Analytic Peculiarities of the Polaron Mass Operator in the First Two Orders of the Coupling Constant

The exact analytic expression for the polaron mass operator in the second order of the coupling constant is established for the rst time. It holds in the whole energy scale. The peculiarities of the mass operator are analysed. The renormalized polaron energy spectrum is obtained and analysed in this approximation.


Introduction
Polaron as a quasiparticle was studied by Feynman, Landau, Pekar, Pines, Rashba 1] and others.Nevertheless, a lot of physical and mathematical problems have not been solved yet.As far as the physics of polaron is concerned, the most important problem concerns the existence of bound complexes 2].The mathematical problems of the correct calculation of the electron and phonon spectra in the region of high energies, quasimomentum and electron-phonon binding are tightly connected with physical ones.There is an important question about the ranges of diagram technique convergence for the polaron Green function.
The actual problem is the analytical calculation of the polaron mass operator (MO) in the higher orders of the binding constants, because the exact numerical results are obtained only in one-phonon approximation 1-3] while the contribution of any higher order diagrams is done, as a rule, only evaluatively.
In this paper the exact analytical calculation of the polaron MO is performed for the rst two orders of the coupling constant in the arbitrary range of energies at T = 0 K. On this basis the dependence of the renormalized energy of the polaron zone bottom on the coupling constant is derived.Some peculiarities of the MO behaviour in the region of energies higher than the phonon creation threshhold are studied.

Frohlich Hamiltonian. Analytic expression for M 2 (k; !)
It is known that the polaron is described by the Frohlich Hamiltonian where E k = E + ~2k 2 2m ; q = !; '(q) = q are the dispersion laws of the electron and optical phonons, respectively, and their binding function is expressed within the Frohlich constant = e 2 ~ 1 1 ? 1 0 r m 2 : (2) At T = 0 K the renormalization of the electron-phonon spectrum is de ned by the polaron Green function Fourier image poles.It is given by the Dyson equation with the total MO given by an in nite range of diagrams 4].
The expression for the rst diagram corresponding to one-phonon approximation reads: M 2 (k; ! 0) = X q ' 2 (q) ! ?E k+q ?: Introducing the convenient dimensionless quantities m 2 = M 2 ; = !?E ; K = p2m k; Q = p2m q (5)   one can obtain Integration in (6) can be performed exactly and gives the known 2,3] result: m 2 (K; 0 ) = ?K Analytical expressions for one-phonon MO give an opportunity to study the renormalized polaron spectrum in this approximation.
3. Analytical calculation of M a 4 (k; !) MO of the fourth order in the powers of the binding function which corresponds to the diagrams without crossing the phonon lines 1,4] is de ned as M a 4 (k; !) = X q1;q2 ' 2 (q 1 )' 2 (q 2 ) (! ?" k+q1 ? ) 2 (! ?" k+q1+q2 ? 2 ) ; (10) according to the rules of a diagram technique.Performing in (10) a transition from summation to integration with taking into account the dispersion laws, binding functions and by introducing the dimensionless quantities, (10) one can obtain (11) Integration in ( 11) is performed in the spherical coordinate system.Herein, in the internal integrals 2 is the angle between vectors K + Q 1 and Q 2 and in the external integrals 1 is the angle between K and Q 1 .Insertion of cos 1 = x 1 and cos 2 = x 2 and integration over the angles ' 1 and ' 2 give the following result: Two internal integrals in (12) are taken in a general case and the results read: (13) As at K 6 = 0 integrating in (13) cannot be performed exactly, then m a 4 ( ; K = 0) can be calculated.From (13) it is clear that in the < 1 region m a 4 ( < 1) is a real function of variable and is given by the expression : Using 1 and taking into account the known integral 5] I(q; p) = 1 Z 0 arctan (qx) dx x(p 2 + x 2 ) = 2p 2 ln(1 + qp); (p > 0; q > 0) (16) from ( 16) the expression for m a 4 ( < 1) containing only the real part can be obtained m a 4 ( < 1) = ? 2 (1 ? ) Finally, formulas (17)-( 19) completely de ne m a 4 ( ) as a complex function of the real dimensionless energy in the whole region of its variation.

Figures 1a, 1b
show the frequency dependences of the MO terms (real and imaginary) calculated by the formulas obtained in the previous sections at = 1.From the gures one can see: , respectively.
In the energy region 6 1 all the three terms of MO (m 2 ; m a As far as near this point m 2 ?p 1? then lim !1+" m a 4 ( ) + m b 4 ( ) m 2 ( ) ? lim !1?" 1 ?= ?" : (32) So, in the region 6 1 MO diagrams of order 2 have a bigger discrepancy than one-phonon diagrams.Therefore, one has to consider more than the rst three diagrams in the total MO.The region where the account of the rst three MO diagrams is not enough can be evaluated from the condition 0 < < 1 where 0 is the solution of the equation m 2 ( 0 ) = m a 4 ( 0 ) + m b 4 ( 0 ): (33) satisfactory results only at 6 0:4.Taking into account both diagrams of the second order in the region ( 6 0:9) gives even better results than all other theories 1].In the region > 1 these terms make the magnitude of the renormalized energy obtained in one-phonon approximation much more exact.The obtained peculiarities of the MO behaviour in the vicinity of = 1 show an important role of MO diagrams of a higher order and not only those having \dangerous crossings " 2].The analysis of this problem is to be made in future taking into account partially summed in nite series of MO diagrams.

4 ; m b 4 )
are real and negative independently of the magnitude.Herein, jm a 4 j < jm b 4 j and only when reaches value 1 from the left-hand side, the relation holds m

Figure 2 .
Figure 2. Dependence of the polaron zone bottom shift on .