Spectral functions of the spinless Holstein model

1.4 MB·English

by  J. Loos

6
0 Spectral functions of the spinless Holstein model
0
2

n J Loos†, M Hohenadler‡, and H Fehske§
a †InstituteofPhysics,AcademyofSciences oftheCzechRepublic,Prague
J
‡InstituteforTheoretical andComputational Physics,TUGraz,Austria

1 §InstituteforPhysics,Ernst-Moritz-ArndtUniversityGreifswald,Germany
3
E-mail: [email protected]

]
el Abstract. An analytical approach to the one-dimensional spinless Holstein
- modelisproposed,whichisvalidatfinitecharge-carrierconcentrations. Spectral
r functionsofchargecarriersarecomputedonthebasisofself-energycalculations.
t
s AgeneralizationoftheLang-Firsovcanonicaltransformationmethodisshownto
. provideaninterpolationschemebetweentheextremeweak–andstrong-coupling
t
a cases. Thetransformationdependsonavariationallydeterminedparameterthat
m characterizes the charge distribution across the polaron volume. The relation
betweenthespectralfunctionsofpolaronsandelectrons,thelattercorresponding
- to the photoemission spectrum, is derived. Particular attention is paid to the
d
distinction between the coherent and incoherent parts of the spectra, and their
n
evolutionasafunctionofbandfillingandmodelparameters. Resultsarediscussed
o
andcomparedwithrecentnumericalcalculationsforthemany-polaronproblem.
c
[

2
PACSnumbers: 71.27.+a,63.20.Kr,71.10.Fd,71.38.-k,71.10.-w
v
2
1 1. Introduction
7
8
Experiments on a variety of novel materials, ranging from quasi-one-dimensional
0
(1D) MX solids [1], organics [2] and quasi-2D high-T cuprates [3] to 3D colossal-
5 c
0 magnetoresistivemanganites[4,5],provideclearevidencefortheexistenceofpolaronic
/ carriers, i.e., quasiparticles consisting of an electron and a surrounding lattice
t
a distortion. This has motivated considerable theoretical efforts to achieve a better
m
understanding of strongly coupled electron-phonon systems in the framework of
- microscopic models.
d
Unfortunately, even for highly simplified models, such as the spinless Holstein
n
o model [6] considered here, no exact analytical solutions exist, except for the Holstein
c polaron problem with a relativistic dispersion [7] or in infinite dimensions [8]. As
: a consequence, numerous numerical studies have been carried out, focussing either
v
i on the empty band limit (i.e., one or two electrons only; see [9–13] and references
X
therein), oron the half-filled bandcase in one dimension, where the Peierlstransition
r takes place [14]. In contrast, very little work has been done at finite carrier densities
a
away from half filling [15,16], which are, however, often realized in experiment [1–5].
Recently, this so-called many-polaron problem has been addressed numerically [17–
19]. The results have led to a fairly good understanding of many aspects, but their
interpretationisnotalwaysstraightforward,whichmakesanalyticalcalculationsalong
these lines highly desirable.
Spectral functions of the spinless Holstein model 2
In this paper, we propose an analytical approach to the 1D spinless Holstein
model, capable of describing finite charge-carrier concentrations at arbitrary model
parameters, including the important adiabatic intermediate-coupling (IC) regime.
First,usingstandardperturbationtheorybasedontheself-energycalculation,the
spectral functions of charge carriers will be determined in the weak-coupling (WC)
and the strong-coupling (SC) limits at zero temperature (T = 0). In the SC regime,
the relation between the spectral function of polarons, determining the equilibrium
properties(inparticular,thechemicalpotential),andtheelectronicspectralfunction,
determining the photoemission spectrum, will be discussed. Special emphasis will be
laid on the distinction between the coherent and the incoherent parts of the spectra,
which may be calculated separately within the present approach.
