Validity of t–J approximation for extended Hubbard model with strong repulsion
It is shown that for finite cyclic systems described by two band Hubbard Hamiltonian with strong electron repulsion the reduction to effective t–J model may give incorrect description of the ground state symmetry due to neglect of the correlated hopping terms.
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irk-123456789-1753202021-02-01T01:26:54Z Validity of t–J approximation for extended Hubbard model with strong repulsion Cheranovskii, V.O. Klein, D.J. Ezerskaya, E.V. Tokarev, V.V. Специальный выпуск. К 80-летию со дня рождения А.И. Звягина It is shown that for finite cyclic systems described by two band Hubbard Hamiltonian with strong electron repulsion the reduction to effective t–J model may give incorrect description of the ground state symmetry due to neglect of the correlated hopping terms. 2017 Article Validity of t–J approximation for extended Hubbard model with strong repulsion / V.O. Cheranovskii, D.J. Klein, E.V. Ezerskaya, V.V. Tokarev // Физика низких температур. — 2017. — Т. 43, № 11. — С. 1622-1625. — Бібліогр.: 10 назв. — англ. 0132-6414 PACS: 75.10.Jm, 75.40.Сx http://dspace.nbuv.gov.ua/handle/123456789/175320 en Физика низких температур Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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Специальный выпуск. К 80-летию со дня рождения А.И. Звягина Специальный выпуск. К 80-летию со дня рождения А.И. Звягина |
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Специальный выпуск. К 80-летию со дня рождения А.И. Звягина Специальный выпуск. К 80-летию со дня рождения А.И. Звягина Cheranovskii, V.O. Klein, D.J. Ezerskaya, E.V. Tokarev, V.V. Validity of t–J approximation for extended Hubbard model with strong repulsion Физика низких температур |
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It is shown that for finite cyclic systems described by two band Hubbard Hamiltonian with strong electron repulsion the reduction to effective t–J model may give incorrect description of the ground state symmetry due to neglect of the correlated hopping terms. |
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Article |
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Cheranovskii, V.O. Klein, D.J. Ezerskaya, E.V. Tokarev, V.V. |
| author_facet |
Cheranovskii, V.O. Klein, D.J. Ezerskaya, E.V. Tokarev, V.V. |
| author_sort |
Cheranovskii, V.O. |
| title |
Validity of t–J approximation for extended Hubbard model with strong repulsion |
| title_short |
Validity of t–J approximation for extended Hubbard model with strong repulsion |
| title_full |
Validity of t–J approximation for extended Hubbard model with strong repulsion |
| title_fullStr |
Validity of t–J approximation for extended Hubbard model with strong repulsion |
| title_full_unstemmed |
Validity of t–J approximation for extended Hubbard model with strong repulsion |
| title_sort |
validity of t–j approximation for extended hubbard model with strong repulsion |
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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2017 |
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Специальный выпуск. К 80-летию со дня рождения А.И. Звягина |
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http://dspace.nbuv.gov.ua/handle/123456789/175320 |
| citation_txt |
Validity of t–J approximation for extended Hubbard model with strong repulsion / V.O. Cheranovskii, D.J. Klein, E.V. Ezerskaya, V.V. Tokarev // Физика низких температур. — 2017. — Т. 43, № 11. — С. 1622-1625. — Бібліогр.: 10 назв. — англ. |
| series |
Физика низких температур |
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AT cheranovskiivo validityoftjapproximationforextendedhubbardmodelwithstrongrepulsion AT kleindj validityoftjapproximationforextendedhubbardmodelwithstrongrepulsion AT ezerskayaev validityoftjapproximationforextendedhubbardmodelwithstrongrepulsion AT tokarevvv validityoftjapproximationforextendedhubbardmodelwithstrongrepulsion |
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2025-07-15T12:34:14Z |
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2025-07-15T12:34:14Z |
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1837716319805374464 |
| fulltext |
Low Temperature Physics/Fizika Nizkikh Temperatur, 2017, v. 43, No. 11, pp. 1622–1625
Validity of t–J approximation for extended Hubbard model
with strong repulsion
V.O. Cheranovskii1, D.J. Klein2, E.V. Ezerskaya1, and V.V. Tokarev1
1V.N. Karazin Kharkiv National University, 4 Svoboda Sq., Kharkiv 61022, Ukraine
E-mail: cheranovskii@i.ua
2Texas A&M University at Galveston, Galveston, TX, 77554 USA
Received June 1, 2017, published online September 25, 2017
It is shown that for finite cyclic systems described by two band Hubbard Hamiltonian with strong electron re-
pulsion the reduction to effective t–J model may give incorrect description of the ground state symmetry due to
neglect of the correlated hopping terms.
PACS: 75.10.Jm Quantized spin models, including quantum spin frustration;
75.40.Сx Static properties (order parameter, static susceptibility, heat capacities, critical exponents, etc.).
