Pekar's Ansatz and the Strong Coupling Problem in Polaron Theory
aa r X i v : . [ c ond - m a t . o t h e r] M a y Pekar Ansatz and the Strong Coupling Problem inPolaron Theory.
V.D. LakhnoInstitute of Mathematical Problems of Biology, RAS
Abstract
A detailed consideration is given to the translation-invariant theory of Tulub polaronconstructed without the use of Pekar ansatz. A fundamental result of the theory is thatthe value of the polaron energy is lower than that obtained on the basis of Pekar ansatzwhich was considered as an asymptotically exact solution in the strong coupling limit.In the case of bipolarons the theory yields the best values of the coupling energy andcritical parameters of their stability. Numerous physical consequences of the existence oftranslation-invariant polarons and bipolarons are discussed.
PACS numbers : 71.38.-K, 74.20-z, 74.72.-hTable of contents1. Pekar ansatz.2. Coordinate-free Hamiltonian. Weak coupling.3. Coordinate-free Hamiltonian. General case.4. Weak coupling limit in Tulub theory.5. Strong coupling.6. TI-bipolaron.7. Functional of the ground state. Tulub ansatz.8. Discussion of the completeness of Tulub theory.9. Consequences of the existence of translation-invariant polarons and bipolarons.10. Conclusive remarks.Appendix 1.Appendix 2.Appendix 3.References. 1
Introduction. Pekar ansatz.
As is known, polaron theory was among the first to describe the interaction between a particleand a quantum field. Various aspects of the polaron theory are presented in numerous reviewsand books [1]-[10]. Being non-relativistic, the theory does not contain any divergencies andfor more than sixty years has been a testing area for approbation of various methods of thequantum field theory. Though no exact solution of the polaron problem has been found up tonow, it has been believed that the properties of the ground state are known in detail. Thisprimarily refers to the limit cases of weak and strong coupling. A solution in the weak couplinglimit was given by Froehlich [11], and that in the strong coupling one was found by Pekar[1],[12]. By now rather an exact solution has been obtained for the energy of the polaronground state in the weak coupling limit [13],[14] : E = − (cid:0) α + 0 . α + 0 . α + · · · (cid:1) ~ ω , (1.1)where ~ ω is the energy of an optical phonon, α is the constant of electron-phonon coupling.A solution of the problem in the opposite strong coupling limit was given by Pekar on theassumption that the wave function Ψ of the electron + field system has the form:Ψ( r, q , ..., q i , ... ) = ψ ( r )Φ( q , ..., q i , ... ) , (1.2)where ψ ( r ) is the electrons wave function depending only on the electron coordinates, Φ –is the wave function of the field depending only on the field coordinates. Pekar himself [1]considered ansatz (1.2) ) to be an approximate solution. In the pioneer works by Bogolyubovand Tyablikov [15],[16] it was shown that in a consistent translation-invariant theory the useof ansatz (1.2) (for decomposed coordinates introduced in [15],[16] ) gives the same results forthe polaron ground state energy as the semiclassical Pekar theory does [1],[12]. With the useof (1.2) the value of the ground state energy has been found to a high precision. According to[17],[18] it is equal to:: E = (cid:0) − . α − . (cid:1) ~ ω . (1.3)The idea that Pekar ansatz (1.2) is an exact solution of the strong coupling polaron problemwas completely established after publication of [19] where asymptotics (1.3) was strictly provedby path integral method, i.e. without the use of ansatz (1.2) (see also review [20]).2efore the publication of paper [19] many attempts were made to improve the strong cou-pling theory [21]-[26]. The reason why Pekar ansatz caused the feeling of disappointment wastranslation invariance of the initial polaron Hamiltonian. When ansatz for the wave function ψ ( r ) (1.2) is used the wave equation has a localized solution. The electron is localized in apotential polarization well induced by it. In other words, the solution obtained does not possessthe symmetry of the initial Hamiltonian. Self-trapping of the electron in the localized potentialwell leads to a spontaneous breaking of the systems symmetry. Attempts to restore the initialsymmetry were based on the use of degeneration of the system with broken symmetry. Sincein a homogeneous and isotropic medium nothing should depend on the position of the polaronwell center r , one can ”spread” the initially localized solution over all the positions of thepolaron potential well by choosing the wave function in the form of a linear combination in allthe positions of the well.In the most consistent form this program was carried out in [24]. With this end in viewfor the wave function which is an eigen function of the total momentum, the authors used asuperposition of plain waves corresponding to the total momentum multiplied by wave functionsobtained from (1.2) to which a translations operator is applied. In other words, they took anappropriate superposition with respect to all the positions of the polaron well r . The mainresult of paper [24] is that calculation of the polaron ground state energy with such a delocalizedfunction yields the same value as calculations with localized function (1.2) do. The authorsof [24] also reproduced the value of the polaron mass which was earlier obtained by Landauand Pekar [27] on the assumption that polaron moves in medium in the localized state (1.2).The results derived in [24] were an important step in resolving the contradiction between therequirement that the translation-invariant wave function be delocalized while the wave functionof the self-trapped state be localized.Notwithstanding the success achieved with this approach it cannot be considered fully ade-quate since it has quite a few inconsistencies. They follow from the very nature of the semiclas-sical description used. Indeed, the superposition constructed in [24] on the one hand determinesthe polaron delocalized state, but on the other hand, without changing this state, one can mea-sure its position and find out a localized polaron well with an electron localized in it. Thereason of this paradox is a classical character of the polaron well in the strong coupling limitand, as a consequence, commutation of the total momentum operator with the position of the3olaron well r . To remedy this defect some approaches were suggested in which the quantity r ( r, q , ..., q i , ... )which is not actually an additional degree of freedom was considered to be that though withsome additional constraints. Discussion of these challenges associated with solution of the prob-lem of introducing collective coordinates is given in review [30]. Since the results obtained byintroducing collective coordinates into the polaron theory are polemical it seems appropriateto describe strict results of the translation-invariant theory without recourse to the conceptof collective coordinates. The aim of this review is to present an approach used in the strongcoupling limit which does not use Pekar ansatz.A solution possessing these properties in the case of a strong coupling polaron was originallyfound by Tulub [31],[32]. During nearly half a century the result obtained in [31],[32] was notrecognized by specialists working in the field of polaron theory. The reason why the importanceof that result was not appreciated was an improper choice of the probe wave function in [32]to estimate the ground state. As a result the ground state energy was found in [32] to be E = − . α ~ ω which is larger than in (1.3). An appropriate choice of the wave functionhas been made quite recently in paper [33]. This has yielded a lower than in (1.3) value ofthe polaron ground state energy equal to E = − . α ~ ω . Hence, actually we haveto do with inapplicability of adiabatic approximation in the case of a polaron, though it isfundamental for solid state physics.In this review we present the main points of the translation-invariant polaron (TI-polaron)theory. In § § At the rise of quantum mechanics, the founders of the science were fully aware of the difficulties arising here.Thus, for example, in [28] Bethe notices that for a proper quantum-mechanical description of an interactionbetween a field and particles, quantizing of the field is required, i.e. quantum theory of the field: ” Thefact is that, when quantizing mechanical parameters (coordinates and momenta) one should also quantize theassociated fields. Otherwise, as Bohr and Rosenfeld showed [29], an imaginary experiment can be suggestedwhich consists in simultaneous measurement of the coordinate and the momentum of a particle from examinationof the field induced by it. This contravenes Heisenberg’s Uncertainty Principle.” § § § § § § §
10 is devoted to discussion of some fundamental problems of the strong coupling theorywhich are still to be solved.In Appendices 1-3 we give proofs of some important statements basic to the approach underconsideration. 5
Coordinate-free Hamiltonian. Weak coupling.