Furthermore, using a generalizationof the Lang-Firsov canonical transformation
method [20], an interpolation scheme between the extreme WC and SC cases will be
proposed. In particular, the canonical transformation will depend on the distance
R characterizing the charge distribution across the polaron volume. For a given
set of model parameters and carrier concentration n, R will be determined from
the minimum of the total energy given by the transformed Hamiltonian in the
first, Hartree-like approximation. With R found in this way, the polaronic and
electronicspectralfunctionswillbecalculatedtostudytheirdependenceonthemodel
parametersandthecarrierdensity. Theresultswillbediscussedwithregardtorecent
numerical calculations [18,19], which have revealed a cross-over from a system with
polaronic carriers to a rather metallic system with increasing band filling n in the
intermediate electron-phonon coupling regime.
The paper is organized as follows. In section 2.1 we introduce the model used,
whereas in section 2.2 we derive the analytical results for WC, SC and IC regimes.
Our results are discussed in section 3, and section 4 contains our conclusions.
2. Theory
2.1. Model
In this paper, we are exclusively concerned with the Holstein model (HM) of spinless
fermions,whichdescribes electronscoupledlocally to Einsteinphonons. Althoughwe
shall use different canonical transformations to describe the IC and SC regime later
on, the Hamiltonian can be written in the general form
H =η c†c C c†c +ω (b†b + 1), (1)
i i − ij i j 0 i i 2
i i,j i
X X X
where the definition of η and C will be different depending on the approach used.
ij
Here c† (c ) creates (annihilates) a spinless fermion at site i, b† and b are bosonic
i i i i
operators for the dispersionless phonons of energy ω (~ = 1), and the strength of
0
the electron-phonon interaction is specified by the dimensionless coupling constants
λ=E /2tandg = E /ω intheadiabatic(ω /t 1)andanti-adiabatic(ω /t 1)
p p 0 0 0
≪ ≫
regimes,respectively,whereE isthewell-knownpolaronbindingenergyintheatomic
p p
limit [C =0 for i=j in equation (1)].
ij
6
In the WC case, in which we use the original, untransformed Holstein
Hamiltonian, we have η = µ, where µ denotes the chemical potential, and non-zero
−
coefficients
C =gω (b†+b ), C =t. (2)
ii 0 i i hiji
Spectral functions of the spinless Holstein model 3
In contrast, the starting point in the SC regime will be the Hamiltonian with
η = g2ω µ= E µ and
0 p
− − − −
Cii =0, Chiji =te−g(b†i−bi−b†j+bj). (3)
2.2. Green functions approach
WetreattheHM(1)usingtheformalismofthegeneralizedMatsubaraGreenfunctions
introduced by Kadanoff and Baym [21] and Bonch-Bruevich and Tyablikov [22], and
applied to the single-polaron problem by Schnakenberg [23]. The Green function
equation of motion deduced from H may be converted into an equation for the self-
energyofthespinlessfermions,andcanbesolvedbyiteration(see[23–25]fordetails).
In the second iteration step, the self-energy is obtained in the form
Σ(j τ ;j τ )= C δ(τ τ )
1 1 2 2 −h j1j2i 1− 2
+ GM(j′τ ;j′′τ )[ T C (τ )C (τ ) C C ], (4)
1 2 h τ j1j′ 1 j′′j2 2 i−h j1j′ih j′′j2i
j′j′′
X
where GM(j τ ;j τ ) represents the first-order fermionic Matsubara Green function,
1 1 2 2
andthesymbolT denotesthetimeorderingoperatoractingontheimaginarytimesτ .
τ i
Fourier-transformingbothsidesofequation(4),carryingoutthestandardsummation
over the Matsubara boson frequencies [26], and using the analytical continuation in
thecomplexfrequencyplane,theretardedmomentum–andenergy-dependentfermion
Green function follows as
1
GR(k,ω)= , (5)
ω (ξ +η) Σ(k,ω)
k
− −
where ξ is the fermionic band dispersion in the first approximation, ω = ω + iǫ
k
(ǫ 0+), andΣ(k,ω)denotesthe Fouriertransformofthe collisionalpartofthe self-
→
energygivenbythesecondtermonther.h.sofequation(4). Therelated,normalized
spectral function is given by
1
A(k,ω)= ImGR(k,ω+i0+). (6)
−π
AlthoughtheinsulatingPeierlsphasewithlong-rangecharge-density-waveorder,
which is the ground state of the half-filled spinless Holstein model above a critical
coupling strength depending on ω [14], can in principle be incorporated into the
0
present theory, this has not been done here, as we are interested in intermediate
band fillings away from n = 0.5. Furthermore, we neglect any extended pairing for
0 < n < 0.5, as well as the possible formation of a polaronic superlattice in the half-
filled band case (n = 0.5). Finally, the possibility of phase separation, which can in
principle occur in the present model, is not considered.
