Condensed Matter Physics, 2000, Vol. 3, No. 1(21), pp. 183–200
Raman light scattering for systems with
strong short-range interaction
I.V.Stasyuk, T.S.Mysakovych
Institute for Condensed Matter Physics
of the National Academy of Sciences of Ukraine,
1 Svientsitskii Str., 79011 Lviv, Ukraine
Received September 29, 1999
Various type contributions to Raman light scattering are investigated for the
Hubbard, t − J and pseudospin-electron models. To construct the polar-
izability operator the microscopic approach is used, which is based on the
operator expansion in the terms of the Hubbard operators using t and J
as formal parameters of the expansion. Two different contributions to the
dipole momentum are taken into account: one is connected with the non-
homeopolarity of filling of the electron states on a site, another – with the
dipole transitions from the ground state to the excited ones (for the case of
the Hubbard model) and with the dipole momentum of the pseudospins (for
the case of the pseudospin- electron model). The general expressions for
the scattering tensor components describing the magnon, electron (intra-
and interband) and pseudospin scattering are obtained. The resonant and
nonresonant contributions are separated; their role at the change of the
hole concentration due to doping is studied. The dependence of the Ra-
man scattering tensor on the polarization of the incident and scattered light
is investigated.
Key words: Raman scattering, Hubbard correlation, pseudospin-electron
model, polarizability operator
PACS: 72.10.Dp, 72.10.Di, 74.20
1. Introduction
The problem of nonphonon contributions to Raman light scattering in the crys-
tals with the strong short-range Hubbard-type interaction between electrons remains
a subject of interest during the last years in spite of the success achieved in describing
the magnetic and electron Raman scattering in the systems with antiferromagnetic
ordering [1–3]. The approach used by [1] was based on the techniques of construct-
ing the effective Hamiltonian of interaction between the system and the incident
light. It was shown in [1] that the main contribution to Raman scattering in antifer-
romagnets is due to the two-magnon scattering; the magnons which participate in
c© I.V.Stasyuk, T.S.Mysakovych 183
I.V.Stasyuk, T.S.Mysakovych
this scattering are the edge magnons of the Brillouin zone. The effective scattering
Hamiltonian has the structure similar to the operator of the exchange interaction
and is proportional to the scalar product of spin operators on the neighbouring lat-
tice sites; scattering intensity is determined by the square of the parameter of the
electron exchange through the excited states of atoms. This idea was partially used
in [2] where in the case of Hubbard model the electron scattering contributions,
connected with the lower Hubbard subband, were considered. At the half filling
(n = 1) and at strong correlation (U ≫ t) an antiferromagnetic state is the ground
one. The transitions with the participation of the upper subband make it possible
to achieve the two-magnon scattering similar to that considered in [1]. Besides such
kind of scattering that can be referred to the resonant type (the scattering operator
is proportional to t2/(U− h¯ω1), the contributions of the higher order with respect to
t/(U−h¯ω1) were analyzed. The corresponding components of the scattering operator
being projected on the states of the lower Hubbard subband (at the homeopolarity
condition ∑σ niσ = 1) are expressed in terms of spin operator products; their form
depends on the geometry of scattering (i.e the polarization of the incident and the
scattered light) [2,3]. The attempt was made to also describe the so-called nonreso-
nant contributions to the scattering which manifest themselves in the doped case (at
the presence of holes, n < 1) and are connected with the intraband electron transi-
tions. There have been separated contributions proportional to t2; the conditions at
which such terms remain nonzero ones in the limit ~k2 −~k1 → 0 were considered (~k1
and ~k2 are the wave vectors of the incident and the scattered light, respectively).
The aim of this work is to develop a scheme that will make it possible to make
a deeper and a more consistent investigation of the various type nonphonon contri-
butions to the Raman scattering in strongly correlated electron systems. A micro-
scopic approach which was proposed in [4–6] is applied. This method is based on
the construction of the polarizability operator Pˆ of the system by means of a direct
solving of the equations of motion with the use of the operator expansion. With-
in this scheme the electron contributions to the Raman scattering in the Hubbard
model are considered, the magnon component of the scattering in the t− J model
is separated (at the mean-field type consideration of the doping level) as well as the
additional contributions appearing in these models due to the exchange interaction
via the excited states are investigated. On the same basis the possible mechanisms
of the Raman scattering in systems described by the pseudospin-electron model are
studied. This model was proposed in connection with the investigation of locally an-
harmonic phenomena in high-Tc superconducting crystals (anharmonic subsystem
of ions in the double-minimum potential wells is described by means of pseudospin;
the electron subsystem possesses a strong short-range correlation similar to that in
the case of Hubbard model [7,8]).
We start from the explicit expression for the cross-section of Raman light scat-
tering ([4,5]):
∂2σ
∂Ω∂ω2
= 1(4πε0)2
√
ε1
ε2
ω32ω1
h¯2c4
∑
αβα′β′
e1αe2βe1α′e2β′Hβ
′α′,β,α
k2,k1 (ω1, ω2), (1)
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Raman light scattering
here ~e1, ~e2 are polarization vectors; ω1, ω2 are incident and scattered light frequencies;
ε1,2 ≡ ε(ω1, ω2); Hβ
′α′,β,α
k2,k1 (ω1, ω2) is the Raman scattering tensor:
Hβ′α′,β,αk2,−k1;−k2,k1(ω1, ω2) =
1
2π
+∞∫
−∞
dtei(ω1−ω2)t〈Pˆ β′α′~k2−~k1(−ω1, t)Pˆ
βα
−~k2~k1
(ω1, 0)〉, (2)
where Pˆ is the polarizability operator
Pˆ βα~k′~k(ω, t) = −
+∞∫
−∞
dseiω(t−s){{Mˆβ(~k′, t)|Mˆα(~k, s)}}, (3)
Here Mˆα(~k) is a dipole momentum of a crystal unit cell in the ~k-representation
and the symbol {{Mˆβ(~k′, t)|Mˆα(~k, s)}} stands for “unaveraged” Green’s function
defined in the following way [6]:
{{A(t)|B(t′)}} = −iΘ(t− t′)[A(t), B(t′)]; (4)
operators A(t), B(t′) are given in the Heisenberg representation.
