Second Order Corrections to the Variational Approximation to Frolich's Polaron Model Citation Valainis, John Reinhold (1983) Second Order Corrections to the Variational Approximation to Frolich's Polaron Model. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/e8k0-8q33. https://resolver.caltech.edu/CaltechTHESIS:09082025-213518247 Abstract Using Feynman's path integral variational principle with a general quadratic trial action, we obtain equations for the absorption function of Frolich's polaron model. We evaluate numerically this absorption function in several cases. To test the accuracy of the variational absorption function, we develop formulas for the second order corrections to the absorption function and evaluate these numerically. Rather than evaluating the correction directly in the double path integral formalism, we make analytic continuations in time which reduce the amount of labor involved in deriving the expressions for the corrections. The method of analytic continuation in time is generalized in such a way as to allow application of the variational principle to nonlinear transport problems and time dependent problems. Finally, we present the variational equations and the second order corrections to them for a somewhat more realistic model of an electron in a crystal. Item Type: Thesis (Dissertation (Ph.D.)) Subject Keywords: (Physics) Degree Grantor: California Institute of Technology Division: Physics, Mathematics and Astronomy Major Option: Physics Thesis Availability: Public (worldwide access) Research Advisor(s): Feynman, Richard Phillips Thesis Committee: Feynman, Richard Phillips (chair) Frautschi, Steven C. Goodstein, David L. Hitlin, David G. Defense Date: 28 June 1982 Record Number: CaltechTHESIS:09082025-213518247 Persistent URL: https://resolver.caltech.edu/CaltechTHESIS:09082025-213518247 DOI: 10.7907/e8k0-8q33 ORCID: Author ORCID Valainis, John Reinhold 0000-0002-9805-815X Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 17673 Collection: CaltechTHESIS Deposited By: Benjamin Perez Deposited On: 12 Sep 2025 10:22 Last Modified: 12 Sep 2025 10:22 Thesis Files PDF - Final Version See Usage Policy. 14MB Repository Staff Only: item control page

SECOND ORDER CORRECTIONS TO TI-IE VARIATIONAL APPROXIMATION TO FROI.JCH'S POLARON MODEL Thesis by John Reinhold Valainis In Partial Fnlfillm P.Ilt of the Requirements for the Degree of Doctor of Philosophy California Institute of Technology Pasadena, California 1983 (Submitted June 28 .1982) -iiAcknowle~rrm cnts 1 t hank Richard Feynman for suggesting the topic of this thesis and for his advice while the work was in progress . J also thank him for the many things I have learned from him and for the perspective on the practice of science which J have gained . J am grateful to Marshall Dixon for encouraging my interest in physics and many other disciplines and for providing my first education in most areas of physics. Finally I want to ex-press my gratitude to my friends and family and especially my wife,Lenora,for their company,encouragernent,and support during my stay at Caltech. --ii,i - Abslract Using Feynman's path integral variational principle v.ith a general quadratic trial action, we obtain equations for tbe absorption function of Frolic h's polaron model. We evaluate numerically this absorption function m several cases. To test the accuracy of the variational absorption function, we develop formulas for the second order corrections to the absorption function and evaluate these numerically. Rather than evaluating the correction directly in the double path integral formalism, we make analytic continuations in time which reduce the amount of labor involved in deriving the expressions for the corrections . The method of analytic continuation in time is generalized in such a way as to allow application of the variational principle to norilinear transport problems and time dependent problems. Finally, we present the variational equations and the second order corrections to them for a somewhat more realistic model of an electron in a crystal. -·.v- Ta.bl e or Contents Ack:noVriedgrnenls ii Abslracl iii Table of Conlenls iv Chapter 1. Inlroduction 1.1 Hislory 1 1.2 Frolich's Model 2 Chapter 2. Formulation of tbe Problem as a Path Integral 2.1 Response Function 9 2.2 Response from Path Jntegral : General Density }.fatrix 13 2.3 Response from Path Integral: Thermal Density }latrix 15 Chapter 3. Development of tbe Approximation 3.1 Variational Melbod and Perturbation Expy.,ri.sion 20 3.2 The Variational Equations 27 3.3 Second Order Calculation 31 Cb.apter 4. Numerical Results 4 .1 Solution of the Variational Equations 41 4.2 Second Order Correction 50 Chapler 5. Further Developments 5.1 Possibility of Improvement of the Method 61 5.2 More General Metbod of Analytic Continus.lion 63 5.3 }fore ReaUstic Model 70 Cb.apter 6. References Conclusion 84 -11. INTRODUCI10N 1.1. History The problem of getting reliable numbers from field theories when _perturbative methods are inapplicable is of immense interest. A successful exa~ple or such a calculation is Feynman's [1] application of his path integral variational principle to find the energy and effective mass of an electron in a polarizable crystal. The calculation was actually done on an idealized model of a polaron • developed by Frolich [2]. This model has become a standard on which to try various methods of computation. We discuss the details of this model below. Others extended Feynman's calculation to find the polaron free energy at nonzero temperatures. Feynman,Hellworth,Iddings,and Platzman [ 4] (referred to as FffiP in the following) used path integral techni.,#es to obtain an expression for the electron's response to weak external fields (that is, it's impedance function) at all frequencies,temperatures, and coupling strengths. A check on the accuracy of Feynman's original calculation was made by Marshall and Mills [5] who computed the second order corrections to the polaron's energy. They found that the correction was never more ~han two _percent at any value of the coupling. The purpose of this thesis is to improve the FffiP impedance calculation by using a more general trial action in the variational method and to obtain an estimate of the reliability of the variational impedance function by calculating the second order corrections to it. The variational equations we obtain for the impedance function have recently been obtained by others by a nonvariational method [6]. • Polaron is the term used for the quasipart.ic1e consisting of an electron w:ith its associated clou d of virtual phonons. -21.2. Fr-olich's Model Frolich's model takes the electron part of the Lagrangian to be just ½mi-2 ·where m is an effective mass and r is the electron's velocity vector. This approx: imation has some justification for slow electrons in a crystal with sufficient sym: metry. As far as the path integral approach is concerned, the electron kinetic , energy could be generalized to say ~ ~11Ltj.xixj where ::i; is a component of the \.J , electron velocity and Tn;.j is an effective mass tensor. The effect of the crystal lattice on the electron could be better accounted for by including a periodic : potential V(r) in the electron Lagrangian. However,this would make it more difl'iculVto find a calculationally tractable trial action for the variational method. The main idealization which Frolich's model makes about the lattice portion of the ~rangian is that the harmonic approximation is ad/ate. This is impor- tant for the path integral approach since it permits the lattice variables to be integrated away exactly. However.applying the variational inequality twice,we can approximately integrate the lattice variables and get an upper bound on the . free energy even in the anharmonic case. We will say more about this later. For : now;, we •stay with the Frolich model. The electron interacts with the lattice via its electric field which attracts ! the positive ions and repels the negative ions . The electron interacts most : strongly with lattice modes in which nearby ions of opposite sign move in oppo: site directions (optical modes). The Frolich model assumes that the electron ; interacts only with the optical phonons. To simplify further, the model assumes : that the crystal has two ions per unit cell and that the optical phonon frequency , does not vary with the wavevector k. This wavevector independence of the phonon· frequency is equivalent to no coupling between different unit cells. Let ou be •the ·deviation from equilibrium of the vector from the negative to the positive ion -3- r in a unit cell. The kinetic ene.r;-gy of the optical modes per unit volume of the crystal is then ½nM[ d:tu Here n is the number density of unit cells and M is the reduced mass of the ions in a unit cell. For convenience we introduce rescaled phonon variables by w=VnlJ:ou.. w can be separated into its longitudinal and transverse pieces: w=w1 +w,. Here V·w, =O and Vxw1 =O. We shall shortly see that only the longitudinal optical :phonons (LO phonons) interact with the electron. Since we assume dispersionless LO phonons, the LO phonon Lagrangian can now be written as .z f.:..2 -w,-w, ?~ 2] d3 L1..0 -_ •1 f l"' r (1.2.1) Here w1 is the frequency of the longitudinal optical phonons . It now remains to write down the electron-phonon interaction. Frolich's model divides the polarization of the crystal into the sum of two pieces P1 and P,. P 1 is due to distortions of the lattice and is proportional to w. Pi is due to distortion of the ions themselves. It is assumed that frequencies of interest are much less than the excitation fre_q uencies of the ions. In this case we can write & -1 P, = ~rr E where e.., is the dielectric constant of the crystal with the ions held fixed and Eis the total electric field. From V· (E+ 4rrP)=O and the expression for Pi we now obtain (1.2.2) Since P1 is proportional to w,we have (1.2.3) Here r,o is the electr ostatic potential generated by a given deformation field w -4- and C is a constant to be determined. For a particle in the crystal at r with charge q the interaction term in the Lagrangian Vti.11 be -q 1(r) . To determine C we consider two heavy charged particles in the crystal. Their interaction energy is given by (1.2.4) Here e0 is the static dielectric constant of the crystal. If the ions are immobilized, the interaction energy of the two particles is (1.2.5) If we allow the ions to move, the effect of the optical phonons must be to restore (l.2.5) to (1.2.4) . Thus if we write down the path integral for this system and integrate away the phonon variables, we should recover (l.2.4) as the interaction term between the two particles. The action for several charged particles in a crystal is in Frolich's model + ... S = j(Lpan+Lw-Liqi.rp(ri,))dt - (1.2.6) i where (1.2.7) In (l.2 .6) rp is linear in Wt, as can be seen from (l.2.3). To find the effective action for the particles alone.the LO phonon variables must be integrated away. (units are such that '/r= 1) e iSeJ/ = J D(phonon variables)eiS (1 .2.8) -5- To carry this out.define first new phonon variables u (k) by _ j (Zrr) dk3 tkr k l'"kf u (k) 3 e w, (r ) - (1.2.9) Note that u •(k) = -u (-k) . Also let (1.2.10) (1.2.3) gives -k2s"o(k) = 4rri0u(k). (1.2.6) now becomes +J.. I 1 S = _ ½)art+ 2 J (dk3 [ •( ) ( ) '"f dk3 u(k) fh'} 2rr) u k u k -r..