Keywords: Hubbard model, t–J model, correlated hopping.
Introduction
One of the basic models in the theory of strongly corre-
lated electron systems is so called t–J model [1]. It can be
derived from one-band Hubbard model with strong repul-
sion in second order of perturbation theory (PT) for the
electron transfers allowing double occupancies in the limit
of a small concentration of holes in half-filled band. If
number of holes is comparable with number of electrons,
the additional correlated hopping terms of PT expansion
should be taken into account [2]. For linear Hubbard chain
these hopping terms renormalized coupling parameter J of
the corresponding t–J model only. For more complicated
lattices like two-leg ladder or diamond chain similar
renormalization cannot be done. Nevertheless, there is a
big experience in successful application of t–J model to the
study of different correlated systems, for instance, in the
theory of high-Tc superconductivity [1,3–5] and people
believe that this model reproduces adequately significant
part of physics even at intermediate concentration of holes.
In this work we study the effect of correlated hopping
terms, omitting in t–J model, on the lowest states of two-
band Hubbard model for finite lattice fragments with the
periodic boundaries (cycles). We show the importance of
these terms for the correct description of the model ground
state and lowest excitations.
1. Two band 1D Hubbard model with strong repulsion
Let us consider the Hubbard model defined on the line-
ar chain formed by N unit cells with two different types of
sites “a” and “b” (Fig. 1). The corresponding Hamiltonian
has the form
( ), , 1, , , ,
, , ,
, , , , , , , ,
, ,
H.c.
.
ab i i i i i i
i j i
a i i i i b i i i i
i i
t
U U
+ + +
σ σ + σ σ σ σ
σ σ
+ + + +
σ σ −σ −σ σ σ −σ −σ
σ σ
= + + + ∆ +
+ +
∑ ∑
∑ ∑
H a b a b a a
a a a a b b b b
(1)
Here abt is the hopping integral, describing the electron
transfer between neighbor sites of the chain; Δ is a difference
between the orbital energies for “a” and “b” sites; ,aU bU
are Hubbard energies for “a” and “b” sites, respectively.
Well known realization of this model is, so called 1D
Emery model for high-Tc superconducting copper oxides
[3–7] (in this case, instead of electrons, we should consider
holes in copper and oxygen bands). Such a model may also
describe the common electron system of some stacked do-
nor-acceptor salts.
Let us suppose that total number of electrons equals
to 1N + and model parameters satisfy the condition
, .b abU t∆ >> In this case each “b” site is occupied by one
electron and only one electron belongs to “a” sites. In the
limit 0abt → we have degenerate ground state of the
chain with respect to the localization of electron on a sites.
Fig. 1. Fragment of the chain system formed by two unit cells and
described by the Hamiltonian (1).
© V.O. Cheranovskii, D.J. Klein, E.V. Ezerskaya, and V.V. Tokarev, 2017
mailto:cheranovskii@i.ua
Validity of t–J approximation for extended Hubbard model with strong repulsion
The doubly occupied sites have significantly bigger en-
ergy and effect on the lowest energy states of (1) in second
order of PT in hopping integral abt . The corresponding
second order PT processes result the translation of electron
located on the given a site along the chain. First type of
these processes is two consequent hops of “a” and “b”
electrons, which may be depicted by following diagrams:
During these consequent hops the spin exchange is ab-
sent and there is no restriction on initial spin configuration
of two electrons involved into the process.
For second processes the movement of the “a” electron is
performed through intermediate doubly occupied “b” site.
Therefore, such a movement is forbidden for parallel
configurations of neighbor “a” and “b” electron spins. It
corresponds to correlated hops of “a” electron along the
chain. These processes are absent in case bU = ∞ studied
in context of hole movement in 1D Emery model [4–7]. On
the other side, the corresponding effective Hamiltonian for
Emery model is essentially a t–J model for the Hamiltoni-
an (1). So, it is of interest to study the effect of the corre-
lated hopping terms on the low-energy spectrum of initial
Hamiltonian.
2. Effective low-energy Hamiltonians
Let enumerate all N+1 electron spins in succession over
the chain sites independently of type. In second order of
PT in abt the Hamiltonian (1) for this chain can be written
in the following form:
( ) ( ) ( )
( )
1 2 1,2 1 1 1, , 1
2
1
2 , 1 2 3 1
1
1 2
H.c. .
N
n n n n n n
n
N
n n n n
n
J J
J J J
+ +
− +
=
−
+
+ +
=
= + − + + − +
+ − − +
∑
∑
H P c c P P c c
P c c
(2)
Here n
+c is a spinless Fermi operator which create electron
on ith “a” site, ,n mP is operator of the transposition of spin
variables of two electrons with numbers n and m
( )
( )
2
1 ,ab
a
t
J
U
=
∆ +
( )
( )
2
2
ab
b
t
J
U
=
− ∆
,
( )2
3
abt
J =
∆
.
