Exact solution of a variety of X-ray probes in the Falicov-Kimball model with dynamical mean-field theory
We examine the core-level X-ray photoemission spectroscopy (XPS), X-ray absorption near-edge spectroscopy (XANES) and X-ray emission spectroscopy (XES) in the Falicov-Kimball model by using the exact solution from dynamical mean-field theory. XPS measures the core-hole propagator, XANES measures the...
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irk-123456789-1203072017-06-12T03:04:50Z Exact solution of a variety of X-ray probes in the Falicov-Kimball model with dynamical mean-field theory Shvaika, A.M. Freericks, J.K. We examine the core-level X-ray photoemission spectroscopy (XPS), X-ray absorption near-edge spectroscopy (XANES) and X-ray emission spectroscopy (XES) in the Falicov-Kimball model by using the exact solution from dynamical mean-field theory. XPS measures the core-hole propagator, XANES measures the absorption of X-rays when the core electron is excited to an unoccupied electronic state of the solid and is not emitted, and XES measures the spectra of light emitted as electrons fill the core-hole state created via some form of X-ray excitation. These three spectra are closely related to one another and display orthogonality catastrophe behavior at T=0. We show an efficient way of evaluating these spectra at finite temperature, with a primary focus on the details of XANES. Дослiджуються рентґенiвськi спектри фотоелектронної емiсiї (XPS), передкрайового поглинання (XANES) та фотоемiсiї (XES) для моделi Фалiкова-Кiмбала, використовуючи точнi розв’язки теорiї динамiчного середнього поля. XPS вимiрює пропагатор дiрки йонного залишку, XANES — поглинання X-променiв, коли електрон йонного залишку збуджується в незаповнену електронну зону твердого тiла, i XES — спектр випромiненого свiтла при заповненнi електроном наперед створеної X-променями дiрки йонного залишку. Всi цi три типи спектрiв тiсно пов’язанi мiж собою i проявляють особливостi типу катастрофи ортогональ-ностi при T = 0. Показано, як ефективно розраховувати такi спектри для скiнчених температур, зокрема з детальнiшим наголосом на спектри XANES. 2012 Article Exact solution of a variety of X-ray probes in the Falicov-Kimball model with dynamical mean-field theory / A.M. Shvaika, J.K. Freericks // Condensed Matter Physics. — 2012. — Т. 15, № 4. — С. 43701:1-10. — Бібліогр.: 22 назв. — англ. PACS: 71.10.-w, 71.27.+a, 71.30.+h, 02.30.Rz DOI:10.5488/CMP.15.43701 arXiv:1207.2744 http://dspace.nbuv.gov.ua/handle/123456789/120307 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
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We examine the core-level X-ray photoemission spectroscopy (XPS), X-ray absorption near-edge spectroscopy (XANES) and X-ray emission spectroscopy (XES) in the Falicov-Kimball model by using the exact solution from dynamical mean-field theory. XPS measures the core-hole propagator, XANES measures the absorption of X-rays when the core electron is excited to an unoccupied electronic state of the solid and is not emitted, and XES measures the spectra of light emitted as electrons fill the core-hole state created via some form of X-ray excitation. These three spectra are closely related to one another and display orthogonality catastrophe behavior at T=0. We show an efficient way of evaluating these spectra at finite temperature, with a primary focus on the details of XANES. |
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Shvaika, A.M. Freericks, J.K. |
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Shvaika, A.M. Freericks, J.K. Exact solution of a variety of X-ray probes in the Falicov-Kimball model with dynamical mean-field theory Condensed Matter Physics |
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Shvaika, A.M. Freericks, J.K. |
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Shvaika, A.M. |
| title |
Exact solution of a variety of X-ray probes in the Falicov-Kimball model with dynamical mean-field theory |
| title_short |
Exact solution of a variety of X-ray probes in the Falicov-Kimball model with dynamical mean-field theory |
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Exact solution of a variety of X-ray probes in the Falicov-Kimball model with dynamical mean-field theory |
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Exact solution of a variety of X-ray probes in the Falicov-Kimball model with dynamical mean-field theory |
| title_full_unstemmed |
Exact solution of a variety of X-ray probes in the Falicov-Kimball model with dynamical mean-field theory |
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exact solution of a variety of x-ray probes in the falicov-kimball model with dynamical mean-field theory |
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Інститут фізики конденсованих систем НАН України |
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2012 |
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Exact solution of a variety of X-ray probes in the Falicov-Kimball model with dynamical mean-field theory / A.M. Shvaika, J.K. Freericks // Condensed Matter Physics. — 2012. — Т. 15, № 4. — С. 43701:1-10. — Бібліогр.: 22 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT shvaikaam exactsolutionofavarietyofxrayprobesinthefalicovkimballmodelwithdynamicalmeanfieldtheory AT freericksjk exactsolutionofavarietyofxrayprobesinthefalicovkimballmodelwithdynamicalmeanfieldtheory |