Let us proceed from Pekar-Froehlich Hamiltonian: H = − ~ m ∆ r + X k V k (cid:0) a k e ikr + a + k e − ikr (cid:1) + X k ~ ω k a + k a k , (2.1)where a + k , a k are operators of the birth and annihilation of the field quanta with energy ~ ω k = ~ ω , m is the electron effective mass, V k is a function of the wave vector k .Interest in the study of this Hamiltonian is also provoked by the fact that, as distinctfrom many model Hamiltonians considered in the condensed matter theory, Pekar-FroehlichHamiltonian (2.1) in the limit of long waves asymptotically exactly describes the behavior of anon-relativistic electron in a continuous polar medium.Electron coordinates can be excluded from (2.1) via Heisenberg transformation [34]: S = exp ( i ~ ~P − X k ~ ~ka + k a k ! ~r ) , (2.2)where ~P is the total momentum of the system. Application of S to the field operators yields: S − a k S = a k e − ikr , S − a + k S = a + k e ikr . Accordingly the transformed Hamiltonian ˜ H = S − HS takes on the form:˜ H = 12 m ~P − X k ~ ~ka + k a k ! + X k V k ( a k + a + k ) + X k ~ ω k a + k a k . (2.3)Since Hamiltonian (2.3) does not contain electron coordinates, it is obvious that solution of thepolaron problem obtained on the basis of (2.3) is translation-invariant. Lee, Low and Pines[35] studied the ground state (2.3) with the probe wave function | Ψ i LLP : | Ψ i LLP = S | i , (2.4)where: S = exp (X k f k ( a + k − a k ) ) , (2.5) f k are variational parameters having the meaning of the value of displacement of the fieldoscillators from their equilibrium positions, | i is the vacuum wave function. The quantity f k in S (2.5) is determined by minimization of energy E = h | S − ˜ HS | i , which for P = 0 yields: E = 2 X k f k V k + ~ m "X k ~kf k + X k ~ k m f k + X k ~ ω k f k , (2.6)6 k = − V k ~ ω k + ~ k / m . (2.7)In the case of an ionic crystal: V k = ek r π ~ ω ˜ εV = ~ ω ku / (cid:18) παV (cid:19) / , u = (cid:18) mω ~ (cid:19) / , α = 12 e u ~ ω ˜ ε , ˜ ε − = ε − ∞ − ε − , (2.8)where e is an electron charge, ε ∞ and ε are high-frequency and static dielectric permittivities, α is a constant of electron-phonon coupling. With substitution of (2.8) into (2.6), (2.7) theground state energy becomes E = − α ~ ω , which is the energy of a weak coupling polaron inthe first order with respect to α .A solution of the problem of transition to the strong coupling case in coordinate-free Hamil-tonian (2.3) was found on the basis of the general translation-invariant theory constructed inTulubs work [32]. The main points of this theory are given in the next section. To construct the general translation-invariant theory in works of [31], [32] was used a canonicaltransformation of Hamiltonian (2.3) with the use of operator S (2.5) which leads to a shift ofthe field operators: S − a k S = a k + f k , S − a + k S = a + k + f k . (3.1)The resultant Hamiltonian ˜˜ H = S − ˜ HS has the form:˜˜ H = H + H , (3.2)where: H = ~P m + 2 X k V k f k + X k ~ ω k − ~ ~k ~Pm ! f k + 12 m X k ~kf k ! + H , (3.3) H = X k ~ ω k a + k a k + 12 m X k,k ′ ~k~k ′ f k f k ′ (cid:0) a k a k ′ + a + k a + k ′ + a + k a k ′ + a + k ′ a k (cid:1) , (3.4) ~ ω k = ~ ω k − ~ ~k ~Pm + ~ k m + ~ ~km X k ′ ~ ~k ′ f k ′ . (3.5)Hamiltonian H contains terms linear, triple and quadruple in the birth and annihilation op-erators. With an appropriate choice of the wave function diagonalizing quadratic form (3.4)7athematical expectation H becomes zero (Appendix 1). In what follows we believe that ~ = 1, ω = 1, m = 1. To transform H to a diagonal form we put: q k = 1 √ ω k ( a k + a + k ) , p k = − i r ω k a k − a + k ) , ~z k = ~kf k √ ω k . (3.6)With the use of (3.6) expression (3.4) is written as: H = 12 X k ( p + k p k + ω k q + k q k ) + 12 X k ~z k q k ! − X k ω k . (3.7)This yields (3.7) the following motion equation for operator q k :¨ q k + ω k q k = − ~z k X k ′ ~z k ′ q k ′ . (3.8)Let us search for a solution of system (3.8) in the form: q k ( t ) = X k ′ Ω kk ′ ξ k ′ ( t ) , ξ k ( t ) = ξ k e iν k t . (3.9)As a result we express matrix Ω kk ′ as:( ν k ′ − ω k )Ω kk ′ = ~z k X k ′′ ~z k ′′ Ω k ′′ k ′ . (3.10)Let us consider determinant of this system which is derived by replacing the eigenvalues ν k in(3.10) with the quantity s which can differ from ν k . The determinant of this system will bedet (cid:12)(cid:12) ( s − ω k ) δ kk ′ − ~z k ~z k ′ (cid:12)(cid:12) = Y k ( s − ν k ) . (3.11)On the other hand, according to [36] :det (cid:12)(cid:12) ( s − ω k ) δ kk ′ − ~z k ~z k ′ (cid:12)(cid:12) = Y k ( s − ω k ) − X k ′ ~z k ′ s − ω k ′ ! . (3.12)It is convenient to introduce the quantity ∆( s ) :∆( s ) = Y k ( s − ν k ) / Y k ( s − ω k ) . (3.13)With the use of (3.11) and (3.12) ∆( s ) is expressed as:∆( s ) = − X k ~z k s − ω k ! . (3.14) In Wentzel’s work z k is not a vector function, but a scalar one, therefore a ”cube” in (3.12) is lacking.Generalization to the vector case is given in [31]. ν k renormalized by interaction are determinedby a solution to the equation: ∆( ν k ) = 0 . (3.15)The change in the systems energy ∆ E caused by the electron-field interaction is equal to:∆ E = 12 X k ( ν k − ω k ) . (3.16)To express the quantity ∆ E via ∆( s ) we use Wentzel approach [36]. Following [36] we writedown the identity equation: X k (cid:8) f ( ν k ) − f ( ω k ) (cid:9) = 12 πi I C dsf ( s ) X k (cid:18) s − ν k − s − ω k (cid:19) == 12 πi I C dsf ( s ) dds ln ∆( s ) = − πi I C dsf ′ ( s ) ln ∆( s ) , (3.17)where integration is carried out over