2.2.1. Weak coupling In the WC limit the self-energy is given by
ω E W dξ 1 n (ξ µ) n (ξ µ)
0 p F F
Σ(k,ω)= − − + − , (7)
πW Z−W 1−(ξ/W)2 (cid:20)ω−ω0−(ξ−µ) ω+ω0−(ξ−µ)(cid:21)
with the Fermi functiopn nF(ω) = (eβω +1)−1 and the bare bandwidth W = 2t. At
T =0, and defining ξ =max(µ, W), we get
0
−
ω E ξ0 dξ 1
0 p
ReΣ(k,ω)=
πW "PZ−W 1−(ξ/W)2ω+ω0−(ξ−µ)
p
Spectral functions of the spinless Holstein model 4
W dξ 1
+ (8)
PZξ0 1−(ξ/W)2ω−ω0−(ξ−µ)#
and p
ω E ξ0 dξ
0 p
ImΣ(k,ω)= δ[ω+ω (ξ µ)]
0
− W "Z−W 1−(ξ/W)2 − −
W pdξ
+ δ[ω ω (ξ µ)] . (9)
0
Zξ0 1−(ξ/W)2 − − − #
Inthe sequel,weshpalldistinguishbetweencoherentandincoherentcontributions
to the single-particle spectral functions, as defined by a zero and non-zero imaginary
part of the self-energy, respectively. The coherent part of the spectrum is given by
Ac(k,ω)=z δ[ω (E µ)], (10)
k k
− −
where the renormalized band energy is the solution of
ω E ξ0 dξ 1
0 p
E =ξ +
k k
πW "PZ−W 1−(ξ/W)2Ek+ω0−ξ
W p dξ 1
+ , (11)
PZξ0 1−(ξ/W)2Ek−ω0−ξ#
with ξk = Wcosk (thepbare band dispersion), and the spectral weight takes the
−
form
ω E ξ0 dξ 1
z−1 = 1+ 0 p
k (cid:12) πW "PZ−W 1−(ξ/W)2(Ek+ω0−ξ)2
(cid:12)
(cid:12) W p dξ 1
(cid:12)
+ . (12)
PZξ0 1−(ξ/W)2(Ek−ω0−ξ)2#(cid:12)
(cid:12)
For the incoherent paprt of the spectral function we fi(cid:12)nd
(cid:12)
Aic(k,ω < ω )= 1 ω0Ep W2−(ω+ω0+µ)2 21 −ξ0W dξδ(ω+ω0+µ−ξ) (13)
− 0 π[W2 (ω(cid:2) +ω0+µ)2](ω+µ(cid:3) ξRk ReΣ(k,ω))2+(ω0Ep)2
− − −
and
Aic(k,ω >ω )= 1 ω0Ep W2−(ω−ω0+µ)2 21 ξW0 dξδ(ω−ω0+µ−ξ) . (14)
0 π[W2 (ω(cid:2) ω0+µ)2](ω+µ (cid:3)ξkR ReΣ(k,ω))2+(ω0Ep)2
− − − −
Finally, the chemical potential µ for a given electron density n is determined by
1 ∞
dωA(k,ω)n (ω)=n. (15)
F
N
k Z−∞
X
2.2.2. Strong coupling Hamiltonian (1) with the coefficients (3) represents the
Hamiltonian of small polarons, which are the correct quasiparticles in the SC limit.