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2025-07-08T17:37:59Z |
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1837101251527966720 |
| fulltext |
Condensed Matter Physics, 2012, Vol. 15, No 4, 43701: 1–10
DOI: 10.5488/CMP.15.43701
http://www.icmp.lviv.ua/journal
Exact solution of a variety of X-ray probes
in the Falicov-Kimball model with dynamical
mean-field theory
A.M. Shvaika1 , J.K. Freericks2
1 Institute for Condensed Matter Physics of the National Academy of Sciences of Ukraine,
1 Svientsitskii St., 79011 Lviv, Ukraine
2 Department of Physics, Georgetown University, 37th and O Sts. NW, Washington, DC 20057, USA
Received July 11, 2012
We examine the core-level X-ray photoemission spectroscopy (XPS), X-ray absorption near-edge spectroscopy
(XANES) and X-ray emission spectroscopy (XES) in the Falicov-Kimball model by using the exact solution from
dynamical mean-field theory. XPS measures the core-hole propagator, XANES measures the absorption of X-
rays when the core electron is excited to an unoccupied electronic state of the solid and is not emitted, and
XES measures the spectra of light emitted as electrons fill the core-hole state created via some form of X-
ray excitation. These three spectra are closely related to one another and display orthogonality catastrophe
behavior at T = 0. We show an efficient way of evaluating these spectra at finite temperature, with a primary
focus on the details of XANES.
Key words: X-ray photoemission spectroscopy (XPS), X-ray absorption near-edge spectroscopy (XANES),
dynamical mean-field theory, Falicov-Kimball model
PACS: 71.10.–w, 71.27.+a, 71.30.+h, 02.30.Rz
1. Introduction
X-ray-based spectroscopy is one of the most important tools in understanding condensed matter
physics. Experiments in X-ray spectroscopy are quite sophisticated and are capable of measuring a
large range of different properties. The basic idea, however, is quite simple. X-rays are shone onto a
sample and one measures either photons or electrons that are emitted after the scattering. Depending on
what is measured, in what energy range, and how the initial X-ray energy is tuned relative to different
absorption edges, one finds a “zoo” of different probes that are used within X-ray scattering. Here we
focus on three of these probes, assuming we are using monochromatic X-rays shining on our sample.
The first is X-ray photoemission spectroscopy (XPS) of the core levels. In this experiment, the X-ray is
tuned far enough above an absorption edge for the photoexcited core electron to have enough energy
to rapidly escape from the crystal (without interacting with it) and to effectively leave the sample as
a free particle. One measures the rate at which the electrons hit a detector as a function of the final
energy of the electron, integrating over all angles (if one measures this as a function of the angle, then
one can determine the angle-resolved photoemission spectroscopy, but this is usually unimportant for
core-level spectra which have essentially no momentum dependence). In this sense, it is a photon in,
an electron out spectroscopy. The second is X-ray absorption near edge spectroscopy (XANES) which
measures the attenuation of X-rays as they are shone onto a material as a function of the incident X-ray
energy. The X-ray energy is tuned such that the photoexcited electron does not have enough energy
to leave the crystal, and hence it is similar to an optical absorption measurement, but with a different
frequency of light. In order to get a spectrum, one needs to change the X-ray frequency and measure the
absorption as a function of the color of the X-ray light, just like one measures the optical conductivity
© A.M. Shvaika, J.K. Freericks, 2012 43701-1
http://dx.doi.org/10.5488/CMP.15.43701
http://www.icmp.lviv.ua/journal
A.M. Shvaika, J.K. Freericks
(after processing the reflectivity data to extract the absorption). If the electron does leave the crystal but
with a low-enough energy to strongly interact with the surface, then one has an X-ray absorption fine
structure (EXAFS) experiment, which we are not going to discuss hereinafter. The third is X-ray emission
spectroscopy (XES), where one has a system with an excited core-hole, and we measure the spectra of
the light emitted from the material as the conduction electrons fill the hole and cause the sample to
fluoresce. It is the stimulated emission of the core hole back down to the ground state, which is enhanced
due to the presence of the X-ray radiation, and hence this is also an elastic process where one measures
the light emitted at the same frequency as the incident light, but in different directions.
One can contrast these probes with inelastic scattering such as nonresonant inelastic light scattering
(NIXS) and resonant inelastic light scattering (RIXS) which are coherent second-order processes similar
to Raman scattering at optical wavelengths. XPS is a single-particle process that can be analyzed via
the sudden approximation, XANES and XES are first-order processes in the matter-light interaction and
correspond to two-particle processes, while NIXS and RIXS are second-order processes in the matter-light
interaction which correspond to four-particle correlation functions and will not be further discussed
here (see [1] for further details in the Falicov-Kimball model).