>ru k u k - 4rrqCi7' (Zrr) Ikl e :> __ •( ) 3 ( )] 3 (1.2.11) Now use the path integral (1.2.12) to find that . I 47r2,...:, J d'-q eflr:·(ri(t)--r,(s)) fDueiSocexpiflpartdt--t.,--Lif dtdsf ( K)3 c.>i ;.,; 2rr k:2 - i'-li!t-=.sl l - (1.2.13) . Usmg dk3 eik·r J (2rr)s k2 1 we have 4rrl rl (1.2.14) For a slow heavy particle r(t) is nearly a constant for times of the order of c.>i- 1. -6- If such particles labeled by 1 and 2 are not too close to each other, then by using Je--ir.il• I= ~we find from s.11 .that their interaction energy is CJ 'l, (1.2.15) Comparing to (1.2.4) we have 1 - -::::: 1 4rrc2 - (1.2.16) ~ This determines C. For further work it will be convenient to choose the units of time and distance in such a way that m =1 and G.>t = 1. Following Frolich, we define the dimensionless interaction strength or coupling constant (1.2.17) The effective action for one particle in the lattice can now be written as 1 Sell= 2 J·i:2dt+ -v'8 ia. ff dtdslr(t)-r(s)I e--ilt-sl (1.2.18) The path integrals used to obtain S 011 are real time path integrals. They are matrix elements of e -iHt where H is the Hamiltonian of the system. One may also consider imaginary time path integrals for the matrix elements of e --rH. These are related to the real time path integrals by analytic continuation. It is these imaginary time path integrals to which Feynman's variational principle applies . = The free energy F of a system at temperature 13- 1 is given by e - fJF Tr e -fJH. If q denotes the set of all system coordinates, then {1.2.19) -7- Thus e -pF may be evaluated by imaginary time path integrals over paths which close on themselves after time /3. The "action" which appears in these integrals is the time integral of the classical Hamiltonian of the system. Thus if H= % 2 I V(r),then (1.2.20) The imaginary time action is obtained from the real time action by the replacement oft by -it. Thus (1.2.7),the particle Lagrangian, becomes• (1.2 .21) The LO phonon Lagrangian (1.2.1) becomes (1.2.22) The interaction term is (1.2.23) Finally.the action (1.2.11) becomes (1.2.24) Now using the path integral •We shall use the italic L for real time Lagrangians and the roman L for the correspondmg imaginary time Legrangiens. The same convention will also apply to actions. - B- (1.2.25) where e= l ~ (1.2.26) the u(k)'s may be integrated away and we obtain (after rescaling r and T as before) for one particle ( 1.2.28) We have dropped the subscript from S811 . The action (1.2.28) will be the starting point for our further discussion and calculations. -'.9 2. FURMULATION OF THE PROBLEM AS A PATH INTEGRAL 2.1. Response Function We can calculate such things as the absorption spectrum.mobility and effective mass for an electron in a polar crystal if we know the electron's response to a weak external force. We take the external field to be constant in space but variable in time. In this chapter we will present the details , of this idea. We will also review how FHIP calculate the response function. Finally, we will present a method which allows a less laborious calculation of the response (when compared to the FHIP method) in the case that the crystal is initially in thermal equilibrium. Weakness of the applied field implies that we can expand the response in powers of the field and retain only the terms which are linear in the field. This linearity in turn implies that we need only consider the effect of a brief pulse. Frolich's polaron model and in particular the action given in (1.2 .28) are invariant under rotations. The substitution r(t) ➔ Rr(t) makes no change in the value of S in (1.2.28). This implies that correlations such as <.:i; (T)x; (a)> are zero for i~j and that there is no response in they direction to an applied force in the x direction. Thus,in this chapter, we look at the response in the x direction to an impulse in the x direction. In chapter five we will give the equations which apply to an anisotropic crystal. Letting the applied force be -y(t) = 7 0 o(t),a term ,(t)x is added to the Hamiltonian. The effect of the pulse is to multiply the wavefunction of the system at a time just before t = 0 by e 'i?o:z: to produce the wavefunction at a time just after t = 0. Subsequently the wavefunction evolves in accordance with the unperturbed system Hamiltonian H. If the system is initially in the state I--Iri> with probability A,then at a time t >0 we have for the expected position of the electron - 10 - <x (t )> = ~Pi <'1', Ie -i-ro= e i.Ht x e-illt e i-ro= I'1',> (2.1.1) i Let x ( t) = e ilit x e -illt and expand (2.1.1 ) to first order in ')'o <x(t)> = I;Pt: <'1',lx(t)1'1r,>+i10I;<'1-tl [x(t),x(O)] 1'1-i>+ • • • (2.1.2) i i We assume that there are no net currents flowing in the unperturbed system. In this case I;Pt: <'1-, Ix (t) I '1',> = constant, which can be taken as zero. Now we define a response function R(t) = ~(t) I; p,<'1ri l[x(t),x(O)]l'1r,> (2.1.3) i ~(t) 'is 1 for t >0 and O for t <0. The linear response to a general time varying force 1(t) is now given by + ... <x(t)> =if R(t-s)-y(s)ds {2.1.4) Now we investigate some properties of R(t). From (2.1.3) we see that R(t) is pure imaginary. Consider the Fourier transform R ("),.,) of R ( t). Using the causality condition.,R(t) =0 fort <0,we have - R(>..) = f ei>.t R(t )dt (2.1.5) 0 If our system is stable in the presence of a weak external force -y(t),then R(t) will not rise exponentially as t ➔ 00 . In this case)?("),.) will be analytic for all Jm")\>0 since the integral (2.1.5) then converges absolutely. Using this information and Cauchy's theorem one finds that R(>-..) may be represented as (2.1.6) - 11 We also note that since R'•(t) -=-R(t) we have from (2 .1.5) for real>-.. that f?•(>-..) = -R(->-..) and Re.R (>-) = -Re R ( -A) ,A real (2.1. 7) This allows us to write (2. 1.8) In the following we denote 2r..>ReR((',..)) bys (r..>). To get a better idea of the significance of R ( t) and ~ J '{t,><'11, I = 1 into (2 :1.3). Letting E;, be the eigenvalue of I R(>-..) insert '11,> and defining i R(t) = -Zi ~Pi 1<'11, Jx l'11n>l 2 sin((.Jnit)4(f) (2.1.9) i,n Taking the Fourier transform of (2.1.9) yields (we put in a convergence factor e-d and later let t: ➔ 0) (2.1.10) -- 'L. ~ I ,T, I 1,T, 2 2(',..)ni ~Pi <Ti X Tn> ('+ '0)2- 2. J t.,n We see from (2.1.10) using ReR (A) = s "- 1 . X+'L 0 'L (',..)ni - P ~rro(x) that X J~> =·~Pi I<'Yi Ix I'i'n > I (rro (A 2 -(',..)n( )-m5(.>-. +C,Jni >) (2.1.11) 1i,n Now apply a weak external force -y(t) = ToCos(wt) to the x coordinate of the system and ask at what rate the system absorbs energy. The answer is given by - 12 first order time dependent perturbation theory. The transition rate from a state I '1'i > to a state I 'Yn > is given by • (2.1.12) For each transition, the energy of the system changes by c.,m. Thus c.,ni Wn ...i(c.,) gives the energy absorbed per unit time at frequency c., by the system in transitions from l'1ri> to l'1rn>- Averaging over initial states and summing over final states, the total energy absorption rate at c.> is ~Pi c.,m Wn---i or n,t (2.1.13) Thuss (CiJ) is proportional to the energy absorption rate, and (2.1.8),(2.1.5) show that the entire respo~se function can be expressed in terrt<s" of it. Therefore we can concentrate on finding s(c.,). Next we derive a sum rule which will be important later. First observe that J s (c.,)dCiJ = 0 Choose units so that the 21T~ Pi I <'Yi IX I '1rn i,n mass associated > I 2r..>ni with (2.1.14) coordinate x is 1. 2 H = T+ · · · ,where p is the momentum conjugate to x. We suppose that the rest of H commutes with x, as it does for our case. This implies that [[x ,H],x] = 1. Forming ~Pi <'Yi I[[x ,H],x] I'1ri> and inserting ~ I'1tn><'1rn I = 1 n i gives the result 2~A I <'1',: Ix l'1rn> l2 c.>ni = 1 i Comparing (2.1.15) with (2.1 .14) gives (2.1.15) - 13 - (2.1.16) If we examine (2.1.8) in the light of (2.1.16) we see that as ;\ ➔ co (2.1.17) This says that at sufficiently high frequencies the electron responds as though it were free. Time is needed for the electron to interact appreciably with the phonons. Also.if we knew that R(>,..) had the behavior given in (2.1.17), then (2.1.16) would follow. 2.2. Response from Path Integral.:General Density Matrix R(t),as given in (2.1.3),can be rewritten as R(t) = ~Pi <'t', leilit x e--illt x l't'i>- complex conjugate (2.2.1) i Take Ta. «0, Tb »0. The first term of (2.2.1) can be written as (2.2.2) Let q represent the coordinates of the system including x. Now (2.2.3) t, where if L is the Lagrangian associated with H,then S[t1 ,td = f Ldt and the ,, path integral is over all paths q (t) which begin at qi q1 = q (ti) and end at =q (t1 ). The notation Dq1i. means that we integrate over all paths with ends fixed at q1 and at q-;,. On the other hand, taking the complex conjugate of (2 .2.3) gives - 14- (2.2.4) Note that the paths still run from CJi. at time ti. ·to q1 at time t 1 . Finally (2.2.5) Combining (2.2.3),(2.2.4),(2.2.5) and letting P =I;i. l-lri>Pi <-lri. I (2.2.6) gives (2.2.7) Dq'ba' is associated with the primed action. The paths associa-ted with the primed action run from CJa· at Ta to CJb at Tb while the paths of the unprimed action run from CJa at Ta to qb at Tb. Noting that ix(t) =[o,,~t) ei.f-r:z: t= ,we see that 0 ( 2.2. 