The equations of motion for this function have the following form
h¯ω1{{A|B}}ω1,ω2 =
h¯
2π [A,B]ω1−ω2 + {{[A,H ]|B}}ω1,ω2, (5)
h¯ω2{{A|B}}ω1,ω2 =
h¯
2π [A,B]ω1−ω2 − {{A|[B,H ]}}ω1,ω2 , (6)
where corresponding Fourier transforms are introduced. The equations (5), (6) are
applied to construct the polarizability operator; the solutions of these equations are
built in the form of operator series in powers of certain parameters of Hamiltonian
H . In this work in the case of the models of the Hubbard model type the expansion
in terms of the electron transfer parameter t is used.
2. The Hubbard model
First let us consider the case of the Hubbard model
Hˆ =
∑
i,j
ti,j cˆ†i,σ cˆj,σ + U
∑
i
nˆi,↑nˆi,↓ −
∑
i
µnˆi, (7)
where the first and the second terms describe the electron transfer and short-range
electron correlation, respectively; µ is chemical potential. We will restrict ourselves
to the case of large U , so we can make expansion in powers of t at the construction
of the Pˆ -operator, considering only the terms which are linear and quadratic in t.
In the case of the Hubbard model the dipole momentum of a unit cell has a form
~ˆMi = e~Ri(nˆi,↑ + nˆi,↓). (8)
185
I.V.Stasyuk, T.S.Mysakovych
In this expression the nonhomeopolarity of filling of the electron states on lattice
sites is taken into account. It is useful to consider the following single-site basis of
the states |ni,↓, ni,↑〉
|1〉 = |0, 0〉, |2〉 = |1, 1〉, |3〉 = |1, 0〉, |4〉 = |0, 1〉 (9)
and to introduce the Hubbard operators Xr,s = |r〉〈s|.
To calculate Green’s function {{Mˆαk |Mˆβl }} we first use the equation of motion
(5):
{{Mˆαk |Mˆβl }} =
eRαk
h¯ω1
∑
i,j,σ
ti,j(δi,k − δj,k){{cˆ†i,σcˆj,σ|Mˆβl }}. (10)
Then we use the equation of motion written in the form (6):
{{cˆ†i,σcˆj,σ|Mˆβl }} =
[
h¯
2π (δj,l − δi,l)cˆ
†
i,σcˆj,σ
−
∑
s,pσ′
ts,p(δs,l − δp,l){{cˆ†i,σcˆj,σ|cˆ†s,σ′ cˆp,σ′}}

 eR
β
l
h¯ω2
. (11)
To calculate the function {{cˆ†i,σcˆj,σ|cˆ†s,σ′ cˆp,σ′}} we again write the equation of motion
(5) and neglect the terms which are proportional to t. Using this scheme, the fol-
lowing expression for Green’s function {{Mˆαk |Mˆβl }} is obtained up to the terms of
the second order of t:
{{Mˆαk |Mˆβl }} =
[∑
i,j,σ
ti,j(δi,k − δj,k)(δj,l − δi,l)cˆ†i,σ cˆj,σ
− 1h¯ω1 + U
Aˆ†kl +
1
h¯ω1 − U
Aˆkl +
1
h¯ω1
Bˆkl
] e2RαkRβl
2πh¯ω1ω2
, (12)
here U plays the role of the energetic distance between the two levels which are
responsible for scattering,
Aˆkl =
∑
i,j,s
ti,jts,j(δi,k − δj,k)(δj,l − δs,l)
×
[
X31i (X22j + X44j )(X13s −X42s ) + X41i (X31s −X24s )X12j
− (X41s + X23s )(X11j + X44j )X32i + X41i X34j (X42s −X13s )
+ X41i (X33j + X22j )(X32s + X14s ) + X21j (X13s −X42s )X32i
− (X31s −X24s )X43j X32i −X31i X43j (X14s + X32s )
−X31i (X41s + X23s )X12j X21j (X14s + X32s )X42i +
+ (X41s + X23s )X34j X42i + (X31s −X24s )(X11j + X33j )X42i
]
,
Bˆkl = (Aˆ†kl + Aˆkl)
(
X11 → X22
X34, X43 → −X34,−X43
)
. (13)
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Raman light scattering
Comparing with the results obtained in [2,3], we can see that the second order terms
in our expression, when i = s and ni,↑ + ni,↓ = 1, are the same as the scattering
Hamiltonian obtained in [2]:
{{Mˆαk |Mˆβl }} ∼
∑
i,j
(δi,k − δj,k)(δi,l − δj,l)
t2i,j
h¯ω1 − U
~ˆSi ~ˆSj (14)
(considering only the resonant term ∼ (h¯ω1 − U)−1). Such a structure of the scat-
tering Hamiltonian leads to the magnon Raman scattering. However, there are some
additional terms in (12) which were not presented in [2]. These terms are connected
with electron transitions between the next nearest neighbours and are responsible
for the scattering with the participation of holes or extra electrons, which is actual
in the doped case (n 6= 1) [9].
