Alexandrov A.S.,Theory of Superconductivity - From Weak to Strong Coupling

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106

Strong-coupling theory

TTC

NTTC

∼γ





Figure 4.3. Two-phonon scattering responsible for the damping of the polaron band.

Moreover, the polaron bandwidth shrinks with increasing temperature because the

phonon-averaged hopping integrals depend on temperature (appendix E):

ˆσ



= T (m − n)δ



exp



−



q,ν

|γ(q,ν)|

[1 −cos(q · m)]coth

qν



For high temperatures (T  ω

/2) the band narrows exponentially:

w ≈ De

−T / T

(4.42)

where

−1



q,ν

|γ(q,ν)|

−1

qν

[1 −cos(q · m)].

However, the two-phonon scattering of polarons (ﬁgure 4.3) becomes more

important with increasing temperature.

We can estimate the scattering rate by applying the Fermi–Dirac golden rule:

= 2π



q,q



δ(

− 

k+q−q



)

(4.43)

where the corresponding matrix element is





i, j

k + q − q



, n

− 1, n



+ 1|ˆσ

i, j

†

|k , n

, n



.

For simplicity we drop the phonon branch index ν and consider the momentum

independent γ(q) = γ

and ω

= ω

. Expanding ˆσ

-operators in powers of the

phonon creation and annihilation operators, we estimate the matrix element of the

two-phonon scattering as



≈

wγ





+ 1). (4.44)

Using this estimate and the polaron density of states,

(ξ) ≡



δ(ξ − 

) ≈

(4.45)

Polaron dynamics

107

we obtain

≈ wγ

(1 + n

) (4.46)

where n

=[exp(ω

/T ) −1]

−1

is the phonon distribution function. The polaron

band is well deﬁned if

<w (4.47)

which is satisﬁed for a wide temperature range

T ≤ T

min

≈

ln γ

(4.48)

about half of the characteristic phonon frequency for relevant values of γ

Phonons dominate in the scattering at ﬁnite temperatures if the number of

impurities is sufﬁciently low. Therefore, the polaron mobility decreases when the

temperature increases from zero to T

min

due to an increasing number of phonons.

At higher temperatures, the incoherent thermal activated hopping dominates in

the polaron dynamics [66–68, 71] and the polaron states are no longer the Bloch

states. Hence, the mobility increases above T

min

, where it is at minimum, due

to thermal activated hopping. There is a temperature range around T

min

where

the thermal activated hopping still makes a small contribution to the conductivity

but the uncertainty in the polaron band is already signiﬁcant [71]. The polaron

transport theory requires a special diagrammatic technique in this region [74,75].

The optical phonon frequencies are exceptionally high, about 1000 K or even

higher and polarons are in Bloch states in the whole relevant range of temperatures

in novel superconductors.

4.3.3 Small Holstein polaron and small Fr

ohlich polaron

The narrowing of the band and the polaron effective mass strongly depend on

the radius of the electron–phonon interaction [100]. Let us compare the small

Holstein polaron (SHP) formed by the short-range e–ph interaction and a small

polaron formed by the long-range (Fr¨ohlich) interaction, which we refer as the

small Fr¨ohlich polaron (SFP). Introducing a normal coordinate at site n as



(2NMω

)

−1/2

iq·n

+ H.c. (4.49)

and a ‘force’ between the electron at site m and the normal coordinate ξ

f (m) = N

−1



γ(q)(Mω

)

1/2

iq·m

(4.50)

we rewrite the e–ph interaction, equation (4.3), as

e−ph



n,i

f (m − n)ξ

ˆn

. (4.51)

108

Strong-coupling theory

For simplicity we consider the interaction with a single phonon branch and

γ(−q) = γ(q). In general, there is no simple relation between the polaron level

shift E

and the exponent g

of the mass enhancement. This relation depends on

the form of the electron–phonon interaction. Indeed, for dispersionless phonons,

= ω

, using equations (4.34) and (4.39) we obtain

2Mω



(m) (4.52)

and

2Mω



(m) − f (m) f (m + a)] (4.53)

where a is the primitive lattice vector. In the nearest-neighbour approximation,

the effective mass renormalization is given by

∗

/m = e

where m is the bare band mass and 1/m

∗

= ∂



/∂k

at k → 0istheinverse

polaron mass.