7) is obtainable from (2.2.8) When the system is in thermal equilibrium at temperature /r1. e -{JH p = Tr e-/JH (2.2.9) Going back to (2.2.2), consider the factor e -t.EIT~ p e -mra _If the density matrix was p at time Ta, then this factor is the density matrix at time t = 0. However,as we let Ta ➔ - 00 , we can just as well start with the iienslty matrix of the noninteracting system (a = 0). We expect that the electron comes into equilibrium with the effectively infinite lattice in a finite time so that ~y t '= 0 the d ensity matrix is exactly that of the int eracting system (a#0) in thermal equilibr ium at the - 15 - original lattice temperature . This replac ement allows the path integrals over the phonon variables to be carried out in a simple fashion. The approach described in this section is used by FHIP to calculate the response function However,in the special case that the system is initially in thermal equilibrium.it is possible to do the calculation more simply. We do it by first calculating some other functions (fj(X) and f (T) below) and working back from them to s ((.) ). This method is described in the next section. 2.3. Response from Path Integral:Thermal Density Matrix When the system is initially in thermal equilibrium.so that the density matrix is given by (2.2.9),we need to compute (2.3.1) Note that this is the analytic continuation to T g ( T) = Tr =it of f e -(ft---r)H X e ---rH X ] Tr e -pH (2.3.2) Equation (2.3.2) may be reexpressed in terms of path integrals as g(T) = <x(T)x(O)> = jDqx(T)x(O)e - s Dq e-s J (2.3.3) p Here S = f LdT where Lis the imaginary time Lagrangian of the system and the 0 path integration is over close d paths. For re al T ands in the range O to {3 we have <x(T)x(s)> = g(IT-s J). From (2.3.2) we can e asily verify that for all complex T (2.3.4) - 16 Also since g (,) is real on the real axis g (,•) = g •(T) . From this and (2 .3.4) with T replaced by -,, (2.3.5) On the interval 0 to {J we can expand g (,) in terms of functions e -£XciT with Ao.= 2 ;a and a= ... ,-2,-1,0, 1,2,.... It is important to note that while g(,) = p- 1 ~{j(Aa.) e -iX.XT for 0~%/J, for -{J~,~{J the sum gives g ( 1,1). The a Fourier sum is periodic in, with period {J while g (T) does not have this property. From (2 .3.2) we obtain by inserting~ I'fi><'ft I = 1 where necessary ( g(T)::: e.SF~e-.SEne--T"'m l<'f,lxl'fn>l 2 (2.3.6) i.n Multiplying bye tX.XT and integrating over, from Oto {3,we have (2 .3.7) .s ·x.x where §'(A.a) = f g (,)e,. d, This may be re expressed as 0 {f(X ) = 1"" s (C,)) a 0 X 2+,,2 a "" de.> n (2 .3.8) where s(G.>) is given by (2.1.11). The analytic continuation of (2.3 .7) to general>.. which behaves as x-2 whenX ➔ 00 is (2 .3.9) Comparing this to (2 .1.8) we see that {j(---iX) = R(X) and that s (G.>) may be obtained by (as e ➔ 0) - 17 - s ( c.,) =lrn2c.,g (t:-i c.>) (2.3.10) Define now a function/ (T).~r-:;;,8, by f ( T) = p-l ~ f (>..a) e -i~T {2.3.11) a In the following we shall obtain the solution to our problem as a series for f (T). This will allow us to find f (>..a). However, to find s (>..) or the response func- tion R(t),we need to know j for general values of>... §"(>..) will then ·b e given ·by §(>..) = (>.. 2 +f (>..))- 1 . The problem is that specifying a function at a discrete set of points does not uniquely determine the analytic continuation of the function. However.the condition that 9(>..) ➔ >..- 2 as l>..I ➔ o:> for all co~ex A is sufficient to fix the physically correct analytic continuation. We now use this condition to continue f (>..a) to f (>..). We assume that we are given f (T) in the complex T plane (at least for ReT>O). If we start with .-. =J e ft f (A.a) i~T f (T)dT (2.3.12) 0 then an obvious attempt at j (>..) is (2.3.13) This has the problem that for some directions in the complex X plane f (>..) rises ex-ponentially as IX I ➔ 00 • !J(X) then falls more rapidly than x-2 . It is possible to see how to correct the problem by changing the path of integration in (2.3.13). Suppose that ReX>O and consider the following contour in the T plane : lS _ I Ii~,, +,8 - 18AB s ➔ 00 ,the piece from is to is +{3 goes to zero. We can now write .. p f (>...)=if e->-t f (it)dt -i ei>-.fl f e-M f ((3+it)dt 0 (2.3.14) 0 So far this f (>...) is the same as in (2.3.12). However,we have isolated the bad behavior. It is given by the e<>-.fl in the second term of (2 .3.14). For X = ~ this factor is just 1. lf we remove this factor, the new f (>...) will still have the correct values at X == Xa and it will have the correct behavior as X goes to infinity. For Rer..<0 we must construct a different expression.but we shall not need this. On the real axis/ (T) is real and/ (/3-T) = / (T). The first property implies f (T•) = f \r). The second property can be extended to the whole complex plane for f (T) analytic. Thus / (T) shares properties (2.3.4) and (2.3.5) with g (T). In particular/ ({3+it) = f •(it) for real t. Thus p f (r..) = -2 f e->-t Imf (it )dt (2.3 .15) 0 We can now summarize our calculational procedure. First we shall obtain f ( T) as a series (2.3.16) We shall obtain f O from a variational calculation. In this case f 1 will vanish and f 2 will be the result of calculating the second order perturbation about the trial action. It will be possible to continue the f's to complex T (and therefore to find f (>...) from (2.3 .15)). From (2.3.10) we are actually interested in X = E:-1-(;) for(;) real. Using (2.3.15) . f(E:-i(;)) = -2/ e - tt+ic.,t Imf (it) 0 (2.3.17) - 19 Finally, we computes (c.>) by (2.3.18) - 20 - 3. DEVELOPMENT OF THE APPROXIIl.ATION 3.1. Variational Method and Perturbation Expansion The path integral for g(-r) in (2.3.3) cannot be performed for the polaron action S given by (1.2.28). Thus we shall imitate S by a trial action S0 and use Feynman's variational principle to pick out the best S0 If A is a functional of the path r(t) of the system, then averages <A> 0 are defined by f DrA e -so <A>o Define S 1 = f Dre -s 0 (3 .1.1) = S-S0 and note that JDre-s = <e-81 >0 JDre - 50. The variational ine- quality states that• (3.1.2) The best S0 is now taken to be the one which maximizes the right hand side of (3.1.2). In terms of free energies F 0 and F defined by e -{JF =JD re -s e -{JFO =J Dre -s 0 •This follows from the fact that if X is a random variable wit.ri probability distribution P(x) and/ (x) is a..'l.y function which lies above any of its tangent lines (a convex :hLTJ.ction),then f P(x)f (x)dx = <f>'?:.f(<x>) To see this,note that f (x )2.f (xo)+ f '(xo)(x-xo) for any Xo. Averaging this over the distribution P(x) gives </ >?:./ (xo)+ f '(xo)( <x >-xo)- Let Xo = <x > to :find <f >2.f ( <x >) Since e;z: always lies above its tangent lines, <ez >2.e <z>_ FinaDy note that is a probabDity d.is:rfou tio~ for the values of S1. (3.1.3) - 21 we can rewrite (3 .1.2) as (3 .1. 4) In the limit of {3-+ 00 this reduces to an upper bound for the ground state energy of the system. The path integrals involving the action S given by (1.2.28) diverge as the volume of the system becomes infinite. This is because S is invariant under the transformation r(T) ➔ r(t)+Ar. To avoid this problem we impose a weak harmonic oscillator potential tn r2 2 on the electron. Although the free energies F and F 0 diverge as O ➔ O, we can obtain finite results by subtracting from each of them the free energy of an electron which is in a harmonic oscillator potential ½o 2 r but which is not interacting with the lattice . In this way we o.btain as 0-+0 !::,.F and !::,.F0 which are exact and approximate free energies of a polaron relative to an electron which does not interact with the optical phonons. The variational principle now applies to !J.F and b.F0 (3 .1.5) The best S0 is taken to be the one which minimizes the right hand side of (3.1.5). If S0 is a trial action which minimizes the right hand side of (3.1.5) and we make a small variation in S0 then first order change in the right hand side must be zero. We now let the action be (3.1.6) For the trial action we take - 22 p p So = J J d-rdar(T) ·r(a)g 0 ( IT-a j ) 1 (3.1. 7) 0 0 Here g 0- 1 is inverse to a function g 0 in the sense that p Jg o- 1 ( IT 1 - a i) go ( Ia-T 2 I)d a = 6 (T1 -T2) (3.1.8) 0 The path integral !Dre -S 0 + J rr may be performed by analogy to the discrete Gaussian integral (3.1.9) Here N is the dimension of the symmetric matrix A. g 0 1 corresponds to ,¾ and go to At. The result is (3.1.10) fJ ]rrot <e o fJ fJ ½-fJcttd.s7( t) ·?'(s )g 0(1t - s I) >o = e o o Here e - PFo is by (3 .1. 3) the value of the path integral in (3.1.10) with 7 = 0. We shall evaluate e -PFo later. c52 - j ?' r >o Noting that fl oy(T)o-y(a) <..e l = = <r(T)r (a)> we have from (3.1.10 ) 0, 7 0 that (3.1.11) ln this equation the X;. are t he components of r . (We use x and x 1 interchangeably.) - 23 Since <x (T) x (a) >o = g 0(I T-a l) ,g 0(T) is for t h e action S0 what g(T) is for the action S . We define (3.1. 12) As we did for g (T),we can for g 0(T) defin e re lated func tions f C>,.,a) and f (T) by ... 1 2 I o(>-.a) = §'o(>-.a) - ~ (3.1.13) f o(T) = (3-I~f 0C>1.a)e --iXaT a In the time domain, the relations between g (T) and/ (T) and between g 0(,) and f 0(T) following from (3 .1.1 3) and (2 .3.11) are (3.1. 14) Comparing the second of the equations (3 .1.14) and the equation (3. 1. 8) we see that go 1( I ,-al) = - dd~ 6(T-a)+ f 0( IT-a I) . The trial action ca n be written as (3.1. 15) Note that the kinetic terms in S0 and S coincide. If one leaves the mass in the trial action as a free parameter, the variational method forces it to have the same value as in the true action. For this reason we have defined f 0 ( T) in such a way that the kinetic terms in the two actions agree. When we say that the kinetic terms agree.what we really mean is that for very short times the kinetic term is - 24 t he dominant term in the action and these dominant terms in S and S0 agree "\\ith each ot~er. Equivalently.we may Fourier analyze the actions, rewriting p . them in terms of variables r(Xa) = JeiX,,.-r-r(T)dT (>-.a= kinetic term in S is now ;{3 2 ;0. ,o.integer) . The ~ r(Xa)·r·(>1.a)\i. If we used a mass m ,,q in the trial action, the kinetic term there would be the S kinetic term just given multiplied by m. The large Aa pieces of these kinetic terms make such a large contribution to <S-S 0 >0 that we are forced to m = 1 in the trial action. That m = 1 in the trial action implies th.at as Aa ➔ 00 .§ 0 (Xa) ➔\;: 2 .(For m ;,t. 1 it goes to mA; 2 .) An action like S0 in (3.1.15) can be gotten from the action of a system which consists of a particle coupled by linear interaction terms to a set of harmonic oscillators . S0 is obtained by integrating away the oscillator variables in the path integral for the particle-oscillator system. Since the parti~oscillator system is described by a Hamiltonian, we can repeat the discussion of the las t ch.apter and derive dispersion relations.sum rules.etc . for the response function R 0 (t) and its Fourier transform R0 (>...) associated with the trial action S 0 . From the comments in the preceding paragraph we see that R 0 (X) ➔ : 2 as ;\➔ 00 so that the absorption function s 0 (>..) for the system v.