Non-resonant Raman scattering through a metal-insulator transition: an exact analysis of the Falicov-Kimball model
For years, theories for Raman scattering have been confined to either the insulating or fully metallic state. While much can be learned by focusing attention on the metal or insulator, recent experimental work on the cuprate systems points to the desirability of formulating a theory for Raman...
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Інститут фізики конденсованих систем НАН України
2001
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| Назва видання: | Condensed Matter Physics |
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| Цитувати: | Non-resonant Raman scattering through a metal-insulator transition: an exact analysis of the Falicov-Kimball model / J.K. Freericks, T.P. Devereaux // Condensed Matter Physics. — 2001. — Т. 4, № 1(25). — С. 149-160. — Бібліогр.: 11 назв. — англ. |
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irk-123456789-1197732017-06-09T03:04:29Z Non-resonant Raman scattering through a metal-insulator transition: an exact analysis of the Falicov-Kimball model Freericks, J.K. Devereaux, T.P. For years, theories for Raman scattering have been confined to either the insulating or fully metallic state. While much can be learned by focusing attention on the metal or insulator, recent experimental work on the cuprate systems points to the desirability of formulating a theory for Raman response which takes one through a quantum critical point – the metalinsulator transition. Using the Falicov-Kimball model as a canonical model of a MIT, we employ dynamical mean-field theory to construct an exact theory for non-resonant Raman scattering. In particular we examine the formation of charge transfer peaks and pseudogaps as well as the low-energy dynamics. The results are qualitatively compared to the experimental B₁g Raman spectra in the cuprates, which probes the hot quasiparticles along the Brillouin zone axes. The results shed important information on normal state electronic transport and the pseudo-gap in the cuprates. Упродовж років теорія комбінаційного розсіяння (КР) обмежувалася розглядом або ізоляторів або суто металічного стану. Хоча можна багато довідатися, зосередивши увагу тільки на металах чи ізоляторах, останні експериментальні роботи з купратних систем вказують на бажаність формулювання теорії раманівського відгуку, яке придатне при проходженні через квантову критичну точку - перехід метал-ізолятор (ПМІ). Використовуючи модель Фалікова-Кімбала як канонічну модель ПМІ, ми застосовуємо теорію динамічного середнього поля для побудови точної теорії нерезонансного КР. Зокрема, ми розглядаємо утворення піків, зумовлених переносом заряду, та псевдощілин, а також низькоенергетичну динаміку. Результати якісно зіставимі з експериментальними B₁g спектрами КР у купратах, в яких фіксують “гарячі” квазічастинки вздовж осей зони Брілюена. Результати дають важливу інформацію про електронний транспорт у нормальному стані та псевдощілину в купратах. 2001 Article Non-resonant Raman scattering through a metal-insulator transition: an exact analysis of the Falicov-Kimball model / J.K. Freericks, T.P. Devereaux // Condensed Matter Physics. — 2001. — Т. 4, № 1(25). — С. 149-160. — Бібліогр.: 11 назв. — англ. 1607-324X PACS: 78.30.-j, 71.30.+h, 74.72.-h DOI:10.5488/CMP.4.1.149 http://dspace.nbuv.gov.ua/handle/123456789/119773 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| language |
English |
| description |
For years, theories for Raman scattering have been confined to either the
insulating or fully metallic state. While much can be learned by focusing attention
on the metal or insulator, recent experimental work on the cuprate
systems points to the desirability of formulating a theory for Raman response
which takes one through a quantum critical point – the metalinsulator
transition. Using the Falicov-Kimball model as a canonical model
of a MIT, we employ dynamical mean-field theory to construct an exact theory
for non-resonant Raman scattering. In particular we examine the formation
of charge transfer peaks and pseudogaps as well as the low-energy
dynamics. The results are qualitatively compared to the experimental B₁g
Raman spectra in the cuprates, which probes the hot quasiparticles along
the Brillouin zone axes. The results shed important information on normal
state electronic transport and the pseudo-gap in the cuprates. |
| format |
Article |
| author |
Freericks, J.K. Devereaux, T.P. |
| spellingShingle |