Using the procedure outlined above, we obtain the polaron self-energy as
W W dξf(k,ξ,s) 1 n (ξ+η) n (ξ+η)
F F
Σ(k,ω)= − + . (16)
2π s≥1Z−fW 1 (ξ/W)2 (cid:20)ω−sω0−(ξ+η) ω+sω0−(ξ+η)(cid:21)
f X −
f q
f
Spectral functions of the spinless Holstein model 5
At T =0, we have W =We−g2,
W W dξ
ReΣ(k,ω)= f f(k,ξ,s)
2π s≥1PZ−fW 1 (ξ/W)2
f X −
θf(ξq+η) θ( ξ η)
f+ − − (17)
× ω sω (ξ+η) ω+sω (ξ+η)
(cid:20) − 0− 0− (cid:21)
and
W W dξ
ImΣ(k,ω)= f(k,ξ,s) θ(ω sω )δ[ω sω (ξ+η)]
0 0
− 2 s≥1Z−fW 1 (ξ/W)2 { − − −
f X −
+θ( ω fsqω )δ[ω+sω (ξ+η)] . (18)
0 0
− − f − }
Here θ(x) is the Heaviside step function and we have used the definition
(2g2)s (g2)s ξ 2 (g2)s
f(k,ξ,s)= + 2 1 + cos(2k). (19)
s! s! " (cid:18)W(cid:19) − # s!
The coherent part of the spectrum, non-zero for ω <ω , is given by
0
f| |
Ac(k,ω)=z δ[ω (E +η)] (20)
k k
−
with the renormalized band energy E being the solution of
k
E =ξ +ReΣ(k,E +η), (21)
k k k
where ξ = W cosk, and the spectral weight takes the form
k
−
W W dξ
z−1 = 1+ f f(k,ξ,s)
k (cid:12) 2π s≥1PZ−fW 1 (ξ/W)2
(cid:12) f X −
(cid:12)(cid:12) fθ(ξq+η) θ( ξ η)
f+ − − . (22)
× (E sω ξ)2 (E +sω ξ)2
(cid:20) k− 0− k 0− (cid:21)(cid:12)
(cid:12)
The imaginary part of the self-energy, determining t(cid:12)he incoherent excitations, is
(cid:12)
non-zero only for ω >ω . Consequently, we get
0
| |
W f∓(k,ξ,s) W
ImΣ(k,ω ≶ ω )= θ( ω sω ) dξδ(ω sω η ξ), (23)
∓ 0 − 2 s≥1 ∓ − 0 (Xs±)12 Z−fW ± 0− −
f X
with f
2
ω sω η
X± =1 ± 0− . (24)
s − W
(cid:18) (cid:19)
Ifω >2W,foragivenω,onlyonetermofthesuminequation(23)contributes. The
0
f
corresponding index σ is determined by the conditions
f
W
θ( ω σω )=1 and dξδ(ω σω η ξ)=1 (25)
0 0
− Z−fW − − −
or
f
W
θ( ω σω )=1 and dξδ(ω+σω η ξ)=1. (26)
0 0
− − Z−fW − −
f
Spectral functions of the spinless Holstein model 6
The incoherent spectrum then consists of non-overlapping parts:
Aic(k,ω ≶ ω )= W (Xσ±)21f∓(k,ω,σ) . (27)
∓ 0 2π 2
X±(ω ξ η ReΣ(k,ω))2+ Wf∓(k,ω,σ)
f σ − k− − 2
(cid:20) (cid:16)f (cid:17) (cid:21)
Here we have defined
f±(k,ω,s)=f(k,ξ =ω sω η,s). (28)
0
∓ −
The polaron spectral function determines the equilibrium properties of the
spinless HM in the SC regime. The photoemission spectra, however, are
determined by the electron spectral function which is related to the retarded Green
function containing electronic operators. According to the canonical Lang-Firsov
transformation [20], which defines small polaron states, the relation between the
polaronic operators c entering the Hamiltonian (1) with the coefficients (3), and
i
the transformed electron operators c , reads
i
c =exp[g(b† b )]c , c† =exp[ g(b† b )]c† (29)
j j − j j j − j− j j
or, in the Bloch representation, e
1 e e 1
c = e−ikRjexp[g(b† b )]c , c† = eikRjexp[ g(b† b )]c†. (30)
k √N j − j j k √N − j − j j
j j
X X
e We start our derivation of the relaetion between the polaronic and electronic
spectra from the time-ordered Green function [26] for the (transformed) electron
operators
GT(k,t ,t )= i T c (t ) c†(t ) = i eikd T c (t )c† (t ) (31)
1 2 −h t k 1 k 2 i − h t j 1 j+d 2 i
d
X
aend factorize the staetisticaleaverages with respecteto polearon and phonon variables
GT(k,t1,t2)= eikdGT(t,d) Tteg[b†j(t)−bj(t)]e−g[b†j+d(0)−bj+d(0)] , (32)
h i
d
X
weheret=t t andGT(d,t)= i T c (t)c† (0) representsthetime-orderedGreen