Before we discuss the formal developments in detail, we need to define our model. We work with the
Falicov-Kimball model [2], with an additional core-hole interaction. For simplicity, we work in a hole pic-
ture, where the fermionic operators are all hole rather than particle operators. The spinless version of the
Falicov-Kimball model involves the interaction of itinerant holes (described by the creation operator d†
i
at
site i ) which can hop between nearest neighbors and localized holes (described by the creation operator
f †
i
at site i ) which do not hop, but interact with the conduction holes via a Coulomb interaction U . When
we add the core-hole (described by the creation operator h†
i
at site i ), it interacts via an on-site Coulomb
repulsion Qd with the conduction holes and Q f with the localized holes. The Hamiltonian is then
H =−
∑
i j
ti j d†
i
d j −µ
∑
i
d†
i
di +E f
∑
i
f †
i
fi +U
∑
i
d†
i
di f †
i
fi +Eh
∑
i
h†
i
hi
+
∑
i
(
Qd d†
i
di +Q f f †
i
fi
)
h†
i
hi (1.1)
=H
kin+
∑
i
H
loc
i , (1.2)
where ti j is a real symmetric matrix that is nonzero only if i and j are nearest neighbors and it equals
t∗/2
p
d on a d -dimensional hypercubic lattice (we will take the limit d →∞ and use t∗ as the energy
unit). The symbol µ is the chemical potential for the conduction holes, while E f and Eh are the site
energies for the localized hole and core hole, respectively. The term H
kin denotes the first term of the
Hamiltonian which includes the kinetic energy of the conduction electrons, while H
loc
i
includes all of
the remaining terms at site i , which are the local terms. Since the localized hole number operators f †
i
f
i
and h†
i
h
i
both commute with H , they are conserved quantities, and hence this model maps onto a
generalized Falicov-Kimball model with spinless conduction and localized holes, which can be solved
exactly within dynamical mean-field theory (DMFT) as described in detail in the review in reference [3].
The retarded local Green’s function for the conduction holes is defined via
Gr
d (t − t ′) =−iθ(t − t ′)
〈{
di (t),d†
i
(t ′)
}
+
〉
, (1.3)
where the hole operators are expressed in the Heisenberg picture d
i
(t) = exp(iH t)d
i
exp(−iH t) and
the angle brackets imply a trace over all states weighted by the density matrix ρ = exp(−βH )/Z with
Z = Trexp(−βH ) and β = 1/T the inverse temperature. The symbols {·, ·}+ denote the anticommuta-
tor. Since we have time-translation invariance in equilibrium, the Green’s function depends only on the
time-difference, and hence it is convenient to take a Fourier transform and represent it as a function of
frequency ω. In addition, we need to define the self-energy Σ(ω), effective medium G0(ω), and dynam-
ical mean field λ(ω). This is done as follows: (i) the local Green’s function can be expressed in terms of
a Hilbert transform of the self-energy with a weighting function given by the noninteracting density of
states ρ(ǫ) = exp
(
−ǫ2
)
/
p
πwhich is a Gaussian for the infinite-dimensional limit of a hypercubic lattice
Gr
d (ω)=
∫
dǫρ(ǫ)
1
ω+µ−Σ(ω)−ǫ
; (1.4)
43701-2
Exact solution of a variety of X-ray probes
(ii) the effective medium is then defined via the Dyson equation for the local Green’s function
1
G0(ω)
= 1
Gr
d
(ω)
+Σ(ω); (1.5)
and (iii) the dynamical mean field is then
λ(ω) =ω+µ− 1
G0(ω)
. (1.6)
It turns out that the dynamical mean field plays a prominent role in determining different X-ray spectro-
scopies.
Note that we are always working in the limit where the number of core holes in the system is so
small, that the density vanishes in the thermodynamic limit. Hence, in determining the dynamical mean
field, we can set all h†
i
h
i
operators to zero and recover the spinless Falicov-Kimball model. The core-
hole operators, however, are quite important when we calculate the X-ray spectroscopies. The details of
the numerical solution of the Green’s functions, self-energies, and dynamical mean fields of the spinless
Falicov-Kimball model can be found in reference [3].
In the next section, we discuss the many-body correlation functions for different spectroscopies. In
section 3, we describe the exact solution of these correlation functions within the DMFT approach. In
section 4, we show some numerical results, and we conclude in section 5.
2. Many-body formalism for different correlation functions
-i β
0 t
Figure 1. The Kadanoff-Baym-Keldysh contour. The
contour starts at time 0, moves along the real time
axis to time t , moves back along the real axis to time
0, then extends down the imaginary axis to imagi-
nary time −iβ.