Now let us calculate the scattering tensor. To deal with the definition (2) we will
use the formula:
〈Cˆ(ω1 − ω2)Dˆ(ω′1 − ω1)〉 = δ(ω′1 − ω2)
2ℑm〈〈Dˆ|Cˆ〉〉ω+iε
eβh¯ω − 1 ω=ω2−ω1
. (15)
In consequence we obtain the following expression for the scattering tensor:
Hβ′α′,β,αk2,k1 (ω1, ω2) =
∑
n,n′,n1,n′1
i,j,i1,j1
ei(
~k2(~Rni−~Rn1i1 )−~k1(~Rn′j−~Rn′1j1
))
×
e42ℑm〈〈Aˆ|Aˆ†〉〉ω=ω2−ω1RαniR
β
n′jRα
′
n1i1R
β′
n′1j1
(eβh¯ω − 1)h¯ω1ω2(h¯ω1 − U)2
, (16)
here only the resonant term (h¯ω1 ∼ U) is retained. To calculate Green’s function
〈〈Aˆ|Aˆ†〉〉 using the Hamiltonian of the Hubbard model, we will use a decoupling
procedure for Fermi type operators:
〈Xpq(t)Xrs(t)XmnXpl〉 ≈ 〈Xpq(t)Xpl〉〈Xrs(t)Xmn〉 ± 〈Xpq(t)Xmn〉〈Xrs(t)Xpl〉,
(17)
having split Boson operators X pp in the product of two Fermi operators: X pp =
Xp1X1p. When calculating Green’s function 〈〈X pq|Xrs〉〉 (where Xpq, Xrs are Fermi
operators), we use the simplified version of the Hubbard-I approximation, corre-
sponding to the case of independent subbands, and restrict ourselves to the lower
Hubbard subband, neglecting contributions connected with the state |2〉. After some
algebra we obtain the following result:
Hβ′α′,β,αk2,k1 (ω1, ω2) =
32e4a4t4〈X11 + X33〉4
N2ω1ω2(h¯ω1 − U)2
×
{
2
∑
q1,q2,q3,q
δ(k2 − k1 − q1 + q + q2 − q3)
(eβ(µ−〈X11+X33〉t(q)) + 1)(e−β(µ−〈X11+X33〉t(q1)) + 1)
×
δ
(
ω − 〈X11+X33〉h¯ (t(q1)− t(q) + t(q3)− t(q2))
)
(eβ(µ−〈X11+X33〉t(q2)) + 1)(e−β(µ−〈X11+X33〉t(q3)) + 1)
187
I.V.Stasyuk, T.S.Mysakovych
× sin(qα′)
(
sin(qβ1 ) + sin(qβ3 )
)
sin(qβ′1 ) (sin(qα) + sin(qα1 ))
+
∑
q1,q2,q3,q
δ(k2 − k1 − q1 + q)
(eβ(µ−〈X11+X33〉t(q)) + 1)(e−β(µ−〈X11+X33〉t(q1)) + 1)
×
δ
(
ω − 〈X11+X33〉h¯ (t(q1)− t(q))
)
(e−β(µ−〈X11+X33〉t(q2)) + 1)(e−β(µ−〈X11+X33〉t(q3)) + 1) (18)
× (sin(qα)(sin(qβ1 ) + (sin(qα3 ) sin(qβ3 ))(sin(qα
′
2 ) sin(qβ
′
2 ) + (sin(qα
′) sin(qβ′1 ))
}
,
here a is a lattice constant. The average 〈X 11+X33〉 is determined in a self-consistent
way [7,9]. From the expression (18) we can see that the difference between the
frequencies of the scattered and the incident light waves is connected with the two-
electron transitions in the band:
ω = ω2 − ω1 =
〈X11 + X33〉
h¯ [t(q1) + t(q3)− t(q2)− t(q3)],
q1 + q3 = q2 + q + k2 − k1 (19)
as well as with the one-electron intraband transitions:
ω2 − ω1 =
〈X11 + X33〉
h¯ [t(q1)− t(q)], q1 = q + k2 − k1. (20)
Similar one-electron transitions can also be obtained from the nonresonant term in
(12), which is linear in t. The last contribution to the scattering tensor was considered
in [2,3] in the case of hole doping (n < 1). One can see that in the simple single-band
case the frequency change (20) tends to zero due to the inequality |k2 − k1| ≪ q, q1
(that is characteristic of light wave vectors). By the same reason the corresponding
term in (18) vanishes in the T = 0 limit.
Calculating Green’s function, we used the Hubbard Hamiltonian with the ex-
cluded state |2〉, so we have obtained only the electron (hole) scattering.To consider
the magnon scattering we will deal with the t− J model.
3. The extended t − J model
In the case of nearly half filling 〈ni,↑ + ni,↓〉 ≈ 1 and U ≫ t the Hubbard
Hamiltonian can be reduced to the effective Hamiltonian of the so-called t − J
model:
Hˆt−J =
∑
i,j,σ
ti,jˆ˜c
†
j,σ
ˆ˜ci,σ +
∑
i,j
Ji,j( ~ˆSi ~ˆSj −
nˆi,↑nˆj,↓
4 )−
∑
i
µnˆi, (21)
here ˆ˜ci,σ = cˆi,σ(1 − nˆi,−σ), Ji,j = 4t2i,j/U is the constant of the antiferomagnetic
type exchange interaction. Let us extend this model taking into account the excited
atomic states ϕαexc, having different parity with respect to the ground state ϕ0 as
188
Raman light scattering
well as their exchange and Coulomb interaction with the ground state1 [1]:
Hˆ = Hˆt−J + (E − µ)
∑
i,σ,α
aˆ†αi,σaˆαi,σ +
∑
i,j,σ,α,β
Mα,βi,j (X31i X13j + X41i X14j )aˆ†α,j,σaˆ†β,i,σ
−
∑
i,j,α,β
Kα,βi,j (X33i aˆ†αj,↓aˆβj,↓ + X44i aˆ†αj,↑aˆβj,↑ + X34i aˆ†αj,↑aˆβj,↓ + X43i aˆ†αj,↓aˆβj,↑), (22)
the indices α, β refer to the orbitally degenerated excited states; E is the excitation
energy. Kα,βi,j ,Mα,βi,j are additional interaction constants. The Hamiltonian is written
in terms of the Hubbard operators X rs. The relation between the spin operator ~Si
and the operators Xrs is as follows:
Szi =
1
2(X
44
i −X33i ), S+ = Sxi + iSyi = X43i , S− = Sxi − iSyi = X34i . (23)
Let us take the dipole momentum in the form:
Mˆαi = dα(aˆ†α,i,↓X13i + aˆ†α,i,↑X14 + h.c.), (24)
separating the component, which is connected with the dipole transitions between
the ground and the excited states; dα = 〈ϕ0|rα|ϕαexc〉. We make expansion in terms of
t/E,M/E,K/E at the construction of the polarizability operator, considering the
linear terms in M/E,K/E and quadratic in t/E. Using the method described in the
previous section the following formula for Green’s function {{Mˆαk |Mˆβl }} is obtained:
{{Mˆαk |Mˆβl }} =
h¯e2dαdβ
2π(h¯ω1 − E)(h¯ω2 − E)
×
[
−
∑
i
δk,lδα,βti,k(X31i X13k + X41i X14k )−
∑
i
δk,l2Ji,k( ~ˆSi ~ˆSk −
nˆinˆk
4 )δα,β
−Mαβl,k ( ~ˆSl ~ˆSk +
nˆlnˆk
4 )−
∑
i
δl,kKαβi,k ( ~ˆSi ~ˆSk +
nˆinˆk
4 )
]
+ h¯e
2dαdβδk,lδα,β
2π(h¯ω1 −E)2(h¯ω2 − E)
∑
i,j
ti,ktj,k(X31i X13j + X41i X14j ). (25)
Here we include only the resonant terms and leave out the terms which concern the
excited states.