If the interaction is short range, f (m) = κδ

m,0

(the Holstein model), then

= E

/ω.Hereκ is a constant. In general, we have g

= γ E

/ω with the

numerical coefﬁcient

γ =

1 −



f (m) f (m + a)



(n)

(4.54)

which might be less than 1. To estimate γ , let us consider a one-dimensional chain

model with a long-range Coulomb interaction between the electron on the chain

(×) and ion vibrations of the chain (◦), polarized in a direction perpendicular to

the chains [94] (ﬁgure 4.4). The corresponding force is given by

f (m − n) =

(|m − n|

+ 1)

3/2

. (4.55)

Here the distance along the chains, |m − n|, is measured in units of the lattice

constant, a, the inter-chain distance is also a and we take a = 1. For this long-

range interaction, we obtain E

= 1.27κ

/(2Mω

), g

= 0.49κ

/(2Mω

) and

= 0.39E

/ω. (4.56)

Thus the effective mass renormalization is much smaller than in the Holstein

model, roughly m

∗

SFP

∝ (m

∗

SHP

)

1/2

Not only does the small polaron mass strongly depend on the radius of the

electron–phonon interaction but the range of the applicability of the analytical

1/λ expansion theory also does. The theory appears almost exact in a wide

region of parameters for the Fr¨ohlich interaction. The exact polaron mass in a

Polaron dynamics

109

7D

;;;;;

Figure 4.4. A one-dimensional model of a small polaron on the chain interacting with the

ion displacements of another chain.

wide region of the adiabatic parameter ω/T (a) and coupling was calculated with

the continuous-time path-integral quantum Monte Carlo (QMC) algorithm [94].

This method is free from any systematic ﬁnite-size, ﬁnite-time-step and ﬁnite-

temperature errors and allows for an exact (in the QMC sense) calculation of the

ground-state energy and the effective mass of the lattice polaron for any electron–

phonon interaction described by the Hamiltonian (4.51).

At large λ (>1.5), the SFP was found to be much lighter than the SHP, while

the large Fr¨ohlich polaron (i.e. at λ<1) was heavier than the large Holstein

polaron with the same binding energy (ﬁgure 4.5). The mass ratio m

∗

a non-monotonic function of λ. The effective mass of Fr

ohlich polarons, m

∗

(λ)

is well ﬁtted by a single exponent, which is e

0.73λ

for ω

= T (a) and e

1.4λ

for

ω = 0.5T (a). The exponents are remarkably close to those obtained with the

Lang–Firsov transformation, e

0.78λ

and e

1.56λ

, respectively. Hence, in the case

of the Fr¨ohlich interaction the transformation is perfectly accurate even in the

moderate adiabatic regime, ω/T (a) ≤ 1forany coupling strength. This is not the

case for the Holstein polaron. If the interaction is short range, the same analytical

technique is applied only in the non-adiabatic regime ω/T (a)>1.

Another interesting point is that the size of the SFP and the length over which

the distortion spreads are different. In the strong-coupling limit, the polaron is

almost localized on one site m. Hence, the size of its wavefunction is the atomic

size. In contrast, the ion displacements, proportional to the displacement force

f (m − n), spread over a large distance. Their amplitude at a site n falls with

distance as |m − n|

−3

in our one-dimensional model. The polaron cloud (i.e.

lattice distortion) is more extended than the polaron itself [66, 72, 100]. Such a

polaron tunnels with a larger probability than the Holstein polaron due to a smaller

relative lattice distortion around two neighbouring sites. For a short-range e–ph

interaction, the entire lattice deformation disappears at one site and then forms at

its neighbour, when the polaron tunnels from site to site. Therefore, γ = 1and

the polaron is very heavy already at λ ≈ 1. In contrast, if the interaction is long-

ranged, only a fraction of the total deformation changes every time the polaron

tunnels from one site to its neighbour and γ is smaller than 1. The fact that the