Freericks, J.K. Devereaux, T.P. Non-resonant Raman scattering through a metal-insulator transition: an exact analysis of the Falicov-Kimball model Condensed Matter Physics |
| author_facet |
Freericks, J.K. Devereaux, T.P. |
| author_sort |
Freericks, J.K. |
| title |
Non-resonant Raman scattering through a metal-insulator transition: an exact analysis of the Falicov-Kimball model |
| title_short |
Non-resonant Raman scattering through a metal-insulator transition: an exact analysis of the Falicov-Kimball model |
| title_full |
Non-resonant Raman scattering through a metal-insulator transition: an exact analysis of the Falicov-Kimball model |
| title_fullStr |
Non-resonant Raman scattering through a metal-insulator transition: an exact analysis of the Falicov-Kimball model |
| title_full_unstemmed |
Non-resonant Raman scattering through a metal-insulator transition: an exact analysis of the Falicov-Kimball model |
| title_sort |
non-resonant raman scattering through a metal-insulator transition: an exact analysis of the falicov-kimball model |
| publisher |
Інститут фізики конденсованих систем НАН України |
| publishDate |
2001 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/119773 |
| citation_txt |
Non-resonant Raman scattering
through a metal-insulator transition:
an exact analysis of the Falicov-Kimball
model / J.K. Freericks, T.P. Devereaux // Condensed Matter Physics. — 2001. — Т. 4, № 1(25). — С. 149-160. — Бібліогр.: 11 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT freericksjk nonresonantramanscatteringthroughametalinsulatortransitionanexactanalysisofthefalicovkimballmodel AT devereauxtp nonresonantramanscatteringthroughametalinsulatortransitionanexactanalysisofthefalicovkimballmodel |
| first_indexed |
2025-07-08T16:34:31Z |
| last_indexed |
2025-07-08T16:34:31Z |
| _version_ |
1837097259123081216 |
| fulltext |
Condensed Matter Physics, 2001, Vol. 4, No. 1(25), pp. 149–160
Non-resonant Raman scattering
through a metal-insulator transition:
an exact analysis of the Falicov-Kimball
model
J.K.Freericks 1,2 , T.P.Devereaux 3
1 Department of Physics, Georgetown University,
Washington, DC 20057, USA
2 Isaac Newton Institute, Cambridge CB3 0EH, UK
3 Department of Physics, University of Waterloo,
Waterloo, ON, Canada, N2l 3G1
Received August 14, 2000
For years, theories for Raman scattering have been confined to either the
insulating or fully metallic state. While much can be learned by focusing at-
tention on the metal or insulator, recent experimental work on the cuprate
systems points to the desirability of formulating a theory for Raman re-
sponse which takes one through a quantum critical point – the metal-
insulator transition. Using the Falicov-Kimball model as a canonical model
of a MIT, we employ dynamical mean-field theory to construct an exact the-
ory for non-resonant Raman scattering. In particular we examine the for-
mation of charge transfer peaks and pseudogaps as well as the low-energy
dynamics. The results are qualitatively compared to the experimental B1g
Raman spectra in the cuprates, which probes the hot quasiparticles along
the Brillouin zone axes. The results shed important information on normal
state electronic transport and the pseudo-gap in the cuprates.
Key words: Raman scattering, metal-insulator transition
PACS: 78.30.-j, 71.30.+h, 74.72.-h
1. Introduction
Raman scattering in the cuprate materials involves the inelastic scattering of
light by electron-hole excitations of the correlated many-body system. It is a non-
destructive bulk probe of the electron dynamics over a wide range of energy scales
and temperatures. The experiments can discriminate between a number of different
symmetries of the electronic system by polarizing the incoming light and measuring
the reflected light with a polarized detector. Three principal symmetries are exam-
c© J.K.Freericks, T.P.Devereaux 149
J.K.Freericks, T.P.Devereaux
ined in the cuprate materials: A1g, which has the full symmetry of the lattice (i.e.
is s-like), and two d-like symmetries B1g (which probes the Brillouin zone axes) and
B2g (which probes the Brillouin zone diagonals). While the Raman response for the
A1g and B2g symmetries are closer to those of conventional metals, the B1g response
is anomalous, and has large spectral weight shifts as functions of doping and tem-
perature [1]. It is believed by many workers in the field that this B1g response is
arising from the proximity of the cuprates to a quantum-critical point corresponding
to a metal-insulator transition.
