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The effect of recoil on edge singularities
P. Nozières
To cite this version:
P. Nozières. The effect of recoil on edge singularities. Journal de Physique I, EDP Sciences, 1994, 4 (9), pp.1275-1280. �10.1051/jp1:1994188�. �jpa-00246990�
Classification Physics Abstracts
71.45 72.15R 78.70D
The elllect of recoil on edge singularities
P. Nozières
Institut Laue-Langevin, BP. 156, 38042 Grenoble cedex 9, France
(Received 20 June 1994, accepted 7 July 1994)
Résumé Une perturbation locale transitoire induit une singularité de seuil infrarouge. Cette singularité disparaît si la perturbation peut diffuser à l'infini et si d > 1. Si d = 1, ou en présence de localisation si d > 1, les exposants caractéristiques sont réduits par le recul. Le facteur de
réduction fait intervenir une moyenne angulaire sur la surface de Fermi.
Abstract. A transient local perturbation gives rise to infrared edge smgularities. These
smgularities are shown to disappear if trie scatterer is free to diffuse to infinity, as long as the dimension exceeds 1. When d
= 1, or in trie presence of localization when d > 1, trie characteristic exponents are reduced by recoil, by a factor thon involves an angular average over
the Fermi surface.
The sudden application of a localized perturbation m metals leads to threshold singularities
which have been widely studied in a variety of situations. The standard example is X-ray
emission or absorption where edge singularities appear at the Fermi level il, 2], Similar elfects
occur for photoemission [3] or Rarnan scattering [4]. Another example is the hopping of a
scatterer between two sites, whether in a two level system or as part of a diffusion process: the
hop changes the potent1al felt by the Fermi sea hence a "polaronic" reduction of the coherent
hopping amplitude which is singular and con even lead to localization [Si.
Ail these elfects rely on a localized transient perturbation and indeed it is often claimed that the threshold exponent is related to trie charge that must be brought from infinity in order
to restore the ground state in the new potent1al. We are thus led to a crucial question: what
is trie elfiect of scatterer recoil on trie s1~lgularities ? Do the latter survive diffusion ? If they do, are the exponents modified by recoil ? Such issues are not pure semantics. Nowadays edge singularities are often invoked in trie framework of strongly correlated Fermi systems and high temperature superconductors. They are applied to the motion of a given electron coupled to its fellows. It is thus important to have a few solid statements that can be trusted.
A qualitative treatment of the effect of recoil was given long ago [6] in the framework of optical absorption in degenerate semiconductors. The electrons have a Fermi level p. measured from the bottom of the conduction bond. A photon creates a hole m the valence bond, via a
1276 JOURNAL DE PHYSIQUE I N°9
momentum conserving transition. (Here the hole is free, with a well defined k). In the absence of final state interactions, the threshold corresponds to k = kF, with a photon energy
wd=Eg+p+cr (1)
Eg is the gap and cr is the k1~letic recoil e~lergy of a fiole with mome~ltum kF. (Sec Fig. l). wd
is the so called "Burstein edge" where the absorption starts discontinuously. When final state interactions are included, such direct transitions are complemented by Auger type processes
m which further electron-hole pairs are excited in the conduction band. For dimensions > 2, such higher processes can take care of momentum conservation at no energy cost. The absoll~te
threshold wo therefore corresponds to a hole at the top of the conduction band, an amount er below wd (Fig. l). The edge singularity is bll~rred between the two thresholds wo and wd. As a function of the final state potent1al the spectrum evolves as sketched in figure 2: the Burstein
edge is Auger broadened and a peak develops which ultimately becomes the edge singularity if the recoil energy cr goes to zero. The conclusion is that edge s1~lgularities do flot survive recoil
of tue scatterer (here the hole). Note that the argument does not holà for Id systems: there
the momentum of an electron-hole pair is discrete (a multiple of 2 kF). As a result one cannot
dispose of the hole momentum at no energy cost.
tÙo OEd
i i i
Er
k
Fig. 1. The two thresholds for momentum conserving optical absorption in a degenerate semicon- ductor.