the contour presented in Fig.1 Ω Ν Ω Ν s - plane Fig. 1. Contour C .Taking f ( s ) = √ s we get:∆ E = 12 X k ( ν k − ω k ) = − πi I C ds √ s ln ∆( s ) . (3.18)Turning in (3.14) from summing up to integration with the use of the relation: X k = 1(2 π ) Z d k in a continuous case, using for ~z k expression (3.14) for ∆( s ) we obtain:∆( s ) = D ( s ) , D ( s ) = 1 − π ) Z k f k ω k s − ω k d k . (3.19)9s a result the total energy of the electron is: E = ∆ E + 2 X k V k f k + X k f k ω k . (3.20)The results obtained here are general and valid for various polaron models (i.e. any functions V k and ω k ). Below we consider limit cases of weak and strong coupling which follow from generalexpression (3.20) on the assumption that ~P = 0.Notice that for ~P = 0, according to [31], expression (3.20) takes the form: E = P m + ∆ E ( ~P ) + 2 X k V k f k + X k ~ ω k − ~ ~k ~Pm ! f k + 12 m X k ~kf k ! , ∆ E ( ~P ) = − πi I C ds √ s ln Y i =1 D i ( s ) ,D i ( s ) = 1 − X k ( z ik ) s − ω k , where z ik – i -th component of the vector ~z k . Functions f k , ω k and z k are depending on | ~k | andon ( ~k ~P ). Quantities f k in the expression for the total energy E should be found from the minimumcondition: δE/δf k = 0 which yields the following integral equation for f k : f k = − V k / (1 + k / µ k ) , µ − k = ω k πi I C ds √ s s − ω k ) D ( s ) . (4.1)In the case of weak coupling α → α → D ( s ) = 1 and µ − k is equal to: µ − k = ω k πi I C ds √ s s − ω k ) = 1 . (4.2)Accordingly, f k from (4.1) is written as: f k = − V k / (1 + k / . (4.3)10he quantity ∆ E involved in the total energy takes on the form:∆ E = − πi I C ds √ s ln D ( s ) , ln D ( s ) = − π ) Z k f k ω k s − ω k d k . (4.4)With the use of (4.3) integrals involved in (4.4) are found to be: ∆ E = ( α/ ~ ω . Havingcalculated the rest of the terms involved in expression (3.20) we get the first term of theexpansion of polaron total energy in the coupling constant α : E = − α ~ ω .In papers [31], [37], [38] a general scheme of calculating the higher terms of expansion in α was developed. In particular, the eigen energy and effective mass were found to be [38]: E = − ( α + 0 . α ) ~ ω ,m ∗ = (cid:16) α . α (cid:17) m . (4.5)Hence within the accuracy of the terms O ( α ) the polaron energy expression calculated withinTulub approach with the use of perturbation theory coincides with exact result (1.1) (see § The case of strong coupling is much more complicated. To reveal the character of the solution inthe strong coupling region let us start with considering the analytical properties of the function D ( s ) in the form: D ( s ) = D (1) + s − π ∞ Z k f k ω k dk ( ω k − ω k − s ) , (5.1)where D (1) is the value of D ( s ) for s = 1: D (1) = 1 + Q ≡ π ∞ Z k f k ω k ω k − dk . (5.2)From (3.19) also follows that: D ( s ) = 1 − π ∞ Z ω k k f k s − ω k dk . (5.3)Function D ( s ), being a function of a complex variable s , has the following properties: 1) D ( s ) has a crosscut along the real axis from s = 1 to ∞ and has no other peculiarities; 2) D ∗ ( s ) = D ( s ∗ ); 3) as s → ∞ sD ( s ) increases not slower than s . These properties enable us topresent the function [( s − D ( s )] − in the form (Appendix 2):1( s − D ( s ) = 12 πi I C + ρ ds ′ ( s ′ − s )( s ′ − D ( s ′ ) , (5.4)11here contour C + ρ is shown in Fig. 2: - plane Ρ C Fig. 2. Contour C + ρ .The integrand function in (5.4) has a pole at s ′ = 1 and a section from s ′ = 1 to s ′ = ∞ .Having performed integration in (5.4) along the upper and bottom sides of the crosscut we getthe following integral equation for D − ( s ) :1 D ( s ) = 11 + Q + s − π ∞ Z k f k ω k dk ( s − ω k )( ω k − | D ( ω k ) | . (5.5)With the use of integration by parts expression (3.18) for ∆ E can be written as:∆ E = 12 π ∞ Z dkk f k ω k πi I C √ s ( s − ω k ) D ( s ) ds . (5.6)From (5.5), (5.6) we have:∆ E = 12 π ∞ Z k f k dk Q ) + 112 π ∞ Z ∞ Z k f k p f p ω p ( ω k ω p + ω k ( ω k + ω p ) + 1)( ω k + ω p ) ( ω p − | D ( ω p ) | dpdk . (5.7)Equation µ − k (4.1), according to (5.5), can be presented in the form: µ − k = 11 + Q + 13 π ∞ Z p f p ( ω k ω p + 1) dp ( ω p − ω k + ω p ) | D ( ω p ) | . (5.8)Equations (4.1), (5.8) for finding f k as well as expressions (3.20), (5.7) for calculating polaronenergy are very complicated and their exact solution can hardly be obtained. To calculateapproximately the energy E given by (3.20), (5.7) in [32] a direct variational principle wasused. For the probe function, the author used Gaussian function of the form: f k = − V k exp( − k / a ) , (5.9)12here a is a variable parameter, besides, as can be seen in the case of strong coupling, a ≫ D ( s ) (see Appendix 3):Re D ( ω k ) = 1 + λv ( y ) , Im D ( ω k ) = k f k / π ,v ( y ) = 1 − ye − y y Z e t dt − ye y ∞ Z y e − t dt ,λ = 4 αa/ √ π , y = k/a . (5.10)In the limit of strong coupling ( α ≫
1) the expression for energy E given by (3.20) with theuse of (5.7) takes on the form: E = 316 a (cid:20) q (cid:18) λ (cid:19)(cid:21) − αa √ π (cid:18) − √ (cid:19) , (5.11) q (cid:18) λ (cid:19) = 2 √ π ∞ Z e − y (1 − Ω( y )) dy (1 /λ + v ( y )) + πy e − y / , (5.12)Ω( y ) = 2 y (1 + 2 y ) ye y ∞ Z y e − t dt − y . As λ → ∞ , integral (5.12) has maximum for y = 3 λ/
4, if the function f k is chosen in the form(5.9), however if the actual boundedness of the region of integration with respect to y is takeninto account, this peculiarity does not take place (see § /λ = 0. As a result of numerical integration q (0) was found to be q (0) = 5 .