1− 2 −h t j j+d i
function of polaron operators fulfilling
GT(d,t)=N−1 e−ik′dGT(k′,t), GT(k′,t)= i T c (t)c† (0) . (33)
−h t k′ k′ i
k′
X
The averages over the phonon variables in equation (32) will be evaluated using
mutually independent local Einstein oscillators having the time-dependence b (t) =
j
e−iω0tbj. Working in the low-temperature approximation we obtain
GT(k,ω)=e−g2GT(k,ω)+e−g2 (g2)s 1 (34)
s! N
s≥1
X
e ∞ 0
dtG>(k′,t)ei(ω−sω0+iǫ)t+ dtG<(k′,t)ei(ω+sω0−iǫ)t
×
k′ (cid:20)Z0 Z−∞ (cid:21)
X
with G>(k,t) = i c (t)c†(0) and G<(k,t) = i c†(0)c (t) , and the convergence
−h k k i h k k i
factor exp( ǫt), ǫ 0+. Introducing the generalized function ζ(ω) = [ω +iǫ]−1,
− | | →
ǫ 0+, we get
→
GT(k,ω)=e−g2GT(k,ω)+e−g2 (g2)s 1 ∞ dω′G>(k′,ω′)
s! N
s≥1 k′ (cid:20)Z−∞
X X
e ∞
iζ(ω sω ω′) dω′G<(k′,ω′)iζ∗(ω+sω ω′) . (35)
0 0
× − − − −
Z−∞ (cid:21)
Spectral functions of the spinless Holstein model 7
The Green functions G≷(k,ω) are related to the polaron spectral function A (k,ω)
p
through
G<(k,ω)=in (ω)A (k,ω), G>(k,ω)= i[1 n (ω)]A (k,ω). (36)
F p F p
− −
At T =0, we have
GT(k,ω)=e−g2GT(k,ω)+e−g2 (g2)s 1 ∞dω′A (k′,ω′)ζ(ω sω ω′)
p 0
s! N − −
s≥1 k′ (cid:20)Z0
X X
e 0
+ dω′A (k′,ω′)ζ∗(ω+sω ω′) . (37)
p 0
−
Z−∞ (cid:21)
The relation between the time-ordered Green function GT(k,ω) and the associated
retarded Green function GR(k,ω) in the low-temperature approximation reads
ω
ImGR(k,ω)= ImGT(k,ω) (38)
ω
| |
and hence we obtain
1 ω
A (k,ω)= ImGT(k,ω). (39)
p
−π ω
| |
Of course, equations (38) and (39) hold for the electron Green function as well.
Consequently, the electron spectral function A (k,ω) is expressed in terms of the
e
polaron spectral function A (k,ω) as
p
A (k,ω)=e−g2A (k,ω)+e−g2 1 (g2)s
e p
N s!
s≥1
X
[A (k′,ω sω )θ(ω sω )+A (k′,ω+sω )θ( ω sω )] . (40)
p 0 0 p 0 0
× − − − −
k′
X
Similar results have been derived before in [27,28].
2.2.3. Intermediate coupling Asweshallseeinsection3,theresultsofthe WC(SC)
approximation are in good agreement with numerical calculations [18,19] if λ,g 1
≪
(λ,g 1). However, the cross-over between these limiting cases, revealed by the
≫
numericalcalculations,appearstobe outofreachforthe analyticalformulaehitherto
deduced.
To interpolate between WC and SC, we shall modify the method of canonical
transformation by Lang and Firsov [20]. In the latter, the term of the HM (1) linear
in the local oscillator coordinate x = x (b† +b ) is completely eliminated by the
i 0 i i
translational transformation U =exp[ gc†c (b† b )]. As a result, the local lattice
i i i i − i
oscillator at the site i is shifted by ∆x = 2x gc†c , if occupied by a charge carrier,
Pi 0 i i
whereas there is no such deformation at unoccupied sites. To generalize this picture,
weabandonthesitelocalizationofboththechargecarrierandthelatticedeformation
inthetransformation. Physically,theselocalizationswillbedestroyedwithincreasing
hopping rate and charge-carrier concentration in the IC regime. Different canonical
transformations,taking into accountcharge-density-waveorder at n=0.5, have been
proposed, e.g., by Zheng et al [29]. However, in this approach, there is no filling
dependence oftheir variationalparameterandofthe meanlattice deformation,which
is crucial for a correct description of the adiabatic IC regime.
