (cid:19) ∆2 E dε. (93)This equation is represented graphically as well as in Fig.6. In order to calculate a contribution of interaction in freeenergy we can use the formula (10):Ω λ = − β ln R ( β ) = − λV X k w k β n =+ ∞ X n = −∞ F ( k , ω n ) X k w k β n =+ ∞ X n = −∞ F ( k , ω n )= λV (cid:16) ν F (cid:17) Z ω D − ω D tanh (cid:18) βE (cid:19) ∆2 E dε Z ω D − ω D tanh (cid:18) βE (cid:19) ∆2 E dε. (94)In the limit of low temperatures β → ∞ this equation transforms to the equation (64). If we suppose that ∆ = 0,then we shall have Ω λ = 0. D. Free energy.
Starting from the above found results we can write the expression a for free energy of a system:Ω = h W i − β S + Ω λ = − iβ lim τ → − X k n =+ ∞ X n = −∞ G ( k , ω n ) e − iω n t ε ( k )+ 2 β X k [ f ( k ) ln f ( k ) + (1 − f ( k )) ln(1 − f ( k ))] − λV X k w k β n =+ ∞ X n = −∞ F ( k , ω n ) X k w k β n =+ ∞ X n = −∞ F ( k , ω n ) (95)We can see, that the energy depends on the unknown amplitude of pairing ∆, which corresponds to Bethe-Salpeteramplitude η in two-particle problem. The observed value of ∆ must minimize the free energy: d Ω d ∆ = 0 = ⇒ ( − i )∆ = λV β X k n =+ ∞ X n = −∞ w k F ( k , ω n ) , (96)that coincides with (17). As in the previous section the equation of a self-consistence for order parameter in Nambu-Gor’kov formalism is the extremal of the functional of free energy (95), and the order parameter is averaged Bethe-Salpeter amplitude over a system.The functional (95) can be written in an explicit form in quadratures:Ω = Ω n + V ν F Z ω D − ω D (cid:20) tanh (cid:18) β | ε | (cid:19) ε | ε | − tanh (cid:18) βE (cid:19) ε E (cid:21) dε + 2 Vβ ν F Z ω D − ω D [ f S ln f S + (1 − f S ) ln(1 − f S ) − f ln f − (1 − f ) ln(1 − f )] dε + V ν F g Z ω D − ω D tanh (cid:18) βE (cid:19) ∆2 E dε Z ω D − ω D tanh (cid:18) βE (cid:19) ∆2 E dε, (97)where Ω n is the energy of a normal phase, g = λ ν F is the effective interaction constant, V is volume of a system. g can be expressed via critical temperature β C with help of the equation ∆( β C ) = 0 as following:1 = − g Z ω D − ω D tanh (cid:18) β c | ε | (cid:19) | ε | dε. (98)If to suppose ∆ = 0, then we shall have Ω = Ω n . The equilibrium value of ∆ is determined by balance of kineticenergy, entropy and energy of interaction, that corresponds to a minimum of the free energy.Let’s consider a low-temperature limit of the free energy (96): ∆ β ≫ T →
0. This means, that the value∆ − ∆ can be parameter of expansion, where ∆ = ∆( T = 0) is equilibrium value of the gap (amplitude of pairing) at0zero temperature. Due a rapid convergence of integration elements in (96), the limits of integration can be −∞ , + ∞ .Then a low-temperature expansion has a form:Ω = Ω n + V (cid:0) α ( T ) + b ( T )∆ + d ∆ (cid:1) , (99)where coefficients of the expansion are α ( T ) = − ν F + 2 ν F T − ν F p π ∆ T e − ∆0 T + ν F g + 1)∆ − ν F g + 1) q π ∆ T e − ∆0 T b ( T ) = − ν F g + 1)∆ + ν F g + 1) p π ∆ T e − ∆0 T , d = ν F g + 1) (100)Since in the expression b ( T )2 d the multipliers g + 1 is cancelled and, as a rule, | g | ≪
1, we can suppose that g + 1 ≈ α ( T ) = ν F + 2 ν F T − ν F p π ∆ T e − ∆0 T + − ν F q π ∆ T e − ∆0 T b ( T ) = − ν F + ν F p π ∆ T e − ∆0 T , d = ν F (101)Let’s consider a high-temperature limit of the free energy: ∆ β C ≪ T → T C . Expansion in powers of ∆ gives:Ω = Ω n + V (cid:18) α ( T )∆ + 12 b ∆ + 13 d ∆ (cid:19) , (102)where the coefficients of the expansion are α ( T ) = ν F T − T c T c b = ν F ζ (3)8 π T c (103) d = ν F (cid:18) . ζ (5) π + 4 . (cid:19) T c . This expansion has a form of Landau expansion of free energy in powers of order parameter.From the all considered above we can see, that averaged over a system Bethe-Salpeter amplitude η and η ∗ - theamplitude of pairing ∆ and ∆ + have the properties, which are analogous to the properties of a order parameter . Thus,we have solved the basic problem of statistical mechanics: the calculation of a partition function (free energy) and, inparticular, of a vacuum amplitude in a system of interacting particles for case, when the interaction causes a phasetransition, that is symmetry of a system changes at a switching of the interaction. VI. THE PAIRING WITH NONZERO MOMENTUM OF CENTER OF MASS OF A PAIR.
Let fermions with momentums k + q and − k + q pair up, so that the momentum of a center of mass of a pair is q . The free propagators, corresponding to these states are G (cid:16) k + q , ω (cid:17) = 1 ω − ε (cid:16) k + q (cid:17) ≡ ω − ε + G (cid:16) − k + q , − ω (cid:17) = 1 − ω − ε (cid:16) − k + q (cid:17) ≡ − ω − ε − . (104)We can suppose that ε + = 12 m (cid:16) k + q (cid:17) − µ ≈ ε + kq m , ε − = 12 m (cid:16) − k + q (cid:17) − µ ≈ ε − kq m . (105)The corresponding mass operator describing an interaction of particles with fluctuations of pairing is:( − i )Σ q = − i ∆ iG (cid:16) − k + q , − ω (cid:17) i ∆ + = ( − i ) ∆∆ + ω + ε − . (106)1Then can find the dressed propagator from Dyson equation: G S = 1 G − − Σ q = ω + ε − ( ω − E + )( ω − E − ) , (107)where the specters of quasi-particles are E + = ε + − ε − s(cid:18) ε + + ε − (cid:19) + | ∆ | ≈ kq m + p ε + | ∆ | E − = ε + − ε − − s(cid:18) ε + + ε − (cid:19) + | ∆ | ≈ kq m − p ε + | ∆ | . (108)We can see, that if to assume q = 0, then the specter (108) turn into the usual specter of boholons: E = ± p ε + | ∆ | .The critical momentum q cr exists when a minimum of the specter(108) touches Fermy surface. Then for excitationof a system it is necessary infinitely small energy. This means, that superfluidity of Fermy gas is absent. The criticalmomentum is: E + ( q = q cr , ε = 0) = E − ( q = q cr , ε = 0) = 0 ⇒ q cr = 2 v F | ∆ | . (109)Similar pairing can take place in high-temperature superconductors (cuprates) with a mirror nesting ε (cid:16) k + q (cid:17) = ε (cid:16) − k + q (cid:17) of regions of Fermy contour [26, 27]. VII. FREE ENERGY IN A CASE OF SLOW SPATIAL INHOMOGENEITY.