The spin part of the expression for {{Mˆαk |Mˆβl }} is similar to that obtained in
[1] for antiferromagnet. The formula transforms into the simple product of spin
operators ~ˆSi ~ˆSj in the case of homeopolarity: ni,↑ + ni,↓ = 1 (when the hole doping
level is equal to zero) and the terms which are linear and quadratic in t/E arise
from the pure band transitions.
Let us pass to the scattering tensor. A spin-polaron approach can be used for the
t− J model in the region of small hole concentrations [10,11]. We introduce for the
1We take into account the possibility of the orbital degeneration of the excited state. In the
case of p-like function ϕαexc and the quasi-two-dimensional structure α = x, y considered below.
189
I.V.Stasyuk, T.S.Mysakovych
electron operators ˆ˜ci,σ the following representation in the sublattices with spin up (i
∈↑) and spin down (i ∈↓) ground states (that correspond to the antiferromagnetic
ordering)
ˆ˜ci,↑ = hˆ†i1, ˆ˜ci,↓ = hˆ†i1bˆi1(i ∈↑); ˆ˜ci,↓ = hˆ†i2, ˆ˜ci,↑ = hˆ†i2bˆi2(i ∈↓), (26)
here hˆi1 and hˆi2 are hole spinless operators and bˆi1, bˆi2 are magnon operators on two
sublattices. Performing the canonical transformation for Fourier components
bk1 = vkαk + ukβ+−k, bk2 = vkβk + ukα+−k, (27)
we obtain the following Hamiltonian of the spin-polaron model:
Ht−J =
∑
k,q
(h+k1hk−q2[g(k, q)αq + g(q − k, q)β+q ] + h.c.)
− µ
∑
k
(h+k1hk1 + h+k2hk2) +
∑
q
ωq(α+q αq + β+q βq). (28)
Here
g(k, q) = 4t√
N/2
(uqγk−q + vqγk), uk =
√
1 + νk
2νk
,
vk = − sign(γk)
√
1− νk
2νk
, νk =
√
1− γ2k,
γk =
1
4
∑
r
eikr, ωk = 2J(1− 2δ)2νk, 2δ = 〈h+i1hi1〉+ 〈h+i2hi2〉. (29)
This representation excludes doubly occupied states and takes into account strong
antiferromagnetic spin correlations at the electron hopping.
Now we rewrite the Raman scattering tensor in terms of the hole spinless oper-
ators and the magnon operators:
Hβ′α′,βαk1,k2 (ω1, ω2) =
∑
n,n′,n1,n′1
i,i1,j,j1
ei
(
~k2(~Rni−~Rn1i1)−~k1(~Rn′j−~Rn′1j1
)
)
×
e4h¯2dαdα′dβdβ′2ℑm〈〈T αβn1i1,n′1j1 |T
α′β′
ni,n′j〉〉
(eβh¯ω − 1)(h¯ω1 − E)2(h¯ω2 −E)2
, (30)
here the operator T αβn1i1,n′1j1 has the form:
T αβn1i1,n′1j1 =
=
[
δαβδn1n′1δi1j1
∑
m,l
2Jml,n1i1(bmlbn1i1 + b+mlb+n1i1 − nmlnn1i1 − 1)hmlh
+
mlhn1i1h+n1i1
+ Mαβn1i1,n′1j1(bn1i1bn′1j1 + b
+
n′1j1
b+n1i1 + nn1i1 + nn′1j1)hn1i1h
+
n1i1hn′1j1h
+
n′1j1
+
∑
m,l
Kαβn1i1,ml(bn1i1bml + b
+
mlb+n1i1 + nn1i1 + nml)hmlh
+
mlhn1i1h+n1i1
]
(1− δ)2. (31)
190
Raman light scattering
We will consider the low values of magnon concentration and will not include the
term nmlnn1i1 ; we will also use the approximation: hmih+mi → (1− δ). In this case
T αβn1i1,n′1j1 =
[
δαβδn1n′1δi1j1
∑
m,l
2Jml,n1i1(bmlbn1i1 + b+mlb+n1i1)
+ Mαβn1i1,n′1j1(bn1i1bn′1j1 + b
+
n′1j1
b+n1i1 + nn1i1 + nn′1j1)
+
∑
m,l
Kαβn1i1,ml(bn1i1bml + b
+
mlb+n1i1 + nn1i1 + nml)
]
(1− δ)2. (32)
So we have to find Green’s function built on the magnon operators. In the case
when the hole-magnon scattering is not taken into account it can be done using
the diagonal part of the magnon Hamiltonian. For instance, let us calculate Green’s
function 〈〈bq1bq22|b+q31b+q42〉〉 (here the magnon operators bqi, b+qi are Fourier compo-
nents of bni, b+ni, respectively). Using the above mentioned canonical transformation,
we can write
〈〈bq1bq22|b+q31b
+
q42〉〉 =
= 〈〈αq1α+−q2 |α
+
q3α−q4〉〉uq1uq3vq2vq4 + 〈〈β
+
−q1βq2 |β−q3β
+
q4〉〉uq2uq4vq1vq3
+ 〈〈αq1βq2|α+q3β
+
q4〉〉uq1uq3uq2uq4 + 〈〈β
+
−q1α−q2|β−q3α−q4〉〉vq1vq3vq2vq4. (33)
The functions obtained can be easily calculated using the standard technique of the
equations of motion:
〈〈αq1α+−q2|α
+
q3α−q4〉〉 =
h¯
2πδq1q3δq2q4
〈nq2 − nq1〉
h¯ω − ωq1 + ωq2
,
〈〈β+−q1βq2|β−q3β
+
q4〉〉 =
h¯
2πδq1q3δq2q4
〈nq1 − nq2〉
h¯ω + ωq1 − ωq2
,
〈〈αq1βq2|α+q3β
+
q4〉〉 =
h¯
2πδq1q3δq2q4
〈nq2 + 1 + nq1〉
h¯ω − ωq1 − ωq2
,
〈〈β+−q1α
+
−q2|β−q3α−q4〉〉 = −
h¯
2πδq1q3δq2q4
〈nq2 + 1 + nq1〉
h¯ω + ωq1 + ωq2
. (34)
Similar to this procedure we can find all other functions.