-ith action S 0 satisfies the same sum rule j de> s (e.>) = 1 as does s (e.>) for the system with action S. 0 0 7T Next we note that the righthand side of equ ation (3. 1.4) is the beginning of a series for the free energy F. This series is obtained by VITiting and then taking logarithms to second order on S 1 we have (3.1.17) - 25 From this expansion it is also possible to get an expo.nsion for g (T). First define h(T-a) = 0 2 o(T-a)-f 0 (T-a). Then Sin (3.1 .6) can be rewritten as (3.1.18) Now generalize this by letting h(T-a) be independent off 0 . Denote the piece in bracket in (3. 1.1 8) by A. Now vary f O leaving h(T-u) constant. Then -RF J o(e ,., ) = o Dre - A - -21 J Jr.:r )r{r;)fo(-r-a) (3.1.1 9) On the other h and o(e - f1F) = o(-{3F)e-fW _ Combining this and (3.1.1 9) using <x1 (T)xi(a)> = oii g( IT-ul) we find that (3.1.20) lf we vary {3F0 with respect to f O we find (3.1.21) If we vary both sides of the expansion (3.1.17) (with S given by (3.1.1B) and h independent of f O) we obtain a series for g (IT-a I). Once we have obtained the series we set h(T-a) = 02 o(T-u)-f 0 (T-o-). This procedure gives us a perturbation expansion for g about any S0 . We shall call the expansion for g the g se ries . (3.1.22) Experience i-vi.th simple examples shows that it is better to transform the series - 26 (3.1.22) for g (T) into a seri es for f ( T). Thus if S is a harmonic oscillator action and So is a different harmonic oscillator action, then the g series is an infinite series. The f series in this case terminates at/ 1 and the result is exact. Other examples such as a single anharmonic oscillator show that the f series gives a qualitatively better picture of the spectrum. Suppose that / = / 0 +f 1 + f 2 + f 3+ • • • . Denote an equation such as (3. 1. 8) sch ematically by g 0- 1g 0 = 1. From (3.1.14) we have and Thus we can write (3.1. 23) Jnverting (3.1.24) Comparing this -with the g series (3.1.22) and equating items of equal order give s (3.1 .25) Solving for f 1 and f 2 gives - 27 (3. J.26) The next section will discuss the variational calculation of f 0 . The section after that will give the explicit construction of lhe g series and its reduction to the f series . 3.2. The variational equations We begin by computing <S-S 0 >0 =<S 1>0 . Using (3.1.18) for Sand (3.1.15) for S0 , then S 1 is (3.2.1) 1 1 4rr Ir ! has been used to rewrite the a term. From (3.1.11) <r(T) ·r(a)> = 3g 0(IT-a !). To evaluate <eik·(r{r) -:r{u))>o use (3.1.10) with o/(µ) = iko(µ-T)--iko(µ-a) . The result is (3.2.2) whe re D 0(T) is defined by D 0(T) = g 0(0)-g 0(T) . This combination occurs repeatedly and also has the nice property that as the O2 :r-2 potential used to confine the electron to a finite portion of the crystal vanishes D0(T) stays finite . So far we have (3.2.3) 00 Making the replacement k- 2 = Je-uk.2du, the k2 integral can be done. Tb.is 0 - 28 leaves au-integ ral . The necessary integral is 1 du ----,3-= 2A 2 0 (u+A) 2 J (3.2.4) Finally , we may use the time tr anslation invariance of the integrals t o re duce the double time integr als t o single integrals . (3 .2.5) Now we are almost ready to vary f 0( T) to find the c onditions for a minimum of {3F +<S > 0 when/ 1 0 0 First replace h(,) by 0 6(,)-/ 2 . is varied the change in g ( , ) 0 . From (3 .1. 14) we find that is given by 6g 0 = -g 0 of ag 0 1 0 1 or in more detail by fJ fJ ogo(l,-o-j) = -JJgo (1,-,' l)6/o(!,'-o-' l)go (!a'-o-l)d , 'do-' 1 1 (3.2.6) 0 0 Combining this with (3 .1. 21) which gives 6({3F0 ) we obtain finally (3.2.7) This e quation makes sense wit h 0 2 = 0, so ·we s hall dr op the 0 2 t erm in t h e following. 1t is c onvenient to define a functio n ,c 0(,) by (3.2.8) so that f 0(,) will be given by fJ f o(,) = - tco (,) + o(,)f,co(,) 0 (3. 2.9) - 29 - From this we have (3.2.10) Follo¥iing the methods of chapter 2 this may be continued away from the discrete valves Aa . We now write down the cycle of equations which self consistently determine the quantities D0 (i.t ), s 0 (c.;), f 0 (-1,A) ( s 0 (GJ) is for the action S0 whats (c.,,) was for S - the energy absorption function) . Suppose we start with some s 0 (c.>) ( this should obey the sum rule j drrc.> s (e.>) = 1 ) using the equivalent 0 0 1 --iA,iT of (2 .2.8) for s 0 (c.>) and § 0 (Aa) and the relation ,i"L e 2 2 ,., a Aa +r..> G"' (T) where Gr.,(T) is given by (l.2.26) and noting D0 (it) = g 0 (0)-g 0 (it), we can write Do(it) = f... ~ (e.>) 1 J1-e-i"'t + 1-e~"'t o ~ 1-e -~"' 1T e "'-1 (3.2.1 1) From this D0 (it) we may construct K, 0 (it) which we then use in the equivalent of (2.2.8) to find •') = f... 0 ( -ii\ .. G (. ) l is ds _2a f (e i>.s - 1-,ll m s... 3v2rr O (. ) D 2 is = (3.2.12) 0 Finally, referring lo (2.2.18), we may complete the cycle by (3 .2.13) Of course, we can start this cycle of equations (3.2.11) - (3.2.13) with D0 (it) or J0 (-iA) just as well as 'with s 0 (e.>). These equations have been also derived in I 6] by another method not connected with a variational principle . - 30 - Since one may also ask to know the free energy associated i•:ith a particular sa(c.J ) , we briefly strelch how {3F0 may be calculated. It is sufficient lo consider the path integral over closed paths x (0) = x (/3) (3.2.14) To avoid worrying about the normalization of Dx divide (3.2 .1 4) by the corresponding path integral for the free energy of a harmonic oscillator of frequency c.J. (3.2.1 5) Introduce new variables Xa by x(T) = ~ ~e - i>,a-rxa ( >-o. =,}:;a., a integer). Since x(T) is real x; =!La, so that the indep endent variables are Rex 0 and for Xa>0 1~ Re.fa, Im.fa, Also, /(Ir-al)= p~e -iX -r~ a J(>-o.) and the /(>-o.) are real. Letting£ denote integration over the independent variables, the ratio of path integrals is .• (3.2 .1 6)· Nun1erator and denomi...riator are each just a product of Gaussian integrals. Carryi.ng out the integrations and using the formula s in.,_½(rrz) = ,.z z2 TI.. (1 + ~ wi th j=l J z = flE.._we finally obtain 2-;; (3 .2.1 7) - 31 -{3F0 is the logarithm of (3.2.17). Using this and our result (3.2.5) for <S 1 >0 we have an expression for {3F0 +<S 1 >. Let h(T-a) = 0 26(T-a)-f o(T-a) . Now we may treat the f oCAo.) as variational parameters. In particular, if we find the bes·t value off (o) by (3.2.18) we find that j (0) = 0 2 . For now we leave the other f ("Ao.)' ( a.~O) arbitrary. If from f3Fo+<S 1 >0 we subtract {3 times ( free energy for electron in potential 2 ~ r·r but not interacting with the phonons ) and then take the limit as o➔o we get a finite result which is just an upper bound on difference in free energy between a polaron and an electron not interacting with the optical phonons. If we denote the actual value of this energy difference by b.F, then we have (3.2.19) The factors of 3 in the first two terms of the right hand side of (3.2.19) are due to the three degrees of freedom of the electron. 3.3. Second Order Calculation First we work out the second order contribution to the free energy given by (3.3.1) f3 P d sk Jo Jo d Td a J ~( -2rr)s ( IT-a I) B = -vBm:x 1 e ik·(r(T)-r{c) ~ - 32 - <AB>o-<A>o<B> 0 . A usefµl theorem in these evaluations is gotten by integration by parts (see [7]) oQ -s jDqQ os -s J Dq 6q(r)e = og(r)e (3 .3.2) or < 6Q oq(r) > = <Q 6S > s oq(r) s Here S is any action with variables q and Q is a functional of the q. For the Gaussian action S0 in (3.1. 7) we find (3.3.3) or Consider now the term <B 2 >0 . To evaluate this we first need to compute (3.3.4) 2 Letting -y(µ) = I; iki[o(µ-ri)-6(µ-ai)] j:1 and using (3.1.10) and D 0 (r) = g 0 (0) - g 0 (r) we find that (3.3 .4) is given by (3.3.5) The Ai; are defined by - 33 (3.3.6) .. To do the k integrals we use J d'l.4. e -u.ikt -~ = +, The k integrals are then Gausk.i o sian integrals. Evaluating these and then performing the resulting integrals over u 1 and u 2 , we finally obtain (3.3.7) ln <AB >0 we encounter Applying (3.3.3) several times and using 'v2 I 1 I r-a -4rr6 3 (r-a), we find that (3.3.8) is given by (3.3.9) The expectation of the delta function is found by using 63 (r) =J (~:~3 eikr and (3.1.10) to give 1 - ;!_ <6 3 (r(T)-r(a))>o = ~ o 2 (IT-al) 8rr 2 (3.3.10) Also note that (3.3.11) - 34- Combining (3.3.1), (3.3.7), (3 .3.9), (3.3.10), (3.3.11), and previous results for <S>o we can finally write down {36.F to second order. Before doing this, however, we shall extend the schematic notation which we introduced earlier in equations (3.1.23) to (3.1.26) . Thus symbols such as a,b, • • • will refer to .a(jT-a j),b(jT-aj), • • • If we ·wish to be more explicit about some of the variables, we will write ( a )T<1 for a (IT-a I) . A product notation ab will be used to denote integrals, so that (ab )-rp will stand for p fa ( IT -a I )b ( I a-p I )d a. If we wish to explicitly display integration over a vario able, we will J write /1 for f d,. The notation [a] will denote 0 T o(T-a)f a( I ,-a' l)da'. With this notations the free energy to first order is (3.3.12) The second order free energy is ( ,c0 is defined in (3.2.B) ) -31 (g chg oh )-r-r + 3f (g chg o!Co)-r-r 4 T 2 (3.3.13a) T The function q is defined by (3.3.13b) - 35 ·with ( from (3 .3.6) ) (3.3.1 3c) and The next step is to find the variation of the series (3.3.1 2) as we vary the function f O with h fixed. After setting h(T) = 0 2 6(T)-f 0(T) in the result ( we shall take O = 0 ) , we obtain a series for g (T). This series will then be transformed to an expansion for f. These expansions do-not require that f 0 satisfy the variational equation (3 .2. 9), but we shall find that many terms drop out if (3 .2. 9) is satisfied. From gg 0 1 = 1 ( 1 will represent the delta function (1)ra = o(T-a) ) and - d: 6(,-a) +f 0 ( I,-a I)= 9o- 1 ( I,-a I) we d1 have shown that ogo = -goof o9o - Since many of the terms involve D0 'sit is helpful to see what contributions such terms give when D 0 ➔ D 0 + oD 0 as a result off 0 ➔ f O + of 0 . Let A represent such a (3.3.14) Here the functional derivatives are to be taken as if (Do)rc is not symmetric in, and a. Since 6({3F) = ~ f (g 6/ o)rr• (3 .3.1 4) leads to a contribution to the series T for (g )ra of - 36 - (3.3 .15) We note that the delta function term in (3.3 .1 5) is completely determined by the other term. Applying (3.3.15) and og 0 = -g 0 6f 0g 0 to (3.3.12) we find the g series to first order. Setting h = -f O and using the first of equations (3. 