In the previous sections we supposed, that amplitudes of pairing ∆ and ∆ + don’t depend on spatial coordinates.This takes place in interminable, homogeneous, isotropic and isolated from external fields superconductor. Howeverin a total case these conditions are not realized. For example, in a sufficiently strong magnetic field the inclusionsof normal phase can exist in volume of a superconductor. Another example - a contact of a superconductor and anormal metal. In this case the order parameter is suppressed in a boundary layer of a superconductor, however itappears in boundary layer of a normal metal.Let’s consider some region of a superconductor, where a distribution of ∆ is inhomogeneous and the amplitudeof pairing can be both smaller and larger than its equilibrium value ∆ - Fig.6. When a quasi-particle propagatesalong the axis Ox its energy is constant E = p ε ( k ) + | ∆ | , but its momentum changes. If a pair moves into region,where | ∆ | < | ∆ | , then forces appear tearing the pair. Each element of the pair gets some increment of momentum q :( k , − k ) → ( k + q , − k − q ). If the pair moves into region, where | ∆ | > | ∆ | , then a forces appear increasing a boundenergy of the pair. Each element of the pair gets some increment of momentum q too. Since the amplitude of pairingis function of coordinates ∆( r ), moreover we suppose that order parameter is real ∆ = ∆ + in momentum space andit has an identical dimension in q -space and in r -space, then we can write the Fourier-transformations:∆( r ) = X q ∆( q ) e i qr = V (2 π ) Z ∆( q ) e i qr d q, ∆ + ( r ) = X q ∆( q ) e − i qr = V (2 π ) Z ∆( q ) e − i qr d q, (110)∆( q ) = 1 V Z ∆( r ) e − i qr d r = 1 V Z ∆ + ( r ) e i qr d r. The mass operator for above-mentioned process is shown in Fig.7. In analytical representation it has a view: − Σ q ( k , ω n ) = ( − ∆ q ) iG +0 ( − k − q , ω n )( − ∆ + q ) = − ∆ q ∆ + q iω n + ε ( k + q ) , (111)where the free propagator G is G = 1 iω n − ε ( k + q ) = i iω n + ε q ( k )( iω n ) − ε q ( k ) . (112)2 - q - q qq - k - k - k kkk x Figure 6: The pairing of fermions in a spatially inhomogeneous system. ∆ is the equilibrium value of a gap in a homogeneoussystem. -k-q k+q- +q = iG + (- k - q , n ) - q - q k+q Figure 7: The diagram for the mass operator Σ describing an interaction of a fermion with fluctuations of pairing in a spatiallyinhomogeneous system.
Then from Dyson equation we can obtain the dressed propagator:1 G = 1 G S − i Σ q ⇒ G S = i iω n + ε q ( iω n ) − E q , (113)where E q is the specter of quasi-particles in a nonhomogeneous system: E q = ε q + | ∆ q | , ε q ≡ ε ( k + q ) ≈ ε ( k ) + kq m , | k | ≃ k F . (114)Dyson equation can be represented in a form of set of Gor’kov equations, from where the expressions for anomalouspropagators follow:( iω n − ε q ) G + ∆ q F + = i ( iω n + ε q ) F + + G ∆ q = 0 ⇒ F + ( k + q , ω n ) = − i ∆ + q ( iω n ) − E q F ( k + q , ω n ) = ( F + ( k + q , ω n )) + = i ∆ q ( iω n ) − E q (115)If to suppose q = 0, then we shall have the expressions (80-82).Now let’s suppose that ∆( r ) changes very slowly on a coherence length l ( T ) which characterizes a size of Cooperpair. Then we can suppose ε ( k + q ) = ε ( k ) in the specter of quasi-particles, such that E q ≈ p ε ( k ) + | ∆ q | . Howeverwe must keep ε ( k + q ) in numerator of the expressions (112) and (113) for G . Hence the normal propagator has aform: G ( k + q , τ ) = − iθ τ (cid:0) g + q A ( k + q ) e − E q τ + g − q B ( k + q ) e E q τ (cid:1) + iθ − τ (cid:0) g − q A ( k + q ) e − E q τ + g + q B ( k + q ) e E q τ (cid:1) , (116)where A ( k + q ) ≈ A q ( k ) + 12 E q kq m , B ( k + q ) ≈ B q ( k ) − E q kq m . (117)The anomalous propagators are F + αβ ( k + q , τ ) = ig αβ ∆ + q E q (cid:2)(cid:0) g + q e − E q τ − g − q e E q τ (cid:1) θ τ − (cid:0) g + S e E q τ − g − q e − E q τ (cid:1) θ − τ (cid:3) ,F αβ ( k + q , τ ) = ig αβ ∆ q E q (cid:2) − (cid:0) g + q e − E q τ − g − q e E q τ (cid:1) θ τ + (cid:0) g + q e E q τ − g − q e − E q τ (cid:1) θ − τ (cid:3) . (118)3We can see, that in the approximation of slowness of changes of ∆( r ) the anomalous propagators depend on q bymeans of ∆( q ) only.Kinetic energy of a system is determined by the following way: h W i = − i X k ε ( k + q ) G ( k + q , τ → − ) = 2 X k ε ( k + q ) (cid:0) g − q A ( k + q ) + g + q B ( k + q ) (cid:1) = 2 X k ε ( k ) (cid:0) g − q A q + g + q B q (cid:1) + 2 X k (cid:0) g − q − g + q (cid:1) E q ( kq ) m = W n + V ν F Z ω D − ω D (cid:20) tanh (cid:18) β | ε | (cid:19) ε | ε | − tanh (cid:18) βE q (cid:19) ε E q (cid:21) dε + V ν F v F q Z ω D − ω D (cid:20) tanh (cid:18) β | ε | (cid:19) | ε | − tanh (cid:18) βE q (cid:19) E q (cid:21) dε. (119)We can see, that the term, which is proportional to q , is added to the kinetic energy (87) (with the replacement E → E q = p ε ( k ) + | ∆( q ) | ). In the approximation of slowness of changes the expressions for entropy and vacuumamplitude coincide with the expressions (89) and (94) accordingly, however it should be written ∆( q ) instead of∆ = const . Then we can write the free energy:Ω( q ) = Ω n ( q ) + Ω(∆ q ) + V ν F v F q Z ω D − ω D (cid:20) tanh (cid:18) β | ε | (cid:19) | ε | − tanh (cid:18) βE q (cid:19) E q (cid:21) dε, (120)where Ω(∆ q ) coincides with the expression (96), where the replacement ∆ → ∆( q ) was done.Expanding the free energy (120) in powers of ∆ we can obtain the expression:Ω( q ) = Ω n ( q ) + V (cid:18) α ( T )∆ q + 12 b ∆ q + γq ∆ q (cid:19) , (121)where the coefficient γ is γ = ν F ζ (3) v F π T c = ν F l , (122)where l is a coherence length at T = 0 (Pippard length). The expansion (121) has a form of Landau expansion offree energy in powers of order parameter at the condition ql ≪