75 whence, varyingenergy E (5.11) with respect to a we get: E = − . α ~ ω . (5.13)Comparison of (5.13) with (1.3) shows that the value of E obtained for α → ∞ lies higher thanthe exact value in Pekar theory (1.3). For this reason, until quite recently it was believed thatTulub theory as applied to a polaron does not give any new results.The situation changed radically after publication of [33]. There it was shown that the choiceof the wave function for minimizing energy (3.20) in the form (5.9) is not optimal since it doesnot satisfy virial relations. As is shown in [33], an appropriate function f k should contain themultiplier √ E = − . α ~ ω . (5.14)13esult (5.14) is fundamental. Above all it means that Pekar ansatz does not give an exactsolution. Though result (5.14) refers to a particular case of PekarFroehlich Hamiltonian with V k given by (2.8), the conceptual conclusion should be valid for all types of self-localized states.Of special interest is to consider the case of bipolarons since they can play an important rolein superconductivity. Great attention given to polaron problem in recent times is associated with attempts to explainthe superconductivity phenomenon relying on the mechanism of Bose condensation of bipolarongas. In this connection the study of conditions under which the bipolaron states are stable isof paramount importance. The theory of large-radius bipolarons which are now considered tobe the best candidates for the role of charged bosons forming Bose-Einstein condensate withpairing in real space is considered in detail in reviews [7], [8], [39].The study of the process of formation of a stable two-electron state in a crystal, or abipolaron, generally implies finding a pairwise interaction between two polarons as a functionof a distance between them [8]. For a large-radius bipolaron, the region of its existence isbordered on the part of the coupling constant α by rather a large value of α c below which thepolaron bound state does not exist. In view of the requirement that α c be large, which may notbe met in high-temperature superconductors, some researchers investigated the contribution ofother types of interactions and coupling symmetries [40], [41].In what follows we will deal with only electron-phonon PekarFroehlich interaction since theapproach under consideration can be generalized to other types of interactions as well. It seemsall the more actual in view of the fact that in recent times some reasoned arguments have beenobtained testifying that electron-phonon interaction in high-temperature superconductors isstrong [42]-[44]. There are also arguments in favor of the fact that owing to weak screening ofhigh-frequency optical phonons, the electron-phonon interaction in high-temperature supercon-ductors is more adequately described not in the framework of a contact interaction of Holsteinpolaron model [45], but in terms of a long-range interaction of Froehlich type [46].Before the publication of [47]-[49] the lowest values of the energy of bipolaron states deter-mined by electron-phonon interaction were obtained in [50]-[52] for α < α >
8. Attempts to find a translation-invariant solution of the bipolaron problem by varia-14ional methods using direct variation of the wave function of the two-electron system [39], [56],[57] yielded higher values of the bipolaron ground state energy as compared to those obtainedwith the use of a wave function lacking translation invariance [51], [52], [55], [58]. In this sec-tion we present the results obtained in [47]-[49] for a bipolaron within the translation-invariantapproach.Let us proceed from Pekar-Froehlich Hamiltonian for a bipolaron [8] : H = − ~ m ∆ r − ~ m ∆ r + X k ~ ω k a + k a k + U ( | ~r − ~r | ) + X k (cid:0) V k e ikr a k + V k e ikr a k + H.C. (cid:1) , (6.1) U ( | ~r − ~r | ) = e ε ∞ | ~r − ~r | , where ~r and ~r are coordinates of the first and second electrons, respectively, quantity U describes Coulomb repulsion between the electrons.In the mass center system Hamiltonian (6.1) takes on the form: H = − ~ M e ∆ R − ~ µ e ∆ r + U ( | r | ) + X k ~ ω k a + k a k + X k V k cos ~k~r (cid:16) a k e i~k ~R + H.C. (cid:17) , (6.2) ~R = ( ~r + ~r ) / , ~r = ~r − ~r , M e = 2 m , µ e = m/ . In what follows we will believe that ~ = 1, ω k = 1, M e = 1 (accordingly µ e =1/4).The coordinates of the mass center ~R can be excluded from Hamiltonian (6.2) via Heisenbergcanonical transformation: S = exp ( − i X k ~ka + k a k ) ~R , ˜ H = S − HS = − r + U ( | r | ) + X k a + k a k + X k V k cos ~k~r a k + a + k ) + 12 X k ~ka + k a k ! . (6.3)From formula (6.3) follows that the exact solution of the bipolaron problem is determined by thewave function ψ ( r ) containing only relative coordinates r and, therefore, possessing translationinvariance.Averaging of ˜ H over ψ ( r ) leads to the Hamiltonian ¯ H :¯ H = 12 X k ~ka + k a k ! + X k a + k a k + X k ¯ V k ( a k + a + k ) + ¯ T + ¯ U , (6.4)¯ V k = 2 V k h ψ | cos ~k~r | ψ i , ¯ U = h ψ | U ( r ) | ψ i , ¯ T = − h ψ | ∆ r | ψ i . V k in (2.3) is replaced by¯ V k and constants ¯ T and ¯ U are added. Therefore, repeating the derivation performed in § E bp as: E bp = ∆ E + 2 X k ¯ V k f k + X k f k + ¯ T + ¯ U , (6.5)where ∆ E is given by expression (5.7). From (6.5) we can get expressions for the bipolaronenergy varying E bp with respect to f k and ψ . Since the equations obtained in this way aredifficult to solve, for actual determining of the bipolaron energy we use a direct variationalapproach, assuming [49]: f k = − N ¯ V k exp( − k / µ ) ,ψ ( r ) = (2 /πℓ ) / exp( − r /ℓ ) , (6.6)where N , µ , ℓ are variational parameters. For N = 1, expression (6.6) reproduces the resultsof work [47], and for N = 1 and µ → ∞ , those of work [48].Having substituted (6.6) into the expression for the total energy and then minimized theexpression obtained with respect to parameter N we write E as: E ( x, y ; η ) = Φ( x, y ; η ) α , (6.7)Φ( x, y ; η ) = 6 x + 20 . x + 16 y − p x + 16 y √ π ( x + 8 y ) + 4 p /πx (1 − η ) . Here x , y are variational parameters: x = ℓα , y = α /µ , η = ε ∞ /ε .Let us write Φ min for theminimum of function Φ of parameters x and y . Fig.3 shows the dependence of Φ min on theparameter η . Fig.4 demonstrates the dependence of x min , y min on the parameter η . Η- - - - F min Fig. 3. Graph Φ min ( η ).16 .0 0.1 0.2 0.3 0.4 Η min y min Fig. 4. Graphs x min ( η ), y min ( η ),.Fig.3 suggests that E min ( η = 0) = − . α yields the lowest estimation of the bipolaronground state energy as compared to all those obtained earlier by variational method. Horizontallines in Fig.3 correspond to the energies: E = − . α and E = − . α , where E =2 E p , E p is the Pekar polaron ground state energy (1.3); E = 2 E p where E p is the groundstate energy of a translation-invariant polaron (5.14). Intersection of these lines with the curve E min ( η ) yields the critical values of the parameters η = η c = 0 . η = η c = 0 . η > η c , the bipolaron decays into two translation-invariant polarons, for η > η c , it breaksdown into Pekar polarons. The values of minimizing parameters x min and y min for these valuesof η are x min (0) = 5 . y min (0) = 2 . x min (0 . . y min (0 . . x min (0 . . y min (0 . . α , determined from comparisonof the energy expressions in the weak coupling limit (doubled energy of a weak coupling polaron: E = − α ~ ω ) and in the strong coupling limit ( E = − . α ~ ω ), at which the translation-invariant bipolaron is formed is equal to α c ≈ .