1.26), we find that/ 1, the first order correction to f O is given by (3.3.16) or more fully fJ f 1( I ,-al) = -/ o( IT-a 1)-JCo( I,-a I )+o(T-a)J da'JC( IT -a' I) (3.3 .17) 0 Here JCo is the function defined in (3 .2.8). We note from (3 .2.9) that for the best trial action S , 0 f O satisfies f 0 +,c -[JC 0 ] 0 = 0, so that/ vanishes for the best S 1 . 0 Before giving the second order terms in the f series we discuss the variation of the q - terms in ( 3.3.13a) in order to establish our notation. The variation of q is entirely due to the variations in A 11 , A22 , A12 , so we may write (3.3 .18) The % are given by (3.3 .1 9) q 12 = ~ [ . ~ sin- 1x l Arz v 1- x X - 37 - From (3.3.13a) and (3.3.19) we note some symmetries of the AJ and qi.j · If -r 1 and a1 ( in 3.3.13c) or_ •2 and a2 are interchanged thenA 11 ➔ A 11 , A22 ➔ A22, and A12->-A12 - The behavior of the qii under the same transformations is given by -r2a2 we find that A22, A11 and also q 22 , q 11 are interchanged while q 12 and A12 are not changed. Using (3.3.18), (3.3.19), (3 .3.15) and the symmetries just mentioned we can readily construct the contribution of the q - term in the free energy expansion to the g - series. From (3.1 .26) the contribution to (J)ra is (3.3.20) +delta function terms The rest of the second order contribution to (f )-ru is (3.3.21) + delta June tion terms where (3.3.22) We note the delta term associate d with the second term in (3.3.20) vanishes due to the antisymmetry of the integral in a and a'. TI1e delta function terms associated with the first term in (3.3.21) also vanishes. - 3B - It seems strange that the first term in (3 .3.21) survives . Wh en we do perturbation theory with polynomial Lagrangians such terms always cancel out. This suggests that further simplifications may be possible . Suppose that we expand q 1 2 in powers of x. The lowest order term is 1 = 3 Ar2 3 1 1 A12 (AllA22) (3.3.23) ~ 2 By using the e1..-pressions (3.3.13c) for the A1 and doing some rearranging we find that the contribution to the f series involving (3.3.23) cancels the first term in (3.3.21). Similarly, the lowest order term in q 11 is = 41 Arz Q.. ~ (3.3 .24) Ai\Af2 The contribution to the f series involving this piece of q 11 combined with the second term in (3 .3.21) is -(L )-,-c J (/ 0+1Co-[1Co])-,-,C1,(g c)-,-e7•((g o)a-r•-(g o).,.-,-,) (3.3.25) -r'a' Note the factor+ f 1 = -/ 0 -,c0 +[,c0 ]. We see that when we use the best S0 so that f 1 = 0, the contribution from (3.3.25) vanishes. The corresponding delta func- tion term also vanishes. We can in this case '\lvTite the second order piece off, that is f 2 , as f 2( I,-a I) = -1C2( IT-a I)+o(T-a)f!Cz( IT-a' I) (3.3.26a) a' where the function 1Cz( I,-a I) is defined by (3. 3.26b) - 39 - cl J - G1( I T-a I)g 1( J-,-'-a' J)q ll(Ta j-,-'a') 3 TT ru' where q 12 and q 11 are given by (3.3.26c) where and as before (3 .3.26d) D0(T) is given by J.. D (,) = so(c.>) de.> Tl 1-e-c.no rr 2c.> 1-e -{J<.; O l_-e c.rr J ~ (3.3.26e) Here s 0 (c.,) is the absorption function for the trial action S0 . Note that this e}...-pression allows us to analytically continue D 0(T) into the complex , plane as long as Cr-:;Re,~,B. The absorption functions (i\), correct to second order is given by (3.3.27) - 40 - J 0(-iA) is given by (3.2.12) and J 1(-i>..) is presumed to be zero. [ 2 (-i>..) is given by .. f 2(-i>..) = zf (e;,>-.s -1)Irruc2(is) (3.3.28) 0 ,c2 (is) is the continuation of tc 2 (T) given by (3 .3.26b) for real T to imaginary - 41 4-. NUMERICAL RESULTS 4.1. Solution of the Variational Equations The first method of solution for the variational absorption function s 0 (c.>) which we used is based on the cycle of equations (3.2.11) to (3.2.13). ln the actual program it is possible to start at several different points in this cycle. We shall briefly describe the program. An array containing the values of D0 (is) at equally spaced values of s is formed. From this a table of values of (4.1.1) in the integrand of (3.2.12) is formed . This function falls off,.slowly as s ➔ 00 and ./ has a singularity at s =0. To improve its behavior, we modify it by subtracting a function rfi(s) to get a table of values of f G1 (is) lm------1-rfi(s) I (4.1.2) n{(is) The function IP(s) is chosen so that it has the same singular behavior at s = 0 as the quantity in (4 .1.1) and also falls off at 00 in the same way as (4.1.1) . 4>(s) is also chosen so that the integral (4.1.3) may be carried out analytically. The numerical integration of (4.1.2) multiplied by 3 ~ (ei>-s -1) is carried out b y using a Fast Fourier Transform and the 211 results are added t o the results of (4.1 .3) to form an array off 0 (-i >..) values at - 42 equally spaced values of>... Equation (3.2.13) is now used to form a table of s0(11.). 1f an older VE;rsion of this table exists, the entries are compared to see if they agree well enough that the iteration may be stopped. If the iterations are stopped, a table off 0 (--i>..) is output . Otherwise a table of s 0 (>..) -u (X) .(4.1.4) is constructed, where u(11.) has the same behavior as s 0(11.) as /\ ➔ 00 . The integral (4 .1. 5) is performed analytically while (4.1.6) is done numerically. The results are added to form a table of values of D0 (it ). (see (3.2.11 )) . This is where we started. The program goes on this way until convergence is achieved. Our second approach to the computation of s 0 (c..>) begins from the expression (3.2.19) for the free energy ~F . We only worked out the {3 = case in this 00 approach In this case (3.2.19) becomes a -J"" - e--r --dT ~ r-::11 O L (4.1.7) D02 ( T ) s 0 (w) was assumed to have the form (for c..P-:1) .L. s 0 (c..>) = C(c.;-1) 2 R(CJ) (4.1.8) Here C is a constant and R(w) is a rational function in CJ. For 0<~1 , so(c..>) is - 43 l taken to be zero. The factor (CJ-1) 2 was included because the threshold at CJ = 1 is known to have this form. Also, it is knovm that as CJ ➔ oo, s 0 (c..>) ➔ 4 a c.i- 2 -5 . Thus 3 the form (4.1. 8) can have the correct behavior at CJ ➔ 1 and as CJ ➔ oo. C and the roots of the numerator and denominator of R(CJ) are taken as variational parameters. D0 ( T) is obtained (numerically) from (4 .1. 9) with s 0 given by (4.1.8) . a 0 is just 1-jso(CJ) dw. f(A) is obtained analytically by l 7r . evaluating (see (2.3.11) and (2.3.9)) 1 (4.1.10) The integrals in ( 4.1. 7) are then evaluated numerically to find the right hand side of (4.1.7). A minimization program was then used to find the values of the parameters in so(c..>) which give the smallest value for the right band side of ( 4.1. 7). Some results for a= 5 ( f3 = 00 ) are shown in figure 4.1.l. The value of s 0 (c.>) is plotted against CJ . Curve 1 is the result of the iteration method (the first method described in this chapter). Curves 2 and 3 are both results of the minimization method. Curve 3 is apparently spurious for when the s 0 (c.>) of curve 3 is used as an input to the iteration method, the results rapidly converge lo curve 1. Also, when a fit of the form (~. 1. 8) is made to curve 1 and the resulting parameters used as a first guess in the minimization method, curve 2 is the result. These results illustrates some of the pitfalls of minimization methods in general. The free energy is a functional of some function but is not very sensitive - 44 - t o the form of the function in the vicinity of the minimum . ..... 0 0 en □ ---- i s- ...... :~,--·· a,fJ, . "I."-> . , , ... :-.:'do..t1 • ., Cl(, ... 0 CD □ a r- 0 '\ a CD a '\ '\ '\ \ \ \ ~. \\ \ '\ Ill a /J I a /!. (fl a a N □ a □ a a °o. 00 L 00 2.00 3.00 ome ga Figure 4. 1.1 { . 00 5. 00 6. 00 7.00 - 46 u -. 0 0 Cl □ 0 co □ 0 r- 0 0 (0 □ 0 Ill 0 (T) a 0 N □ 0 □ a a °o. 00 1. 00 2. 00 3.00 {.00 omeg a Figure 4.1.2 5.00 6.00 7.00 - 47 - ...., 0 ...; 0 ID ..... 0 U) 0 ~ - /\ I \ I \ I \ I \ \ I \ I 0 "' ..... Cl Olo fJ~ I I I I I E .... 0 -0 \1l 0 ID a 0 <O a .., 0 a I I I I "'~ !, ~._/{ c .... o(' ~ FHIF" ~,.-1 ... r \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ / 0 "' a \ \ 1/ \ 0 0 9:i. 00 1. 00 2.00 3.00 om ega Figure 4.1. 3 .{_ 00 "- 5.00 6.00 7.00 - 48 0 0 0 en a 0 CD -< ' s:- a ~ . 'i Cl( ::: 3 0 r- a 0 ID I 0 I \ \ I \ I d - \ /['\. Ola ti)~ ED I 0 / I/ fI ~ 0 I.'\ 0 ~ a 0 CT) a 0 N a \ ' '\ I I I fI I I ' I fI II ' If 'I I \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ !1 \ '\ \ '\ lJ ·, !, , \ \ '\ \. ' \. ) 0 '\ ' ' ' ' .... '- .... a .......... '' ........ ' - - ------.......... '- 0 0 °o'. 00 1. 00 2.00 3.00 4. 00 ome ga Figure 4.1.4 5.00 6.00 7.00 - 49 Flgure Caplion s [1] Results from two methods of computing s 0 (c.>) for cx=5. The continuous curve is the result of the iteration method described by equations 4.1.1 to 4.1.6. The dashed curves are both the results of the minimization method using equations 4.1.7 to 4.1.10 ·with different initial guesses for s 0 (w) . When inserted into the iteration program both of these curves converge to the continuous curve. [2] The continuous curve is the result of the iteration method for a.=3. The dashed curve results from starting with an initial s 0 (c.>) = (1-a 1)o(c.>)+a 1 6(c.>-c.> 1) and iterating once. a 1 and c.> 1 are determined by minimizing the free energy. This is the ansatz used by FHIP. [3] This is the same as figure 4.1.2 except that it is for a:=5. Note that the self consistent s 0 (c.>) is quite different from the FHJP s 0 (~. Interpretations of the structure in the dashed curve have been given in [10],but we see that this structure goes away in the self consistent solution. [ 4] Self consistent results of the iteration method for a= 3,4,5. - 50 - 4. 2. Second order correction s (>-). the absorption.function, is given correct to second order by (3 .3.27) . This requires the computation off2 (-it,) as in equation (3.3.28), which is [ 2 (-i>-..) = 2f(ei>-.s_ 1)In1Kdis)ds (4.2.1) C For real T and a, K:z(IT-al) is given by (3.3.26b) and the follo,ving equations. We need to continue these to imaginary T =is . A simple example illustrates how this may be done. If we wish to continue a(T) given by a (T) = f e - r..i 1-r-u Id a (4.2.2) C into the complex T plane, we can rewrite it as T a(T) = e-r..i-rJ e1..1°da 0 + fe e= -(,Juda (4.2.3) T Now since the integrands are analytic and convergent, we are free to move the contours around. Thus a(it) may be computed by a (it) = e-iGJt it .. C it f er.-ada + ei"'t f e -"' d a 0 (4.2.4) Consider the contour C running from Oto some complex T and then to 00 • for ai, a2 on this contour v.Tite a 1>a2 if we encounter a2 first as we traverse the contour starting at 0. For complex T and a on C define lT-a !c to be T-<J if T>a and a-T if T-5°,a. T'nen for complex T (4.2.5) where T must be on the contour C. With this notation K:2 (is) is given by - 51 - (4. 