1. We can see, that a spatial inhomogeneity increasesthe free energy of a superconductor. Hence in most cases we can be restricted by the term ∼ q in the expansion,because more fast changes of ∆ increase the free energy essentially.The full free energy of a system in a spatial inhomogeneous case can be obtained by summation of the expression(120) over all possible q :Ω = Ω n + X q Ω s ( q ) = Ω n + V (2 π ) Z Ω s ( q ) d q = Ω n + V (2 π ) Z (cid:18) α ( T )∆ q + 12 b ∆ q + γq ∆ q (cid:19) d q (123)Let’s pass from momentum space to real space using the expressions (110): Z ∆ q ∆ q d q = Z ∆ q (cid:20) V Z ∆( r ) e − i qr d r (cid:21) d q = 1 V Z ∆( r ) (cid:20)Z ∆( q ) e − i qr d q (cid:21) d r = (2 π ) V Z ∆( r )∆ + ( r ) d r, (124) Z q ∆ q q ∆ q d q = Z q ∆ q (cid:20) V Z e − i qr ( − i ) ∂∂ r ∆( r ) d r (cid:21) d q = − iV Z (cid:20)Z q ∆( q ) e − i qr d q (cid:21) ∂∂ r ∆( r ) d r = (2 π ) V Z (cid:20) ∂∂ r ∆( r ) (cid:21) ∂∂ r ∆ + ( r ) d r. (125)For the term ∆ and terms with more high powers the situation is more difficult. This is because a square of a Fouriertransform is not equal to a Fourier transform of a square: (cid:0) V R ∆( r ) e − i qr d r (cid:1) = V R ∆ ( r ) e − i qr d r . Apparentlythis fact results to some nonlocality of a superconductor’s state in zero magnetic field described in [28], where a valueof gap in a point is determined by a distribution of gap in some neighborhood: ∆( r ) = R d r ′ Q ( r , r ′ )∆ ′ ( r ). However4in first approximation this correlation can be neglected and we can write the expansion of free energy in powers of∆∆ + in real space: Ω = Ω n + Z " α ( T ) | ∆( r ) | + b | ∆( r ) | + γ (cid:12)(cid:12)(cid:12)(cid:12) ∂∂ r ∆( r ) (cid:12)(cid:12)(cid:12)(cid:12) d r. (126)This expansion coincides with Ginzburg-Landau expansion in zero magnetic field. VIII. FREE ENERGY OF A SUPERCONDUCTOR IN MAGNETIC FIELD.
In this section we shall generalize the previous results for the case, when a superconductor is placed in a magneticfield H ( r ) = rot A ( r ). Our aim is to obtain the functional of free energy Ω (cid:0) ∆( r ) , ∂∂ r ∆( r ) , A ( r ) (cid:1) , which is correct foran arbitrary value of the relation ∆( T ) /T , for an arbitrary scale of a change of ∆( r ) in comparison with a coherentlength l ( T ), for an arbitrary value of a magnetic penetration depth λ ( T ) in comparison with a coherent length l (nonlocal electromagnetic response). Thus, the all three restriction on Ginzburg-Landau functional, described insection I, are excluded.Let the microscopic magnetic field exists in a superconductor with a potential A and an intensity H : A ( r ) = X q a ( q ) e i qr ⇒ H ( r ) = rot A ( r ) = i X q q × a ( q ) e i qr . (127)Then the magnetic field inducts a current: J ( r ) = c π rot H ( r ) = − c π X q q × q × a ( q ) e i qr = − c π X q (cid:0) q ( qa ) − a q (cid:1) e i qr ≡ X q j ( q ) e i qr . (128)Energy of the magnetic field is W f = 18 π Z | H ( r ) | d r = V π X q (cid:0) q a q − ( qa q ) (cid:1) . (129)A magnetic field affects on a superconductor essentially. In the first place, a distribution of order parameter becomesinhomogeneous. As it was shown in the section VII, an inhomogeneity leads to same growth of momentum of eachelement of a pair: k → k + q , − k → − k − q , moreover the order parameter depends on the momentum ∆ = ∆( q ).In the second place, an ordinary momentum must be replaced by a canonical momentum: k + q → k + q − ec a q ,moreover the order parameter depends on the momentum ∆ = ∆( q − ec a q ). , q n eiG ck q a q eck q a q eck q a q eck q a q ecq a q ecq a q ecq a = Figure 8: The diagram for the mass operator Σ describing the interaction of a charged fermion with fluctuations of pairing ina spatially inhomogeneous system situated in magnetic field with a potential a ( q ). The mass operator for a process of interaction of a fermion (with charge e ) with a fluctuation of pairing is shownin Fig.8. In analytical representation this diagram has the form: − Σ (cid:16) k + q − ec a q , ω n (cid:17) = − ∆ (cid:16) q − ec a q (cid:17) iG +0 (cid:16) − k − q + ec a q , ω n (cid:17) (cid:16) − ∆ + (cid:16) q − ec a q (cid:17)(cid:17) = − ∆ (cid:0) q − ec a q (cid:1) ∆ + (cid:0) q − ec a q (cid:1) iω n + ε (cid:0) k + q − ec a q (cid:1) , (130)5where the free propagator G is G = 1 iω n − ε ( k + q − ec a q ) = i iω n + ε (cid:0) k + q − ec a q (cid:1) ( iω n ) − ε (cid:0) k + q − ec a q (cid:1) . (131)Then from Dyson equation we can obtain the dressed propagator:1 G = 1 G S − i Σ q ⇒ G S = i iω n + ε (cid:0) k + q − ec a q (cid:1) ( iω n ) − E (cid:0) k + q − ec a q (cid:1) , (132)where E is the specter of quasi-particles in a inhomogeneous system situated in magnetic field: E (cid:16) k + q − ec a q (cid:17) = ε (cid:16) k + q − ec a q (cid:17) + | ∆ (cid:16) q − ec a q (cid:17) | ≡ E q , a ε (cid:16) k + q − ec a q (cid:17) ≈ ε ( k ) + k (cid:0) q − ec a q (cid:1) m ≡ ε q , a , | k | ≃ k F , (133)where we