Theoretical treatments of non-resonant Raman scattering are rather complete
for band metals [2] and insulators [3], but there is no theory that can successfully
interpolate between these two limits to pass through the correlated metal-insulator
transition. In this contribution, we show how an exact solution of the spinless Falicov-
Kimball model [4] using dynamical mean field theory [5,6] provides a theoretical
model that illustrates the generic behaviour of Raman response through the metal-
insulator transition and interpolates between the known limits. We find that our
Raman spectra show charge-transfer peaks and nontrivial low-frequency spectral-
weight transfers that are indicative of the proximity to a quantum critical point and
represent well the experimental data in the cuprates.
The Hamiltonian of the spinless Falicov-Kimball model [4] is
H = − t∗
2
√
d
∑
〈i,j〉
d†idj + Ef
∑
i
wi − µ
∑
i
(d†idi + wi) + U
∑
i
d†idiwi, (1)
where d†i (di) is the spinless conduction electron creation (annihilation) operator at
lattice site i and wi = 0 or 1 is a classical variable corresponding to the localized
f -electron number at site i. The hopping matrix between the nearest neighbours
〈i, j〉 (on a hypercubic lattice in d-dimensions [5], with d → ∞) is −t∗/(2
√
d) with
t∗ chosen as our energy unit, Ef is the localized electron level, µ is the chemical
potential and U is the mutual electron repulsion when a conduction electron and a
localized f -electron both occupy the same lattice site. We will adjust both E f and
µ so that the average filling of the d-electrons is 1/2 and the average filling of the
f -electrons is 1/2.
The Raman response is found from a density-density correlation function
χRaman(iνl) =
∑
k
∫ β
0
dτeiνlτ
{
Tr Tτ 〈e−βHρk(τ)ρk(0)〉
Z
−
[
Tr 〈e−βHρk(0)〉
Z
]2}
, (2)
with the uniform (q = 0) Raman density operator
ρk = γ(k)d†
k
dk, dk =
1
N
∑
j
e−Rj ·kdj, (3)
Z = Tr 〈e−βH〉, the partition function, and iνl = 2iπlT the Bosonic Matsubara
frequency (the τ -dependence of the operators is with respect to the full Hamiltonian).
The Raman scattering amplitude γ(k) is a complicated function of the incoming and
150
Raman scattering through a MI transition
outgoing photon polarizations, of the photon energies, and the polarizability of the
medium. In nonresonant Raman scattering (which we examine here) one neglects
the frequency dependence of the Raman scattering amplitude, and characterizes
the Raman response in terms of the different spatial symmetries of the remaining
function γ(k). One can expand this function in a Fourier series and examine the
contributions of the lowest components of the series, and compare them to the
experiment. More sophisticated approaches would calculate the Raman scattering
amplitude from “first-principles” and would include any possible resonant Raman
scattering effects. We leave those pursuits to future work.