We start with a discussion of XPS. This is a
photoemission process. If we invoke the sudden
approximation and ignore the momentum depen-
dence of matrix elements which connect the free
electron states with the scattering states, then the
XPS signal is simply given by a greater core-hole
Green’s function (see, for example, reference [4]).
This Green’s function can be found from the so-
called contour-ordered Green’s function which is
defined for each time lying on the Kadanoff-Baym-
Keldysh contour [5, 6] depicted in figure 1. The
contour starts at time t̄ = 0, runs out to time t̄ = t ,
then runs backwards in time and finally moves
along the imaginary axis a distance of β = 1/T .
The greater Green’s function has the first time ly-
ing on the lower branch and the second time lying
on the upper branch, so the first time is ahead of the second one on the contour.
Before we define the contour-ordered core-hole propagator, we need to extend the definition of the
dynamical mean field from real frequencies to times on the contour. This is done by defining the following
function
λc (t̄ , t̄ ′) =− i
π
∞
∫
−∞
dω Im[λ(ω)]e−iω(t̄−t̄ ′) [ f (ω)−θc (t̄ , t̄ ′)
]
, (2.1)
where f (ω) = 1/[1+exp(βω)] is the Fermi-Dirac distribution function and θc (t̄ , t̄ ′) is the generalization of
the Heaviside unit step function to the contour, which equals one if t̄ is ahead of t̄ ′ on the contour, equals
zero when t̄ is behind t̄ ′ on the contour and equals 1/2 when t̄ = t̄ ′. Note that on the contour, two times
which numerically have the same value are only considered equal if they also lie on the same branch of
the contour, and the designation ahead or behind refers to this ordering along the contour as illustrated
by the arrows in figure 1.
The contour-ordered Green’s function for the core hole is then defined to be Gc
h
(t , t ′) =
−i〈Tc h
i
(t)h†
i
(t ′)〉, where Tc denotes the time-ordering along the contour. Within DMFT, a local Green’s
43701-3
A.M. Shvaika, J.K. Freericks
function on the lattice can also be calculated as an impurity Green’s function that evolves within the
dynamical mean field. Hence, we can also write the core-hole propagator as
Gc
h(t , t ′) =−
i
Zimp
Tr
e−βH
loc
Tc exp
−i
∫
c
dt̄
∫
c
dt̄ ′d†(t̄)λc (t̄ , t̄ ′)d(t̄ ′)
h(t)h†(t ′)
. (2.2)
Here, H loc is the local Hamiltonian evaluated at one lattice site and Zimp is the impurity partition func-
tion calculated for an impurity evolving with respect to the local Hamiltonian and the dynamical mean
field evolution operator, given by the exponential function above. In the next section, we briefly discuss
the strategies that can be used to calculate the core-hole propagator.
The XANES signal arises when the X-ray is tuned just slightly above the absorption edge. So, a core
hole is created and the photoelectron goes to an unoccupied state in the conduction band. When this oc-
curs, a photon is taken out of the X-ray beam, and we measure this absorption by counting the number
of photons that pass through the sample and comparing this with the number that pass when the sample
is not there. Viewed in this way, the XANES experiment is similar to an optical conductivity experiment,
except that in the latter, it is an optical photon that causes an excitation of an electron from an occu-
pied to an unoccupied state [7]. Nevertheless, the similarity between these two processes tells us that we
can find the XANES signal by evaluating the imaginary part of the Fourier transform of the appropri-
ate current-current correlation function. Here is how we find the corresponding current operator for a
XANES experiment. First, we assume that the core-hole and the conduction electron are created at the
same site, and for the moment, we assume the core hole will only create itinerant electrons, not localized
electrons. Hence, the transition is to one band only. Second, the photons have momentum q, frequency
ωq = cq , and a spatial profile for the vector potential that describes the light field given by an amplitude
A multiplied by a shape function fq(r). We define the dipole matrix element at the lattice site i via
Mq(i )= 〈0|d†
i
ieħ
me c
fq(r)eiq·r
eq ·∇h†
i
|0〉, (2.3)
which is the matrix element between a core-hole state h†
i
|0〉 and an electronic state in the conduction
band d
i
|0〉, both of which need to be represented by their corresponding wavefunctions in coordinate
space. Here, eq is the polarization of the light field. In a translationally invariant system, the matrix
element is independent of the lattice site i , so the current operator becomes
jq = e−iωqt Mqdi h†
i
+eiωqt M∗
q hi d†
i
. (2.4)
The response of the current is given by the current-current correlation function. Noting that for XANES
we have no core hole in the initial state, we find that the XANES spectra become
χXANES(ωq) =−
2|Mq|2
ħ
Im
∞
∫
0
dteiωqt
〈
d†(t)h(t)d(0)h†(0)
〉
, (2.5)
where the local thermodynamic expectation value can be evaluated for the impurity problem, just like
the core-hole propagator (implying we need to introduce the dynamical mean-field evolution operator,
etc.).