Lang–Firsov transformation is rather accurate for the long-range interaction in

a wide region of parameters, allows us to generalize this result. Including all

110

Strong-coupling theory

Figure 4.5. Inverse effective polaron mass in units of 1/m = 2T (a)a

phonon branches in the three-dimensional lattice, we obtain m

∗

SFP

= (m

∗

SHP

)

where the masses are measured in units of the band mass, and

γ =



q,ν

(q,ν)[1 − cos(q · m)]



q,ν

(q,ν)

. (4.57)

For the Fr¨ohlich interaction (i.e. γ(q)

1/q), we ﬁnd γ = 0.57 in a cubic

lattice and γ = 0.255 for the in-plane oxygen hole in cuprates (Part II). A lighter

mass of SFP compared with the non-dispersive SHP is a generic feature of any

dispersive electron–phonon interaction. As an example, a short-range interaction

with dispersive acoustic phonons (γ(q) ∼ 1/q

1/2

, ω

∼ q) also leads to a lighter

polaron in the strong-coupling regime compared with the SHP. Actually, Holstein

[68] pointed out in his original paper that the dispersion is a vital ingredient of the

polaron theory.

4.3.4 Polaron spectral and Green’s functions

The multi-polaron problem has an exact solution in the extreme inﬁnite-coupling

limit (λ =∞) for any type of e–ph interaction conserving the on-site occupation

numbers of electrons (equation (4.3)). For ﬁnite coupling, the 1/λ perturbation

expansion is applied. The expansion parameter is actually [72, 74,98, 99]

2zλ

 1

so that the analytical perturbation theory has a wider region of applicability

than one can expect using a semiclassical estimate E

> D.However,

Polaron dynamics

111

the expansion convergency is different for different e–ph interactions. Exact

numerical diagonalizations of vibrating clusters, variational calculations [83, 85,

87, 89, 92, 93], dynamical mean-ﬁeld approach in inﬁnite dimensions [91], and

quantum Monte Carlo simulations [94] revealed that the ground state energy

(≈−E

) is not very sensitive to the parameters. In contrast, the effective mass, the

bandwidth and the polaron density of states strongly depend on the adiabatic ratio

ω/T (a) and on the radius of the interaction. The ﬁrst order in 1/λ perturbation

theory is practically exact in the non-adiabatic regime (ω/T (a)>1) for any

value of the coupling constant and any type of e–ph interaction. However, it

overestimates the polaron mass by a few orders of magnitude in the adiabatic

case (ω/T (a)<1), if the interaction is short-ranged [83]. A much lower

effective mass of the adiabatic Holstein polaron compared with that estimated

using ﬁrst-order perturbation theory is the result of the poor convergency of the

perturbation expansion owing to the double-well potential [68] in the adiabatic

limit. The tunnelling probability is extremely sensitive to the shape of this

potential. However, the analytical theory is practically exact in a wider range of

the adiabatic parameter and of the coupling constant for the long-range Fr¨ohlich

interaction.

Keeping this in mind, let us calculate the one-particle GF in the ﬁrst

order in 1/λ. Applying the canonical transformation we write the transformed

Hamiltonian as

H = H

+ H

int

(4.58)

where



ξ(k)c

†

(4.59)

is the ‘free’ polaron contribution and



†

+ 1/2) (4.60)

is the free phonon part. For simplicity, we drop the spin and phonon branch

indexes. Here ξ(k) = Z



E(k) − µ is the renormalized polaron band dispersion.

The chemical potential µ includes the polaron level shift −E

. It might also

include all higher orders in 1/λ corrections to the spectrum independent of k.