2. Formalism
The Falicov-Kimball model can be solved exactly in the infinite-dimensional
limit by using dynamical mean-field theory [6,7]. We summarize the main points
to establish our notation. The local Green’s function at the Fermionic Matsubara
frequency iωn = iπT (2n+ 1) is defined by
Gn = G(iωn) = −Tr Tτ
∫ β
0
dτeiωnτ
〈e−βHatd(τ)d†(0)S(λ)〉
Z
, (4)
with
Z = Z0(µ) + e−β(Ef−µ)Z0(µ− U), (5)
the atomic partition function expressed in terms of
Z0(µ) = Trd 〈e−βH0S(λ)〉, H0 = −µd†d. (6)
In the above equations, the atomic Hamiltonian Hat is the Hamiltonian of equa-
tion (1) restricted to one site, with t∗ = 0, and all time dependence is with respect
to this atomic Hamiltonian. The evolution operator S(λ) satisfies
S(λ) = exp
[
−
∫ β
0
dτ
∫ β
0
dτ ′d†(τ)λ(τ − τ ′)d(τ ′)
]
, (7)
with λ(τ − τ ′) a time-dependent atomic field adjusted to make the atomic Green’s
function equal to the local lattice Green’s function. We define an effective medium
by
G−1
0 (iωn) = G−1
n + Σn = iωn + µ− λn, (8)
with Σn the local self-energy and λn the Fourier transform of λ(τ). The trace in
equation (4) can be evaluated directly to yield
Gn = w0G0(iωn) + w1[G
−1
0 (iωn)− U ]−1, (9)
with w0 = 1− w1 and
w1 = exp[−β(Ef − µ)]Z0(µ− U)/Z. (10)
151
J.K.Freericks, T.P.Devereaux
The self-consistency relation needed to determine λn and Gn is to equate the local
lattice Green’s function to the atomic Green’s function via
Gn =
∫ ∞
−∞
dǫ
ρ(ǫ)
iωn + µ− Σn − ǫ
, (11)
with ρ(ǫ) = exp(−ǫ2)/
√
π the noninteracting density of states for the infinite-
dimensional hypercubic lattice.
The iterative algorithm to solve for Gn starts with Σn = 0. Then equation (11)
is used to find Gn, equation (8) is employed to extract the effective medium, equa-
tion (9) is used to find a new local Green’s function, and then equation (8) is used
to find the new self-energy. The algorithm is then repeated until it converges, which
usually requires only about a dozen or so iterations. This algorithm can also be
used on the real axis (with suitably modified equations) to directly solve for the
Green’s function and self-energy on the real axis. Here, we examine the half-filled
case ρd =
∑
i〈ni〉/N = 1/2 and ρf =
∑
i〈wi〉/N = 1/2, which corresponds to
µ = U/2 and Ef = 0.
The dynamical charge vertex is local in infinite dimensions which implies that
correlation functions that have the same symmetry as the lattice are renormalized
due to a nontrivial charge vertex, but correlation functions that are orthogonal to the
lattice, have no vertex corrections, and so they are represented by their bare bubble
diagrams [8]. In two dimensions, the Raman scattering amplitudes are typically
chosen as follows:
γA1g
(k) ≈ ∂2ǫ(k)
∂kx∂kx
+
∂2ǫ(k)
∂ky∂ky
≈ −ǫ(k),
γB1g
(k) ≈ ∂2ǫ(k)
∂kx∂kx
− ∂2ǫ(k)
∂ky∂ky
≈ cos kx − cos ky,
γB2g
(k) ≈ ∂2ǫ(k)
∂kx∂ky
≈ sin kx sin ky, (12)
with ǫ(k) the electronic band structure. Note that the B2g response vanishes for
nearest-neighbour hopping only, which is what we consider here. The above forms
can be generalized to the infinite-d limit by choosing
γA1g
(k) ≈ c− ǫ(k), γB1g
(k) ≈ 1√
d
d
∑
i=1
(−1)i cos ki, (13)
where we include a constant term c in the A1g amplitude, since it is allowed by
symmetry.