have introduced the notations E q , a and ε q , a for convenience. Then Dyson equation can be represented inthe form of Gor’kov equations. From these equations the expressions for anomalous propagators follow:( iω n − ε q , a ) G + ∆ q , a F + = i ( iω n + ε q , a ) F + + G ∆ q , a = 0 ⇒ F + ( k + q − ec a q , ω n ) = − i ∆ + q , a ( iω n ) − E q , a F ( k + q − ec a q , ω n ) = ( F + ( k + q − ec a q , ω n )) + = i ∆ q , a ( iω n ) − E q , a (134)If to suppose q = 0 a = 0, then we shall have the expressions (80-82).In the space ( k , t ) the normal propagator has the form: G ( k + q − ec a q , τ ) = − iθ τ (cid:16) g + q , a A q , a e − E q , a τ + g − q , a B q , a e E q , a τ (cid:17) + iθ − τ (cid:16) g − q , a A q , a e − E q , a τ + g + q B q , a e E q , a τ (cid:17) , (135)where A (cid:16) k + q − ec a q (cid:17) = 12 (cid:18) ε q , a E q , a (cid:19) , B (cid:16) k + q − ec a q (cid:17) = 12 (cid:18) − ε q , a E q , a (cid:19) . (136)The anomalous propagators are F + αβ ( k + q − ec a q , τ ) = ig αβ ∆ + q , a E q , a h(cid:16) g + q , a e − E q , a τ − g − q , a e E q , a τ (cid:17) θ τ − (cid:16) g + q , a e E q , a τ − g − q , a e − E q , a τ (cid:17) θ − τ i ,F αβ ( k + q − ec a q , τ ) = ig αβ ∆ q , a E q , a h − (cid:16) g + q , a e − E q , a τ − g − q , a e E q , a τ (cid:17) θ τ + (cid:16) g + q , a e E q , a τ − g − q , a e − E q , a τ (cid:17) θ − τ i . (137)where g + q , a and g − q , a are statistical multipliers: g − (cid:16) k + q − ec a q (cid:17) = 1 e βE q , a + 1 , g + (cid:16) k + q − ec a q (cid:17) = 1 e − βE q , a + 1 . (138)We can see, that the normal and anomalous propagators have a complicated dependence on vector q , amplitudes ofpairing ∆ , ∆ + and magnetic field a ( q ).Kinetic energy of a system is determined in the following way: h W i = − i X k ε (cid:16) k + q − ec a q (cid:17) G (cid:16) k + q − ec a q , τ → − (cid:17) = 2 X k ε q , a (cid:16) g − q , a A q , a + g + q , a B q , a (cid:17) = X k ε q , a (cid:18) − ε q , a E q , a tanh βE q , a (cid:19) = W n + X k , ( | ε ( k ) | <ω D , | k |≃ k F ) ε q , a (cid:18) ε q , a | ε q , a | tanh β | ε q , a | − ε q , a E q , a tanh βE q , a (cid:19) ≡ W n + W S . (139)6We can see, that the kinetic energy depends on the vectors q and a ( q ) in a complicated way. If we suppose a = 0and q is small, then we shall obtain the expression (119).Free energy of a superconductor is the sum of the following terms:Ω = Ω n + W S − β S S + Ω λ + W field ( a ) , (140)where W S is the kinetic energy of fermions of a system in superconductive phase, S S is the entropy of boholons,Ω λ = − β ln R ( β ) is the energy corresponding to an interaction, W field ( a ) is the energy of the magnetic field (129).The expressions for S and Ω λ are obtained from the expressions (89) and (94) (without transition to integration over ε only) by the replacement ∆ → ∆ q , a , E → E q , a , ε → ε q , a , f → f q , a . Hence the free energy of a superconductorfor given q isΩ( q , a q ) = Ω n ( q , a q ) + X k , ( | ε ( k ) | <ω D , | k |≃ k F ) ε q , a (cid:18) ε q , a | ε q , a | tanh β | ε q , a | − ε q , a E q , a tanh βE q , a (cid:19) + X k , ( | ε ( k ) | <ω D , | k |≃ k F ) 2 β h f S q , a ln f S q , a + (1 − f S q , a ) ln(1 − f S q , a ) − f q , a ln f q , a − (1 − f q , a ) ln(1 − f q , a ) i + λV X k , ( | ε ( k ) | <ω D , | k |≃ k F ) ∆ q , a E q , a tanh βE q , a X k , ( | ε ( k ) | <ω D , | k |≃ k F ) ∆ q , a E q , a tanh βE q , a V π (cid:0) q a q − ( qa q ) (cid:1) ≡ Ω n ( q , a q ) + Ω S ( q , a q ) + w field ( q , a q ) , (141)where f S q , a ( k ) = 1 e βE q , a + 1 , f q , a ( k ) = 1 e β | ε q , a | + 1 (142)are occupation numbers of states by boholons, moreover with help of the term f q , a a normal part of entropy isseparated, such that Ω S (∆ = 0) = 0. The full free energy is sum of the expression (141) over all possible q :Ω = Ω n + X q { Ω S ( q , a q ) + w field ( q , a q ) } . (143) Unlike Ginzburg-Landau functional the obtained functional of free energy (143) is correct for an arbitrary value ofthe relation ∆( T ) /T , for an arbitrary scale of a change of ∆( r ) in comparison with a coherent length l ( T ), for anarbitrary value of a magnetic penetration depth λ ( T ) in comparison with a coherent length l - it describes a nonlocalresponse to magnetic field . However the obtained expression is complicated for analyze. For its simplification let’ssuppose that ∆( r ) changes in space slowly. Then it is necessary to expand the expression (141) in degrees of q − ec a q keeping terms which are proportional to the vector in second degree only. Then supposing E q , a ≈ q ε ( k ) + ∆ q , a we have:Ω( q , a q ) = Ω n ( q , a q ) + V ν F Z ω D − ω D (cid:20) tanh (cid:18) β | ε | (cid:19) ε | ε | − tanh (cid:18) βE (cid:19) ε E (cid:21) dε + 2 Vβ ν F Z ω D − ω D [ f S ln f S + (1 − f S ) ln(1 − f S ) − f ln f − (1 − f ) ln(1 − f )] dε + V ν F g Z ω D − ω D tanh (cid:18) βE (cid:19) ∆2 E dε Z ω D − ω D tanh (cid:18) βE (cid:19) ∆2 E dε + V ν F v F (cid:16) q − ec a q (cid:17) Z ω D − ω D (cid:20) tanh (cid:18) β | ε | (cid:19) | ε | − tanh (cid:18) βE (cid:19) E q (cid:21) dε + V π (cid:0) q a q − ( qa q ) (cid:1) , (144)where g = λ ν F is the effective interaction constant, and the gap is ∆ = ∆ (cid:0) q − ec a q (cid:1) .Let’s consider a high-temperature limit of the free energy: ∆ β