2.6) Here -r', a' are on a contour C such as: t c) We also have (. ql2 'IS,T I I O, a ') = with x ftilrv1-x~ 2 A12 X - ~ • -1 Slil X - ~ 3] (4. 2.?a) A12 = --;== =-and -v'A11A22 (4. 2.?b) I wr.J.le for q 11 we have - ( ' O'j T,G' I') =2A X A q111.S, ->-111 - A12 again with z = 1 but with "\i A11A22 12 rl1+ -2x h1 V ~ 1-:r (4.2. Ba) - 52 - A 11 = Do(is) (4.2.Bb) (4.2.9) It is convenient to put these equations into a form in which the limit {3 ➔ 00 is easily taken. Consider G1(T) as f3 becomes very large. (4.2.10) For fixed finite a, G 1(a) and G 1({3-a) have finite limits as {3 ➔ 00 • Both limits are given by {3 ➔ 00 (4.2.11) We must also consider the behavior of D0(T) as {3 ➔ 00 . From (4.2.9) we see that for finite a (4.2.1-2) At {!= 00 (zero temperature) there are no optical phonon present initially in the crystal. The electron can absorb energy from the field at zero frequency (electron accelerated by a time constant field) or it can absorb energy from a time varying field and emit a phonon. However, this cannot happen until the frequency of the applied field is equal to or greater than the optical phonon frequency. From these considerations (c.;) has a delta function at CJ= 0 and is zero for O<c.><l. Letting - 53 - f"" so(c..>) d al - I c..> (4.2.13) 1T and defining a 0 = 1-a 1 we find that at {3 = =, D 0 (,) has the form _ aoT , j.. so(r..>) d 1 f -,.rr] D o(T ) - -2-, l TT CJ 2c..> l1 -e (4.2.14) Note that this becomes 00 as T ➔ 00 , Consider now the integrals in (4.2.6) and (4.2.7) . Using (4.2.14), (4.2 .12), and (4.2.11) we find that the integrand is now zero as {3 ➔ 00 only if T' or {3-T' and a' or {3-a' remain finite as {3 ➔ 00 , Denote the integrand in the Slli'TI of (4.2.6) and (4.2.7) by !(is ,T',a') so that (4.2.15) When {3 ➔ 00 only the pieces of C near to O or f3 give any contribution. Let C1 run from O to f If T is on C1 take C2 to be the set of all points ({3-T) . C1 and C2 together form C. Rewrite each integral over C as (4.2.16) so that (4.2.15) becomes (4.2.17) 2 (is) !(; k = f dT'j d a" (is ,T',a')+ I (is ,{3-,' ,a')+ J(is ,, ' ,,B-a')+ J(is ,{3-T',{3-a')] c1 C1 Finally use D 0({3-T) = D 0(,), G1(p-T) = Gi(T) to rewrite the integrands. The {3 ➔ = - 54 limit may now be taken. Of cours e , we may also use the form (4. 2.17) for {Jc/- 00 . ( 4.2.17) is the form we actually used in the numerical computations. To compute the second order corrections we did not use directly the table of values for D0(it) computed by the iteration method. Rather a fit was made to the s 0 (c..,) computed by the iteration method of the form (4.2.18) Here the 4- and b are parameters which we are free to vary. Figures 4.2.1 and 4 .2.2 show the fits for ex= 3 and a:= 5 respectively. The D 0 (it) arising from (4. 2.1 B) is given by (4.2.19) where a 0 for T = 1-a 1 and a 1 = jso(w) dc.>. An important feature of this D0(r) is that l 7T' =it and large t, the oscillating term in D0(r) =D0(it) ➔ 0 as t ➔ 00 , To evaluate ic2 (is) along the interval from s =O to s =a we use a contour in the ,=is plane such as: 0 (b We split this cont our into s tr aight line segment s. An integ r al a long any segment may be done numerically by Gaussian integration. For each segme nt of contour we have a s et of points and a weight for each point. Combining t he points and - 55 weights for all of the segments gives an array of points and an associated array of weights des~ribing the contour. The integrals in (4.2.17) are carried out using these points and weights . Using a set of such contours a table of ,c2 (is) is built up . Similar ly, a table of ,c2 (s) may be constructed using a contour lying along the real axis in the , plane. Finally f z( - i">-) is evaluated using (3.3.28) and s (A) is con structed from f 0(--i>,..)+ f 2(-i/\). In the path integral variational me thod, the trial action S0 (or s 0 (CJ) from which we may co nstruct the trial action) plays a role analogous to the wave function in the usual Hamiltonian variational method. At a.=3 t he se cond order correction to the polaron energy is one percent while at a=5 the correction is 1.5 percent. The second order corrections to the absorption function shown in figures 4.2.3 and 4.2.4 are quite reasonable in relation to the size of the energy corrections. For example.the difl'erence in area under the va riational curves and the corrected curves between CJ= 1 and CJ= 00 is in each case less than 10 percent. For a=3 the correction is quite small except between CJ =1.5 and CJ= 2.5 and in this range its maximum is 25 percent. For a= 5 the location of the peak is shifted by about 10 percent and its width at half maximum is changed by about 20 percent. - 56lJ 0 0 en 0 0 CI) 0 0 r- e 0 lD 0 d Olo 0~ ED 0 lll 0 ~ 0 0 (Tl 0 0 N 0 0 0 0 0 °o. OD 1. OD 2.DD 3.0D ome qa Figure 4.2.1 ·C 00 5.DD 6. 00 7.DD - 57'-J -. 0 D O') D D co 0 D r0 0 tO 0 Vl 0 (Tl 0 0 N 0 ~ 0 0 ........ ' .... 0 0 0 o. 00 I 1. 00 2. 00 3.00 ome ga. Figure 4-. 2 .2 ,{. 00 5.00 6.00 7.00 - 58 0 0 0 en a 0 [D a 0 i- 0 0 lD a 0 (Tl a 0 N a 0 a 0 0 °o. 00 1. 00 2. 00 • 3.00 .I,. 00 omeo a Figure 4.2.3 5.00 6.00 7.00 - 59 0 0 0 en o 0 [I) o 0 r- 0 0 lD o I I I I I I I I c1 o,0 l)~ EO 0 l.n \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ I 1/ / 0 0 \ I I er) \ \ ,. I 0 N o "" ""- ' I I I I ~ 0 0 0 °o. 00 1. 00 2.00 3.00 orne oa. F'igl.lJ'"e 4 .2 .4 4.00 5.00 6 . 00 7.00 - 60 Flgure Captions [ 1} Fit of form (4.2.18) to variational results for a:= 3. [2] Fit of form (4.2 .18) to variational results for a:= 5. [3] Second order correction to the variational result for a= 3 computed using the fit shown in figure 4.2 .1. [4] Second order correction to the variational result for a= 5 computed using the fit shown in figure 4 .2.2. - 61 - 5. FURTHER DEVELOPMENTS 5.1. Possibility of Improving the Method To what extent can we remove the idealizations made in the Frolich model and still retain the ability to perform an accurate variational calculation for the free energy and the response functions? The treatment of the crystal as a continuum is nonessential, as is the restriction to one type of phonon. The phonon dispersion relations can also be generalized. Of course, dropping these simplifications makes the numerical work more laborious, but this is a quantitative and not a qualitative difference. At least the solution of the variational equations could be carried out in a reasonable amount of time if one were sufficiently clever in formulating the numerical algorithms . The most crucial idealizations which the Frolich model makes are that the lattice oscillations are harmonic and that the electron-phonon interaction is linear in the phonon variables. Of these, the latter is perhaps the more important approximation. The great virtue of Feynman's method of treating the polaron problem is that the phonon variables are intergrated away exactly. A variational calculation in which one tries to imitate the whole Frolich action by a Gaussian trial action cannot be expected to work as well as the Feynman method does. No coupling linear in both the electron and lattice variables can simulate correctly the distortion of the lattice by the electron. If the electron-lattice coupling remains linear in the lattice variables but the lattic e anharmonicity is taken into account in the lattice Lagrangian itself, we may still approximately integrate away the phonons in such a way that the variational inequality is preserved. Let Si be the lattice action. Let the q denote the lattice variables and let the I be functionals of the electron path - 62 such that J-y-q is the electron lattice coupling term. We need to do the path integral J Dqe -s, J-r·q + (5 .1.1 ) When SL is Gaussian this can be done . If SL is not Gaussian then take a trial action of the form (5.1.2) where S0 is a Gaussian in the q's. The difference between S 1 -:-- f -y- q and So- J1 ·q is independent of the ,·s. Finally, -choose S by minimizing 0 {3F3o + <St -S 0 >s0 . Here Fs0 is the free energy of S 0 alone ( with the , in (5.1.2) set to zero). This seems like a reasonable procedure and gives the exact result if Si is Gaussian. Of course, it would be nicer to replace (5. Y2) by (5.1.3) where So is again Gaussian but J is different from 1 . Now choose S0 and J so as to minimize f3Fs 0 + <S-f,-q -So>s0 (5.1.4) The result is a set of nonlinear equations which, if solved, would give S0 and J as functionals of SL and, more importantly, of y. While we may solve these equations numerically to find S0 and J for a given 1 , this is not of much use in writing down an effective action for the electron. Either the nonline ar equations resulting from the minimization of (5.1.1=) must be solved e:x-plicitly for general I or we must find an indirect method. To carry out a variational approximation to the electron effective action resulting from the approximate integration of the phonon variables, we need to calculate averages of quantities such as S0 and J which - 63 - are given as implicit functionals of the electron path ' ( since ')' is a function.al of the electron path ). If this problem we re solved, it would also be possible to treat the situation of an electron-phonon coupling nonlinear in the phonon variables . In fact it might be possible to construct sequences of increasingly better variational approximations to a ·wide variety of problems. However, we have not discovered any way to solve the problem. 