E(k) =



T (m) exp(−ik · m) is the bare dispersion in the rigid lattice and





T (m)e

−g

(m)

exp(−ik · m)



T (m) exp(−ik · m)

(4.61)

is the band-narrowing factor (Z



= exp(−γ E

/ω)) as discussed earlier). The

interaction term H

int

comprises the polaron–polaron interaction, equation (4.32),

and the residual polaron–phonon interaction

p−ph

≡



i=j

[ˆσ

−ˆσ



†

. (4.62)

112

Strong-coupling theory

We can neglect H

p−ph

in the ﬁrst order in 1/λ  1. To better understand the

spectral properties of a single polaron, let us also neglect the polaron–polaron

interaction. If H

int

= 0, the energy levels are

{˜m}



+ 1/2] (4.63)

and the transformed eigenstates |˜m are sorted by the polaron Bloch-state

occupation numbers, n

= 0, 1, and the phonon occupation numbers, n

0, 1, 2,...,∞. The spectral function (equation (D.10)) is deﬁned by the matrix

element n|c

|m. It can be written as

n|c

|m=

√



−ik·m

˜n|c

|˜m (4.64)

by the use of the Wannier representation and the Lang–Firsov transformation.

Here

= exp





(q)d

− H.c.



Now, applying the Fourier transform of the δ-function in equation (D.10),

δ(ω

+ ω) =

2π



∞

−∞

dt e

i(ω

+ω)t

the spectral function is expressed as

A(k,ω) =



∞

−∞

dt e

iωt



m,n

ik·(n−m)

×{c

(t)

(t)c

†

 + c

†

(t)

(t)}. (4.65)

Here the quantum and statistical averages are performed for independent polarons

and phonons, therefore

c

(t)

†

 = c

(t)c

†



(t)

†

. (4.66)

The Heisenberg free-polaron operator evolves with time as

(t) = c

−iξ

(4.67)

and

c

(t)c

†

 =







i(k



·m−k



·n)

c



(t)c

†









[1 −¯n(k



)]e



·(m−n)−iξ



(4.68)

c

†

(t) =





¯n(k



·(m−n)−iξ



(4.69)

Polaron dynamics

113

where ¯n(k) =[1 + exp ξ

/T ]

−1

is the Fermi–Dirac distribution function of

polarons. The Heisenberg free-phonon operator evolves in a similar way,

(t) = d

−iω

and



(t)

†

 =



exp[u

(q, t)d

− H.c.]exp[−u

(q)d

− H.c.] (4.70)

where u

i, j

(q, t) = u

i, j

(q)e

−iω

. This average is calculated using the operator

identity (appendix E)

A+

= e

−[

B]/2

(4.71)

which is applied for any two operators

A and

B, whose commutator [

B] is a

number. Because [d

, d

†

]=1, we can apply this identity in equation (4.70) to

obtain

(q,t)d

−H.c.]

[−u

(q)d

−H.c.]

= e

(α

∗

†

−αd

)

× e

(q,t)u

∗

(q)−u

∗

(q,t)u

(q)]/2

where α ≡ u

(q, t) − u

(q). Applying once again the same identity yields

(q,t)d

−H.c.]

[−u

(q)d

−H.c.]

= e

∗

†

−αd

−|α|

× e

(q,t)u

∗

(q)−u

∗

(q,t)u

(q)]/2

. (4.72)

Now the quantum and statistical averages are calculated by the use of

(appendix E)

e

∗

†

−αd

 = e

−|α|

(4.73)

where n

=[exp(ω

/T ) − 1]

−1

is the Bose–Einstein distribution function of

phonons. Collecting all multiplies in equation (4.72), we arrive at



(t)

†

 = exp



−



|γ(q)|

(m − n, t)



(4.74)

where

(m, t) =[1 −cos(q · m) cos(ω

t)]coth

+ i cos(q · m) sin(ω

t). (4.75)

Here we have used the symmetry of γ(−q) = γ(q), and, hence, the terms

containing sin(q · m) have disappeared. The average 

†

(t), which is a

multiplier in the second term in the brackets of equation (4.65), is obtained by

replacing u

(q, t) u

(q) in the previous expressions. The result is



†

(t) = 

(t)

†



∗

. (4.76)