c ≪ T → T c . This means, that the expression(144) can be expended in degrees of ∆ q , a :Ω = Ω n + V X q (cid:18) α ( T )∆ q , a + 12 b ∆ q , a + 13 d ∆ q , a + γ (cid:16) q − ec a q (cid:17) ∆ q , a (cid:19) + V π X q (cid:0) q a q − ( qa q ) (cid:1) , (145)7 H(x) H ext H=H ext Figure 9: The screening of the external magnetic field H ext , impressed in parallel to a superconductive cylinder, by inductedclosed currents. The inducted currents are directed so as to compensate the external field. The resultant current goes arounda lateral area of cylinder, but strives to zero inside of the cylinder. The area where the resultant current is not equal to zero issurface layer with thickness ∼ λ . The magnetic field penetrates in a superconductor in the depth ∼ λ too. where the coefficients α ( T ) , b, d are determined by the formulas (103), and the coefficient γ is determined by theformula (122). The expansion (145) has a form of Ginzburg-Landau expansion of free energy in degrees of orderparameter. Observed configuration of the order parameter ∆ q , a and the magnetic field a ( q ) minimizes the freeenergy: δ Ω δ ∆ = 0 ⇒ α ( T )∆ q , a + b ∆ q , a + d ∆ q , a + γ (cid:16) q − ec a q (cid:17) ∆ q , a = 0 (146) δ Ω δ a = 0 ⇒ j ( q ) = 2 eγ q ∆ q , a − γ e c ∆ q , aa ( q ) , (147)where j ( q ) is Fourier component of a current: j ( q ) = − c π q × q × a ( q ) = − c π (cid:0) q ( qa ) − a q (cid:1) . (148)These equations take more simple form in a transverse gauge q · a ( q ) = 0. This gauge gives a condition of closureof a current (special case of conservation of charge): q · a ( q ) = 0 ⇒ j ( q ) = − c π a ( q ) q ⇒ q · j ( q ) = 0 ⇔ div J ( r ) = 0 . (149)The closed currents (149) screen a magnetic field in a superconductor - Fig.9. The currents is analogy to molecularcurrents of Ampere, their resulting gives rise to observed magnetic effects. In transverse gauge the functional of freeenergy has a form:Ω = Ω n + V X q (cid:18) α ( T )∆ q , a + 12 b ∆ q , a + 13 d ∆ q , a + γ (cid:18) q + e c a q (cid:19) ∆ q , a (cid:19) + V π X q q a q . (150)8The equations of extremals are α ( T )∆ q , a + b ∆ q , a + d ∆ q , a + γ (cid:18) q + e c a q (cid:19) ∆ q , a = 0 (151) j ( q ) = − γ e c ∆ q , aa ( q ) . (152)From (152) one can see, that the value Q = − γ e c ∆ q , a is a Fourier transform of a kernel in the integral law of amagnetic response (Pippard law). The order parameter is function of q and a ( q ):∆ q , a ( T ) = | α ( T ) | b (cid:18) − γ | α ( T ) | (cid:16) q + ec a q (cid:17)(cid:19) ≡ | α ( T ) | b (cid:16) − l ( T ) (cid:16) q + ec a q (cid:17)(cid:17) , (153)where smallness of 1 /q in comparison with the coherent length l ( T ): ql ( T ) ≪ d = 0 for simplification. From the formula (153) one can see, that the kernel of the magnetic response Q is function of magnetic field. Hence the electrodynamics of a superconductor is nonlinear. If to suppose ∆ = const at given temperature, then we shall obtain London equation: j ( q ) = − γ e c ∆ ( T ) a ( q ) ≡ − c πλ ( T ) a ( q ) ⇒ λ ( T ) = c πe b | α ( T ) | γ , (154)where λ ( T ) is the magnetic penetration depth in a superconductor. It is necessary to note, that λ ∝ me n S = m (2 e ) n S / .This means that the exchange of mass, charge and concentration of superconductive electrons to corresponding valuesof Cooper pairs doesn’t change the observed values.However it is necessary to note, that in a high-temperature limit a gap (and a kernel Q ) depends on vector q andfield a ( q ) strongly. From the formula (153) one can see, that the gap decreases at an increase of q , hence a magneticpenetration depth λ increases. Moreover with a rise of temperature this dependence becomes stronger (at T = T C we have λ = ∞ ). Besides at temperature T = T C the critical magnetic field is zero H C = 0. This means, that inthe limit T → T C any magnetic field H ( q ) can not be considered as weak. Therefore it suppresses order parameteressentially and penetrates in a superconductor deeply (in macroscopic distant even). For example, the penetration ofmagnetic field along a core of Abrikosov vortex in a type II superconductor. Such structure exists in infinitely weakmagnetic field at T → T C .For research of the nonlocal characteristics of the functional of free energy (141,143) let’s consider a low-temperaturelimit ∆ β ≫ T →
0. A value of gap is close to the value at zero temperature ∆( T ) ≤ ∆ . Moreover, magneticfield is weak, such that it changes a value of gap lightly, that is the magnetic field is much smaller than critical field H ≪ H C . Either as above, we assume that a change of a gap in space is slow. Starting from aforesaid and using theexpansion (99) we obtain the free energy:Ω = Ω n + V X q (cid:18) α ( T ) + b ( T )∆ q , a + d ∆ q , a + γ (cid:16) q − ec a q (cid:17) ∆ q , a (cid:19) + V π X q (cid:0) q a q − ( qa q ) (cid:1) , (155)where coefficients α ( T ) , b ( T ) , d are determined by the formulas (101), and coefficient γ is determined by the formula(122). The observed configurations of order parameter ∆ q , a and magnetic field a ( q ) minimized free energy: δ Ω δ ∆ = 0 ⇒ b ( T ) + 2 d ∆ q , a + γ (cid:16) q − ec a q (cid:17) ∆ q , a = 0 (156) δ Ω δ a = 0 ⇒ j ( q ) = 2 eγ q ∆ q , a − γ e c ∆ q , aa ( q ) . (157)In the transverse gauge q · a ( q ) = 0 the functional of free energy and the equations for the