5.2. More General Method of Analytic Continuation In this section we present a generalization of the method described in section 2.3. These generalizations allow us to treat nonthermal density matrices and to extend the path integral variational principle to a wider range of problems. To motivate the generalization we consider a particular problem. A system consisting of an electron and a crystal has a Mamiltonian H. ( H could be Frolich's Hamiltonian. ) The system is initially in thermal equilibrium at temperature p- 1. At time t =O a steady force Fis applied to the electron. The Hamiltonian becomes H = H - F- r ( r is the electron position ) . No assumption is made that the force Fis weak. We wish to compute <r(t )> . This function tells us how the electron responds to the force. In particular, it can tell us whether a steady state is reached in which Lhe energy pumped into the electron by Fis balanced by the energy lost by the electron due to interactions ,\Tith phonons, etc.. We can also discover such things as the rate of energy dissipation at a given velocity, etc .. If the system is initially in state I-I'i > with probability Pi we have (5.2.1) • Tne variar.io~ equatio:is corning from the rrJ:n:mizatio:i of (5.1.4) :;i;-ovide :he irr.plich relatio:is be:wce:1 So ,'l, e.:i.d fae elec':.::-o:i. path. - 64This problem has been discussed by Thornber & Feynman [8) . For a thermal density matrix we have 7r [e -PH e illt re -illt ] <r(t )> = Tr e-PH (5.2.2) ( 5.2.1) is the analytic continuation of (5.2.3) to T 2 = -it and T 1 =it while (5.2 .2) is the analytic continuation of Tr [e -pH e --r2'J re --riR] (5 .2.4) Tr e-PH = -it and T 1 =it. In terms of path integrals (~3) may be written again to T 2 as T1+Tz JDq,ip (qf ,qi)e - JO I.d-r (5 .2.5) r(T 1) Here the q are all the coordinates of the system including the electron coordinates. L is the classical energy associated 1vith H. p(qJ ,qi) = ~ <qt Iti>A <ti Iqi> and the path integral is over the path beginning i at qi and ending at q1 . We can compute the path integral in (5 .2.5) from path integrals of the form Tl+Tz f DqJiP(qf ,qJe Tl+Tz - j L+ j 'y·r o o (5 .2.6) Here L is the imaginary time Lagrangian of the system with Hamiltonian H. To use (5 .2 .5) or (5.2.6) we need an e:x--pression for the density matrix p(q1 ,qi). In - 65 order to use a variational principle for the approximate calculation of (5.2.6) it The nun1erator of (5.2.4) requires the evaluation of the path integral (5 .2.7) over all closed paths q(0) =q({3+T +T2). Lhas the same meaning in (5.2.7) as in 1 (5.2.6). In the case that Lis the imaginary time Lagrangian of the Frolich model. we integrate the phonon variables in (5 .2. 7) and find an effective action for the electron alone. The integral over the electron paths is now approximated using a trial action and the variational principle. The results must then be continued back to real times . The continuation process given in section 2.3 will not work in the present case because the action in (5.2. 7) has explicit time dependence due '1'1 +T2 to the term J F- r even when -y = 0. The best trial action will also have explicit -o time dependence. To see how to carry out the analytic continuation, we shall return to the problem of the weak response of an electron to a time varying field which we formulated in chapters 2 and 3. Although the effective action and the best trial action no longer have explicit time dependence in that case, the method of continuation in time v.'i.U apply also to the path integral in (5.2. 7). The condition for the trial action given in (3.1.15) to be the best one with which to imitate the action (3.1.6) (v.ith D = 0) was found to be from (3.2. 7) (5.2.8) with D0(,) =g 0 (0)-g 0 (T) . In addition g 0 (,) and f 0 (T) are related by (3.1.14) - 66 - To compute the real time response function R 0(t) we need g 0(it) since R 0 (t) = g 0 (it)-g;(it) . (5.2.8) and (5.2.9) enable us to solve for g 0 (-r) for -r real and in the interval 0~-r~(3 . In principle this is enough information to obtain the analytic continuation of g 0(-r) to all complex,. Ho wever, it is difficult to see how to do this directly by means of (5.2.8) and (5.2.9). It is not clear how to treat the absolute values and especially the delta function 6(-r-a) as -r , a, and -r' become complex. To see what to do recall from equation (2.2.1) that we e>-.rpressed the response function R(t) in terms of (5.2.10) or in terms of (5.2.11) In chapter 2 we continued this to (5.2.12) and expressed it as a single path integral. However (5.2.11) is also e:x-pressable as a single path integral if we merely allow the time variable to follow some contour in the complex t-plane (or, plane) . Thus (5.2.11) is -;.fw f Dxe c x(t 1)x(O) (5.2.13) Here the contour C in the t-plane start s at t = 0 , moves along the positive real a.xis tot = t 1• a nd t hen fin ally follows some c ontour tot = -i{3. - 67 L in (5.2 .13) is the continuation of the real time Langrangian to times which lie along C. We must also impose on C the condition that Irnt is decreasing as we traverse C from its beginning at t = 0 to its end at t = -i.(3 since otherwise the path integrals in (5.2.13) diverge. lt is convenient for us to work in the complex T -plane where T =it. The contour C above is now C and the path integral (5.2.13) is now (5.2.14) where L is the continuation of the imaginary time Lagrangian (which is the classical energy) of the system to T along C. Order the contour C with O considered the least point and f3 the greatest point. If T and u are two points on this contour, write T<a if T is encountered first as the contour is traversed from O to {3, otherwise write T~u. IT-CJ Ic denotes T-a if 2a and a-T if T<u. Consider now the Gaussian action (5.2.15) - 68 We "l'.ri sh t o evaluate (5.2.16) By analogy to a discrete Gaussian integral such as (3 .1. 9) we need to find 9o( I T-cr I c) inverse tog 0 1 ( IT-a I z:,) in the sense that Jgo(!T-cr' lc)9o 1 ( 1a'-a lc)da' = oc(T-a) (5.2.17) C Here oc(T-a) is a representation of the identity. Jt satisfies a(T) = f 6c(T-a}a(a)da (5.2.18) C for T and a .on C and a (T) a test function on C. (5 .2.17) ma~so be written as (5.2.19} And this is the generalization of (5.2.9) which we need. Let's check that we get its correct results. Define C to the contour shmvn below Let T and a be on the left piece of the contour and let T = ii , a =is . a' =is' on left piece of contour while a· = is'+(3 on right piece of contour. By u sing (5. 2.19) and it s complex con- f ;(T) = f 0(T•) which are true for complex T, we arrive at an equation 2 ' ~o(lt-s l)-2fimf 0 (ii -'is ')R 0 ( ls '-s l) ds ' = - Zi o(t-s) dt s (5. 2.20) Here R 0 is the response functi on given by R 0 (t) = g 0 (ii )- gc (it) for t >0 and R 0 (t) = 0 f or t <0. (5. 2.20) implies that 2 ~ 2 R o(t) - dt t J 2 lmf 0 (it - is )R 0 (s )ds = - io(t) - o, (5.2 .21) - 69 +oo Substituting Ro(t) = £e---i."At ~;Ro(>--) into (5.2.21) we find that i Ro(r..) = - .. - - - - r..2 +2f[Imf o(is )]eiM ds (5.2.22) 0 so that from (2.1.11), the absorption function is - (>-) = I m - - -2r.. s 0 (>,) = Re 2AR -----0 (5.2.23) ->-.2-2 j[lmf o(is )]ei>-s ds 0 comparing to (2.3.18) we see that fo(-i>-.) =-2fei>-simf (is)ds 0 (5.2.24) 0 which is just (2.3.17). Thus we have recovered the results of chapter 2 on analytical continuation by means of a different approach. The utility of this approac h is that it allows the application of the variational method to problems such as the electron in a constant arbitrary strength electric field discussed earlier in this section. Tne method of analytic continuation in section 2.3 cannot be used for this problem. Also, the idea of working with the analytically continued path integrals such as given in (5.2. 7) in order to evaluate expressions such as (5.2.2) seems to have the advantage that the computational labor involved working with the analytically continued path integrals is less thar1 the labor involved on working with a direct path integral representation of the original problem. - 70 5.3. M ore Realistic Model 1n this section we present the variational and second or der equations for a generalized version of Frolich's model. We retain the assumption of a harmonic lattice. However we do not make the continuum approximation nor do we make any assurnpticn s about the phonon dispersion relations or about the types of phonons coupled to the electron. We do assume that the electron-phonon coupling is linear in the phonon variables. For the electron we initially assume that the energy is of the form (5.3.1) Here r is the positron vector of the electron and the x 1 are the components of r . V(r) may be a periodic potential or it may include contributions due to impurities, etc. By a combination of an orthogonal transformation and a r e scaling of coordinates one may always bring (5.3.1) into the form Lawe = ~ ~(:i;\) 2 + V'(r') (5.3.2) \ Here r', x\ are the new coordinates . In the following we shall drop the primes on the new coordinates and take (5.3 .2) as our s tarting point for t he electron portion of the energy. We work vvi th a ver y large crystal ·with N unit cells and u se p eriodic boundary conditions. N vri.11 denot e t h e position of a unit cell and ·,JXf;,.X-tfo. v-ri.11 be the displacement from equilibrium of the a. th atomic c oordinate in the unit cell at N. M o. is the mass associated with the a. th coordinate. In terms of the XNo.• the lat- tice energy is ( relc.ining only quadratic terms in the potential ) - 71 (5.3.3) (5.3.4) The k sums will always be over a unit cell ink space. Because J ap(N) is real and Jpo.(N) =Jap(-N) , J p(k) has the properties 0 Jo.p(k) = J;_B(-k) (5 .3.5) Note that Jap(k) is a Hermitian matrix. For each k we have e~(k) and CJ>.(k) such that (5 .3.6) We now define new variables q)c'),. by (5 .3.7) Note that qk>. = q -le>.· CJ>,.(k) is the frequency of a p honon of t ype "A. as a functio n of its wave vector k and e;(k) is a phonon "polarization''. In terms of the q's the lattice energy is (5.3.8) The most general electron-lattice intera ction ·which dues not involve time derivatives and is linear in the Xmi. and is invariant U...'l.der simultaneous displac em ent of the ele ctr on and the displacements field XNo. by a lattice vector 1I is - 72 - Ltnt == ~ Ua(r-N)XNa Na (5.3.9) The reciprocal lattice vectors G satisfy (5.3.10) for all lattice vectors N. Once we chose a unit cell in k -space any vector q can be uniquely decomposed into the sum of a reciprocal lattice vector G and a vector k within the unit cell. Any periodic function (/ (r) == f (r+ N) for all N) can be \'.Titten as/ (r) == LJf ceiG-r. Now define G BJc.,.(G) == ~(e~(k)f a J d re - i(c+k) ·rua(r) 3 (5.3.11) Note that .BkA (G) = .B _kA(-G). The interaction energy can now be rewritten as T. == '-'int t _1_~q -vN -k.X B k\ (G)e i(G+k)·r (5.3.12) Now we have established our notation and formulated a total energy L == Laiec + Liat, + l...mt for the system. The next step is to integrate away the lattice variables and arrive at an effective imaginary time action S for the electron. That is, (5.3.13) where the integration is over all closed q paths qw,.