A simple analysis in the B1g case shows that the B1g response does not have any
vertex corrections, as expected, and that it is equal to the bare bubble. The basic
argument is that we must evaluate a summation over k of the form
∑
k
1√
d
d
∑
i=1
(−1)i cos ki
1
X + 2√
d
∑d
j=1 cos kj
, (14)
152
Raman scattering through a MI transition
which arises when examining the Dyson equation for the B1g response (and we
can assume the imaginary part of X is greater than zero). Writing the fraction in
equation (14) as the integral of an exponential
1
X + 2√
d
∑d
j=1 cos kj
= −i
∫ ∞
0
dz exp
iz
X +
2√
d
d
∑
j=1
cos kj
, (15)
allows one to decouple the summation over momentum to the sum over d identical
terms, each multiplied by (−1)i. This then vanishes for all even d and for odd-d in
the limit d → ∞ (due to the 1/
√
d term). So the evaluation of the B1g response
reduces to the evaluation of the bare bubble. A straightforward calculation then
yields
χB1g
(iνl) = −T
2
∑
n
G(iωn)−G(iωn+l)
iνl + Σ(iωn)− Σ(iωn+l)
. (16)
This formula can be easily analytically continued to the real axis by following the
same procedure outlined in the calculation of the dynamical charge susceptibility [9]:
rewrite the sum over Matsubara frequencies by a contour integral of advanced or
retarded Green’s functions and self-energies multiplied by the Fermi factor, and
then deform the contours to the real axis picking up any poles in the complex plane.
Under the assumption that there are no extra poles when the contours are deformed,
one ends up with the following expression for the B1g response:
χB1g
(ν) =
−i
4π
∫ ∞
−∞
dω
{
f(ω)
G(ω)−G(ω + ν)
ν + Σ(ω)− Σ(ω + ν)
− f(ω + ν)
G∗(ω)−G∗(ω + ν)
ν + Σ∗(ω)− Σ∗(ω + ν)
− [f(ω)− f(ω + ν)]
G∗(ω)−G(ω + ν)
ν + Σ∗(ω)− Σ(ω + ν)
}
, (17)
with f(ω) = 1/[1 + exp(βω)] the Fermi function. We verify that this expression
is indeed accurate, by using the spectral formula to calculate the Raman response
on the imaginary axis and comparing it to the result directly calculated from the
expression in equation (16). We find that the results rarely differ by more than one
part in a thousand confirming the accuracy of the analytic continuation.
The A1g response is more complicated, because it requires a proper treatment
of the vertex contributions. Fortunately, the charge vertex for the Falicov-Kimball
model is well-known [9] and assumes a simple form (for ν l 6= 0)
Γ(iωm, iωn; iνl) = δm,n
1
T
Σ(iωn)− Σ(iωn+l)
G(iωn)−G(iωn+l)
. (18)
Hence, the Raman response in the A1g channel can be found by solving the relevant
Dyson’s equation, using the above form of the charge vertex. The steps to do this
are straightforward, but the algebra is somewhat cumbersome and will be presented
153
J.K.Freericks, T.P.Devereaux
elsewhere. The result is
χA1g
(iνl) = −T
∑
n
χ̄0(iωn; iνl)− [χ0(iωn; iνl) +G(iωn)G(iωn+l)]TΓ(iωn, iωn; iνl)
1 + χ0(iωn; iνl)TΓ(iωn, iωn; iνl)
,
(19)
where the charge vertex is found in equation (18), the bare susceptibility χ0 satisfies
χ0(iωn; iνl) = − G(iωn)−G(iωn+l)
iνl + Σ(iωn)− Σ(iωn+l)
, (20)
and the other bare susceptibility χ̄0 (which is where all of the c dependence lies) is
χ̄0(iωn; iνl) = [−c2(Gn −Gn+l) + 2c(ZnGn − Zn+lGn+l) + Zn − Zn+l
− Z2
nGn + Z2
n+lGn+l]/[iνl + Σn − Σn+l]. (21)
Here we used the notation Zn = iωn + µ− Σ(iωn).
It is a straightforward exercise to perform a similar analytic continuation of this
expression, but we will not write down the result here.
−3 −2 −1 0 1 2 30
Frequency [t]
0
0.5
1
D
O
S
[1
/t]
U=0.5t
1
1.5
2
4
Figure 1. Interacting density of states for various values of U as indicated. The
MIT occurs for U = 1.5t. Note the interacting DOS is independent of temperature
for the Falicov-Kimball model in d → ∞.