The above argument is physically transparent, but its validity is not easily assessed. The purpose of this note is to approach the question from a more rigorous vantage point, along the fines of reference [2]. We first recall the argument for a recoilless scatterer. For simplicity
we consider a l1~le spectrum m which the scatterer jumps between two discrete states: a final state potent1al V excites electron-hole pairs and broadens the fine. The generalization
to band spectra m which the transition creates a photoelectron in the conduction baud is
straightforward. Assume the pote~lt1al V is created at time 0 and destroyed at time t. We want to calculate the transient propagator for the scatterer
G(t) = (0 d e~~~d* 0)
~,i
j i
'
~ / j
1jj
Ùlo 0ld 0l
Fig. 2. -Broadenmg of the edge singularity due to recoil: evolution
as a function of final store interaction strength.
where H is the transient Hamiltonian and 0) is the init1al ground state (without V). The spectrum is the Fourier transform of V. As argued in [2] the problem is a one body transient
problem. Wuen it exists the potent1al V is structureless: it does flot change l~po~l scatter1~lg by condl~ction electrons. If we adopt a time space represe~ltation, bubbles in the perturbation
representation of G are decoupled from each other (Fig. 3). Hence they exponentiate vithin a linked cluster expansion. G is exactly given by
G(t) = expjc(t)j (2)
where C(t) is the contribution of a single closed loop integrated over the range (0, t).
Fig. 3. The factorization of closed loops in a lime space representation.
In the long time limit one con calculate C exactly in ternis of phase shifts à off the transient potential at the Fermi level. Such a calculation is elegant, but not necessary for a simple picture: a Bom approximation using only second order loops is enough! In that order, and for
a contact potential, C(t) reduces to
c(t) = /~ dr dr' v2g(o, r r'i g(o, ri r) (3) g(0, r r') is the on site free conduction electron propagator (we work on a lattice in order to
avoid ultraviolet divergences). In the limit of long times
g(o,r) m -iv/r (4)
1278 JOURNAL DE PHYSIQUE I N°9
where v is the density of states at Fermi level. When we carry the time integration in (3), we
obtain a logarithmically divergent closed loop, C(t) oc Log t: exponentiation yields the power law behaviour:
G(t) oc 1/t'~ with n
=
v~V~
= ô~/7r~ (5)
(for a single channel). The edge singularity follows from the logarithmic divergence of C.
We assume now that the scatterer recoils. It is created and destroyed at site 0 (by some
externat probe), but in between it can wander, monitored by the hopping from site to site,
Wd] d~. It does not matter whether trie scatterer motion is free or dilfusive. Within a Feynman
picture the propagator G may be expressed as a sum over ail possible trajectories R(r) of the scatterer, subject to the boundary conditions R
= 0 when r
= 0,t. Summing over trajectories
is equivalent to a perturbation expansion in powers of the hopping W in a time representation.
For a given history, the transient potent1al is once again independent of the conduction electron response. In contrast to the recoilless case it is time dependent, monitored by the position R(r) but it is the same for everybody. It follows that exponentiation remains true: (2) is still exact, the only difference being the replacement in (3) of the local propagator
g(0, r r') by
g(p, r r') with p = R(r) R(r') (6)
For r > r', g is Fourier expanded as
g(p,T) " ~j e~~k~ ~k ~ (7)
k > kF
In (7) the summation over k can be broken into an energy and an angular integration. The
former involves both p and r, but it is dominated by the time factor in the relevant limit p < vfr (where vF is the Fermi velocity): it yields g(0,r). What remains is the angular
average over tue Fermi surface of exp[ik pi, denoted as À. In the end we have
g(P, r)
= Àg(0,r)
(It is actually easy to calculate g(p,r) explicitly: the above simple argument is more transpar-
ent). The asymptotic behaviour of C and G is controlled by À.
The angular average is easily carried out for any dimension, yielding
cos kFP Id
j =
Jo(kFP) 2d (~)
É~~ 3d
FP
The edge singularities involve À~, which should be averaged over the oscillations in p. We thus find
= 1/2 Id
lÀ~) '~ ~~ (9)
r- ~y 3d
P
(9) is our central result.