The probability for the charge carrier to be found at a distance r r from
j i
| − |
the center r of the polaron will be assumed to be proportional to exp( r r /R).
i j i
−| − |
Spectral functions of the spinless Holstein model 8
In one dimension, and setting the lattice constant to unity, we use the normalized
distribution
p(j i)=e−|jR−i| tanh 1 . (41)
| − | 2R
Accordingly, the shift of the local oscillator with coordinate x is assumed to be
i
∆x =2x gKc†c +2x gγ(1 c†c ) (42)
i 0 i i 0 − i i
with
1
K =tanh , γ =2Ke−R1n. (43)
2R
The last term in equation (42), characterizing the mean lattice deformation
background, takes into account the influence of nearest-neighbor sites only.
The canonical transformation leading to the oscillator shift (42) reads
U =exp g(γc†c +γ)(b† b ) , γ =K γ. (44)
" i i i − i # −
i
X
Carrying out the transformation U†HU = H for the Hamiltonian (1) with (2), the
terms of H containing polaron operators are modified with the following coefficients
[cf. equations (2) and (3)] e
e
Cii =gω0(1−γ)(b†i +bi), Chiji =te−γg(b†i−bi−b†j+bj). (45)
Moreover,we have η = µ E [γ(2 γ)+2γ(1 γ)].
p
− − − −
Owing to numerical problems which occur in certain parameter regimes when
using the full Hamiltonian with the coefficients (45), the variational parameter R of
the transformation will be determined in the first approximation, which is analogous
to the Hartree approximation. The corresponding polaron spectral function is given
as A (k,ω) = δ[ω (ξ +η)], where ξ = Wcosk with W = W exp[ (γg)2], and
p k k
− − −
η as defined above. R is then defined by the position of the minimum of the total
energy per site E/N in the first approximatiofn, i.e., f
E 1 W ξ+η
= dξ θ( ξ η)+µn+E γ2, (46)
p
N πW Z−fW 1 (ξ/W)2 − −
−
with the condition for µ f q
f
f
1 W θ( ξ η)
dξ − − =n. (47)
πW Z−fW 1 (ξ/W)2
−
f q
The last term in efquation (46) arises in H from the lattice deformation background
f
at finite concentration n.
Rdeterminedinthiswayforeachsetoefmodelparameterswillbeusedtocalculate
boththeelectronandpolaronspectralfunction,takingintoaccountthemulti-phonon
processes included in H [cf. equation (45)].
e
Spectral functions of the spinless Holstein model 9
Using the same procedure as in the SC case, we calculate the self-energy and
spectral function at T =0, finding
W W dξ
ReΣ(k,ω)= f(k,ξ,s)
2π s≥1PZ−fW 1 (ξ/W)2
f X −
θf(ξq+η) θ( ξ η)
f+ − −
× ω sω (ξ+η) ω+sω (ξ+η)
(cid:20) − 0− 0− (cid:21)
E W dξ ξ
p
2γ(1 γ) +cosk
− − π PZ−fW 1 (ξ/W)2 (cid:18)W (cid:19)
−
θ(ξ+η) f q θ( ξ η)
−f− f
× ω ω (ξ+η) − ω+ω (ξ+η)
(cid:20) − 0− 0− (cid:21)
E ω W dξ
+(1 γ)2 p 0
− πW PZ−fW 1 (ξ/W)2
−
θ(ξ+η) f q θ( ξ η)
f + −f− (48)
× ω ω (ξ+η) ω+ω (ξ+η)
(cid:20) − 0− 0− (cid:21)
and
W f∓(k,ω,s) W
ImΣ(k,ω ≶ ω )= θ( ω sω ) dξδ(ω sω η ξ)
∓ 0 − 2 s≥1 ∓ − 0 (Xs±)12 Z−fW ± 0− −
f X
∓2γ(1−γ)Ep(X1±)−12 ω±Wω0−η +cosfk
(cid:18) (cid:19)
W
θ( ω ω0) dξδ(ω fω0 η ξ)
× ∓ − Z−fW ± − −
E ω W
−(1−γ)2 Wp 0(Xf1±)−12θ(∓ω−ω0)Z−fW dξδ(ω±ω0−η−ξ). (49)
Here W = We−(γg)2 and f, f± are defined as in equations (19) and (28) but with g
f
f
replacedbygγ. NotethattheaboveequationsreproducethecorrespondingSCresults
in theflimit R=0 (γ =1), and also the WC ones in the limit R= (γ =0).