(O) = qJcA(f3) . With the aid of the path integral given in (1.2.25) we may carry out the integration in (5.3.13) and find - 73 p S = ~ j [r+ V(r)]dr (5.3.14) 0 where (5.3.15) The second term in (5.3.14) can also be written as (5.3.16) If a lattice vector N is added both arguments of W(a,b) then its value remains unchanged W(a,b) = W(a+N,b+N) (5.3.17) This is not true if we add on an arbitrary vector R to a+b. This is due to the G1 ¥0 and G2 ¥0 terms in (5.3.14). The presence of these terms and the periodic potential V(r) makes it difficult to formulate a good trial action for the problem. We have formulated some possible trial actions, but have not derived the variational equations or done any other calculations ¥iith these. For this reason we only consider those cases in which it is reasonable to take V(r) = 0 and to drop the G1 ,G2 ~0 terms in the second term of (5.3.14) . Thus the action we consider is (5.3.18) For a trial action we no,v take - 74 - (5.3.19) As in equation (3.1.18), it is convenient to rewrite S as (5.3.20) Here we eventually set l¼i = -f o,ii. For the purpose of computing the series e)..'-pansion of g1.;(IT-al) = <xi(T)x1 (a)>, the I¼; are initially allowed to be arbitrary functions as h was in chapter 3. For the trial action S0 , note that if we define g 0_ij(T) by then (5 .3.22) where <> 0 denotes an average t aken with e -So as a weight. (As in (3. i .1). ) S1 = S-S0 = A+B is now given by (5.3. 23) - 75 To evaluate <S 1>o and <Sf>o - <S 1>5we need the results (5.3.24) (5.3.25) where now (5.3.26) Note that, in particular (5.3.27) If F 0 is the free energy associated with S0 and F is the free energy associ- ated with S, then to first order in S 1 we have (5.3.28) lf we s et l¼J = - f o,ii and then set the first order variation in the right hand side of (5 .3 .28) to zero for an. arbitrary variation inf o:ii , we find the condition p J o.ij(,) = -ko,ij(T) + o(T)Jko.ij(a)du 0 where (5.3.29a) - 76 - (5.3.29b) for the free energy to be a minimum. If the states of the whole system (before we integrate away the phonon variables) can be taken as real (which is true for the Hamiltonian associated with Lciect + Lw+ L1att ), then we find that RiJ (t ), the response of the coordinate :z; to a kick in the j direction at zero time, satisfies (5.3.30) Ri1(t) is given by (5.3.31) The same comments apply to Ro:i.j(t), the response function associated with the trial function S0 . (5.3.32) If we apply a weak external force 10 ncosc.>t to the electron ( n is a unit vector), so that there is is an extra term --y0 n·rcos(;Jt in the Hamiltonian, we find by an application of time dependent perturbation theory that the rate of energy absorption is proportional to ( at frequency /\ ) ~~n1si;(r..) i,j where (5 .3.33) - 77 (5.3 .34) If "' ( ) _ 1 ffJ iXa-rfo,iJ (-) f o,ii Ao. - Poe , dT 2rra a an7,nteger • Acx = -{3-, (5.3.35 ) then the analytic continuation off o.iJ (Xa) to general values of A may be obtained by methods discussed in section 2.3 or in the previous section 5.2. We now write down the cycle of equations for the solution of the variational problem. Starting with s 0 _ij ((.,,)) as in equation {3.2.11), we have (5.3.36) From this we construct ko.iJ (11) by means of (5 .3.29b). Then ... !o,ij(-iA) = 2j(ei'>•.s-1)Imko.ij(is) (5.3.37) 0 Finally, we recover so.i;((.,,)) from (5.3.38) the inverse_ referring to the matrix inverse off 0 (-ic.>)-c,.>2 . Now we turn to the second order corrections to the variational results . AB in chapter 3, the first order corrections vanish. First we extend the notation discussed on page 34 of the third chapter. Let Uo , h, f 0 , etc. stand for the matrices 9o.iJ, ~i• f o.ij, etc .. Extend the meaning of the product notation ab so that (ab ).,.p ,ik = LJJ~i ( IT-a I)bjl, ( Ia-p I) j 0 (5.3.39) - 78 - Using the expansion (3.1.17) -..•,i.th S 1 as given in (5.3.23) and applying (5.3.25) we find after some work that the free energy to second order is given by the first order free energy in (5.3.28) plus -!~J \ (gohgoh )-rr,ii + ~ B(gohgoko)rr,ii - T \ J P(T1a1 I T2a2) (5.3.40a) T1C1TzC2 where (5.3.40b) and where P is given by (5.3.40c) Now f O is varied while keeping h constant. 1Ne use the relation (5.3 .41 ) and for a term T which depends on D 0 's we use the analogue of (3.3 .14) 6T = J ~ o(D)6T .. (go6fo9c),a ,ii - f r5(D)6T .. ,gc6fo9o)rr.v TU ij O TC,t) TC:,ij (5.3.42) O TU,\] The contribution to the f series from su8h a term comes out to be (specifically the contribution to (/ ),,.o,ij ) - 79 - (5.3.43) After the second order contribution to the f ser ies is worked out, we specialize to the case that (see (3.3.16)) (5 .3.44) where ,c0 is given now by (5 .3.29b) . This is just the s tat em ent that f O is the variational/ 0 . Taking into account the vanishing of terms due to (5.3.44) and other cancellations discussed in chapter 3, we may now write down the second order contribution to the f series in the case that f O is given by the variational equations (5.3.29) and (5.3 .36) to (5.3 .38). (f 2)-ru,ij = -(1C2)-ru,ij + o( T-a) f (,c2)-rp,ij (5.3.45a) p and ,c2 is given by two terms. First we have where (5. 3.45c) (A 12\ ; = ½{Do.iJ ( IT-a' I)+ Do.iJ ( IT. - a I) - D o.if ( 1, - T' I) - Do,iJ ( i a-a' I)] - 80 - The second term is given by where now (5.3.45e) To evaluate ,c2 (it) we proceed as in chapter 4. The inte~s from O to {3 are replaced by integrals along a contour C which runs from O to {3 through it. The absolute values IT-a I are replaced by IT-a I c as defined in section 5.2 or in chapter 4. - B1 6. CONCLUSION In this thesis we have treated Frolich's polaron model by Feynman's path integral variational method and have calculated corrections to the variational approximation. Following Feynman, we first integrate away the lattice variables in a path integral for the free energy of the system and obtain an efl'ective imaginary time action for the electron alone. This effective action is then imitated by general quadratic trial action. The path integral variational inequality gives an upper bound on the true free energy of the system as a functional of the trial action. Minimizing this upper bound yields a set of variational equations for the functions which appear in the trial action. Rather than working with real time double path integrals as FHIP [ 4] do, we work with the imaginary time path integrals for the free energy and obtain results for real time functions by carrying out an analytical continuation. In / chapter 2 this analytical continuation is carried out by working with the absorption functions(:>..). s (X) is proportional to the rate of energy absorption by the electron in a weak spatially uniform time varying field of frequency X. Other functions of interest such as the response function of the electron R (t) may be expressed in terms of this absorption function. The method of working with imaginary time path integrals rather than real time double path integrals saves a great deal of computational labor. We obtain equations which may be solved numericaliy for the variational approximation s 0 (>--) to the absorption function. Some numerical results for s 0 (X) are p resented. We also compute corrections to s 0 (>--) which are second order L11. the difl'erence b etween the trial action and the true action. The same equations but different numeric al results for sc(>--) h ave been obtained by Klyukanov, }lu..1tyan, and Pokatilov [6] usiP..g a nonvariational method. Our numerical results disagree 'rvith theirs in that they obtain an - 82 - absorption function with two p eaks at, fo r example , a= 4 and temp eratures as high as f3 =4 ( {3- = tempera ture ) while we only h ave a single peak for a.. =4 1 even at zero temperature. We next compute the second order corrections to s 0 (t,) as a test of the accuracy of s 0 (11..) . We only obtain results for moderate a.. ( up to a = 5 ) . For these values of a the corrections e.re reasonable in comparis on to the second order corrections of the free energy. This is discussed more fully at the end of chapter four. It would be of interest to see how large the corr ections are for large values of ex. From other examples to which we have applie d the varia tional method, we expect that for large a, the corrections are of the same relative size as for moderate a. There are many problems in which much greater accuracy is needed than the variational method seems capable of attaining. It would be very useful to find ways of system atically increasing the accuracy of the variational method. At present it is a one shot approximation, and if the accuracy is insufficient, there is not very much which can be done to improve it. • In section 5 .2 we present a generalization of the method of analytic continuation used in chapter 2. This generalizat ion would allow us to use the path integral variational principle to choose the best trial action in the problem of a polaron in a space-time constant electric field (this field m ay be strong) discussed by Thornber and Feynman [8]. In their paper Thornber and Feynman lacked a principle by which to choose a quadratic action to im itate the true action. We h ave not yet a pplied the m ethod of s ection 5.2 t o this proble m of a polaron in a s trong c onstant electric field, n or have we invesUgated in wh at other problem s t his me t hod may be of u se. Also, the reasoning of section 5.2 is T:1is is not true fo;:- systems of a finite m1rr1cer of degrees o-; freedorr. at nonzero te::r.perat.1.ll"e. In that case there ex:;sts a se:ies of :::etter a..--i.d :::e~.. er ve.riatio:iU: beq:.iS.:ities. - 83 very nonrigorous and it is necessary to investigate under ·what conditions the methods discussed there might fail. In section 5.3 we present the variational equations and the second order corrections to them for a more realistic version of an electron in a crystal lattice. We retain the assumplion that the lattice is harmonic. Within the frame work of this assumption the biggest flaw in the treatment of section 5.3 is that we cannot deal effectively "l'r'ith the periodicity of the crystal lattice. It would be useful to find a trial action which could in some way imitate the effect of periodic crystal potential V(r) in which the electron moves. - 84 - References [ 1] R. P. Feynman, Phys. Rev. 97, 660 (1955) [2] H. Frolich, Adv. in Phys. 3, 325 (1954) [3] Y. Osaka, Prog. Theor. Phys. (Kyoto) 22, 437 (1959) [ 4] R. P . Feynman, R. vr Hellworth, C. K. Iddings, P. M. Plalzman, Phys. Rev. 127, 1004 (1962) [5] J. T. Marshall, L. R. Mills, Phys. Rev. B 2, 3143 (1970) [6] A. A. Klyukanov, A. P. Muri.tyan,-E. P. Pokatilov. Sov. Phys. Solid State 22, 750 (1980} [7] R. P. Feynman and A. Hibbs, Quantum Mechanics and Path Integrals McGraw Hill, 1965. [8] K. K. Th.ornber, R. P. Feynman, Phys. Rev. B 1, 4099 (1970) [9] R. P. Feynman, Statistical Mechanics: A Set of Lecture Notes, W. A. Benjamin, 1972. [ 10] J .Devreese, J. De Sitter, M.Goovaerts, Phys. Rev. B 5, 2367 (1972)