Let us first assume that the scatterer is Dot localized: p goes to infinity if (r r') does.
Then
1) The edge singularity persists in Id, but with an exponent halved compared to the recoilless
case. Such a singular behaviour confirms well known results. The factor 1/2 is unexpected.
Note that in the strong coupling unitanty limit (à
= 7r/2), the edge exponent becomes ô~/27r~ = 1/8
The relationship to exact results for Id systems is not clear.
ii) In 2d and 3d, a typical p goes to infinity as r~/~ (the free propagator is Gaussian,
g oc exp[ip~/r]). The time integration in (3), which used to be marginal (logarithmically divergent) is now convergent. There is no singularity whatsoever: tue infrared singularity is
washed ol~t by recoil. We thus confirm the results of the simple qualitative argument.
The situation is dilferent if the partiale is localized. Then (À~) remains finite, albeit con-
siderably reduced. We expect a power law behaviour of G(t), with a much smaller exponent.
The whole issue is to decide about that localization. One reason for that is the coupling to the electron gas the sc-called "ohmic dissipation". Note that it is the saine infrared singularity
that produces edge exponent (switch on and off of the potential) and localization (shift of the
potent1alfrom one site to the next). A convenient approach is to scale the conduction electron bond width A down, as for as possible. Upon scaling the scatterer hopping amplitude W is
reduced, according to a power law
w' # wjA'/àj~ (io)
where p is an exponent that depends on phase shifts and lattice spacing (see Ref. [Si for
details). Scaling proceeds as long as A' > W' (otherwise scatterer recoil blurs the logarithmic singularity). Hence two possibilities [7]:
1) p < 1: W' crosses A' at a definite crossover A*, which is the low temperature effective
bond width of the scatterer. The latter is not localized: its mean square displacement (p~) alter a time t diverges as t - oJ. Note that A* defines an effective time scale t*
= &/A* which separates two regimes
~~~~ '~
Î ÎÎ Î ~
~~ ~~
At very low temperatures a free electron coherent diffusion is recovered.
ii) p > 1: then W' scales down to 0 at zero temperature. In that limit the partiale is localized and (p~) is finite. In the latter case the edge singularity persists and indeed it should be possible to formulate the two problems localization and edge elfects in a unified picture.
The purpose of this note is not to dwell on the localization problem, but only to stress the elfect of scatterer motion on the edge singularities. For a given history, the logarithmic divergence is reduced according to [9]. Of course the real difliculty is the sum over histories,
which cannot be done in contrast to the recoilless case there is no exact solution! But
qualitatively it looks reasonable to average over the distribution of probability P(p, r). The closed loop divergence responsible for the edge singularities should be reduced by a factor
A(r) =
f d~pP(p,r) je~~'P)n
j~ (12)
All the physics is hidden in the probability P.
Our main conclusion is that infrared singulanties should not occur m the presence of recoil when the dimension exceeds 1, except if the scatterer is localized a fairly obvions restriction.
Extrapolating results obtained for a fixed transient impunty is dangerous. The real issue is not the X-ray edge elfect: it is the self-localization a problem of major interest, but physically
dilferent.
1280 JOURNAL DE PHYSIQUE I N°9
References
[1] Mahan G-D-, Phys. Rev. 163 (1967) 612;
Rculet B., Gavoret J., Nozières P., Phys. Rev. 178 (1969) 1072, 1084.
[2] Nozières P., De Domimcis C., Phys. Rev. 178 (1969) 1097.
[3] Doniach S., Sunjic M., J. Phys. IYance 31 (1970) C3-285.
[4] Ting C.S., Birman J-L-, Abrahams E., Sofia State Commun. 10 (1972) 1101.
[Si The eflect was first discussed by J. Kondo, Physica 848 (1976) 40, 207. Ii was extensively discussed by Yamada et al.: see for instance Yamada K., Sakurai A., Takeshige M., Progr.
Theor. Phys. 70 (1983) 73.
[6] Gavoret J., Nozières P., Rouler B., Combescot M., J. Pllys. IFance 30 (1969) 987.
[7] See for instance Aslangul C., Pottier N., Saint James D., Phys. Lett. lllA (1985) 175.
