Evolution of the spectrum of the Hubbard model with filling
The diagram technique for the one-band Hubbard model is formulated for the case of moderate to strong Hubbard repulsion. The expansion in powers of the hopping constant is expressed in terms of site cumulants of electron creation and annihilation operators. For Green’s function an equation of the...
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| Zitieren: | Evolution of the spectrum of the Hubbard model with filling / A. Sherman // Condensed Matter Physics. — 2006. — Т. 9, № 3(47). — С. 535–544. — Бібліогр.: 29 назв. — англ. |
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irk-123456789-1213552017-06-15T03:03:27Z Evolution of the spectrum of the Hubbard model with filling Sherman, A. The diagram technique for the one-band Hubbard model is formulated for the case of moderate to strong Hubbard repulsion. The expansion in powers of the hopping constant is expressed in terms of site cumulants of electron creation and annihilation operators. For Green’s function an equation of the Larkin type is derived and solved in a one-loop approximation for the case of two dimensions and nearest-neighbor hopping. With decreasing the electron concentration in addition to the four bands observed at half-filling, a narrow band arises near the Fermi level. The dispersion of the new band, its bandwidth and the variation with filling are close to those of the spin-polaron band in the t-J model. Сформульовано дiаграмну технiку для однозонної моделi Хаббарда для випадку промiжного та сильного хаббардiвського вiдштовхування. Розклад за степенями константи перескоку виражено через вузловi кумулянти електронних операторiв народження та знищення. Отримано рiвняння Ларкiна для функцiї Грiна та розв’язано його у однопетлевому наближеннi для двовимiрного випадку та переносу мiж найближчими сусiдами. При пониженнi електронної концентрацiї крiм чотирьох зон, що iснують при половинному заповненнi, з’являється вузька зона поблизу рiвня Фермi. Дисперсiя цiєї нової зони, її ширина та змiна при змiнi концентрацiї є подiбними до таких же величин для випадку спiн-поляронної зони в t-J моделi. 2006 Article Evolution of the spectrum of the Hubbard model with filling / A. Sherman // Condensed Matter Physics. — 2006. — Т. 9, № 3(47). — С. 535–544. — Бібліогр.: 29 назв. — англ. 1607-324X PACS: 71.10.Fd, 71.10.-w DOI:10.5488/CMP.9.3.535 http://dspace.nbuv.gov.ua/handle/123456789/121355 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
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The diagram technique for the one-band Hubbard model is formulated for the case of moderate to strong
Hubbard repulsion. The expansion in powers of the hopping constant is expressed in terms of site cumulants
of electron creation and annihilation operators. For Green’s function an equation of the Larkin type is derived
and solved in a one-loop approximation for the case of two dimensions and nearest-neighbor hopping. With
decreasing the electron concentration in addition to the four bands observed at half-filling, a narrow band
arises near the Fermi level. The dispersion of the new band, its bandwidth and the variation with filling are
close to those of the spin-polaron band in the t-J model. |
| format |
Article |
| author |
Sherman, A. |
| spellingShingle |
Sherman, A. Evolution of the spectrum of the Hubbard model with filling Condensed Matter Physics |
| author_facet |
Sherman, A. |
| author_sort |
Sherman, A. |
| title |
Evolution of the spectrum of the Hubbard model with filling |
| title_short |
Evolution of the spectrum of the Hubbard model with filling |
| title_full |
Evolution of the spectrum of the Hubbard model with filling |
| title_fullStr |
Evolution of the spectrum of the Hubbard model with filling |
| title_full_unstemmed |
Evolution of the spectrum of the Hubbard model with filling |
| title_sort |
evolution of the spectrum of the hubbard model with filling |
| publisher |
Інститут фізики конденсованих систем НАН України |
| publishDate |
2006 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/121355 |
| citation_txt |
Evolution of the spectrum of the Hubbard model with filling / A. Sherman // Condensed Matter Physics. — 2006. — Т. 9, № 3(47). — С. 535–544. — Бібліогр.: 29 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT shermana evolutionofthespectrumofthehubbardmodelwithfilling |
| first_indexed |
2025-07-08T19:42:29Z |
| last_indexed |
2025-07-08T19:42:29Z |
| _version_ |
1837109085037658112 |
| fulltext |
Condensed Matter Physics 2006, Vol. 9, No 3(47), pp. 535–544
Evolution of the spectrum of the Hubbard model with
filling
A.Sherman
Institute of Physics, University of Tartu, Riia 142, 51014 Tartu, Estonia
Received February 27, 2006, in final form May 12, 2006
The diagram technique for the one-band Hubbard model is formulated for the case of moderate to strong
Hubbard repulsion. The expansion in powers of the hopping constant is expressed in terms of site cumulants
of electron creation and annihilation operators. For Green’s function an equation of the Larkin type is derived
and solved in a one-loop approximation for the case of two dimensions and nearest-neighbor hopping. With
decreasing the electron concentration in addition to the four bands observed at half-filling, a narrow band
arises near the Fermi level. The dispersion of the new band, its bandwidth and the variation with filling are
close to those of the spin-polaron band in the t-J model.
