Two channels of scattering of Hubbard chain electrons with spin and orbital interactions (exact results)
A simple way to determine the channel of scattering of the Hubbard chain electrons with additional quantum number, which means the orbital or band characteristic, is demonstrated on the integrable 1D modified Hubbard model. Exact solutions are obtained for each scattering channel of the electrons. T...
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України
2001
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| Назва видання: | Вопросы атомной науки и техники |
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| Цитувати: | Two channels of scattering of Hubbard chain electrons with spin and orbital interactions (exact results) / A.F. Zhuravlyov // Вопросы атомной науки и техники. — 2001. — № 6. — С. 380-382. — Бібліогр.: 5 назв. — англ. |
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irk-123456789-800542015-04-10T03:02:52Z Two channels of scattering of Hubbard chain electrons with spin and orbital interactions (exact results) Zhuravlyov, A.F. Quantum fluids A simple way to determine the channel of scattering of the Hubbard chain electrons with additional quantum number, which means the orbital or band characteristic, is demonstrated on the integrable 1D modified Hubbard model. Exact solutions are obtained for each scattering channel of the electrons. The preferable channel is determined by means of numerical calculations of the ground-state energies and comparison its values. The dependence of the scattering matrices on the parameters of Coulomb, orbital (band), and spin interactions gives the possibility to present the properties of the system considered in different ranges (reasonable in physics sense) of parameters which correspond to the electron coupling. The ratio between magnetic susceptibility and heat capacity of the model is presented. 2001 Article Two channels of scattering of Hubbard chain electrons with spin and orbital interactions (exact results) / A.F. Zhuravlyov // Вопросы атомной науки и техники. — 2001. — № 6. — С. 380-382. — Бібліогр.: 5 назв. — англ. 1562-6016 PACS: 71.10.Fd; 71.10.Pm http://dspace.nbuv.gov.ua/handle/123456789/80054 en Вопросы атомной науки и техники Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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Quantum fluids Quantum fluids Zhuravlyov, A.F. Two channels of scattering of Hubbard chain electrons with spin and orbital interactions (exact results) Вопросы атомной науки и техники |
| description |
A simple way to determine the channel of scattering of the Hubbard chain electrons with additional quantum number, which means the orbital or band characteristic, is demonstrated on the integrable 1D modified Hubbard model. Exact solutions are obtained for each scattering channel of the electrons. The preferable channel is determined by means of numerical calculations of the ground-state energies and comparison its values. The dependence of the scattering matrices on the parameters of Coulomb, orbital (band), and spin interactions gives the possibility to present the properties of the system considered in different ranges (reasonable in physics sense) of parameters which correspond to the electron coupling. The ratio between magnetic susceptibility and heat capacity of the model is presented. |
| format |
Article |
| author |
Zhuravlyov, A.F. |
| author_facet |
Zhuravlyov, A.F. |
| author_sort |
Zhuravlyov, A.F. |
| title |
Two channels of scattering of Hubbard chain electrons with spin and orbital interactions (exact results) |
| title_short |
Two channels of scattering of Hubbard chain electrons with spin and orbital interactions (exact results) |
| title_full |
Two channels of scattering of Hubbard chain electrons with spin and orbital interactions (exact results) |
| title_fullStr |
Two channels of scattering of Hubbard chain electrons with spin and orbital interactions (exact results) |
| title_full_unstemmed |
Two channels of scattering of Hubbard chain electrons with spin and orbital interactions (exact results) |
| title_sort |
two channels of scattering of hubbard chain electrons with spin and orbital interactions (exact results) |
| publisher |
Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| publishDate |
2001 |
| topic_facet |
Quantum fluids |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/80054 |
| citation_txt |
Two channels of scattering of Hubbard chain electrons with spin and orbital interactions (exact results) / A.F. Zhuravlyov // Вопросы атомной науки и техники. — 2001. — № 6. — С. 380-382. — Бібліогр.: 5 назв. — англ. |