As a result, considering a two-dimensional volume centred square lattice, we can
write the diagonal components of the scattering tensor as follows:
Hxx;yy(ω1, ω2) =
(1− δ)42πe4h¯2dx4
(eβh¯ω − 1)(h¯ω1 −E)2(h¯ω2 − E)2
×
[∑
k
64(v2k + u2k)
2δ(ω − 2ωk) cos2(ky/2) cos2(kx/2)(Mxx + J + Kxx)2(2nk + 1)
+
∑
k
4v2ku2kδ(ω − 2ωk)(Mxx + Kxx)2(2nk + 1)
]
(35)
(here the polarization directions coincide with the crystallographic axes). Here we
have put the wave vectors ~k2, ~k1 equal to zero because they are small in comparison
191
I.V.Stasyuk, T.S.Mysakovych
with the edge vector of the Brillouin zone, which plays the most significant role in
the scattering. The nondiagonal components of the tensor have the form
Hxy;xy(ω1, ω2) =
(1− δ)42πe4h¯2dx4
(eβh¯ω − 1)(h¯ω1 − E)2(h¯ω2 − E)2
×
[∑
k
64(v2k + u2k)2δ(ω − 2ωk) sin2(ky/2) sin2(kx/2)(Mxy + Kxy)2(2nk + 1)
]
.(36)
The main contribution comes from Green’s function 〈〈S+S−|S+S−〉〉; Green’s func-
tion 〈〈S+S−|SzSz〉〉, 〈〈SzSz|S+S−〉〉 gives contribution only to the diagonal com-
ponents of the tensor. The nondiagonal components of the tensor have the terms
which are proportional to sin(kx/2) sin(ky/2); these terms lead to the appearance of
the peak at the edge of the Brillouin zone [1].
The results obtained describe the two-magnon scattering and are similar to those
obtained for antiferromagnetics [1] but the condition of homeopolarity is not valid
for the t − J model and so the hole doping level is not equal to zero. It has led in
our approximation to the appearance of the factor (1− δ)4 in the expression for the
scattering tensor. In general, the influence of the doping on the scattering at the
hole concentration increase should be more complicated due to a rapid destruction
of the antiferromagnetic state. The changes in the scattering spectrum in this case
can be investigated even based on the expression (32) at the proper consideration
of the hole-magnon scattering.
4. General case
Let us return now to the Hubbard model and consider the consequences of in-
troducing the excited atomic states [9] into this model. Into the Hamiltonian (7) we
insert the additional terms which are similar to the ones used in the case of t − J
model:
Hˆ =
∑
i,j
ti,j cˆ†i,σcˆj,σ + U
∑
i
nˆi,↑nˆi,↓ −
∑
i
µnˆi
+ (E − µ)
∑
i,ν
aˆ†νi,σaˆνi,σ +
∑
i,ν
Mλνij aˆ†λj,σ′ aˆνi,σ′ cˆ†i,σ′ cˆj,σ′. (37)
Here the operators c, n refer to the ground state, the operators aλ, nλ refer to the
excited states. We do not include the term connected with the interaction K αβij be-
cause it does not lead to the new contributions to the scattering tensor in comparison
with that obtained from the term connected with the interaction M αβij . The dipole
momentum is taken in the form:
P αi = eRαi [n↑i + n↓i] +
∑
λ
[nλ↑i + nλ↓i] + dα
∑
σ
[a+αiσciσ + h.c.], (38)
which takes into account both types of the contributions considered above.
192
Raman light scattering
Similar to the previous cases we can obtain the following expression for the
operator Green’s function {{Mα|Mβ}}:
{{Mαk |Mβl }} = h¯
e2dαdβ
2π
1
(h¯ω1 −E)(h¯ω2 −E)
δα,β
×
[
−
∑
i
δk,lti,k(X31i X13k + X41i X14k )
−Ml,k(X33k X33l + X44k X44l + X43k X34l + X34k X43l )
]
+ h¯e
2RαkRβl
2πh¯2ω1ω2
∑
i,j,s
ti,jts,j(δi,k − δj,k)(δj,l − δs,l)
× (X31i X44j X13s + X41i X33j X14s −X41i X34j X13s −X31i X43j X14s )
− h¯e
2dαdβ
2π
∑
ij
[ 1
(h¯ω1 − E + U)(h¯ω2 −E)(h¯ω2 −E)
− 1(h¯ω1 + E − U)(h¯ω1 + E)(h¯ω2 + E)
]
δkltiktjl
× (X31i X14j X43k + X41i X13j X34k −X41i X14j X33k −X31i X13j X44k )
− 2h¯e
2dαdβ
2π
∑
ij
[ 1
(h¯ω1 −E + U)(h¯ω2 − E)(h¯ω2 − E + U)
+ 1(h¯ω1 + E − U)(h¯ω2 + E − U)(h¯ω1 + E)
]
δkltiktjl
× (X31i X14j X43k + X41i X13j X34k −X41i X14j X33k −X31i X13j X44k ) (39)
(only resonant contributions are presented). One can see the presence of the resonant
terms with the frequencies:
h¯ω1 ∼ E; h¯ω1 ∼ U ; h¯ω1 ∼ E − U ; h¯ω1 ∼ U − E. (40)
The formula (39) for the function {{Mα|Mβ}} includes new terms in comparison
with the expression (12). One of them (proportional to the interaction constant Mk,l)
has the structure of the scalar product ~Sl~Sk. Similar to (25) it leads to the magnon
scattering. A similar contribution can be obtained from the resonant term propor-
tional to (h¯ω1 − U)−1 in the case i = j. Considering such terms and maintaining
only the resonant contributions (with factors (h¯ω1 − U)−1 and (h¯ω1,2 −E)−1) we
can write the formula for the diagonal components of the magnon scattering tensor
as follows:
Hαα,αα(ββ,ββ)(ω1, ω2) =
(1− δ)42πe4h¯2
(eβh¯ω − 1)
∑
k
(v2k + u2k)2δ(ω − 2ωk)
×
{
cos2(ky/2) cos2(kx/2)(2nk + 1)
×
[ 4dx2Mxx
(h¯ω1 − E)(h¯ω2 − E)
+ 2a
2t2
h¯ω1h¯ω2(h¯ω1 − U)
]2
+ sin2(ky/2) sin2(kx/2) sin2(2γ)(2nk + 1)
193
I.V.Stasyuk, T.S.Mysakovych
×
[ 4dx2Mxy
(h¯ω1 −E)(h¯ω2 −E)
+ 2a
2t2
h¯ω1h¯ω2(h¯ω1 − U)
]2}
.(41)
(magnon Green’s functions were calculated here similar to (34)). The components
Hαα,ββ, Hββ,αα differ from Hαα,αα only by the sign of the second term. The nondi-
agonal components can be written as follows:
Hαβ,αβ(βα,βα)(ω1, ω2) = 2π
e4h¯2
(eβh¯ω − 1)
∑
k
(v2k + u2k)
2
× δ(ω − 2ωk)
[
4dx2Mxy
(h¯ω1 − E)(h¯ω2 − E)
+ 2a
2t2
h¯ω1h¯ω2(h¯ω1 − U)
]2
× (2nk + 1) cos2(2γ) sin2(ky/2) sin2(kx/2)(1− δ)4. (42)
Here the case is considered, when the directions of light polarization do not coincide
with the crystallographic axes; γ is the corresponding angle (γ = 0 in the case con-
sidered in the previous section). We can see that if γ 6= 0, the diagonal components
of the tensor Hαα,αα(ββ,ββ)(ω1, ω2) also lead to the appearance of the peak at the
edge of the Brillouin zone [1]. The factor (1− δ)4 is present in the expression for the
scattering tensor contrary to the results obtained in [1]. This factor gives a rough
estimate of the hole effect on the scattering (in [1] the case of pure antiferromagnets
was considered).