Key words: Hubbard model, diagram technique, energy spectrum
PACS: 71.10.Fd, 71.10.-w
1. Introduction
For the last two decades the discovery of high-Tc superconductors, heavy-fermion compounds
and organic conductors has revived interest in strongly correlated electron systems. One of the
simplest and still realistic models in this field is the one-band Hubbard model [1–3] the two-
dimensional version of which has been extensively studied in connection with the cuprate perovskite
superconductors. The Hamiltonian of the model reads
H =
∑
nmσ
tnma†
nσamσ +
U
2
∑
nσ
nnσnn,−σ, (1)
where tnm is the hopping constants, the operator a†
nσ creates an electron on the n site of a plane
square lattice with the spin projection σ = ±1, U is the on-site Coulomb repulsion and the electron
number operator nnσ = a†
nσanσ. If the Coulomb repulsion dominates it is reasonable to treat this
interaction exactly and the kinetic energy with the use of a perturbation expansion. Apparently
the first expansion of this kind was considered in reference [4]. The further development of this
approach was given in references [5–9] where the diagram technique for Hubbard operators was
developed and used in investigating the Mott transition, the magnetic phase diagram and the
superconducting transition in the Hubbard model.
Another, more compact version of the diagram technique was proposed in references [10–13].
In this approach the power expansion for the electron Green’s function is expressed in terms of
cumulants of electron operators anσ and a†
nσ. Based on this diagram technique the equations
of the Larkin type [15] for Green’s function were derived [10,12,13]. However, the application
of this approach runs into problems. In particular, at half-filling the spectral weight obtained
after a resummation of diagrams appears to be negative near frequencies ωd = ±U/2 [13]. This
drawback is connected with divergencies in cumulants at these frequencies [14]. As can be seen
from formulas given below, all higher-order cumulants have such divergencies at ωd with sign-
changing residues, which are expected to compensate the negative spectral weight in the entire
series. On the other hand, at frequencies close to ωd cumulants are regular. If a selected subset
of diagrams is expected to give a correct estimate of the entire series for these frequencies the
c© A.Sherman 535
A.Sherman
values at ωd can be corrected by an interpolation using the results for the regular regions. This
procedure was applied in reference [14] for the case of two dimensions, half-filling, nearest-neighbor
hopping and with the use of the one-loop approximation. The spectrum was shown to consist of
four bands. These band structures and the calculated shapes of the electron spectral function are
close to those obtained in the Monte-Carlo [16–18], cluster perturbation [19] and the two-particle
self-consistent [20] calculations, provided that the temperature is high enough to ensure a short
magnetic correlation length. The four-band structure of the spectrum of the Hubbard model was
also considered in reference [21].
In the approach of reference [14] the mentioned four-band structure of the spectrum at half-
filling has its origin in the regions of large damping which separate the low- and high-frequency
bands. In the major part of the Brillouin zone both these types of the bands owe their origin
to the same terms of the irreducible part, i.e. to the same interaction processes. This brings up
the questions: How this spectrum changes with filling and how the quasiparticle peak, which
determines the photoemission leading edge, arises in the spectrum? As expected, this peak differs
in nature from other spectral features. In the present work it is shown that at certain deviation
from half-filling an additional narrow band arises near the Fermi level on the background of the
above-mentioned four bands. The dispersion of the new band, its bandwidth and the variation with
filling are close to those of the spin-polaron band in the t-J model [22,23].