extremals have a form:Ω = Ω n + V X q (cid:18) α ( T ) + b ( T )∆ q , a + d ∆ q , a + γ (cid:18) q + e c a q (cid:19) ∆ q , a (cid:19) + V π X q q a q (158) b ( T ) + 2 d ∆ q , a + 2 γ (cid:18) q + e c a q (cid:19) ∆ q , a = 0 (159) j ( q ) = − γ e c ∆ q , aa ( q ) . (160)9If in the equation (160) to assume ∆ = const at given temperature, then we shall have London equation again: j ( q ) = − γ e c ∆ ( T ) a ( q ) ≡ − c πλ ( T ) a ( q ) ⇒ λ ( T ) = c πe d b ( T ) γ , (161)The set of equations (159,160) allows to generalize London equation. From the equation (159) we can find value of agap: ∆ q , a ( T ) = − b ( T )2 d + 2 γ (cid:0) q + e c a q (cid:1) (162)Then the equation for current has a form: j ( q ) = − γ e c b ( T ) (cid:0) d + 2 γ (cid:0) q + e c a q (cid:1)(cid:1) a ( q ) ≡ Q ( q , a ) a ( q ) (163)This equation is the nonlocal and nonlinear generalization of London equation in a long wavelength limit q → Q ( q , a ) is function of q and magnetic field a ( q ). However the equation (163) is correct when amagnetic field is much weaker than the critical field H ≪ H C .Let’s neglect by the nonlinearity, that is we suppose that the kernel Q is function of q only. Then we have: j ( q ) = − γ e c b ( T )(2 d + 2 γ q ) a ( q ) = − γ e c ∆ ( T ) (cid:16) γd q (cid:17) a ( q ) ≈ − γ e c ∆ ( T ) (cid:0) − l q (cid:1) a ( q ) ≡ Q ( q ) a ( q ) , (164)where we took into account a slowness of changes of a gap in space: l q ≪
1. 2 γd = 2 l ν F / ν F = l is a coherentlength at temperature T = 0. Thus we have the nonlocal kernel Q ( q ), where radius of a nonlocality is equal to thecoherent length l . This result corresponds to nonlocal Pippard electrodynamics (long wavelength limit).
This factproves nonlocality of the obtained functional of free energy of a superconductor (141, 143). For generalization in caseof a large value l q it is necessary to expand the free energy (141) in degrees of q . Then we can obtain a shortwavelength limit of Q : Q ∼ /ql . Starting from correctness of the asymptotics (145) and (155) of the functional(141, 143) we can make a conclusion about its correctness for description of a superconductive phase. IX. CONCLUSION.
In this paper on the example of superconductivity we described the type II phase transition on a microscopiclevel, namely starting from first principles. This means, that the method of calculation of a free energy Ω(
T, N/V ))has been developed in a range of temperatures, which includes a point of pase transition, without introducing anyartificial parameters of type of order parameter and sourses of ordering, but starting from microscopic parameters ofHamiltonian only. Moreover, the theorems about connection of a vacuum amplitude with thermodynamics potentialsare realized.Microscopic picture of a phase transition lies in the following. At switching of attraction between particles of a Fermysystem the instability relatively formation of bound states of two fermions rises. The given states are characterizedby Bethe-Solpiter amplitudes - amplitudes of pairing, which can be found exactly in the case of an isolated pair.However, in consequence of statistical correlations between pairs the amplitudes are determined by dynamics of allparticles of a system. Their observing value is result of averaging over the system. Thus, a collective (condensate)of pairs exists. A particle propagating through a system interacts with fluctuations of pairing. As a result of suchinteraction a dispersion law of quasi-particles is changed and anomalous propagators appear. This means, that aspontaneous symmetry breakdown takes place. After consideration of interaction of particles with fluctuations ofpairing all characteristics of a system must be calculated over the new vacuum with broken symmetry. So, calculationof a vacuum amplitude over new ground state gives a possibility to use the theorem about connection of a vacuumamplitude with a ground state energy (with free energy at nonzero temperature). As a result, the free energy isfunction of amplitudes Ω = Ω(∆∆ + ), and their observed value minimizes the free energy. Analyze of the obtainedfunctional of free energy shows, that the amplitude of pairing plays a part of an order parameter. Namely, the orderparameter is Bethe-Solpiter amplitude averaged over a system due to statistical and dynamical correlations. Since inNambu-Gor’kov formalism (the method of anomalous propagators) any phase transition can be described [5], thenour method can be generalized to the rest transitions (ferromagnetism and antiferromagnetism, waves of charge andspin density, ferroelectricity and so on).0The functional of a superconductor’s free energy (141, 143) has been obtained in this paper using the developedmethod of microscopic description of phase transitions and generalizing its in the cases of spatial inhomogeneity andpresence of magnetic field. The functional generalizes Ginzburg-Landau functional for cases of arbitrary temperatures,arbitrary spatial inhomogeneities and a nonlocality of a magnetic response. The equations of superconductor’s stateare extremals of the functional obtained by variation over the gap ∆ and the magnetic field a . The equationsdetermined equilibrium configurations of a gap and a magnetic field at given conditions. Appendix A: The method of uncoupling of correlations and Dyson equation.