5. The pseudospin-electron model
In this section we consider the main contributions to Raman scattering in a more
complicated case of the pseudospin-electron model. This model can be considered
as an extension of the Hubbard model by the inclusion of the interaction with
pseudospin degrees of freedom. The Hamiltonian of the model has the following
form [8]:
H =
∑
i
Hi +
∑
i,j,σ
ti,j cˆ†i,σcˆj,σ, (43)
here the Hamiltonian Hi describes (besides the Hubbard electron correlation) the
interaction with the local anharmonic vibrational modes, described by pseudospins:
Hi = Uni↑ni↓ − µ(ni↑ + ni↓) + g(ni↑ + ni↓)Szi − ΩSxi − hSzi , (44)
New terms in Hi have the following meaning: interaction with the local vibrations
(g-term), splitting of the vibrational mode by tunnelling (Ω-term), asymmetry of
the vibrational mode (h-term).
In the case of narrow electron bands (t ≪ U), the single-site Hamiltonian H i
plays the role of a zero order approximation. Therefore let us introduce the following
single-site basis of states |ni↑, ni↓, Szi 〉 [8]:
|1〉 = |0, 0, 12〉, |1˜〉 = |0, 0,−
1
2〉, |2〉 = |1, 1,
1
2〉, |2˜〉 = |1, 1,−
1
2〉,
|3〉 = |0, 1, 12〉, |3˜〉 = |0, 1,−
1
2〉, |4〉 = |1, 0,
1
2〉, |4˜〉 = |1, 0,−
1
2〉. (45)
194
Raman light scattering
It is useful to use the Hubbard operators XRS = |R〉〈S|, acting in the space spanned
by the vectors (45). The Hamiltonian can be reduced to a diagonal form, using the
transformation
|R〉 = cos(φr)|r〉+ sin(φr)|r˜〉, |R˜〉 = cos(φr)|r˜〉 − sin(φr)|r〉, (46)
where
cos(2φ1) =
−h√
h2 + Ω2
, cos(2φ2) =
2g − h
√
(2g − h)2 + Ω2
,
cos(2φ3) = cos(2φ4) =
g − h
√
(g − h)2 + Ω2
. (47)
Thus we get for Hi in terms of the operators X rs :
Hi =
∑
r
εrXrri , (48)
where
ε1,˜1 = ±
1
2
√
h2 + Ω2, ε2,˜2 = 2E0 + U ±
1
2
√
(2g − h)2 + Ω2,
ε3,˜3 = ε3,˜3 = E0 ±
1
2
√
(g − h)2 + Ω2. (49)
The total Hamiltonian is given by the expression
H =
∑
i,r
εrXrri +
∑
i,j,σ
ti,j cˆ†i,σcˆj,σ, (50)
with
cˆ†i,↑ = cos(φ4 − φ1)(X41i + X 4˜1˜i )− sin(φ4 − φ1)(X41˜i −X 4˜1i )
+ cos(φ2 − φ3)(X23i + X 2˜3˜i )− sin(φ2 − φ3)(X23˜i −X 2˜3i ),
cˆ†i,↓ = cos(φ3 − φ1)(X31i + X 3˜1˜i )− sin(φ4 − φ1)(X31˜i −X 3˜1i )
− cos(φ2 − φ4)(X24i + X 2˜4˜i ) + sin(φ2 − φ4)(X24˜i −X 2˜4i ). (51)
Let us choose the dipole momentum of a unit cell in the form
Mαi = de(n↑,i + n↓,i) + dsSzi , (52)
or in terms of the operators X rs
Mαi = de(X22i + X 2˜˜2i −X11i −X 1˜1˜i + 1)
+ ds
∑
r
[
cos(2φr)(Xrri −X r˜r˜i ) + sin(2φr)(Xrr˜i + X r˜ri )
]
, (53)
The expression (52) was used in [8] in the case of quasi-two-dimensional crystal
structure of the YBa2Cu3O7−δ type systems and corresponds to the “transverse”
195
I.V.Stasyuk, T.S.Mysakovych
component of polarization vector. Two equilibrium positions of anharmonic ion de-
scribed by the pseudospin values Szi = ±12 are oriented normally to the conducting
(a, b) plane. The first term in (52) is connected as the (8) term with the nonhome-
opolarity of filling of the electron states on the site; the second term has an ionic
origin and is determined by the particle localization (that is, the orientation of
pseudospin). To calculate the unaveraged Green’s function {{M αk |Mβl }} we make
an operator expansion, using t as a formal small parameter [12]. At first we use the
equation of motion, which is presented in the form (5):
{{Mαk |Mβl }} =
δk,ld2s
h¯ω1
∑
r
sin(4φr)(Xrr˜ −X r˜r)
+
∑
r
δk,ld2s
(h¯ω1 − Er + Er˜)
[
sin(4φr)X r˜r − sin2(2φr)(Xrr −X r˜r˜)
]
−
∑
r
δk,ld2s
(h¯ω1 + Er − Er˜)
[
sin(4φr)Xrr˜ − sin2(2φr)(Xrr −X r˜r˜)
]
+ deh¯ω1
∑
i,j,σ
ti,j(δi,k − δj,k){{cˆ†i,σcˆj,σ|Mˆβl }}
+
∑
r
ds
h¯ω1
∑
i,j,σ
ti,j cos(2φr){{[Xrr −X r˜r˜, cˆ†i,σcˆj,σ]|Mˆβl }}
+
∑
r
ds
(h¯ω1 − Er + Er˜)
∑
i,j,σ
ti,j sin(2φr){{[X r˜r, cˆ†i,σcˆj,σ]|Mˆβl }}
+
∑
r
ds
(h¯ω1 + Er −Er˜)
∑
i,j,σ
ti,j sin(2φr){{[Xrr˜, cˆ†i,σcˆj,σ]|Mˆβl }}. (54)
We can see that contrary to the Hubbard and t − J models, in the case of the
pseudospin-electron model Green’s function {{M αk |Mβl }} possesses the terms, which
are of the zero order with respect to the electron transfer parameter. The appearing
of these terms is caused by the dipole momentum dynamics, connected with the pseu-
dospin reorientation. The calculation of Green’s functions 〈〈X rr|Xpp〉〉, 〈〈Xrr|X p˜p˜〉〉,
〈〈Xrr|Xpp˜〉〉 etc. and the investigation, on this basis, of corresponding contributions
to the Raman scattering tensor was performed in [13]. The frequency dependence
of scattering at various values of electron concentration and temperature was con-
sidered.