In the following section the perturbation expansion for the electron Green’s function is formu-
lated in the form convenient for calculations and Larkin’s equation is derived. In section 3 the
equations of the previous section are used in calculating the spectral function and the obtained
results are discussed. Concluding remarks are presented in section 4.
2. Diagram technique
The present work considers the electron Green’s function,
G(n′τ ′,nτ) = 〈T ān′σ(τ ′)anσ(τ)〉 . (2)
Here the angular brackets denote the statistical averaging with the Hamiltonian H = H−µ
∑
nσ nnσ,
µ is the chemical potential, T is the time-ordering operator which arranges other operators
from right to left in ascending order of times τ , anσ(τ) = exp(Hτ)anσ exp(−Hτ) and ānσ(τ) =
exp(Hτ)a†
nσ exp(−Hτ). Choosing
H0 =
U
2
∑
nσ
nnσnn,−σ − µ
∑
nσ
nnσ and H1 =
∑
nmσ
tnma†
nσamσ (3)
as the unperturbed Hamiltonian and the perturbation, respectively, and using the known expansion
[24] for the evolution operator we get
G(n′τ ′,nτ) =
∞
∑
k=0
(−1)k
k!
∫
. . .
∫ β
0
dτ1 . . . dτk
∑
n1n
′
1
σ1
. . .
∑
nkn′
k
σk
tn1n
′
1
. . . tnkn′
k
× 〈T ān′σ(τ ′)anσ(τ)ān′
1
σ1
(τ1)an1σ1
(τ1) . . . ān′
k
σk
(τk)ankσk
(τk)〉0c, (4)
where β = T−1 is the inverse temperature, the subscript “0” near the angular bracket indicates
that the averaging and time dependencies of operators are determined with the Hamiltonian H0.
The subscript “c” indicates that terms which split into two or more disconnected averages should
be dropped out.
In equation (4), the average in the k-th order term which contains k+1 creation and annihilation
operators can be represented by the sum of all possible products of cumulants [25] with the sum of
orders equal to k + 1 [10,11,14]. All possible distributions of operators between the cumulants in
these products should be taken into account. The sign of a term in this sum is determined by the
number of permutations of fermion operators, which bring the sequence of operators in the initial
average to that in the term. The order of a cumulant is determined by the numbers of creation or
536
Spectrum of the Hubbard model
annihilation operators in it. These numbers should be equal for the cumulant to be nonzero. Since
the Hamiltonian H0 in equation (3) is diagonal in the site representation, operators in cumulants
belong to the same lattice site. Therefore, due to the translation symmetry of the problem, the
cumulant does not depend on the site index. Cumulants of the first and second order read
K1(τ
′σ′, τσ) = 〈T āσ(τ ′)aσ(τ)〉0δσσ′ ,
K2(τ
′σ, τσ, τ ′
1σ1, τ1σ1) = 〈T āσ(τ ′)aσ(τ)āσ1
(τ ′
1)aσ1
(τ1)〉0
−K1(τ
′σ, τσ)K1(τ
′
1σ1, τ1σ1) + K1(τ
′σ, τ1σ1)K1(τ
′
1σ1, τσ), (5)
where the site indices are dropped.
= - + +
1
2
- - - +
-
1
6
+
1
2
+
1
2
+
+ + -
1
2
-
- - - -
- +
(a) (b) (c) (d)
(e) (f) (g) (h)
(i) (j) (k)
(l)
(m) (n)
(o)
(p)
(q) (r) (s) (t)
(u) (v)
Figure 1. Diagrams of the first four orders of
expansion (4).
Actually the above statements determine
the rules of the diagram technique. Addition-
ally one has to take into account the presence
of topologically equivalent terms i.e., the terms
which differ only by permutation of operators
H1(τi) in equation (4). Since these terms are
equal, in the expansion only one of them can be
taken into account with the prefactor ν = j/k!
where j is the number of topologically equiva-
lent terms. Following reference [11] in diagrams
a cumulant is denoted by a circle and the hop-
ping constant tnn′ by a line directed from n
′ to
n. The external operators ān′σ(τ ′) and anσ(τ)
are denoted by directed lines leaving the cumu-
lant and entering it. The order of a cumulant
is equal to a number of incoming or outgoing
lines. Summations and integrations over the in-
ternal indices ni, n
′
i, σi and τi are carried out.