The XES spectra are evaluated in a similar way, but this time there is a core hole present in the initial
state, so different terms from the current-current correlation function contribute and we find
χXES(ωq) =−
2|Mq|2
ħ
Im
∞
∫
0
dte−iωqt
〈
d(t)h†(t)d†(0)h(0)
〉
. (2.6)
In the remainder of this work, we will assume that the matrix element Mq is independent of momen-
tum (and hence independent of the frequency ωq).
43701-4
Exact solution of a variety of X-ray probes
3. Formalism to evaluate the XANES correlation function
We will focus primarily on showing the details of an exact evaluation of the XANES correlation
function. The XPS function can be evaluated using the techniques similar to those developed earlier
for the f -electron spectral function, which employ either a matrix-based formalism or a Wiener-Hopf
sum-equation approach [8–10]. We are preparing a separate publication on this problem [11] which also
examines the numerical renormalization group approach [12] for the core-hole propagator.
To calculate the XANES signal, we need to evaluate the four-point function that is defined by the
functional derivative of the core-hole Green’s function with respect to the dynamical mean field
δGc
h
(t , t ′)
δλc (t2, t1)
=
〈
Tc h(t)h†(t ′)d(t1)d†(t2)
〉
+Gc
h (t , t ′)Gc
d (t1, t2), (3.1)
where the symbol c denotes the contour-ordered quantities and the itinerant hole Green’s function Gc
d
is defined analogous to equation (2.2) but with h(t)h†(t ′) replaced by d(t)d†(t ′); this Green’s function
can also be easily constructed from the retarded Green’s function and the Fermi-Dirac distribution as a
function of frequency because it is in equilibrium, butwe do not give the details here.We need to evaluate
the four-point function for t ′ = 0, t1 = 0 and t2 = t before we take the Fourier transform.
Calculation of this four-point function is straightforward, but tedious. To do so, requires us to define a
fair amount of symbols before we can get to the final formula. In doing so, we closely follow the work in
reference [10], which shows how to represent these objects in terms of determinants of Toeplitz matrices,
which can either be evaluated directly numerically, or can be solved in the large time limit asymptotically
using Szegö’s theorem [13–16] within the Wiener-Hopf approach [17, 18] as developed by McCoy and
Wu [19]. We do not provide details of those approaches here.
We start by defining the fermionic Matsubara frequencies via iωn = iπT (2n +1). Then, the impurity
partition function can be represented as Zimp =Z00 +Z01 +Z10 +Z11, with
Znh n f
= e−β(E f n f +Eh nh+Q f n f nh )
[
1+eβ(µ−Un f −Qd nh )
] ∞
∏
m=−∞
[
1− λ(iωm )
iωm +µ−Un f −Qd nh
]
(3.2)
with n f = 0,1 and nh = 0,1 the two possible fillings for each localized particle. Here, λ(iωm) is the dy-
namical mean field evaluated at the Matsubara frequencies [which can be found by using the spectral
formula given λ(ω)]. Alternatively, these different partition functions can be defined over the contour
via
Znh n f
= Tr
e−βH loc(nh ,n f )
Tc exp
−i
∫
c
dt̄
∫
c
dt̄ ′d†(t̄)λc (t̄ , t̄ ′)d(t̄ ′)
, (3.3)
so that the functional derivative satisfies
δZnh n f
δλc (t2, t1)
= iTr
e−βH loc(nh ,n f )
Tc exp
−i
∫
c
d t̄
∫
c
d t̄ ′d†(t̄)λc (t̄ , t̄ ′)d(t̄ ′)
d(t1)d†(t2)
(3.4)
=−Znh n f
G0d ,nh n f
(t1, t2),
where G0d ,nh n f
is the effective medium projected onto the sectors of fixed nh and n f , and is defined in
the equation above.
Using these definitions, one has, for example, 〈 f † f 〉 = (Z01 +Z11)/Zimp. Since βEh → ∞ for any
reasonable temperature, we usually have Z1n f
→ 0 (T is typically less than 0.1 eV, while Eh is typically
larger than 100 eV).
Now we are ready to start the evaluation of the functional derivative. Note that we always have
t Ê t ′ = 0, so we are evaluating the derivative of the greater core-hole Green’s function, which has nh = 0
in the initial state and the greater core-hole Green’s function satisfies
G>
h (t ,0) =G>
h,00(t ,0)+G>
h,01(t ,0) (3.5)
43701-5
A.M. Shvaika, J.K. Freericks
with
G>
h,00(t ,0) =−i
Z00
Zimp
e−iEh t+C00(t ), (3.6)
G>
h,01(t ,0) =−i
Z01
Zimp
e−i(Eh+Q f )t+C01(t ), (3.7)
and
Cnh n f
(t)= ln detc
[
Ic −Qc
d (t ,0)G0d ,nh n f
]
. (3.8)
The determinant is taken over the contour of the continuous matrix operator defined inside the brackets.