54, being the lowest estimate obtained byvariational method. It should be emphasized that this value is conventional. Hamiltonian (6.4)coincides in structure with one-electron Hamiltonian (2.3), therefore, as in the case of a polaron,the bipolaron energy, by [20], is an analytical function of α . For this reason at the point α = α c the bipolaron energy does not have any peculiarities and the bipolaron state exists over thewhole range of α and η variation: 0 < α < ∞ ; 0 < η < − / √
2, for which
E <
0. To solvethe problem of the existence of α c value at which the bipolaron state can decay into individualpolarons, one should perform calculations for the case of intermediate coupling. In particular, ascenario is possible when the bipolaron energy for some values of η will be lower than the energy17f two individual polarons for all values of α , i.e. the bipolaron state exists always. Notice,that for the derived state of the translation-invariant bipolaron, the virial theorem holds to ahigh precision.The problems of arising high-temperature superconductivity (HTSC) and explaning thisphenomenon by formation of bipolaron states were dealt with in a number of papers and reviews[7], [8], [59], [60]. In these works the existence of HTSC is explained by Bose condensation ofa bipolaron gas. The temperature of Bose condensation T = 3 . ~ n / /k B m bp , which isbelieved to be equal to the critical temperature of a superconducting transition T c for m bp ≈ m , depending on the bipolarons concentration n , varies in a wide range from T = 3 K at n = 10 cm − to T ≈ K at n ≈ cm − . In the latter case the bipolarons concentrationis so high that for a bipolaron gas as well as for Cooper pairs, the compound character of abipolaron when it ceases to behave like an individual particle should show up. In the caseof still higher concentrations a bipolaron should decay into two polarons. According to (6.6)the characteristic size of a bipolaron state is equal to ℓ and in dimension units is written as: ℓ corr = ~ ˜ εx ( η ) /me . Here ℓ corr has the meaning of a correlation length. The dependence x ( η )is given by Fig.4. Fig.4 suggests that over the whole range of η variation where the bipolaronstate is stable the value of x changes only slightly: from x ( η = 0) ≈ x ( η = 0 . ≈ η = η c , the critical value of the concentration at which the bipolarons multipiececharacter is noticeable is of the order of n c ∼ = 10 cm − . This result testifies that a bipolaronmechanism of HTSC can occur in copper oxides. To diagonalize quadratic form (3.4) we can use Bogolyubov-Tyablikov transformation [61]. Letus write α k for the operators of physical particles in which H is a diagonal operator.Let us diagonalize the quadratic form with the use of the transformation: a k = X k ′ M kk ′ α k ′ + X k ′ M ∗ kk ′ α + k ′ ,a + k = X k ′ M ∗ kk ′ α + k ′ + X k ′ M kk ′ α k ′ , (7.1)so that the equalities: [ a k , a + k ′ ] = [ α k , α + k ′ ] = δ kk ′ , [ H , α + k ] = ω k α + k . (7.2)18e fulfilled.Relying on the properties of unitary transformation (7.1) we have: M M +1 = M ∗ M T , ( M +1 ) − = M − M ∗ ( M ∗ ) − M . (7.3)With the use of (7.3) transformation of the operators reciprocal to (7.1) takes on the form: α k = X k ′ M ∗ kk ′ a k ′ − X k ′ M ∗ kk ′ a + k ′ ,α + k = X k ′ M kk ′ a + k ′ − X k ′ M kk ′ a k ′ . (7.4)According to [31], [32] matrices M and M are:( M , ) kk ′ = 12 ( ω k ω k ′ ) − / ( ω k ± ω k ′ ) (cid:20) δ ( k − k ′ ) + ( ~k ~k ′ ) f k f k ′ ω k ω k ′ ) / ( ω k ′ − ω k ± iε ) D ± ( ω k ) (cid:21) , (7.5) D ± ( ω p ) = 1 + 13 π ∞ Z f k k ω k ω k − ω p ∓ iε dk , where the superscript sign in the right-hand side of (7.5) refers to M and the subscript signto M . As a result of diagonalization quadratic form (3.4) changes to: H = ∆ E + X k ν k α + k α k . (7.6)Functional of the ground state Λ | i is chosen from the condition: α k Λ | i = 0 . (7.7)The explicit form of functional Λ is conveniently derived if we use Fock representation foroperators a k and a + k [62], [63] which associates operator a + k with some c -number ¯ a k and operator a k with operator d/d ¯ a k . Then, with the use of (7.4) condition (7.7) takes on the form: X k ′ M ∗ kk ′ dd ¯ a k ′ − X k ′ M ∗ kk ′ ¯ a k ′ ! Λ | i = 0 . (7.8)It is easy to verify by direct substitution in (7.8) that solution of equation (7.8) is written as:Λ = C exp ( X k,k ′ a + k A kk ′ a + k ′ ) , (7.9)were C is a constant. To this end it is sufficient to return in (7.9) to quantities ¯ a k instead of a + k . Matrix A satisfies the conditions: A = M ∗ ( M ∗ ) − , A = A T . (7.10)19ence, the ground state energy corresponding to functional Λ is equal to: h | Λ +0 H Λ | i = ∆ E . (7.11)In Appendix 1 we show that h | Λ +0 H Λ | i ≡ | ψ i p has the form: | ψ i p = Ce − i ~ P k ~ ~ka + k a k ~r e P k f k ( a + k − a k ) Λ | i . (7.12)Accordingly the bipolaron wave function | ψ i bp , with regard to (5.3), (5.4) is: | ψ i bp = Cψ ( r ) e − i ~ P k ~ ~ka + k a k ~R e P k f k ( a + k − a k ) Λ | i . (7.13)Formula (7.12), (7.13) imply that the wave functions of a polaron and bipolaron are delocalizedover the whole space and cannot be presented as an ansatz (1.2).From formulae (7.12), (7.13) it follows that the reason why the attempt of Lee, Low andPines [35] to investigate the polaron ground state energy over the whole range of α variationfailed was an improper choice of probe function (2.4) which lacks the multiplier correspondingto the functional Λ .However, it should be stressed that notwithstanding a radical improvement of the wavefunction achieved by introducing the multiplier Λ in Lee, Low, Pines function enables one totake account of both weak and strong coupling, the results obtained by its application are notexact. The fact that Tulub function is an ansatz follows from its properties: h | Λ +0 H Λ | i = 0 , E = h | Λ +0 H Λ | i , H Λ | i = E Λ | i . (7.14)Being an ansatz, Tulubs solution presents a solution of the polaron problem in a specific classof functions having the structure of Λ | i . That Tulub ansatz is not an exact solution of theproblem follows at least from the fact that the use of expression (3.20) alone for calculation ofthe energy, for example, in the case of weak coupling yields for E : E = − α − (cid:0) − π (cid:1) α [31].To get an exact coefficient at α in the expansion of the energy in powers of α (1.1) we shouldtake into account the contribution of Hamiltonian H , as the perturbation theory suggests [38].The fact that wave functions (7.12), (7.13) are delocalized has a lot of important conse-quences which will be discussed in Section 9. 20 Discussion of the completeness of Tulub theory.