| series |
Вопросы атомной науки и техники |
| work_keys_str_mv |
AT zhuravlyovaf twochannelsofscatteringofhubbardchainelectronswithspinandorbitalinteractionsexactresults |
| first_indexed |
2025-07-06T03:59:23Z |
| last_indexed |
2025-07-06T03:59:23Z |
| _version_ |
1836868555610521600 |
| fulltext |
TWO CHANNELS OF SCATTERING OF HUBBARD CHAIN ELECTRONS
WITH SPIN AND ORBITAL INTERACTIONS (EXACT RESULTS)
A.F. Zhuravlyov
Institute for Magnetism of NAS of Ukraine, Kiev, Ukraine
A simple way to determine the channel of scattering of the Hubbard chain electrons with additional quantum
number, which means the orbital or band characteristic, is demonstrated on the integrable 1D modified Hubbard
model. Exact solutions are obtained for each scattering channel of the electrons. The preferable channel is
determined by means of numerical calculations of the ground-state energies and comparison its values. The
dependence of the scattering matrices on the parameters of Coulomb, orbital (band), and spin interactions gives the
possibility to present the properties of the system considered in different ranges (reasonable in physics sense) of
parameters which correspond to the electron coupling. The ratio between magnetic susceptibility and heat capacity
of the model is presented.
PACS: 71.10.Fd; 71.10.Pm
1. INTRODUCTION
Michel Gaudin admitted that the expression "exactly
soluble model" means no more than suitable
mathematical expression for some physical value or at
least a possibility to reduce the problem to the problem,
which can be considered in a frame of classical
analyses. These solutions illustrate common principles
and permit to control the approximations of realistic and
sophisticated models. Therefore a way to sophistication
of initially exactly solved model [1] aims both the
mathematical beauty and physical reality.
The interest to the exactly solvable Hubbard's chain
of the electrons is high especially. It is stipulated for
modern achievements both as the fundamental
investigations of the nature of the super conductivity,
and as the direct applications of these investigations.
For example strong correlations in solids are effectively
described by the Hubbard's model. The real objects for
this model in condensed matter are narrow-zone
transition metals. Hubbard's model describes effectively
the correlations between spin and Coulomb interactions
in these solids.
It is well known that at the weak Coulomb
interaction limit Hubbard's model describes the non-
localized spin state and Fermi-liquid behavior of the
electron system. At the limit of strong Coulomb
interaction Hubbard's model describes the state of
localized magnetic moments of an electrons. Hence the
situation which arisen in the case of intermediate ratio
between the parameters of Coulomb interaction and a
hopping integral is most interesting from a physical
point of view. In this case it is possible to describe the
phenomena of metal-dielectric phase transition, arising
of local magnetic moments, correlation between the
charge carriers and magnetic ordering, etc. Exact
solutions of the based on the Hubbard's model problems
are promising in these aspects.
The classical exact solution of the Hubbard's chain
of electrons [1] does not describe the effective attraction
of the electrons. Therefore it is impossible to construct
the mechanism of the super conductivity in the frame of
the model [1]. Some modifications of the Hubbard's
model are presented in a number of works. The
modifications, which are most interesting for physical
interpretations permit to change effective constant of
Coulomb interaction.
The effective attraction between the electrons arises
in the many-band Hubbard's model, which was
considered the two-band Fermi-gas model with square
dispersion and δ-like potential. It takes into account the
orbital and spin interactions between bands and neglects
the Coulomb one.
The most interesting field of the applications of the
integrable models results is high-temperature
superconductivity investigations.
The models [2, 3] permit to describe more physical
properties than the initial exactly solved model [1]. In
the work [3] exact solution of fermionic gas two-bands
model describing a parabolic band of conducting
electrons and a band of local pairs interacting via δ-
function was presented.
Listed above modifications of the Hubbard model
got hopeful results on the way of description of
effective attraction in the electron systems. Therefore it
is interesting to investigate other many channel
modifications of Hubbard chain of electrons.