In case of periodic boundary conditions this Hamiltonian
should contain the additional term
( ) ( )1 1 1 1 2 1, 1 1NJ J+
+= + − +H c c P
( ) ( ){ }1 2 1, 2 3 1, 11 H.c. .N
N N NJ J J+
+ + − − + + c c Q Q (3)
Here 1, 1N+Q is the cyclic permutation operator of all
spin variables iσ in spin part of wave function
1 2 3 1( , , , ).N+Φ σ σ σ σ
( ) ( )1, 1 1 2 3 1 2 3 1 1, , , , , ,N N N+ + +Φ σ σ σ σ = Φ σ σ σ σQ .
The Hamiltonian (2) has translation symmetry. There-
fore, the eigenfunctions of (2) should be characterized by
hole quasi-momentum 2 / ,k l N= π 0,1, 2 1.l N= − The
symmetry adapted basis functions corresponding to a fixed
value of k can be constructed by the combination of stand-
ard group theory approach and cyclic spin permutation
technique [8,9].
( ) ( ) ( ) ( )1/2 1
, 1, 1
1
, exp 1 0 , ,
N
n
m k N n m
n
S M N ik n S M− − +
+
=
Ψ = − Φ ∑ Q c
2 / , 0, 1 1k l N l N= π = − . (4)
Omitting simple but cumbersome manipulations with
the cyclic permutations, we can rewrite the Hamiltonian
(2) in the basis of functions (3) in the following form:
corrt J−= +H H H , (5)
( ) ( )1 2 1,2 1, 1 2t J NJ J− += + + − +H P P
( )( )3 1, 1exp H.c. ,NJ ik ++ +Q
( ) ( )corr 2 1, 1 1, 1exp 1 H.c.N NJ ik + + = − + H P Q .
(Here we used the unitary transformation ( 1)n
n n= −c c in
order to derive the same form of the effective low-energy
Hamiltonian for even and odd N.)
The Hamiltonian t J−H does not contain correlated
hopping terms and corresponds to t–J like model. For
2 0J = the Hamiltonian (5) coincides with the effective
Hamiltonian for 1D Emery model with one hole in the ox-
ygen band, which was introduced in [6,7]. The Hamiltoni-
an (5) commutes with the operator of square of total spin
2S and the corresponding operator of z-projection of total
spin .zS Therefore the eigenstates of (5) can be classified
by quantum numbers: total spin S and z-projection of total
spin .M The eigenvalue problem for (5) can be studied
analytically for ( 1)/2S N= ± . In particular, for the states
with maximal value of S there is simple analytical formu-
la for the eigenvalues
( )32 coskE J k= − .
Low Temperature Physics/Fizika Nizkikh Temperatur, 2017, v. 43, No. 11 1623
V.O. Cheranovskii, D.J. Klein, E.V. Ezerskaya, and V.V. Tokarev
The eigenvalue problem for the states with one inverted
spin ( ( 1)/2)M N= − can be reduced to finite difference
equations studied in Appendix. The case of the states with
more than one inverted spin is studied numerically only.
3. Results of numerical simulation
We study numerically the lowest energy states of the
Hamiltonian (5) for finite lattice clusters with periodic
boundaries containing 4–10 unit cells by means of Davidson
method. Similar to [7] we put 0,aU = which corresponds to
the condition 1 3.J J= The numerical calculations were per-
formed in each subspace with given value of total spin S and
quasi-momentum k. The Hamiltonian matrix elements in
subspace spanned on spin adapted basis functions were cre-
ated by means of branching diagram technique [10]. The
results of these calculations for the case N = 10 are pre-
sented below on Figs. 2 and 3 (only the lowest energy
levels with the total spin ( )1 /2S N m= + − and 0 k≤ ≤ π
are given).
We found numerically that the ground state is doubly
degenerate and corresponds to S = 1/2, 2 /5.k = ± π The
lowest excited states with 3/2,S = 5/2 are also doubly
degenerate. The increase of the value of 2J leads to the
change of the symmetry of above energy states (Fig. 3). For
2 14.33J J> the ground state becomes nondegenerate and
corresponds to k = 0. For cluster formed by 8 unit cells the
ground state corresponds to /2k = ±π at 1 2 1.J J= = The
transition to the nondegenerate ground state with 0k = ap-
pear at 2 1~ 2.874J J . Probably, this effect is important only
for finite clusters, because the increase of N leads to more
and more flat character of the lowest excitation bands.