Here, Ic is the identitymatrix on the contour,Qc
d
(t ,0) is a diagonal matrix equal toQd on the top branch of
the contour only [Qc
d
(t ,0) =Qd I
[0,t ], with I[0,t ] equal to the identity matrix for times on the upper branch,
and equal to zero otherwise], and G0d ,nh n f
is the continuous matrix operator for the effective medium
projected onto the sectors of fixed nh and n f . As it was already mentioned above, the greater core-hole
Green’s function (3.5) defines the XPS signal.
The functional derivative of Znh n f
is in equation (3.4). We also need the functional derivative of
Cnh n f
(t), which satisfies
δCnh n f
(t)
δλc (t2, t1)
=−Trc
{
[
Ic −Qc
d (t ,0)G0d ,nh n f
]−1
Qc
d (t ,0)
δG0d ,nh n f
δλc (t2, t1)
}
. (3.9)
The functional derivative of the projected effective medium is straightforward to calculate and it yields
δG0d ,nh n f
(t̄ , t̄ ′)
δλc (t2, t1)
=G0d ,nh n f
(t̄ , t2)G0d ,nh n f
(t1, t̄ ′). (3.10)
Substituting equation (3.10) into equation (3.9) yields
δCnh n f
(t)
δλc (t2, t1)
=−
{
G0d ,nh n f
[
Ic −Qc
d (t ,0)G0d ,nh n f
]−1
Qc
d (t ,0)G0d ,nh n f
}
t1t2
=G0d ,nh n f
(t1, t2)−G
Q
0d ,nh n f
(t1, t2), (3.11)
where we have defined
G
Q
0d ,nh n f
(t1, t2) =
{
G0d ,nh n f
[
Ic −Qc
d (t ,0)G0d ,nh n f
]−1
}
t1t2
, (3.12)
which satisfies the equation
G
Q
0d ,nh n f
(t1, t2) =G0d ,nh n f
(t1, t2)+Qd
t
∫
0
dt̄G0d ,nh n f
(t1, t̄)G
Q
0d ,nh n f
(t̄ , t2). (3.13)
Plugging this result into equation (3.1) and taking the limits t1 = t ′ = 0 and t2 = t gives our final result
χXANES(t) =
〈
Tc h(t)d†(t)d(0)h†(0)
〉
=−iG>
h,00(t ,0)G
Q
0d ,00
(0, t)− iG>
h,01(t ,0)G
Q
0d ,01
(0, t). (3.14)
What is remarkable about this formula is that once one has projected onto the different static particle
number sectors, the expectation values can be evaluated by factorization similar to Wick’s theorem, even
though we make no such assumption and the above result is exact. Since the core-hole propagator dis-
plays the phenomena of the orthogonality catastrophe, where, in the limit as T → 0, the spectral function
diverges with a power law that depends on the coupling strength [7, 20], we expect the same behavior to
occur in the XANES (and XES) signals because they involve matrix determinants that are very similar to
the XPS signal and should have the similar asymptotic behavior.
In order to calculate this expression, we solve equation (3.13) via Cramer’s rule, and then manipulate
the determinants to get a uniform expression.We start by discretizing the interval [0, t ] with N time slices
43701-6
Exact solution of a variety of X-ray probes
(t1 = 0 and tN = t ), with a step size of ∆t = t/(N −1). Then, we find that we can approximate the XANES
correlation function by taking determinants of discrete (N −1)× (N −1) matrices (which can be scaled to
∆t → 0 by repeating with different ∆t values). The formula is
χXANES(t) =− Z00
Zimp
e−iEh t (3.15)
×det
G0d ,00(t1 − tN ) −∆tQdG0d ,00(t1 − t2) · · · −∆tQd G0d ,00(t1 − tN−1)
G0d ,00(t2 − tN ) 1−∆tQd G0d ,00(0) · · · −∆tQd G0d ,00(t2 − tN−1)
...
...
. . .
...
G0d ,00(∆t) −∆tQdG0d ,00(tN−1 − t2) · · · 1−∆tQd G0d ,00(0)
− Z01
Zimp
e−i(Eh+Q f )t
×det
G0d ,01(t1 − tN ) −∆tQdG0d ,01(t1 − t2) · · · −∆tQd G0d ,01(t1 − tN−1)
G0d ,01(t2 − tN ) 1−∆tQd G0d ,01(0) · · · −∆tQd G0d ,01(t2 − tN−1)
...