The effective attraction between the electrons arises in
the many-band Hubbard's model as it was admitted in the
work [2], which the two-band Fermi-gas with square
dispersion and delta-like potential was considered. It takes
into account the orbital and spin interactions between
bands and neglects the Coulomb one. The exact solutions
were presented in the form of the tensor multiplication of
the spin and orbital two-particles scattering matrixes. This
model permits to describe more physical properties than
the model [1]. The Hubbard's chain of the electrons with
the Coulomb, and orbital, and spin interactions is
considered in the work presented.
Below the Hubbard's chain of the electrons with
Coulomb, orbital, and spin interactions is considered.
The exact solutions are obtained by means of the
coordinate Bethe-anzatz method on anti-symmetrical
wave functions.
2. THE MODEL AND SOLUTION
124 PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY. 2001, № 6, p. 380-382.
Modified on the case of additional type of the
interaction (orbital or band) Hubbard chain Hamiltonian
we shall consider in the form:
∑ ∑ ∑
−=
+
++
+ −+−=
2/
2/ ', ',
11[
a
a
N
Nn mm
nmmnmnnm ccccH
σσ
σσσσ
]4 '''''''' σσσσσσσσ nmnmnmnmnmnmnmnm cccUccccbc ++++ −− . (1)
Here Na +1 denotes the number of the sites; +
σnmc and
σnmc denotes operators of the creation and annihilation
of electrons in the n-th site of the m-th band (m=1,2)
with z-component of the spin σ (σ=↑,↓); value of the
hopping integral we put equal to 1. The problem of the
integrability of the Hamiltonian (1) is complicated in
comparison with the problem [1] because a degeneracy
on the second quantum number. Common approach to
solve this problem leads to necessity to introduce new
system of quantum characteristics, which are analogous
to that of charge and spin rapidities, and a construction
of three-particle system of quantum equations, which
would describe the excitations of the system considered.
It is way to get the wave function of the system (1) by
the coordinate Bethe-anzatz in this case using two-particle
scattering matrices. The procedure is described near.
We have found the solution of the Schrödinger
equation for two-particle wave function
21
21
mm
σσ (n1n2)= f 21
21
mm
σσ (n1n2)c +
111 σmn c +
222 σmn ,
where function f 21
21
mm
σσ (n1n2) is the amplitude of the wave
function, which satisfies the equation
−f 21
21
mm
σσ (n1+1n2)−f 21
21
mm
σσ (n1−1n2)−f 21
21
mm
σσ (n1n2+1)−
−f 21
21
mm
σσ (n1n2−1)+2b(1+Pm)f 21
21
mm
σσ (n1n2)12−
+2U f 21
21
mm
σσ (n1n2)12=E f 21
21
mm
σσ (n1n2).
Here permutation operators (Pm)2=1, Pm= − P, and E is
the energy of the system (see below). The solution of
this equation gives the following cases of exact
solubility of the problem and the following scattering
matrices, (a) the case U=b
,
2sinsin
2sinsin
),(
ibkk
ibPkk
kkS
ji
m
ijji
ji
m
ij +−
+−
=
ibkk
ibPkk
kkS
ji
ijji
jiij 2sinsin
2sinsin
),(
+−
−−
=
σ
σ ; (2)
and (b) the case U= −b
,
2sinsin
2sinsin
),(
ibkk
ibPkk
kkS
ji
m
ijji
ji
m
ij −−
+−
=
ibkk
ibPkk
kkS
ji
ijji
jiij 2sinsin
2sinsin
),(
−−
−−
=
σ
σ . (3)
Here Pa
ij denotes the permutation operator of the i-th
and j-th electrons with the spin σ in the case when a=
σ , and is the number of the band (orbital number) m,
when a=m. All four matrices satisfy the Yang-Baxter
triangle equations; therefore we can declare the model is
integrable. Energy of the considered system is
E= − 2(cosk1+cosk2).
The whole system wave function is:
∑ ∑
=
=
P
N
j
QjPj
e
xkiPQAQ
1
}exp{)/()(ψ .