Now for the function (54) we use the equation of motion in the form (6). The
expression obtained after this procedure is very cumbersome, so here we write out
only one term thereof:
∑
i,j,σ
ti,j(δi,k − δj,k){{cˆ†i,σcˆj,σ|(n↑l + n↓l)}} =
= 1h¯ω2
∑
i,j,σ
s,p,σ′
ti,j(δi,k − δj,k)ts,p(δp,l − δs,l){{cˆ†i,σcˆj,σ|cˆ†s,σ′ cˆp,σ′}}
− 12πω2
∑
i,j,σ
ti,j(δi,k − δj,k)(δi,l − δj,l)cˆ†i,σ cˆj,σ. (55)
196
Raman light scattering
The structure of the other terms is similar. Then we again write the equation of
motion in the form (6) in order to find Green’s function {{cˆ†i,σcˆj,σ|cˆ†s,σ′ cˆp,σ′}} up to
the terms of the second order in t. Considering the resonant term, which describes
the one-electron transition 1˜ 4˜ → 1 4, we have
∑
i,j,
s,p,σ′
ti,j(δi,k − δj,k)ts,p(δp,l − δs,l) cos2(φ4 − φ1){{X 4˜˜1i X14j |cˆ†s,σ′ cˆp,σ′}} =
=
∑
i,j,
s,p
ti,j(δi,k − δj,k)ts,p(δp,l − δs,l)h¯ cos2(φ4 − φ1)
2π(h¯ω1 − E4 + E1 + E4˜ − E1˜)
×
{
X 4˜1˜i [cos(φ4 − φ1)(X11j + X44j )− sin(φ4 − φ1)(X11˜j −X 4˜4j )]cˆs,↑
− cˆ†s,↑X14i [cos(φ4 − φ1)(X 1˜1˜j + X 4˜˜4j )− sin(φ4 − φ1)(X 4˜4j −X11˜j )]
+ X 4˜1˜i [cos(φ4 − φ1)X34j + sin(φ4 − φ1)X 3˜4j ]cˆs,↓ −
− cˆ†s,↓X14i [cos(φ4 − φ1)X 4˜3˜j − sin(φ4 − φ1)X 4˜3j ]
}
. (56)
We have omitted in this expression the terms, connected with the doubly occupied
states |2〉, |2˜〉. If i=s, the expression can be written as follows:
∑
i,j t2i,j(δi,k − δj,k)(δj,l − δi,l)h¯ cos2(φ4 − φ1)
2π(h¯ω1 − E4 + E1 + E4˜ − E1˜)
×
{
cos2(φ4 − φ1)
[
X11i (X 4˜˜4j + X 3˜3˜j )−X 1˜˜1i (X44j + X33j )
]
− sin(φ4 − φ1) cos(φ4 − φ1)
×
[
(X 4˜4i + X 3˜3i )(X11j + X 1˜1˜j ) + X11˜i (X44j + X33j + X 4˜˜4j + X 3˜3˜j )
]}
. (57)
One can see that the terms connected with the electron spin reorientation aren’t
present in this expression because the subband 1˜4˜ is created by the pseudospin
reorientation – for the Hubbard model this subband is not present. This expression
includes the terms which effectively take into account the electron correlations on
the neighbouring lattice sites in connection with the pseudospin dynamics. The
corresponding Raman scattering contributions can be important in the presence of
the charge ordered states with the modulation of the electron density and orientation
of pseudospins (the possibility of such ordering in the pseudospin-electron model
with the unit cell doubling was investigated in [14]).