114

Strong-coupling theory

To proceed with the analytical results, we consider low temperatures, T  ω

when coth(ω

/2T ) ≈ 1. Then expanding the exponent in equation (4.74) yields



(t)

†

 = Z

∞



l=0

{



|γ(q)|

i[q·(m−n)−ω

}

(2N)

(4.77)

where

Z = exp



−



|γ(q)|



. (4.78)

Substituting equations (4.77) and (4.68) into equation (4.66) and performing

summation with respect to m, n, k



and integration with respect to time in

equation (4.65), we arrive at [101]

A(k,ω) =

∞



l=0

(−)

(k,ω)+ A

(+)

(k,ω)] (4.79)

where

(−)

(k,ω) = π Z



,...,q

r=1

|γ(q

(2N)



1 −¯n



k −



r=1





ω−



r=1

−ξ

k−



r=1



(4.80)

and

(+)

(k,ω) = π Z



,...,q

r=1

|γ(q

(2N)

×¯n



k +



r=1





ω+



r=1

−ξ



r=1



. (4.81)

Obviously, equation (4.79) is in the form of a perturbative multi-phonon

expansion. Each contribution A

(±)

(k,ω) to the spectral function describes the

transition from the initial state k of the polaron band to the ﬁnal state k±



r=1

with the emission (or absorption) of l phonons. The 1/λ expansion result

(equation (4.79)) is applied to low-energy polaron excitations in the strong-

coupling limit. In the case of the long-range Fr¨ohlich interaction with high-

frequency phonons, it is also applied in the weak-coupling and intermediate

regimes (section 4.3.3). Differing from the canonical Migdal GF (chapter 3),

there is no damping of polaronic excitations in equation (4.79). Instead the e–ph

coupling leads to the coherent dressing of electrons by phonons because of the

Polaron dynamics

115

energy conservation (section 4.3.2). The dressing can be seen as the phonon ‘side-

bands’ with l ≥ 1. While the major sum rule (equation (D.11)) is satisﬁed,



∞

−∞

dω A(k,ω) = Z

∞



l=0



,...,q

r=1

|γ(q

(2N)

= Z

∞



l=0





|γ(q)|



= Z exp





|γ(q)|



= 1 (4.82)

the higher-momentum integrals,



∞

−∞

dωω

A(k,ω)with p > 0, calculated using

equation (4.79), differ from the exact value by an amount proportional to 1/λ.

The difference is due to a partial ‘undressing’ of high-energy excitations in the

side-bands, which is beyond the ﬁrst-order 1/λ expansion.

The spectral function of the polaronic carriers comprises two different parts.

The ﬁrst (l = 0) k-dependent coherent term arises from the polaron band

tunnelling,

coh

(k,ω) =[A

(−)

(k,ω)+ A

(+)

(k,ω)]=π Z δ(ω − ξ

). (4.83)

The spectral weight of the coherent part is suppressed as Z  1. However,

in the case of the Fr¨ohlich interaction, the effective mass is less enhanced,

= Z



− µ, because Z  Z



< 1 (section 4.3.3). The second incoherent

part A

incoh

(k,ω) comprises all the terms with l ≥ 1. It describes the excitations

accompanied by emission and absorption of phonons. We note that its spectral

density spreads over a wide energy range of about twice the polaron level shift

, which might be larger than the unrenormalized bandwidth 2D in the rigid

lattice without phonons. In contrast, the coherent part shows a dispersion only

in the energy window of the order of the polaron bandwidth, 2w = 2Z



D.It

is interesting that there is some k dependence of the incoherent background as

well, if the matrix element of the e–ph interaction and/or phonon frequencies

depend on q. Only in the Holstein model with the short-range dispersionless

e–ph interaction (γ(q) = γ

and ω

= ω

) is the incoherent part momentum

independent. Replacing k ±



r=1

by k



in equations (4.80) and (4.81), we

obtain the following expression in this case:

incoh

(k,ω) = π

∞



l=1





{[1 −¯n(k



)]δ(ω − lω

− ξ



) +¯n(k



)δ(ω +lω

− ξ



)}

(4.84)