Since site indices of operators included in a cu-
mulant coincide, some site summations disap-
pear. Also some summations over σi get lost,
because in any cumulant, the spin indices of
creation and annihilation operators should match. Taking into account the multiplier (−1)k in
equation (4), the sign of the diagram is equal to (−1)l where l is the number of loops formed by
hopping lines. Figure 1 demonstrates the connected diagrams of the first four orders of the power
expansion (4) with their signs and prefactors. Here the thick line with arrow on the left-hand side
of the equation is the total Green’s function. Notice that if we set tnn = 0, contributions of the
diagrams (b), (d)–(f), (i)–(n), (p), (t), and (u) vanish. However, below a renormalized hopping
parameter will be introduced which is nonzero for the coinciding site indices and therefore the
mentioned diagrams are retained in figure 1.
All diagrams can be separated into two categories – those that can be divided into two parts
by cutting some hopping line and those that cannot be divided in this way [8,15]. These latter
diagrams are referred to as irreducible diagrams. Denoting the sum of all irreducible diagrams by
K(n′τ ′,nτ), after the Fourier transformation we find the following Larkin equation for Green’s
function:
G(k, iωl) =
K(k, iωl)
1 − tkK(k, iωl)
, (6)
where tk =
∑
n
e−ik(n−m)tnm, ωl = (2l + 1)πT is the Matsubara frequency with an integer l.
The partial summation can be carried out in the hopping lines of irreducible diagrams by
inserting them into the lines. In doing so the hopping constant tk in the respective formulas is
substituted by
Θ(k, iωl) =
tk
1 − tkK(k, iωl)
= tk + t2
k
G(k, iωl). (7)
537
A.Sherman
In the approximation used below the total collection of irreducible diagrams K(k, iωl) is substi-
tuted by the sum of the two diagrams (a) and (b) in figure 1, which appear in the first two orders
of the perturbation theory. Due to the form of the latter diagram this approximation is referred to
as the one-loop approximation. In the diagram (b) the hopping line is renormalized in accordance
with equation (7). Thus,
K(iωl) = K1(iωl) − T
∑
l1σ1
K2(iωlσ, iωl1σ1, iωl1σ1)
1
N
∑
k
t2
k
G(k, iωl1), (8)
where K1(iωl) and
K2(iωl′σ, iωlσ, iωl′
1
σ1, iωl1σ1) =
=
∫∫∫∫ β
0
dτ ′dτdτ ′
1dτ1e
−iωl′τ
′+iωlτ−iωl′
1
τ ′
1+iωl1
τ1K2(τ
′σ, τσ, τ ′
1σ1, τ1σ1)
= βδl+l1,l′+l′
1
K2(iωlσ, iωl′
1
σ1, iωl1σ1)
are the Fourier transforms of cumulants (5), N is the number of sites and we set
∑
k
tk = 0. Notice
that in this approximation K does not depend on momentum.
Now we need to calculate the cumulants in equation (8). To do this it is convenient to introduce
the Hubbard operators Xij
n
= |in〉〈jn| where |in〉 are eigenvectors of site Hamiltonians forming
H0, equation (3). For each site there are four states: the empty state |0n〉 with the energy E0 = 0,
the two degenerate singly occupied states |σn〉 with the energy E1 = −µ and the doubly occupied
state |2n〉 with the energy E2 = U − 2µ. The Hubbard operators are connected by the relations
anσ = X0σ
n
+ σX−σ,2
n
, a†
nσ = Xσ0
n
+ σX2,−σ
n
(9)
with the creation and annihilation operators. The commutation relations for the Hubbard operators
are easily derived from their definition. Using equation (9) the first cumulant in equation (5) can
be computed straightforwardly:
K1(iωl) =
1
Z0
(
e−βEσ + e−βE0
iωl − Eσ0
+
e−βE2 + e−βEσ
iωl − E2σ
)
, (10)
where Z0 = e−βE0 +2e−βEσ +e−βE2 is the site partition function and Eij = Ei−Ej . As indicated in
references [8,10,13], if K(k, iωl) is approximated by this cumulant the resulting Green’s function (6)
corresponds to the Hubbard-I approximation [2].