In [64], [65] a question was raised as to whether Tulub theory [31], [32] is complete. Argumentsof [64], [65] are based on the work by Porsch and R¨oseler [37] which reproduces the results ofTulub theory. However, in the last section of their paper Porsch and R¨oseler investigate whatwill happen if the infinite integration limit in Tulub theory changes for a finite limit and thenpasses on to the infinite one. Surprisingly, it was found that in this case in parallel with cuttingof integration to phonon wave vectors in the functional of the polaron total energy one shouldaugment the latter with the addition δE P R which input will not disappear if the upper limittends to infinity [37], [65]. Relying on this result the authors of [64], [65] concluded that Tulubdid not take this addition into account and therefore his theory is incomplete.To resolve this paradox let us consider the function ∆( s ) determined by formula (3.14)(accordingly, (3.19) in continuous case). As formulae (3.14), (3.19) imply, zeros in this functioncontribute into ”polaron recoil” energy ∆ E given by (3.16) and, according to (3.15) are foundfrom the solution of the equation: 1 = 23 X k k f k ω k s − ω k . (8.1)If the cutoff is absent in the sum on the right-hand side of equation (8.1), then the solutionof this equation yields a spectrum of s values determined by frequencies ν k i lying betweenneighboring values of ω k i and ω k i +1 for all the wave vectors k i . These frequencies determine thevalue of the polaron recoil energy: ∆ E = 12 X k i ( ν k i − ω k i ) . (8.2)Let us see what happens with the contribution of frequencies ν k i into ∆ E in the region of thewave vectors k where f k vanishes but nowhere becomes exactly zero. From (8.1) it follows thatas f k →
0, solutions of equation (8.1) will tend to ω k i : ν k i → ω k i . Accordingly, the contributionof the wave vectors region into ∆ E , where f k →
0, will also tend to zero.In particular, if we introduce a certain k such that in the region k > k the values of f k are small, we will express ∆ E in the form:∆ E = 12 X k i ≤ k ( ν k i − ω k i ) , (8.3)21hich does not contain any additional terms. To draw a parallel with Tulub approach, therewe could put the upper limit k , but no additional terms would appear.For example, if in an attempt to investigate the minimum of Tulub functional (3.20), (5.7)we choose the probe function f k not containing a cutoff in the form [49]: f k = − V k exp( − k / a ( k )) ,a ( k ) = a (cid:20) (cid:18) k b − ka (cid:19)(cid:21) , (8.4)where a is a parameter of Tulub probe function (5.9), k b satisfies the condition a ≪ k b ≪ k oc , k oc = a p λ/ a → ∞ , Tulub integral q (1 /λ ) will be writtenas: q (cid:18) λ (cid:19) ≈ .
75 + 6 (cid:18) ak b (cid:19) exp (cid:18) − k b a (cid:19) . (8.5)The second term in the right-hand side of (8.5) vanishes as k b /a → ∞ and we get, as we mightexpect, Tulubs result: q (1 /λ ) ≈ . f k = 0, if k > k it has an isolated solution ν k which differs from the maximum frequency ω k by a finite value. This isolated solution leads to an additional contribution into ∆ E :∆ E = 12 X k i P R = 32 ( ν k − ω k ) , (8.6)where ν k has the meaning of ”plasma frequency” [37]. Hence, here a continuous transitionfrom the case of f k → k > k to the case of f k = 0 for k > k is absent. As is shownby direct calculation [67], of the contribution of the term with ”plasma frequency” δE P R into(8.6), even for k → ∞ , Porsch and R¨oseler theory does not transform itself into Tulub theory.In Tulub theory we choose such f k which lead to the minimum of the functional of thepolaron total energy. In particular, the choice of the probe function in the form (8.4) providesthe absence of a contribution from ”plasma frequency” into the total energy and in actualcalculations one can choose a cutoff f k without introducing any additional terms in Tulubfunctional [66], [67].To sum up, critical remarks in [64], [65] are inadequate. Their inadequacy was demonstratedin papers [66], [67] and in work [49] reproduced here. At the present time Tulub theory andthe results obtained on its basis [33], [47]-[49] give no rise to doubt.22 Consequences of the existence of translation-invariantpolarons and bipolarons. According to the results obtained, the ground state of a TI-polaron is a delocalized state of theelectron-phonon system: the probabilities of electrons occurrence at any point of the space aresimilar. The explicit form of the wave function of the ground state is presented in § 7. Boththe electron density and the amplitudes of phonon modes (corresponding to renormalized byinteraction frequencies ν q i ) are delocalized.It should be noted that according to (3.15) renormalized phonon frequencies ν q i in the caseof a TI-polaron have higher energies than non-renormalized frequencies of optical phonons and,therefore, higher energies than non-renormalized frequencies of a polaron with spontaneouslybroken symmetry [68]. This holds out a hope to find such phonon modes in experiments onlight combination scattering and optical absorption. According to [68], if a polaron (bipolaron)is bound on the Coulomb center, i.e. it forms an F -center ( F ′ -center), then all renormalizedlocal phonon frequencies ω n have lower energies than the frequency of optical phonon ω does.This fact also makes easier experimental validation of the occurrence of delocalized TI-phononmodes with ν q i > ω .The concept of a polaron potential well (formed by local phonons [68]) in which an electron islocalized, i.e. the self-trapped state is lacking in the translation-invariant theory. Accordingly,the induced polarization charge of the TI-polaron is equal to zero. Polarons lacking a localized”phonon environment” suggests that its effective mass is not very much different from that ofan electron. The ground state energy of a TI-polaron is lower than that of Pekar polaron andis given by formula (5.14) (for Pekar polaron the energy is determined by (1.3)).Hence, for zero total momentum of a polaron, there is an energy gap between the TI-polaronstate and the Pekar one (i.e. the state with broken translation invariance). The TI-polaron is astructureless particle (the results of investigations of the Pekar polaron structure are summedup in [68]).According to the translation-invariant polaron theory, the terms ”large-radius polaron”(LRP) and ”small-radius polaron” (SRP) are relative, since in both cases the electron stateis delocalized over the crystal. The difference between the LRP and SRP in the translation-invariant theory lies in the fact that for the LRP the inequality k char a < π is fulfilled, while for23he SRP k char a > π holds, where a is the lattice constant and k char is a characteristic value ofthe phonon wave vectors making the main contribution into the polaron energy. This statementis valid not only for Pekar-Froehlich polaron, but for