3. Results
The Falicov-Kimball model has a ground state that is not a Fermi liquid be-
cause the lifetime of a quasiparticle is finite at the Fermi energy. As U increases,
154
Raman scattering through a MI transition
U=0.5t
U=1.5t
U=2t
U=4t
U=0.5t
U=1.5t
U=2t
U=4t
R
am
an
r
es
po
ns
e
[a
rb
. u
ni
ts
]
(a) B
1g
(b) A
1g
Frequency [t]
0 1 2 3 4 0 1 2 3 5 64
Figure 2. The Raman response (imaginary part of the Raman susceptibility) for
T = 0.5t for four different values of U for the (a) B1g and (b) A1g channels, re-
spectively. Notice how the vertex corrections suppress the low-frequency spectral
weight in the insulating phase for A1g channel.
the system first enters a pseudogap phase, where spectral weight is depleted near
the chemical potential, and then undergoes a metal-insulator transition [10]. The
interacting density of states (DOS) is, however, temperature-independent for fixed
U and fixed electron fillings.
We plot the DOS in figure 1 for values of U ranging from a weakly correlated
metal U < 0.65, to a pseudogap phase 0.65 < U < 1.5 to the insulator phase
U > 1.5. The quantum critical point occurs at U = 1.5 where the interacting
DOS is suppressed to zero at the chemical potential because a pole develops in the
self-energy (the “gap region” actually has an exponentially small DOS because the
hypercubic lattice has infinite tails). The corresponding imaginary part of the Raman
response is plotted in figure 2 at a moderate temperature T = 0.5t and various values
of U . In figure 2a we show the B1g response and in 2(b) we show the A1g response (for
c = 0.1). Note that the Raman scattering is quite similar for the two channels for the
gross features–they both display the classic band-metal behaviour for small U , which
evolves to a charge transfer peak centered at U in the large-U mode, as expected
for an insulator. The B1g response, however, has more low-frequency spectral weight
when we are near the quantum critical point(at U = 1.5). The vertex corrections in
the A1g channel suppress these low-energy features.
In figure 3, we show the Raman response as a function of temperature for a
system on the insulating side of, but close to, the quantum critical point (U = 2).
The B1g response is in figure 3a and the A1g response is in figure 3b. Note how the
155
J.K.Freericks, T.P.Devereaux
0 21 30 1 2 4
T=0.95t
T=0.65
T=0.35t
T=0.05t
T=0.95t
T=0.65t
T=0.35t
T=0.05t
(a) B (b) A
R
am
an
r
es
po
ns
e
[a
rb
. u
ni
ts
]
Frequency [t]
1g 1g
Figure 3. The Raman response (imaginary part of the Raman susceptibility)
as a function of temperature for U = 2t for the (a) B1g and (b) A1g channels,
respectively. Notice how the charge-transfer peaks have similar behaviour in both
channels, but that the low-frequency response is quite anomalous in the B1g
channel.
charge transfer peak feature is present at high temperatures, and steadily increases
as T is lowered. The B1g response is most interesting at low energies. The spectral
feature at low energy has a strong temperature dependence that is sharply reduced
as T → 0. This is similar to the B1g response seen in the cuprates. We study this
phenomenon further in figures 4 and 5 which plot the normalized low-frequency
spectral weight as a function of T and the inverse of the Raman slope (as ν → 0)
for the B1g channel. We arbitrarily define the low-frequency weight to be all spectral
features from ν = 0 to ν = U/2 and the high-frequency weight to be spectral features
with ν > U/2. This division is obvious in the insulator phase, since the low-energy
features and the high-energy features are well separated, but becomes less obvious
in the pseudogap region. Note how, in all cases, the low-frequency spectral weight is
sharply depleted as T is lowered, with the effect being the largest when one is well
into the insulating phase. The Raman inverse slope is even more interesting. Since
the Falicov-Kimball model has a temperature-independent DOS, this implies that
the self-energy is also temperature-independent. Conventional reasoning would then
say that the scattering rate (derived from the imaginary part of the self-energy)
would be a constant, and hence, the Raman inverse slope should also be a constant,
since it measures the scattering rate of the charge excitations. This is indeed true at
low T for the weakly interacting systems, but is violated as we near and pass through
the metal-insulator transition. This rise in the Raman inverse slope is indicative of
the formation of a gap in the quasiparticle spectrum. What is interesting is that we
156
Raman scattering through a MI transition
1
0
0.8
0.4
0.6
0.2
10 0.25 0.5 0.75
U=0.5t
U=1.5t
U=2t
U=4t
Reduced temperature
N
or
m
al
iz
ed
in
te
gr
at
ed
s
pe
ct
ra
l w
ei
gh
t
U=0.25t
U=0.5t
U=t
U=1.5t
U=2t
R
am
an
in
ve
rs
e
sl
op
e
[a
rb
. u
ni
ts
]
0.1 1.0
Temperature [t]
insulating
pseudogap
phase
weakly
interacting
Figure 4. The integrated B1g spectral
weight ratio for low frequencies to high
frequencies normalized to its value at
T = 0.95t plotted versus reduced tem-
perature T/0.95t. Notice how the spec-
tral weight is sharply depleted at low
temperatures as one nears and crosses
the MIT.