Let us consider another term, which is resonant at the transition 1˜ 4˜ → 3˜ 2˜
∑
i,j,
s,p,σ′
ti,j(δi,k − δj,k)ts,p(δp,l − δs,l) cos(φ2 − φ3) cos(φ4 − φ1)
× {{X 4˜˜1i X 3˜2j −X 3˜1˜i X 4˜2j |cˆ†s,σ′ cˆp,σ′}} =∑
i,j h¯ cos(φ4 − φ1) cos(φ2 − φ3)ti,j(δi,k − δj,k)ts,j(δj,l − δs,l)
2π(h¯ω1 −E1˜ + E4˜ + E3˜ − E2˜)
×
{
X 4˜1˜i [cos(φ2 − φ3)X 3˜3˜j + sin(φ2 − φ3)X 3˜3j ]cˆs,↑
197
I.V.Stasyuk, T.S.Mysakovych
−X 4˜˜1i [cos(φ2 − φ3)X 3˜4˜j + sin(φ2 − φ3)X 3˜4j ]cˆs,↓
+ X 3˜1˜i [cos(φ2 − φ3)X 4˜4˜j + sin(φ2 − φ3)X 4˜4j ]cˆs,↓
−X 3˜˜1i [cos(φ2 − φ3)X 4˜3˜j + sin(φ2 − φ3)X 4˜3j ]cˆs,↑
}
, (58)
having omitted the terms, including the states |2〉, | 2˜〉. Separating the terms with
i = s, we obtain the following formula:
∑
i,j h¯ cos(φ4 − φ1) cos(φ2 − φ3)t2i,j(δi,k − δj,k)(δj,l − δi,l)
2π(h¯ω1 −E1˜ + E4˜ + E3˜ −E2˜)
×
{
cos(φ2 − φ3) cos(φ4 − φ1)[X 4˜4˜i X 3˜˜3j + X 3˜˜3i X 4˜4˜j −X 4˜3˜i X 3˜˜4j −X 3˜4˜i X 4˜˜3j ]
+ sin(φ2 − φ3) sin(φ4 − φ1)[X 4˜3i X 3˜4j + X 3˜4i X 4˜3j −X 4˜4i X 3˜3j −X 3˜3i X 4˜4j ]
+ sin(2φ4 − φ2 − φ1)[X 4˜3i X 3˜4˜j + X 4˜˜3i X 3˜4j −X 4˜˜4i X 3˜3j −X 4˜4i X 3˜3˜j ]
}
. (59)
In this expression there are both the terms, connected with the reorientation of the
electron spins and the ones, connected with the reorientation of the pseudospins.
Therefore Raman scattering at the transitions between the band separated by U
is connected with the electron spin dynamics, very similar to that described by
Fleury and Loudon approach [1] (creation of magnon pairs), as well as with the
pseudospin dynamics. As it was shown in [8], the interaction with pseudospins leads
to the modulation of the parameter of the effective exchange interaction, depending
on the state of the pseudospin subsystem. The role of this effect can be studied
by calculating the Green’s functions 〈〈X r˜r˜X p˜p˜|X s˜s˜X q˜q˜〉〉, 〈〈X 4˜˜3X 3˜˜4|X 4˜3˜X 3˜˜4〉〉 and
others, and by investigating, on this basis, the expression for the Raman scattering
tensor.
6. Conclusions
The method of constructing the polarizability operator for systems with a strong
short-range correlation between electrons is developed in this work. The operator
expansions in powers of the parameter of the electron hopping are carried out to cal-
culate the polarizability operator. The expressions for the Raman scattering tensor
in terms of the correlation functions calculated on the spin operators or Hubbard
operators are obtained for the Hubbard, t − J and pseudospin-electron models. It
is shown that for the pure Hubbard model, Raman scattering in the doped case
cannot be reduced to the nonresonant scattering only. The resonant contributions
are present as well having the character of the two-electron transitions in the con-
ductance band. In the case of the t − J model, the scattering due to the magnon
pair creation is modified by the magnon-hole scattering process. In the simplest
approximation the scattering intensity decreases in the doped case proportionally
to the factor (1− δ)4, where δ is the concentration of holes. It is established that
the inclusion of the excited electron states (and the corresponding exchange inter-
actions) leads in the Hubbard model to the appearance, in the explicit form, of the
198
Raman light scattering
scattering components which correspond to the Fleury-Loudon [1] mechanism of the
two-magnon scattering. In general, two different contributions to the polarizability
operator, one of which is connected with the nonhomeopolarity of filling of the elec-
tron states on a site and another is responsible for the dipole transitions to the
excited states, lead to similar contributions to the scattering tensor.
In the case of the pseudospin-electron model the polarizability operator is not
equal to zero if the electron hopping isn’t present. This corresponds to the pure
pseudospin scattering that is caused by the dipole transitions connected with the
reorientation of pseudospins. Analysing the expression for the polarizability oper-
ator, we can see that there exist resonant transitions between the electron energy
subbands which appear due to the pseudospin-electron interaction. Investigating
the terms which are proportional to t2 we have shown that the two main scatter-
ing mechanisms can be separated between others. The first one is connected with
the correlation of the pseudospin dynamics with electron filling on the neighbour-
ing sites, which can be important in the case of the charge and pseudospin spatial
modulation. The second one is analogous to the two-magnon scattering in antifer-
romagnets which is here accompanied by the reorientation of pseudospins.
References
1. Fleury P., Loudon R. Scattering of light by one- and two-magnon excitations. // Phys.
Rev., 1968, vol. 166, No. 2, p. 514–530.
2. Shastry B.S., Shraiman B.I. Theory of Raman scattering in Mott-Hubbard systems.
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Koмбінаційне розсіяння світла в системах з
сильною короткодіючою взаємодією
І.В.Стасюк, Т.С.Мисакович
Інститут фізики конденсованих систем НАН Укpаїни,
79011 Львів, вул. Свєнціцького, 1
Отримано 29 вересня 1999 р.
Досліджено внески різного типу в комбінаційне розсіяння світла для
моделі Хаббарда, t−J та псевдоспін-електронної моделeй. Для по-
будови оператора поляризованості використовується мікроскопіч-
ний підхід, здійснюючи операторні розклади в термінах операторів
Хаббарда і використовуючи t та J як формальні параметри розкла-
ду. До розгляду приймалися два різні внески до дипольного момен-
ту: oдин пов’язаний з негомеополярністю заповнення електронних
станів на вузлах гратки, інший – із дипольними переходами з основ-
ного у збуджені стани ( для випадку моделі Хаббарда) та із дипольним
моментом псевдоспінів (для псевдоспін-електронної моделі). Отри-
мано загальні вирази для компонент тензора розсіяння, які описують
магнонне, електронне (внутрі- та міжзонне) та псевдоспінове роз-
сіяння. Виділено резонансні та нерезонансні внески, вивчається їх
роль при зміні концентрації дірок внаслідок легування. Досліджено
вигляд тензора розсіяння в залежності від співвідношень між поляри-
зацією падаючого і розсіяного світла.
Ключові слова: комбінаційне розсіяння світла, хаббардівська
кореляція, псевдоспін-електронна модель, оператор
поляризованості
PACS: 72.10.Dp, 72.10.Di, 74.20
200