the whole class of polarons whose couplingconstant is independent of the electron wave vector, such as Holstein polaron, for example. Forpolarons whose coupling constant depends on the electron wave vector, these criteria may nothold (as is the case with Su-Schreiffer-Heeger model, for example [69]).These properties of TI-polarons determine their physical characteristics which are qualita-tively different from those of Pekar polarons. When a crystal has minor local disruptions, theTI-polaron remains delocalized. For example, in an ionic crystal containing vacancies, delocal-ized polaron states will form F -centers only at a certain critical value of the static dielectricconstant ε c . For ε > ε c , a crystal will have delocalized TI-polarons and free vacancies. For ε = ε c , a transition from the delocalized state to that localized on vacancies (collapse of thewave function) will take place. Such behavior of translation-invariant polarons is qualitativelydifferent from that of Pekar polarons which are localized on the vacancies at any value of ε .This fact accounts for, in particular, why free Pekar polaron does not demonstrate absorption(i.e. structure), since in this case the translation-invariant polaron is realized. Absorption isobserved only when a bound Pekar polaron, i.e. F -center is formed. These statements arealso supported by a set of recent papers where Holstein polaron is considered [70]-[72]. Theapproach presented is generalized by the author to the case of Holstein polaron in [73].Notice that the physics of only free strong-coupling polarons needs to be changed. Theoverwhelming majority of results on the physics of strong-coupling polarons has been obtainedfor bound (on vacancies or lattice disruptions) polaron states of Pekar type and do not requireany revision.Taking account of translation invariance in the case of a polaron leads to a minor changein the assessment of the ground state, however leads to qualitatively different visions of theproperties of this state. In paper [32], in the section devoted to scattering of a TI-polaron, Tulubshows that as the constant of electron-phonon coupling increases up to a certain critical value,scattering of an electron on optical phonons turns to zero. Hence, when the coupling constantsexceed a critical value a polaron becomes superconducting. Though in ionic crystals the mainmechanism of electron scattering is scattering on optical phonons [74], it might appear that thecontribution of acoustic phonons should also be taken into account in this case. However, as24t follows from the law of conservation of energy and momentum, a TI-polaron will scatter onacoustic phonons only if its velocity exceeds that of sound [75].As distinct from polarons, TI-bipolarons have much greater binding energy. This leads tosome important physical consequences. In particular, when a crystal has minor local disrup-tions, a TI-bipolaron will be delocalized. Thus, in an ionic crystal with lattice vacancies, forma-tion of F ′ -centers by delocalized bipolarons will take place only at a certain critical value of thestatic dielectric constant ε c . For ε > ε c , the crystal will contain delocalized TI-bipolaronsand free vacancies. In the case of ε = ε c TI-bipolarons will pass on from the delocalized stateto that localized on vacancies, i.e. to F ′ -center. Such behavior of TI-bipolarons is qualitativelydifferent from the behavior of polarons with spontaneously broken symmetry of Pekar type [8],which are localized on the vacancies at any value of ε .The fundamental difference between TI-bipolarons and bipolarons with spontaneously bro-ken symmetry is that the former are not separable while the latter are separable. This is due tothe fact that in the case of bipolarons with spontaneously broken symmetry interaction betweenelectrons and polarization has the form: Φ ( ~r , ~r ) = F ( ~r ) + F ( ~r ). For | ~r − ~r | ≫ R , where R is the bipolaron radius, bipolaron equations separate into two independent polaron equations.This fact enables us to treat a bipolaron state with spontaneously broken symmetry as a boundstate of two polarons [8]. In the case of TI-bipolarons: Φ( ~r , ~r ) = Φ( ~r − ~r ). In this case,splitting of the bipolaron interaction functional into the functionals of interaction of individualpolarons is impossible for any | ~r − ~r | and treatment of TI-bipolarons as composite states be-comes invalid. This conclusion corresponds to modern ideas that a quantum-mechanical systemcannot be separated into independent subsystems [76].For zero total momentum of a bipolaron, TI-bipolarons, being delocalized, will be sepa-rated from those with broken translation invariance by an energy gap. As with TI-polarons,in the case when the coupling constant exceeds a certain critical value, TI-bipolarons becomesuperconducting. As is known, interpretation of the high-temperature superconductivity rely-ing on the bipolaron mechanism of Bose-condensation runs into a problem associated with agreat mass of bipolarons and, consequently low temperature of Bose-condensation. The pos-sibility of smallness of TI-bipolarons mass resolves this problem. It should be stressed thatthe above-mentioned properties of translation-invariant bipolarons impart them superconduct-ing properties even in the absence of Bose-condensation, while the great binding energy of25ipolarons substantiates the superconductivity scenario even in badly defect crystals. 10 Conclusive remarks. At the present time Tulub theory and quantitative results obtained on its basis give no rise todoubt. The quantum field theory under consideration is nonperturbative and can reproducenot only the limits of weak and strong coupling but also the case of intermediate coupling.One of the most effective methods for calculating polarons and bipolarons in the case ofintermediate coupling is integration over trajectories [77]. Unless properly modified, this ap-proach is not translation-invariant since in this method the main contribution into the energylevels is given by classical solutions (i.e. extrema points of the exponent of classical action,involved in the path integral). However, such solutions, in view of translation invariance, arenot isolated stationary points, but belong to a continuous family of classical solutions obtainedas a result of action of the translation operator on the initial classical solution. Accordingly,the stationary phase approximation is inapplicable in the translation-invariant system. In thequantum field theory some approaches are developed for restoring translation invariance. Theyare based on introducing collective coordinates into the functional integral [78]. However, theyhave not been used in the polaron theory as yet. Therefore it is not surprising that the methodof integrals over trajectories employed in the plaron theory yields a result coinciding with