Here xQj are the coordinates of electrons, xQ1<xQ2<…
<xQN<L, L is the length of the chain; Q is permutation
operator of the symbols {1,2,…Ne}. Construction of the
transfer matrix from the scattering matrices (Eqs. (2-3)), and
calculation of the eigenvalues of this matrix, which coincide
with the eigenvalues of the Hamiltonian (1), reduce the
problem to the diagonalization of transfer matrix.
Realization of the scattering channel in our
consideration means that the quantum number of the
other one is fixed. In other words the variables will
divide. Each one of the equations (2) and (3) describe
the special channel of scattering. It is clear, that real one
will be determined by the lowest value of the ground-
state energy. Therefore to calculate it we have to get the
Bethe equations. To calculate the ground-state energies
we get the Bethe equations as:
∏
= +−
−−
=
e
aj
N
i ji
jiNik
ibkk
ibkk
e
1 2sinsin
2sinsin
∏
= +−
−−M
j
j
ibk
ibk
1 sin
sin
β β
β
λ
λ
,
∏
= +−
−−eN
j j
j
ibk
ibk
1 sin
sin
α
α
λ
λ
= − ∏
= −−
+−M
ib
ib
1 2
2
β βα
βα
λλ
λλ
. (4)
Here denotes a rapidities; Ne denotes the number of
the electrons; M denotes the number of electrons in the
spin state “down” [2, 3].
We get the system of equations, which describes the
densities of a distribution of wave numbers (k), and
spin rapidities () in the continuum limit. Eqs. (4) are
common for all four channels of a scattering. They
describe the density of the rapidity distribution and may
be written as:
()=b ∫
− −+
Q
Q kb
dkk
22 )sin(
)(
λ
ρ
− 2b ∫
− ′−+
′′B
B b
d
22 )(4
)(
λλ
λλσ .
Here the limits of the integration in the case of half
filled chain (Na =Ne) are determined by the conditions
[1] and are equal toQ, B in the case of ground state,
as it follows from the Lieb-Mattis theorem. We shall
consider this case.
For the functions of the density of the momentum
distribution we get:
21
m =1+2bcosk ∫
∞
∞− −+ 22 )sin(
)(
kb
dm
λ
λλσ ,
21
=1−2bcosk ∫
∞
∞− −+ 22 )sin(
)(
kb
dm
λ
λλσ +
+4bcosk ∫
− −′+
′′
π
π
σρ
22 )sin(sin4
)(
kkb
kkd ,
22
=1−2bcosk ∫
∞
∞− −+ 22 )sin(
)(
kb
dm
λ
λλσ
,
22
m =1+2bcosk ∫
∞
∞− −+ 22 )sin(
)(
kb
dm
λ
λλσ −
125
−4bcosk ∫
− −′+
′′
π
π
σρ
22 )sin(sin4
)(
kkb
kkd .
Here we denote
the densities, which are the orbital
ones if =m, a=1 and if =, a=1, the spin ones in the
case when U=b, and =m, , a=2, in the case when U=−
b correspondingly.
At numerical calculation of the energies of the
ground state we get the following result:
− in the case when the scattering is described by
the Eqs. (2) the spin channel is preferable;
− in the case when the scattering is described by
the Eqs. (3) the orbital channel is preferable.
3. THERMODYNAMIC PROPERTIES OF
THE MODEL
Let us consider the thermodynamic properties of
the model considered in the case when the spin channel
of a scattering is realized, namely the scattering
matrices of Eqs. (2). Thermodynamic potential of the
system will be as:
aN
Ω
=E0T ∫
∞
∞−
))((ln)( 10 λελλ σ nd +T ∫
−
π
π
ρ ))((ln)(0 kKnkdk , (5)
where n(E) = (1+exp{E/T})−1 denotes the Fermi-Dirac
distribution function of the electrons with the energy E,
temperature T; 0(k), 0() are the distribution functions
on the rapidities and momentum k consequently when
Na=Ne and T=0:
0(k) = (2)−1(1+cosk ∫
−
π
π
dk'R(sink−sink')(1+2(k))),
0()=(2)−1 ∫
−
π
π
dks(−sink)(1+2(k))
Here s() = (4bcoshb)−1, and R(x) is the Fourier
transformed function R() = (1+exp{2b})−1.