...
. . .
...
G0d01(∆t) −∆tQdG0d ,01(tN−1 − t2) · · · 1−∆tQd G0d ,01(0)
.
Note that since each time t is independent of each other, this calculation can be easily parallelized; the
t dependence in the determinant comes from Qd which is nonzero only on the interval [0, t ] and deter-
mines the size of the matrix. The calculation for the XES signal is similar, but the details will not be given
here.
4. Numerical results
For the numerical work, we will examine the Falicov-Kimball model on a hypercubic lattice at half-
filling for the conduction holes and for the localized holes. We evaluate the XPS and XANES spectra for
three different cases: U = 0.5 which is a diffusive metal, U = 1 which is an anomalous metal that has
a dip in the density of states at the chemical potential, and U = 2 which is in the Mott-insulating phase
(due to space constraints we do not include XES results). We choose the core-hole interactions to satisfy
Qd = Q f = 2. Both the XPS in equation (3.5) and XANES in equation (3.14) responses contain two contri-
butions (groups of peaks), one from the sites with empty f -states and another from the occupied ones,
separated by the interaction energy Q f and are represented by the blue dashed and red dot-dashed lines,
respectively, in the figures below.
For the cases of the diffusive (figure 2) and anomalous (figure 3) metals, each of the contributions to
the XPS response contain two peaks separated by the interaction energy Qd . At high temperatures for
the case of the diffusive metal (figure 2a), we have in total four peaks (with two peaks merged due to
the parameter selection Qd = Q f ). When the temperature becomes lower (figure 2c), the peaks become
more pronounced with an almost Lorentzian profile and one of them becomes very sharp and at zero
temperature transforms into the power law singularity of the Anderson orthogonality catastrophe [20].
For the case of the anomalous metal (figure 3), the XPS response is almost the same with the appearance
of additional shoulders at the peaks and now the edge singularity is much narrower and sharper due to
the decrease of the density of states at the chemical potential level, but it is still a power law divergence
at zero temperature. The XANES response for these cases of the diffusive and anomalous metal contains
two peaks, one of which transforms into a power law singularity at zero temperature. In other aspects,
the behavior is similar to the XPS response.
The behavior of the XPS and XANES spectra in the Mott-insulating case is completely different (fig-
ure 4). The XPS response at high temperatures is similar to the one in the metallic phase, but when the
temperature becomes lower, the first group of the peaks displays a gapped spectra similar to what is seen
in the f -particle density of states [10]. The other group of the peaks display a very narrow Lorentzian
feature which changes at zero temperature from a power law singularity to a δ-peak. The XANES re-
sponse also displays a sharp edge singularity, but now it is accompanied by a near edge structure which
corresponds to the gapped Mott-insulator density of states.
43701-7
A.M. Shvaika, J.K. Freericks
-4 -2 0 2 4
ω + µ - E
h
0
0.5
1
1.5
2
χX
PS
(ω
)
a)
-4 -2 0 2 4
ω + µ - E
h
0
0.2
0.4
0.6
0.8
χX
A
N
E
S (ω
)
b)
-4 -2 0 2 4
ω + µ - E
h
0
0.5
1
1.5
2
χX
PS
(ω
)
c)
-4 -2 0 2 4
ω + µ - E
h
0
0.2
0.4
0.6
0.8
χX
A
N
E
S
d)
Figure 2. (Color online) XPS (a, c) and XANES (b, d) spectra for U = 0.5, Qd =Q f = 2, and T = 1 (a, b) and
T = 0.3 (c, d). The black solid line gives the total spectra and the blue dashed and red dot-dashed lines
correspond to the first and second contributions in the equations (3.5) and (3.14), respectively.
-4 -2 0 2 4
ω + µ - E
h
0
0.5
1
1.5
2
χX
PS
(ω
)
a)
-4 -2 0 2 4
ω + µ - E
h
0
0.2
0.4
0.6
0.8
χX
A
N
E
S (ω
)
b)
-4 -2 0 2 4
ω + µ - E
h
0
0.5
1
1.5
2
χX
PS
(ω
)
c)
-4 -2 0 2 4
ω + µ - E
h
0
0.2
0.4
0.6
0.8
χX
A
N
E
S (ω
)
d)
Figure 3. (Color online) The same as in figure 2 but for U = 1.
43701-8
Exact solution of a variety of X-ray probes
-4 -2 0 2 4
ω + µ - E
h
0
0.5
1
1.5
2
χX
PS
(ω
)
a)
-4 -2 0 2 4
ω + µ - E
h
0
0.2
0.4
0.6
0.8
χX
A
N
E
S (ω
)
b)
-4 -2 0 2 4
ω + µ - E
h
0
0.5
1
1.5
2
χX
PS
(ω
)
c)
-4 -2 0 2 4
ω + µ - E
h
0
0.2
0.4
0.6
0.8
χX
A
N
E
S (ω
)
d)
Figure 4. (Color online) The same as in figure 2 but for U = 2.