∞
Thecoherentspectrumisagaingivenbyequations(20)and(21),withthespectral
weight
z−1 = 1 [∂ReΣ(k,ω)/∂ω] , (50)
k | − ω=Ek+η|
whereas the incoherent part takes the familiar form
1 ImΣ(k,ω)
Aic(k,ω)= . (51)
e −π[ω (ξ +η) ReΣ(k,ω)]2+[ImΣ(k,ω)]2
k
− −
Finally, equation (40), which determines the relation between the electronic and
polaronic spectral functions, also applies to the present case if g is replaced by gγ
throughout.
3. Numerical results
As in section 2.2, we first discuss the WC and SC limits, before turning to the
important IC regime. Since we use a finite number of momenta k, it is not possible
Spectral functions of the spinless Holstein model 10
AAiicc((kk,,ωω)),, Ac(k,ω) AAiicc((kk,,ωω)),, Ac(k,ω)
(a) n=0.1 (b) n=0.2
14 14
12 12
10 10
8 8
6 6
4 4
2 2
0 0
0.2 0.2
0.4 0.4
k / π k / π
0.6 0.6
0.8 0.8
1 -2 -1 0 1 2 3 4 5 1 -2 -1 0 1 2 3 4 5
ω / t ω / t
AAiicc((kk,,ωω)),, Ac(k,ω) AAiicc((kk,,ωω)),, Ac(k,ω)
(c) n=0.3 (d) n=0.4
14 14
12 12
10 10
8 8
6 6
4 4
2 2
0 0
0.2 0.2
0.4 0.4
k / π k / π
0.6 0.6
0.8 0.8
1 -2 -1 0 1 2 3 4 5 1 -2 -1 0 1 2 3 4 5
ω / t ω / t
Figure1. (colouronline)Coherent(Ac,----)andincoherent(Aic,——)parts
of the spectral function in the weak-coupling approximation for different band
fillingsn. Hereω0/t=0.4andEp/t=0.1.
to tune the band filling n (via the chemical potential µ) to a specific, desired value
with arbitrary accuracy. In order to simplify the discussion of the different density
regimes, we therefore report rounded values of n in the figures and the text. The
largest deviations of the actual n from the value reported occur in the SC case for
which, however,the density dependence is very weak (see section 3.2).
3.1. Weak coupling
Figures 1–3 show the electronic spectral function A(k,ω) obtained from the WC
approximation. The coherent spectrum (ImΣ 0) is given as the solution of
≡
equations (10)–(12) with energies E µ < ω . For ω > ω , Aic, calculated
k 0 0
| − | | |
according to equations (13) and (14), consists of peaks having widths proportional
to E . A comparison of figures 1 (a) and 2(a) shows the spreading of the coherent
p
spectrum with increasing ω . Finally, comparing figure 1(d) [2(b)] with figure 3(a)
0
[3(b)], we observe a broadening of the peaks in Aic with increasing E .
p
The evolution of the spectral function with increasing carrier density n is
illustrated in figure 1(a)–(d). The coherent part is shifted inside the spectrum as
a function of n (i.e., with µ). Additionally, the shape of Aic is also affected by n, due
to the dependence of equations (13) and (14) on the chemical potential µ.
In figure 4, we plot the total spectral weight dk dωA(k,ω), as well as the
coherent weight dk dωAc(k,ω). Note that for any E > 0, the coherent band E
p k
R R
is restricted to the interval [ ω ,ω ], so that the corresponding coherent weight is
0 0
R R −