Functions K(k) and )(λε i satisfies the set of the
integral equations which appears in the classical
Hubbard problem [4] (i=1,2):
K(k) = − 2cosk+H−4 ∫
∞
∞−
ds(sink−) 2)(1 ib+− λ
+ T ∫
∞
∞−
ds(sink−) ))((
))((ln
1
1
λε
λε
′n
n
,
)(λε i = − Tsln(n( )(1 λε +i )n( )(1 λε −i )) +
+ Ti1 ∫
−
π
π
dkcosks(sink−)lnn(−K(k))
Here H denotes the amplitude of external magnetic
field. Boundary conditions for this case we choose as:
H
i
i
i 2)(lim =→ ∞
λε
; )(0 λε = − .
Here symbol denotes the convolution:
s f = dps(−p)f(p).
In the limit of weak external magnetic field H 0
the magnetic susceptibility we get as:
Hub=
)(
)(
1
2
0
aI
aSUI
π
[1+(NlnH)−1− 2)ln(
lnln
HN
H
+ …].
Here I0(a) = dkeasink(k), I1(a) = − (1/2)dkcoskeasink(k),
a = 4/NU and functions (k), (k) calculate from the well-
known procedure [1]. The result is
Hub= )(4
)(
1
0
aI
aUI
HC,
where HC is the heat capacity of the Heisenberg chain
was presented in the work [5]. This capacity satisfy the
following conditions in the case N=2:
0
lim
0
lim
→→ HT HC = 2/3,
0
lim
0
lim
→→ TH HC = (1/3)(1+()1/2).
In our case minimal value of the number N=4 (when we
have two orbits or bands). Therefore we have to
consider 4 integral Bethe anzatz equations. But due to
our approximation we may consider each channel of a
scattering separately and use the above results.
Hence we can write
)
2
(0 b
I
HubHub π
χ
χ
γ
γ == .
4. CONCLUSIONS
Presented way of the separation of the channels of a
scattering of Hubbard chain electrons model permits to
reduce effective constant of Coulomb interaction in
Hubbard's model due to the consideration of orbital
scattering channel of the electrons.
The modification of the Hubbard model on the case
of existing in the system of the strong correlated
electrons two channels of a scattering gives the exact
values of the modification parameter b, which makes
the model Hamiltonian integrable. It is interesting to
admit that the region of integrability is defined by the
equality of the values of a modification parameter and
model Coulomb interaction.
Hence, the mechanisms of the realization of the
transitions from different magnetic states, metal-
insulator transition, etc., in 1D integrable strong
correlated electron quantum systems, which are
described by the Hamiltonian Eq. (1) will be
determinate by the parameters of 4-fermionic
interaction.
REFERENCES
1. E.H. Lieb, F.Y. Wu. Absence of Mott transition
in an exact solution of the short range, one band model
in one dimension // Phys. Rev. Lett. 1968, v. 20, p.
1445-1448.
2. P. Schlottmann. Integrable two-band model
with attractive and repulsive interactions // Phys.
Rev. Lett. 1992, v. 68, p. 1916-1919.
3. I.N. Karnaukhov. Model of fermions with
correlated hopping (integrable cases) // Phys. Rev.
Lett. 1994, v. 73, p. 1130-1133.
4. M. Takahashi. One-Dimensional Hubbard model
at Finite temperature // Prog. Theor. Phys. 1972, v. 47,
p. 69-82.
126
5. Lee Kong-Ju-Bock, P. Schlottmann.
Anomalous low-temperature properties of the
degenerate one-dimensional Hubbard model //
Phys. Rev. Lett. 1989, v. 63, p. 2299-2302.
127
A.F. Zhuravlyov
Institute for Magnetism of NAS of Ukraine, Kiev, Ukraine
REFERENCES
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