thesemiclassical theory of the strong coupling polaron [79].Recently in the polaron theory a powerful computational method, namely Monte-Carlotechnique has been developed [80], [81]. This procedure, being only a calculation tool, cannotreproduce the results of Tulub ansatz unless properly modified. As for Monte-Carlo diagramtechnique, the obstacle to checking Tulub ansatz in the strong coupling limit is presented bythe necessity to calculate diagrams of very high order.To sum up, Pekar ansatz (1.2) provides an original assumption of the form of the solutionwhich was confirmed in the course of numerous examinations. For nearly eighty-year historyof the polaron theory development (if dating from Landau short paper [82]) ansatz (1.2) hasbeen recognized to be an asymptotically exact solution of the polaron problem in the strongcoupling limit.Tulub ansatz ( § 7) provides another assumption of the form of the solution whose structureis determined by the form of the function Λ | i . In terms of this assumption Tulub solution is26lso asymptotically exact. Since Tulub solution yields a lower energy value for a polaron, fromthe variational standpoint, preference should be given to Tulub ansatz.Hence, polaron theory in no way can be considered to be complete. In the framework ofTulub ansatz great work is to be done to revise many concepts (such as superconductivity) incondensed matter physics. Extension of the application area of Tulub ansatz to other divisionsof the quantum field theory can lead to radical revision of many results which nowadays seemdoubtless and vice-versa. Thus, for example, non-separability of a bipolaron state in the polaronmodel of quarks [83] (the role of phonons in [83] is played by a gluon field) provides a naturalexplanation of their confinement. In paper [73] it is noted, that in TI-theory there is no needto use Higgs mechanism of spontaneous symmetry breaking to get the elementary particlesmasses.The author expresses sincere gratitude to Prof. A.V. Tulub for numerous discussions andvaluable advices. The author is also grateful to N. I. Kashirina for valuable discussions.Various aspects of the problems considered here have also been talked over with V.A. Osipov,E.A. Kochetov, S.I. Vinitsky to whom the author also renders his thanks.The work was done with the support from the RFBR, Project N 13-07-00256. Appendix 1. Hamiltonian H involved in (3.2) has the form: H = X k ( V k + f k ~ ω k )( a k + a + k )+ X k,k ′ ~k ~k ′ m f k ′ ( a + k a k a k ′ + a + k a + k ′ a k )+ 12 m X k,k ′ ~k ~k ′ a + k a + k ′ a k a k ′ , ( A . ~ ω k is given by expression (3.5). Let us apply the operator H to functional Λ (7.9).We will show that h | Λ +0 H Λ | i = 0. Indeed, the action of Λ on H terms containing an oddnumber of operators in H (i.e. the first and second terms in H ) will always contain an oddnumber of terms and mathematical expectation for these terms will tend to zero.Let us consider mathematical expectation for the last term in H : h | Λ +0 X k,k ′ ~k ~k ′ a + k a + k ′ a k a k ′ Λ | i . ( A . h | Λ +0 a + k a + k ′ a k a k ′ Λ | i represents the norm of vector a k a k ′ Λ | i and will be posi-tively defined for all k and k ′ . If we replace ~k → − ~k in (A1.2) than the whole expression willchange the sign and, therefore, (A1.2) is also equal to zero. Hence h | Λ +0 H Λ | i = 0.27 ppendix 2. Let us show that (5.4), (5.5) follow from (5.1), (5.2). To this end notice that analytical proper-ties of D ( s ) pointed out in [32] straightforwardly follow from (3.19). Indeed, the pole of D ( s )can lie only on the real axis since in view of positive definedness of ω k k f k in (3.19) equation:1 + 13 π ∞ Z ω k k f k ( ω k − s + iε )( ω k − s ) + ε dk = 0 , ( A . ε = 0. Besides, D ( s ) is a monotonously increasing function s since for s < D ′ ( s ) > 0, and as s → ∞ , D ( s ) turns to unit. Therefore D ( s ) cannot have zeros for −∞ < s < 1. Hence function ( s − D ( s ) can be presented in the form:1( s − D ( s ) = 12 πi I C + ρ ds ′ ( s ′ − s )( s ′ − D ( s ′ ) , ( A . A . 2) is shown in Fig.2. The integrandfunction in ( A . 2) has a pole for s ′ = 1 and a section from s ′ = 1 to s ′ → ∞ . Performingintegration in ( A . 2) with respect to the upper and bottom sides of the crosscut we will getintegral equation (5.5). Appendix 3. Let us perform a detailed calculation of the quantity ∆ E (5.7) with the use of probe function(5.9).To this end, to calculate the real part of D ( ω p ) involved in (5.7) we use Sokhotskys formula:1 ω k − ω p − iε = P ω k − ω p + iπδ ( ω k − ω p ) , Re D ( ω p ) = 1 + 13 π ∞ Z f k k P ω k ω k − ω p dk . It is convenient to present Re D in the form:Re D = 1 + I + I ,I = 13 π ∞ Z f k k dkω k + ω p , I = P ω p π ∞ Z f k k dk ( ω k − ω p )( ω k + ω p ) . f k in the form of (5.9) into these expressions we present I as: I = 8 α √ π ∞ Z e − k /a dk − α ( p + 4)3 √ π ∞ Z e − k /a k + p + 4 dk . Assuming k/a = ˜ k in the strong coupling limit ( a → ∞ ) we write for I : I = 8 αa √ π √ π − π pe ˜ p − √ π ˜ p Z e − t dt . Accordingly, I takes on the form: I = P αω p π √ ∞ Z e − k /a k dk ( ω k − ω p )( ω k + ω p ) . This integral can be expressed as: I = I + I , where: I = 16 αω p π √ (cid:18) − ω p − p + 2 (cid:19) ∞ Z e − k /a k + p + 4 dk ,I = 16 αω p ( ω p − π √ p + 2) P ∞ Z e − k /a k − p dk . The integrals involved in I and I will be: ∞ Z e − k /a k + p + 4 dk = 1 a − √ π ˜ p Z e − t dt π e ˜ p ˜ p , P ∞ Z e − k /a k − p dk = − √ πa e − ˜ p ˜ p ˜ p Z e t dt . As a result, I has the form: I = 23 αa ˜ p √ e ˜ p − √ π ˜ p Z e − t dt − αa ˜ p √ π e − ˜ p ˜ p Z e t dt . Finally, Re D will be written as:Re D = 1 + 4 αa √ π − ˜ pe ˜ p ∞ Z ˜ p e − t dt − ˜ pe − ˜ p ˜ p Z e t dt . D bySokhotskys formula, we get:Im D = 13 π ∞ Z f k k ω k δ ( ω k − ω p ) dk = 16 π f p p . As a result, | D ( ω k ) | is expressed as: | D | = (Re D ) + (Im D ) = 29 α a e − p ˜ p + 82 π − ˜ p ∞ Z ˜ p e − t dt − ˜ pe − ˜ p ˜ p Z e t dt . The first term in formula (5.7) is easily calculated to be:14 π ∞ Z k f k (1 + Q ) dk = 316 a . In calculating the second term in (5.7) we will separate out integral I p in it: I p = ∞ Z e − k /a k ( ω k ω p + ω k ( ω k + ω p ) + 1)( ω k + ω p ) dk . As a → ∞ , it is equal to: I p = a √ π − ˜ p e ˜ p ∞ Z ˜ p e − t dt (2 + 4˜ p ) + 2˜ p = a √ π − Ω(˜ p )) , where: Ω(˜ p ) = 2˜ p (1 + 2˜ p )˜ pe ˜ p ∞ Z ˜ p e − t dt − ˜ p , which corresponds to the expression Ω( y ) in (5.12) As a result, the second term in formula (5.7)will be: 112 π πα √ ∞ Z I p p f p ω p ( ω p − | D ( ω p ) | dp . As a → ∞ , this expression takes on the form: α a π √ π ∞ Z (1 − Ω(˜ p )) e − ˜ p | D ( ω p ) | d ˜ p = 316 a q , where q = q (0) is given by expression (5.12). Hence, finally for ∆ E (5.7) we get:∆ E = 316 a (1 + q ) , which corresponds to the first term in the right-hand side of (5.11).30 eferences [1] Pekar S.I. 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