To calculate K2 it is convenient to use Wick’s theorem for Hubbard operators [5–8]:
〈T Xα1
(τ1) . . . Xαi
(τi)Xα(τ)Xαi+1
(τi+1) . . . Xαn
(τn)〉0 =
=
n
∑
k=1
(−1)Pkgα(τ − τk)〈T Xα1
(τ1) . . . [Xαk
,Xα]±(τk) . . . Xαn
(τn)〉0, (11)
where α is the index combining the state and site indices of the Hubbard operator. If Xα is a
fermion operator (X0σ, Xσ2 and their conjugates), Pk is the number of permutation with other
fermion operators which is necessary to transfer the operator Xα from its position on the left-hand
side of equation (11) to the position on the right-hand side. In this case
gα(τ) =
eEijτ
eβEij + 1
{
−1, τ > 0,
eβEij , τ < 0,
(12)
where i and j are the state indices of Xα. If Xα is a boson operator (X00, X22, Xσσ′
, X02, and
X20), Pk = 0 and
gα(τ) =
eEijτ
eβEij − 1
{
1, τ > 0,
eβEij , τ < 0.
(13)
538
Spectrum of the Hubbard model
In equation (11), [Xαk
,Xα]± denotes an anticommutator when both operators are of fermion type
and a commutator in other cases.
The substitution of equation (9) in 〈T āσ(τ ′)aσ(τ)āσ1
(τ ′
1)aσ1
(τ1)〉 in K2, equation (5), leads to
six nonvanishing averages of Hubbard operators (such averages are nonzero if the numbers of the
operators X0σ and Xσ0 coincide therein, and the same is true for the pair Xσ2 and X2σ). Applying
Wick’s theorem (11) to these averages the number of operators therein is sequentially decreased
until only time-independent operators are left. For H0 in equation (3) these are X00, Xσσ′
, and
X22. Their averages are easily calculated. As a result, after some algebra we find
∑
σ1
K2(iωlσ, iωl1σ1, iωl1σ1) = −Z−1
0 U
{
e−βE0g0σ(iωl)g0σ(iωl1)g02(iωl + iωl1)
[
g0σ(iωl) + g0σ(iωl1)
]
+ e−βE2gσ2(iωl)gσ2(iωl1)g02(iωl + iωl1)
[
gσ2(iωl) + gσ2(iωl1)
]
+ e−βE1
[
g0σ(iωl)gσ2(iωl)
(
g0σ(iωl1) − gσ2(iωl1)
)2
+ g0σ(iωl1)gσ2(iωl1)
(
g2
0σ(iωl) + g2
σ2(iωl)
)]}
− Z−2
0 U2βδll1
(
e−β(E0+E2) + 2e−β(E0+E1) + 3e−2βE1 + 2e−β(E1+E2)
)
g2
0σ(iωl)g
2
σ2(iωl)
+ Z−2
0 U2β
(
2e−β(E0+E2) + e−β(E0+E1) + e−β(E1+E2)
)
g0σ(iωl)gσ2(iωl)g0σ(iωl1)gσ2(iωl1), (14)
where gij(iωl) = (iωl + Eij)
−1 is the Fourier transform of functions (12) and (13).
Equations (10) and (14) can be significantly simplified for the case of principal interest U � T .