Figure 5. Log-log plot of the inverse
Raman slope for the B1g channel at
low frequencies versus temperature for
several values of U . The characteristic
rise at low-T for the pseudogap and in-
sulating phases matches well with the
experimental data on the cuprates.
see a precursor effect of the insulating behaviour as we near the quantum critical
point. This data compares quite well with the results seen in the underdoped cuprate
materials.
4. Discussion and conclusions
There is a simple explanation to the origin of the anomalous low-frequency fea-
tures of the Raman spectra in the insulating mode, but close to the quantum critical
point. If we examine the integral for the B1g Raman response in equation (17), we
note three important points (i) the imaginary part of the Raman response is pro-
portional to the real part of the integrand; (ii) the integrand vanishes if the Green
function (and self-energy) are both real; (iii) all temperature dependence arises from
the Fermi factors, since both G and Σ are temperature-independent. In the insulat-
ing mode, the DOS breaks into two pieces, a lower band centered at −U/2 with a
width of O(1) and an upper band at U/2 with a width of O(1). The Green’s func-
tions are complex only when the frequency argument lies within one of the bands.
157
J.K.Freericks, T.P.Devereaux
Hence there are two main contributions to the Raman response: (i) intraband pro-
cesses, where ω ≈ −U/2 or U/2 and ν ≈ 1; and (ii) interband processes, where
ω ≈ −U/2 and ν ≈ U . The interband processes, with ν ≈ U are what give rise
to the charge-transfer peaks seen in the Raman response. The intraband processes,
with ν ≈ 1, give rise to the low-frequency spectral features. Furthermore, at low
temperatures, the low-frequency features are proportional to f(ω)− f(ω+ ν) which
can be approximated by exp(−U/2T )[1 − exp(ν/T )]. Hence, we expect a Raman
inverse slope to act like T exp(U/2T ) and the low-frequency spectral weight should
go like exp(−U/2T ), which can explain the features seen in our results. In the A1g
channel, the charge vertex makes the integrand more complicated to evaluate, and
the vertex corrections end up suppressing the low-frequency response.
Since the form for the B1g response is identical in other single-band models, like
the Hubbard model, we expect that the B1g Raman response will be essentially the
same as seen for the Falicov-Kimball model. The only modifications are that the
DOS now has T -dependence, which will relax criterion (iii) above, and that the
system has a Fermi-liquid ground state for low temperatures and small to moderate
U . Nevertheless, in the insulating mode, the DOS will separate into an upper and
lower Hubbard band, and the analysis given above will hold for determining the
low-T features, so we expect to see the same anomalous low-frequency behaviour in
the B1g channel as is seen here. We cannot make a similar comment about the A1g
channel, because the charge vertex will no longer assume the simple form for the
Falicov-Kimball model, and the vertex corrections can modify the results, but we
don’t expect there to be much qualitative difference there either.