As it was shown in [20] for propagators G (one-particle), K (two-particle), K (tree-particle) ... K n (n-particle)the set of coupling equations can be written. The set of equations is analogous to BBGKY hierarchy for a s-particleprobability density function. For example, for G and K these equations have a view: G (1 , ′ ) = G (1 , ′ ) + i Z G (1 , U (2 , K (2 ,
3; 1 ′ , + ) dx dx (A1) K (1 ,
2; 1 ′ , ′ ) = G (2 , ′ ) G (1 , ′ ) − G (2 , ′ ) G (1 , ′ ) + i Z G (1 , U (3 , K (3 , ,
4; 1 ′ , ′ , + ) dx dx , (A2)where 3 + ≡ ( ξ , t + 0), U (2 , ≡ V ( ξ , ξ ) δ ( t − t ), dx ≡ dξdt . The cross term G (2 , ′ ) G (1 , ′ ) − G (2 , ′ ) G (1 , ′ )appeared as result of calculation of Fermy symmetry of particles (it would be ” + ” for bosons). The equation for athree-particle propagator K will be is determined by four-particle propagator K and so on: K n = f ( K n +1 ).It is obviously that this set of equations can not be solved. However, in most cases for description of a system it isenough to know functions G and K (less). Then the method of uncoupling of correlations is used. The function K in a zero approximation is K (0)2 = G (1 , ′ ) G (2 , ′ ) − G (1 , ′ ) G (2 , ′ ) . (A3)We can see, that in an absence of interaction the two-particle propagator is represented in a multiplicative form by theone-particle free propagators G . Statistical correlation exists only in the course of Pauli exclusion principle. Thenin first approximation let’s use the free propagator K (0)2 instead of the dressed propagators K in the formula (A1).Hence the correction for the dressed one-particle propagator G is G (1) (1 , ′ ) = i Z G (1 , U (2 , G (3 , + ) G (2 , ′ ) dx dx − i Z G (1 , U (2 , G (3 , ′ ) G (2 , dx dx . (A4)The correction (A4) is represented graphically in Fig.10. Integration is carried over coordinates of the internal lines.The first term corresponds to direct Hartree interaction, the second term corresponds to exchange Fock interaction.In order to obtain the next approximation for G it is necessary to find K (1)2 . In a symbolic representation we have
32 1’1 + G (1) = Figure 10: The correction of first order G obtained by uncoupling of correlations in the equation (A1). the equations: K (1)2 = G (1) G − G G (1) + iG U K (0)3 G (2) = iG U K (1)2 . (A5)The procedure of uncoupling of correlations can be represented in another way. Let a correction for two-particlepropagator K (1) is determined by the matrix elements: V klkl is a direct interaction and V kllk is an exchange interaction.1A two-particle propagator has two entering momentums (represented by lines with corresponding indexes) and twooutgoing momentums. An one-particle propagator has one entering momentum and one outgoing momentum. Theprocedure of uncoupling of correlations consist in the fact, that we connect two lines in K taking into accountconservation of momentum and spin: Fig.11. The connection means integration over intermediate momentums andenergy parameters (in a momentum-energy representation ξ, t → k , ω ) in the formula (A4). As a result, we have thesame diagrams for G as in Fig.10. llll kkkkll kkl ; klk lk Figure 11: The graphical method of uncoupling of correlations. In the diagram for the first order correction K of two-particlepropagator we connect two lines taking into account conservation of momentum and spin. As a result, we have the first ordercorrection G for one-particle propagator. The procedure can be generalized to higher corrections (with two and more lines of interaction). In general casethe rules of diagram technique are:1. The multiplier iG ( k , ω ) is associated with each bold line, the multiplier iG ( k , ω ) is associated with each thinline2. The multiplier − iV klmn is associated with each dashed line of interaction between particles, and the multiplier − iV kl is associated with line of interaction of a particle with an external field.3. The multiplier − k + l = m + n , energy parameter and spin are reserved in every vertex.5. Integration is made over each intermediate momentum and summation is made over each intermediate energyparameter: P k → V R d k (2 π ) ( V is volume of a system) and R dω π .The diagrams of the type Fig.10 are summarized with help of the mass operator Σ - any diagram without externallines. Hence a dressed propagator G can be found from Dyson equation: iG = iG + iG ( − i )Σ iG ⇒ G = 1 G − − Σ (A6)The mass operator Σ has the sense of a mean field of all particles of a system acting on a test (marked) article. Inthis fact the sense of the procedure of uncoupling of correlations is: interaction and propagation of all particle of asystem is reduced to propagation of each particle in the mean field of all rest particles.Other approach exists (more widely represented in literature) for obtaining of the diagram expansion for G . In thisapproach a one-particle propagator is determined as G ( k , k , t − t ) = lim T → −∞ (1 − iδ ) T → + ∞ (1 − iδ ) − i h Φ | T [ e U ( T , T ) b C k ( t ) b C + k ( t )] | Φ ih Φ | e U ( T , T ) | Φ i , (A7)where b C + ( t ) , b C ( t ) is creation and annihilation operators in interaction representation , e U is evolution operator ininteraction representation, Φ is ground state of a system of noninteracting fermions. This definition of a propagatoris equivalent to the definition G ( k , t − t ) = − i h Ψ | T [ C k ,σ ( t ) C + k ,σ ( t )] | Ψ i , (A8)where b C + ( t ) , b C ( t ) is creation and annihilation operators in Heisenberg representation, Ψ is ground state of a systemof interacting fermions, and expansion of G in series of perturbation theory is possible if the condition of adiabaticityis realized (8): h Φ | Ψ i 6 = 0 . (A9)2In the method of uncoupling of correlations we didn’t use the condition (A9) and Wick theorem unlike the standardformulation of the perturbation theory based on (A7). Thus the advantage of stated above method of uncoupling ofcorrelations consists in that we can formulate a perturbation theory without using of the adiabatic hypothesis (A9). Appendix B: The method of uncoupling of correlations for a vacuum amplitude in the case of normalprocesses.