In this case if µ satisfies the conditions
λ < µ < U − λ, (15)
λ � T , the exponent exp(−βE1) is much larger than exp(−βE0) and exp(−βE2). By passing
to real frequencies we can ascertain that terms in (14) with the two latter multipliers have the
same peculiarities as other terms. Therefore the terms with these multipliers can be omitted in the
equations and we get
K1(iωl) =
iωl + µ − U/2
(iωl + µ)(iωl + µ − U)
,
∑
σ1
K2(iωlσ, iωl1σ1, iωl1σ1) = −
1
2
Ug0σ(iωl)gσ2(iωl)
[
g2
0σ(iωl1) + g2
σ2(iωl1)
]
−
1
2
Ug0σ(iωl1)gσ2(iωl1)
[
g0σ(iωl) − gσ2(iωl)
]2
−
3
4
U2βδll1g
2
0σ(iωl)g
2
σ2(iωl). (16)
Let us turn to real frequencies by substituting iωl with z = ω + iη where η is a small positive
constant which affords an artificial broadening possible. Results given in the next section were
calculated with G(k, z) on the right-hand side of equation (8) taken from the Hubbard-I approxi-
mation. As mentioned, this Green’s function is obtained if K(kω) in equation (6) is approximated
by K1 from equation (16) which gives
G(k, z) =
1
2
(
1 +
tk
√
U2 + t2
k
)
1
z − ε1,k
+
1
2
(
1 −
tk
√
U2 + t2
k
)
1
z − ε2,k
,
ε1,k =
1
2
(
U + tk +
√
U2 + t2
k
)
− µ, ε2,k =
1
2
(
U + tk −
√
U2 + t2
k
)
− µ. (17)
Below the two-dimensional square lattice is considered. It is supposed that only the hopping con-
stants between the nearest neighbor sites t are nonzero which gives tk = 2t[cos(kx)+cos(ky)] where
the intersite distance is taken as the unit of length. Due to the electron-hole symmetry in this case
the consideration can be restricted to the range of the chemical potentials µ 6 U/2.
539
A.Sherman
3. Spectral function
Figure 2 demonstrates =K(ω) calculated with the use of equations (8), (16) and (17). The
change to real frequencies carried out in the previous section converts the Matsubara function (2)
into the retarded Green’s function [24]. It is an analytic function in the upper half-plane which
requires that =K(ω) be negative. As seen in figure 2, this condition is violated at ωd = −µ and
Figure 2. The imaginary part of K(ω) calculated using equations (8), (16) and (17) for a 100×100
lattice, t = −U/8 and T = 0.001U (the dashed lines). (a) µ = 0.5U , η = 0.01U . (b) µ = 0.1U ,
η = 0.02U . The solid lines show the corrected =K(ω) (see text).
U −µ. This difficulty of the considered approximation was indicated in reference [13]. The problem
is connected with divergencies at these frequencies introduced by functions g0σ(ω) and gσ2(ω) in
the above formulas. As can be seen from the procedure of calculating the cumulants in the previous
section, these functions and divergencies with sign-changing residues appear in all orders of the
perturbation expansion (4). In the entire series the divergencies are expected to compensate each
other so that the resulting =K(ω) is negative everywhere. However, in the considered subset of
terms such compensation does not occur. Nevertheless, as seen in figure 2, at frequencies close
to ωd =K(ω) is negative. If the used subset of diagrams is expected to give a correct estimate
of the entire series for these frequencies, the values of =K(ω) for the problem frequency domains
where =K(ω) > 0 can be reconstructed using an interpolation and the values of =K(ω) in the
regular region. For this purpose different interpolating functions were tested. Since the regions
where =K(ω) > 0 are connected with poles, their widths can be decreased by decreasing the
artificial broadening η. This provides a way of testing the suitability of interpolation functions
and the choice of supporting points used in determining the free parameters in the functions. The
points were chosen away from the regions where =K(ω) > 0. The functions and supporting points
were adopted in such a way that the sharp minima in =K(ω) were not blurred. It was found that
in this case the choice of the interpolating function and supporting points only weakly effects the
shape of the spectral function discussed below. Examples of the interpolations used are given in
figure 2.
As seen in figure 2a, at half-filling, µ = U/2, =K(ω) has two broad minima. With the chemical
potential decreasing from this value, these minima shift with respect to the Fermi level without a
noticeable change of their shapes until the Fermi level enters one of the minima at µ ≈ 0.17U . As
this takes place, two new sharp minima arise near the frequencies −µ and U −µ on the background
of the above-mentioned broad minima. The appearance of the broad features in figure 2 is connected
with the third term in
∑
σ1
K2 in equation (16), while the sharp minima are related to the second
term in this formula. Its contribution to K(ω), equation (8), grows rapidly when the Fermi level
enters the broad minimum.