The calculations presented here are remarkably similar to those observed in the
B1g channel in the normal state of the cuprates [1]. As the doping in these materials
is reduced from the overdoped side to the underdoped side of the phase diagram, the
B1g Raman spectra deviate strongly from our expectations for metallic behaviour. A
large depletion of spectral weight at low frequencies is observed for B1g with under-
doping, with a concommitant shift of spectral weight out to two-magnon energies
∼ 2700 cm−1. For a given temperature, the integrated low-frequency spectral weight
in the B1g channel falls by over an order of magnitude from overdoped to under-
doped. Both of these features are similar to our observations shown in figures 2–4.
This is indicative of the formation of a pseudogap which affects the B1g quasipar-
ticles and sets in well before the metal-insulator transition is reached. The inverse
Raman slope is found to increase with doping at a given temperature and, moreover,
has a strong upturn with decreasing temperature in the more underdoped systems,
qualitatively similar to our figure 5. Again, since we believe our model captures the
salient features of Raman scattering in the vicinity of a quantum critical point, the
Raman data from the cuprates directly shows the influence on the B1g quasiparticles
by the underlying metal-insulator transition.
In summary, we have examined the Raman response of a system that goes
through a metal-insulator transition exactly, using the dynamical mean field theory
for the spinless Falicov-Kimball model. Our results provide a continuous interpo-
lation between the known metallic and insulating Raman responses. The results of
158
Raman scattering through a MI transition
this model system should be qualitatively similar to those of other correlated models
that require quantum Monte Carlo simulation to solve (such as the Hubbard and
Holstein models), and display the qualitative features seen in the cuprates.
5. Acknowledgments
We acknowledge useful discussions with Lance Cooper, Rudi Hackl, J.C.Irwin,
Miles Klein, Paul Miller, and Andrij Shvaika. J.K.F. acknowledges support from
the National Science Foundation under grant number DMR–9973225. He also ac-
knowledges the hospitality of the Newton Institute where some of this work was
completed in June of 2000. T.P.D. acknowledges support from the National Science
and Engineering Research Council of Canada. This work was supported in part by
Award No. UW0–1012 of the U.S. Civilian Research and Development Foundation
for the Independent States of the former Soviet Union (CRDF).
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Нерезонансне комбінаційне розсіяння при
проходженні переходу метал-ізолятор: точний
аналіз моделі Фалікова-Кімбала
Дж.К.Фрірікс 1,2 , Т.П.Деверо 3
1 Фізичний факультет, Університет Джорджтауну, Вашінгтон, округ
Колумбія 20057, США
2 Інститут Ісаака Ньютона, Кембрідж СВ3 0ЕН, Великобританія
3 Фізичний факультет, Університет Ватерлоо, Ватерлоо, Онтаріо,
Канада, N2l 3G1
Отримано 14 серпня 2000 р.
Упродовж років теорія комбінаційного розсіяння (КР) обмежувала-
ся розглядом або ізоляторів або суто металічного стану. Хоча мож-
на багато довідатися, зосередивши увагу тільки на металах чи ізо-
ляторах, останні експериментальні роботи з купратних систем вка-
зують на бажаність формулювання теорії раманівського відгуку, яке
придатне при проходженні через квантову критичну точку - перехід
метал-ізолятор (ПМІ). Використовуючи модель Фалікова-Кімбала як
канонічну модель ПМІ, ми застосовуємо теорію динамічного серед-
нього поля для побудови точної теорії нерезонансного КР. Зокрема,
ми розглядаємо утворення піків, зумовлених переносом заряду, та
псевдощілин, а також низькоенергетичну динаміку. Результати якіс-
но зіставимі з експериментальними B1g спектрами КР у купратах, в
яких фіксують “гарячі” квазічастинки вздовж осей зони Брілюена. Ре-
зультати дають важливу інформацію про електронний транспорт у
нормальному стані та псевдощілину в купратах.
Ключові слова: комбінаційне розсіяння, перехід метал-ізолятор
PACS: 78.30.-j, 71.30.+h, 74.72.-h
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