Let us consider the processes of direct and exchange interaction of first order, which is described by the matrixelements V klkl and V lkkl , moreover the interaction doesn’t act to spins of particles. Contribution to vacuum amplitudeof such processes is (let’s suppose t > t for definiteness): R ( t ) = 1 + 11! Z t dt X α,β X k , l (cid:18) − i V klkl (cid:19) h Φ | C + l ,β ( t ) C + k ,α ( t ) C k ,α ( t ) C l ,β ( t ) | Φ i + 11! Z t dt X α X k , l (cid:18) − i V lkkl (cid:19) h Φ | C + k ,α ( t ) C + l ,α ( t ) C k ,α ( t ) C l ,α ( t ) | Φ i + 12! Z t dt Z t dt X α,β X k , l (cid:18) − i V klkl (cid:19) X α ′ ,β ′ X k ′ , l ′ (cid:18) − i V k ′ l ′ k ′ l ′ (cid:19) ×h Φ | C + l ′ ,β ′ ( t ) C + k ′ ,α ′ ( t ) C k ′ ,α ′ ( t ) C l ′ ,β ′ ( t ) C + l ,β ( t ) C + k ,α ( t ) C k ,α ( t ) C l ,β ( t ) | Φ i + 12! Z t dt Z t dt X α X k , l (cid:18) − i V lkkl (cid:19) X α ′ X k ′ , l ′ (cid:18) − i V l ′ k ′ k ′ l ′ (cid:19) ×h Φ | C + k ′ ,α ′ ( t ) C + l ′ ,α ′ ( t ) C k ′ ,α ′ ( t ) C l ′ ,α ′ ( t ) C + k ,α ( t ) C + l ,α ( t ) C k ,α ( t ) C l ,α ( t ) | Φ i + 12! 2 Z t dt Z t dt X α,β X k , l (cid:18) − i V klkl (cid:19) X α ′ X k ′ , l ′ (cid:18) − i V l ′ k ′ k ′ l ′ (cid:19) ×h Φ | C + k ′ ,α ′ ( t ) C + l ′ ,α ′ ( t ) C k ′ ,α ′ ( t ) C l ′ ,α ′ ( t ) C + l ,β ( t ) C + k ,α ( t ) C k ,α ( t ) C l ,β ( t ) | Φ i + ... (B1)For approximate calculation R ( t ) we shall use the method of uncoupling of correlations, which lies in the fact that anaverage of four creation and annihilation operators is represented by a product of averages of pairs of the operators: h C + C + CC i → h C + C ih C + C i . The averages correspond to propagators of particles with initial and final statescorresponding to the matrix element of interaction V klmn taking into account conservation of momentum and spin. It isachieved by preliminary transposition of the operators before the uncoupling taking into account Fermy commutation.For Hartree and Fock processes the procedure corresponds to the diagrams in Fig.12. In each process of scattering weconnect incoming and outgoing lines taking into account the laws of conservation. The obtained diagram must nothave free ends. Then we have the amplitude of transition ”vacuum-vacuum”. Analytically it will be so: R ( t ) ≈ − Z t dt X α,β X k , l (cid:18) − i V klkl (cid:19) h Φ | C + l ,β ( t ) C l ,β ( t ) | Φ ih Φ | C + k ,α ( t ) C k ,α ( t ) | Φ i +( −
1) 11! Z t dt X α X k , l (cid:18) − i V lkkl (cid:19) h Φ | C + k ,α ( t ) C k ,α ( t ) | Φ ih Φ | C + l ,α ( t ) C l ,α ( t ) | Φ i +( − Z t dt Z t dt X α,β X k , l (cid:18) − i V klkl (cid:19) X α ′ ,β ′ X k ′ , l ′ (cid:18) − i V k ′ l ′ k ′ l ′ (cid:19) ×h Φ | C + l ,β ( t ) C l ,β ( t ) | Φ ih Φ | C + k ,α ( t ) C k ,α ( t ) | Φ ih Φ | C + l ′ ,β ′ ( t ) C l ′ ,β ′ ( t ) | Φ ih Φ | C + k ′ ,α ′ ( t ) C k ′ ,α ′ ( t ) | Φ i +( − Z t dt Z t dt X α X k , l (cid:18) − i V lkkl (cid:19) X α ′ X k ′ , l ′ (cid:18) − i V l ′ k ′ k ′ l ′ (cid:19) ×h Φ | C + k ,α ( t ) C k ,α ( t ) | Φ ih Φ | C + l ,α ( t ) C l ,α ( t ) | Φ ih Φ | C + k ′ ,α ′ ( t ) C k ′ ,α ′ ( t ) | Φ ih Φ | C + l ′ ,α ′ ( t ) C l ′ ,α ′ ( t ) | Φ i
3+ 12! ( − ( − Z t dt Z t dt X α,β X k , l (cid:18) − i V klkl (cid:19) X α ′ X k ′ , l ′ (cid:18) − i V l ′ k ′ k ′ l ′ (cid:19) ×h Φ | C + l ,β ( t ) C l ,β ( t ) | Φ ih Φ | C + k ,α ( t ) C k ,α ( t ) | Φ ih Φ | C + k ′ ,α ′ ( t ) C k ′ ,α ′ ( t ) | Φ ih Φ | C + l ′ ,α ′ ( t ) C l ′ ,α ′ ( t ) | Φ i + . . . = 1 + ( R Hartree + R F ock ) + 12! ( R Hartree + R F ock ) + . . . = exp( R Hartree + R F ock ) (B2) lkll kkll kkl ; klk lk Figure 12: The procedure of uncoupling of correlations in vacuum amplitude for a correction of first order. In the average ofmatrix element of interaction operator V h C + C + CC i we connect the lines taking into account conservation of momentum andspin. As a result, we obtain the correction of first order R for a vacuum amplitude of a view V h C + C ih C + C i . The uncoupling lets to write expression for R ( t ) via normal propagators: R Hartree = (2 s + 1) Z t dt X k , l (cid:18) − i V klkl (cid:19) iG ( l , t − t ) iG ( k , t − t )= ( − i ) X k , l V klkl B ( l ) B ( k ) t (B3) R F ock = (2 s + 1)( − Z t dt X k , l (cid:18) − i V lkkl (cid:19) iG ( l , t − t ) iG ( k , t − t )= i X k , l V lkkl B ( l ) B ( k ) t. (B4)The multiplier (2 s + 1) is result of summation over spin states (number of spin configurations), s = 1 /
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