The function K(z) should be analytic in the upper half-plane as well and therefore its real part
can be calculated from its imaginary part using the Kramers-Kronig relations. We use this way
with the interpolated =K(ω) to avoid the effect of the divergencies on <K(ω). However, the use
of the interpolation somewhat overrates the values of |=K(ω)| which leads to the overestimation
540
Spectrum of the Hubbard model
of the tails in the real part. To correct this defect the interpolated K(ω) is scaled so that in the
far tails its real part coincides with the values obtained from equation (8).
The spectral function
A(kω) = −
1
π
=G(kω) = −
1
π
=K(ω)
[1 − tk<K(ω)]2 + [tk=K(ω)]2
(18)
obtained in this way for the momenta along the symmetry lines of the square Brillouin zone is
shown in figure 3. The shapes of the spectral function in figure 3a are nearly the same as at half-
Figure 3. Panels (a) and (c): the spectral function A(kω) calculated for momenta along the
symmetry lines of the square Brillouin zone in a 100×100 lattice for t = −U/8, T = 0.001U ,
η = 0.02U , µ = 0.2U (a) and µ = 0.05U (c). The dispersions of maxima in the panels (a) and
(c) are shown in the panels (b) and (d), respectively. Here darker areas correspond to larger
intensities of maxima.
filling – with decreasing µ from 0.5U to approximately 0.17U these curves shift with respect to the
Fermi level without perceptible changes in their shapes. As indicated in reference [14], four bands
can be distinguished in these spectra. For parameters of figure 3a these bands are located near
frequencies −4|t|, |t|, 4|t|, and 9|t| (see figure 3b). For the major part of the Brillouin zone the
peaks forming the bands arise at frequencies which satisfy the equation 1 − tk<K(ω) = 0 and fall
into the region of a small damping |=K(ω)| [see equation (18)]. As seen in figure 2, such regions of
541
A.Sherman
small damping are located between and on the outside of the two broad minima in =K(ω). This
is the reason of the existence of the four well separated bands – two of them are located between
the minima of =K(ω), while two others are on the outside of these minima. Broader maxima of
A(kω) for the momenta near the boundary of the magnetic Brillouin zone are of different nature
– since tk is small for such momenta, the resonant denominator in equation (18) does not vanish
and the shape of the spectral functions is determined by =K(ω) in the numerator of this formula.
More substantial changes in A(kω) occur for µ 6 0.17U when the Fermi level enters one of the
broad minima in =K(ω). As seen from figure 3c, in addition to the mentioned four bands there
appear sharp dispersive features near ω = −µ and U − µ for the momenta in the vicinity of the
boundary of the magnetic Brillouin zone. It is clear that these changes in the spectral function are
connected with the changes in =K(ω) shown in figure 2. The peaks near −µ are more intensive than
those near U −µ and are located in the nearest vicinity of the Fermi level. For µ ≈ 0.17U the width
of the band formed by the former peaks is comparable to the superexchange constant J = 4t2/U
of the effective Heisenberg model which describes magnetic excitations in the limit U � |t|. This
obviously indicates the participation of the spin excitations in the formation of these band states.
The bandwidth decreases with further reduction of the electron concentration. As this takes place,
the peak intensities first grow and then saturate. The maximum energies of the band are located
near the boundary of the magnetic Brillouin zone. For parameters in figure 3c for these momenta,
the band touches the Fermi level and the corresponding peaks disappear above it (see figure 3d).
These properties of the band resembles those of the spin-polaron band in the t-J model. This
latter band is also located near the Fermi level, has the similar dispersion and the bandwidth,
which decreases with the reduction of the electron concentration (with the increase of the hole
doping counted from half-filling) [23]. However, there is one essential difference in the behavior of
these bands in the Hubbard and t-J models. In the former model the narrow band near the Fermi
level appears at a certain deficiency of electrons, while the above-mentioned four bands exist in
the entire range of electron concentrations considered. In the t-J model the situation is opposite
– the spin-polaron band exists in the wide range of hole concentrations 0 6 x 6 0.17, while the
wider band – an analog of the four-band structure – starts to form at x ≈ 0.06 [23].