In order to study the magnetic structure of the model
ground state we calculated the values of the following spin-
spin correlator for the ground state Ψ0 of the Hamiltonian (2):
0 1 1 0
1
N
n n n n
n
+
− +
=
ρ = Ψ Ψ∑S S c c . (6)
This quantity describes the ordering of “b” spins in vicinity
of the “a” spin. It can be shown by spin permutation for-
malism, that the correlator (6) for the ground state 0Φ of
the Hamiltonian (5) can be rewritten as
0 2 1 0N+ρ = Φ ΦS S .
The numerical calculations for cluster with 10N =
showed preferably ferromagnetic character of the ordering
of “b” spins near to “a” spin: ~ 0.2459ρ for 2 1J J= and
~ 0.2499ρ for 2 110J J= .
If similar to t–J model, we neglect the contribution of
correlated hopping terms corrH in model Hamiltonian (5),
the transition to nondegenerate ground state is absent. The
results of the corresponding calculations are presented in
Fig. 4. The corresponding dispersion law is similar to the
case 1 2 1J J= = .
In the result we may conclude that for finite cyclic sys-
tems described by two band Hubbard Hamiltonian with
strong electron repulsion the reduction to effective t–J
model may give incorrect description of the ground state
symmetry due to neglect of the correlated hopping terms.
Fig. 2. (Color online) Lowest energy levels of given spin and sym-
metry for cyclic cluster with N = 10 and J1 = J2 = 1, k = πl/5.
Fig. 3. (Color online) Lowest energy levels of given spin and
symmetry for cluster with N = 10 and J1 = 1, J2 = 10, k = πl/5.
Fig. 4. (Color online) Lowest energy levels of given spin and
symmetry for cluster with N = 10 and J1 = 1, J2 = 10 in the ab-
sence of correlated hopping terms.
1624 Low Temperature Physics/Fizika Nizkikh Temperatur, 2017, v. 43, No. 11
Validity of t–J approximation for extended Hubbard model with strong repulsion
Acknowledgments
VOC acknowledges the support of the Volkswagen-
Stiftung, Germany (via grant 151110).
DJK acknowledges the support of the Welch Founda-
tion of Houston, Texas (via grant BD-0894).
Appendix. Exact solution for the states with one
inverted spin
The stationary states with N + 1 electrons and one invert-
ed spin are determined by Schrödinger equation (for sim-
plicity we shift all the eigenvalues by constant 2(J1 + J2))
( )corr 1 22( ) ,t J J J− + + + ψ = ε ψH H
1
1
0 .
N
m m
m
A S
+
−
=
ψ = ∑
Here is 0 “vacuum” state, ψ is the vector of state.
The wave function in the lattice-site representation mA
obeys the finite difference equations
( ) ( )1 2 3 1 12 e e 0,ik ik
m m mJ J A J A A−
− +ε − + − + =
1, 2, 1m N≠ + (A.1)
with the following “boundary conditions”:
____________________________________________________
[ ] ( ) ( )2 1 1 2 2 3 2 1 2 2 3 12 cos e e 0;ik ik
NJ k A J J J J A J J J J A−
+ ε + − + + + − + + + =
( ) ( )
( ) ( )
1 2 1 1 2 2 3 1 3 3 2 1
1 2 1 1 2 2 3 1 3 2 2
e e e 0;
e e e 0.
ik ik ik
N
ik ik ik
N N
J J A J J J J A J A J A
J J A J J J J A J A J A
−
+
− −
+
ε − + − + + + − + =
ε − + − + + + − + =
(A.2)
Let seek the wave function in the form
( )1 2e , 2,..., 1.ikm m m
mA C x C x m N−= + = +
Omitting simple but cumbersome calculation, we obtain from (A.2) the following dispersion equation:
( )1 2 3
12 ,J J J x
x
ε = + − +
(A.3)
where parameter x obeys the algebraic equation
( ) ( ) [ ]{ } ( )
( ) ( )
2 4
2 12 2 2
1 1
0 0
3
2 2 1 2
1
0
1 1 2 | |
2 1 | | 2 0;
N N
mm
m m
N
m N
m
u u r p u x u x
u r x u p x
− −
+
= =
−
+ −
=
ε − −β ε − −β − +β −β + α −
− α ε − −β − + α β − =
∑ ∑
∑
(A.4)
( )
( )
1 1 1
3 1 2 1
; 2 cos ; 1 e ;
2Re ; ; .
ikJ u k r
p r J J J J
ε = ε = ε + β = +β+ α +β
= α = β =
_______________________________________________
The values of exp( )x iq= correspond to quasi-
continuous band and the real values of , | | 1x x < corre-
spond to local levels.
The formula (A.3) and Eq. (A.4) reproduce the results
of exact diagonalization study for the energy states of finite
lattice clusters with ( )1 /2.S N= ±
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Introduction
1. Two band 1D Hubbard model with strong repulsion
2. Effective low-energy Hamiltonians
3. Results of numerical simulation
Acknowledgments
Appendix. Exact solution for the states with one inverted spin
|