5. Conclusions
In this work, we have shown how to calculate three different X-ray spectroscopies in strongly corre-
lated materials: XPS, XANES, and XES within the Falicov-Kimball model in the limit of large spatial dimen-
sions. In all cases, one can find analytic expressions for these spectra in terms of determinants of continu-
ous matrix operators. These continuous operators can be discretized and evaluated on a computer using
standard LAPACK routines. The XPS and XANES spectra have similar behavior, but the peak locations and
weights differ, indicating how they are measuring different processes. Generically, the XANES signal has
fewer peaks, but the strong-coupling satellites often appear with a higher weight. While these results do
not incorporate complicated multiplet or spin-orbit coupling effects, which are often seen in real materi-
als, they do include all of the strong-correlation effects in a simple model, allowing for an exact solution.
A similar formalism to what was developed here can be used to solve for time-resolved XPS, XANES
and XES experiments, where the system is pumped to a nonequilibrium state via a high-intensity fem-
tosecond pulse, and then the system is probed with lower amplitude X-rays after some time delay with
respect to the pump. The main difference in the formalism is that Green’s functions lose their time-
translational invariance and need to be solved using nonequilibrium DMFT [21]. We will present the re-
sults for that work elsewhere. With the advent of newer high brilliance, short pulse width, X-ray sources
(like the linac coherent light source, LCLS), we expect there to be more studies of pump/probe phenom-
ena in strongly correlated materials, and it is possible that probes like XPS, XANES, or XES, can shed light
on interesting nonequilibrium dynamics. XPS has already been employed [22] to measure the properties
of the order parameter of charge-density-wave systems on ultrafast time scales, indicating that we are at
the frontier of these types of experiments.
43701-9
A.M. Shvaika, J.K. Freericks
Acknowledgements
J.K.F. was supported by the Department of Energy under grant number DE-FG02-08ER46542. The US-
Ukraine collaboration was supported by the Department of Energy under grant number DE-SC0007091.
J.K.F. also acknowledges support from the McDevitt bequest at Georgetown.
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Точний розв’язок теорiї динамiчного середнього поля
для рiзноманiтних дослiдiв з X-променями
в моделi Фалiкова-Кiмбала
А.М. Швайка1, Дж.К. Фрiрiкс2
1 Iнститут фiзики конденсованих систем НАН України, вул. Свєнцiцького 1, 79011 Львiв, Україна
2 Фiзичний факультет, Унiверситет Джорджтауну, Вашингтон, округ Колумбiя 20057, США
Дослiджуються рентґенiвськi спектри фотоелектронної емiсiї (XPS), передкрайового поглинання (XANES)
та фотоемiсiї (XES) для моделi Фалiкова-Кiмбала, використовуючи точнi розв’язки теорiї динамiчного се-
реднього поля. XPS вимiрює пропагатор дiрки йонного залишку, XANES — поглинання X-променiв, коли
електрон йонного залишку збуджується в незаповнену електронну зону твердого тiла, i XES — спектр ви-
промiненого свiтла при заповненнi електроном наперед створеної X-променями дiрки йонного залишку.
Всi цi три типи спектрiв тiсно пов’язанi мiж собою i проявляють особливостi типу катастрофи ортогональ-
ностi при T = 0. Показано, як ефективно розраховувати такi спектри для скiнчених температур, зокрема
з детальнiшим наголосом на спектри XANES.
Ключовi слова: фотоелектронна рентґеноспектроскопiя (XPS), рентґеноспектроскопiя передкрайового
поглинання (XANES), теорiя динамiчного середнього поля, модель Фалiкова-Кiмбала
43701-10
http://dx.doi.org/10.1103/PhysRevB.86.125103
http://dx.doi.org/10.1103/PhysRevLett.22.997
http://dx.doi.org/10.1103/RevModPhys.75.1333
http://dx.doi.org/10.1002/pssb.200881555
http://dx.doi.org/10.1103/PhysRev.163.612
http://dx.doi.org/10.1007/BF01318160
http://dx.doi.org/10.1103/PhysRevB.71.115111
http://dx.doi.org/10.1103/PhysRevB.71.125101
http://dx.doi.org/10.1103/PhysRev.178.1097
http://dx.doi.org/10.1103/PhysRevLett.97.266408
http://dx.doi.org/10.1103/PhysRevLett.105.187401
Introduction
Many-body formalism for different correlation functions
Formalism to evaluate the XANES correlation function
Numerical results
Conclusions
|