The value of the chemical potential for which the Fermi level enters the minimum of =K(ω)
and the narrow band begins to form, depends on the ratio t/U . For example, for t = −U/4 this
happens at µ ≈ 0.27U .
There is some difficulty in comparing our results with the data of Monte-Carlo calculations and
cluster theories. This is due to the fact mentioned in reference [14]: the one-loop approximation
overestimates the spectral weights of the two internal bands of the four-band structure near the
momenta (0, 0) and (π, π). As a result, the electron concentration calculated from the formula
〈n〉 =
2
N
∑
k
∫ ∞
−∞
dω
A(kω)
exp(βω) + 1
appears to be considerably underestimated. For example, for the parameters in figures 3a and 3c
such estimated 〈n〉 is equal to 0.87 and 0.68, respectively. From the comparison of the dispersions in
figures 3b and 3d with the results of Monte-Carlo calculations (figure 9 in [18]) it can be concluded
that these concentrations should be approximately 0.95 and 0.85, respectively. With this in mind
we find that the spectral functions and dispersions in figure 3 are close to the Monte-Carlo spectra
(cf. with figures 10 and 11 in [18]). In these latter spectra, for some deviation from half-filling,
a weakly dispersive feature is also observed near the Fermi level. However, this feature has low
intensity and is lost at the foot of a more intensive maximum on approaching the boundary of
the magnetic Brillouin zone. These differences may be connected with the comparatively high
temperature T = 0.33|t| used in the Monte-Carlo simulations. A similar weakly dispersive feature
near the Fermi level was also obtained in quantum cluster theories, though for much smaller
deviations from half-filling (cf. with figure 2c in [27] and figure 32 in [28]).
542
Spectrum of the Hubbard model
4. Conclusion
The considered diagram technique is very promising in being generalized to many-band Hubbard
models for which energy parameters of the one-site parts of the Hamiltonians exceed or at least
are comparable to the intersite parameters. The expansion in powers of these latter parameters
can be expressed in terms of cumulants in the same manner as discussed above. Now there are
distinct cumulants which belong to different site states characterized by dissimilar parameters of the
repulsion and the level energy. These cumulants are described by formulas similar to equations (10)
and (14). For example, for the Emery model [29] which describes oxygen 2pσ and copper 3dx2−y2
orbitals of Cu-O planes in high-Tc superconductors there are two types of cumulants corresponding
to these states. Diagrams of the lowest orders, e.g., for Green’s function on copper sites resemble
those shown in figure 1 where “oxygen” cumulants are included in hopping lines. Equations of the
type of (6) and partial summations similar to equation (7) can be also derived in this case.
In summary, the diagram technique for the one-band Hubbard model was formulated for the
case of moderate to strong Hubbard repulsion. The expansion in powers of the hopping constant
is expressed in terms of site cumulants of electron creation and annihilation operators. For Green’s
function the equation of the Larkin type was derived and solved for the case of two dimensions
and nearest-neighbor hopping. With the electron concentration decreasing, in addition to the four
bands observed at half-filling, a narrow band arises near the Fermi level for the momenta near the
boundary of the magnetic Brillouin zone. On the occurrence, the width of the band is comparable
to the superexchange constant J = 4t2/U which indicates the participation of the spin excitations
in the band formation. The bandwidth decreases with the electron concentration decreasing. The
maximum energies of the band are located near the boundary of the magnetic Brillouin zone.
For some deviation from half-filling in these points, the band touches the Fermi level. With these
properties the band resembles the spin-polaron band of the t-J model.
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Еволюцiя спектру моделi Хаббарда при змiнi концентрацiї
О.Шерман
Iнститут фiзики, Унiверситет Тарту, Рiа 142, 51014 Тарту, Естонiя
Отримано 27 лютого 2006 р., в остаточному виглядi – 12 травня 2006 р.
Сформульовано дiаграмну технiку для однозонної моделi Хаббарда для випадку промiжного та силь-
ного хаббардiвського вiдштовхування. Розклад за степенями константи перескоку виражено через
вузловi кумулянти електронних операторiв народження та знищення. Отримано рiвняння Ларкiна
для функцiї Грiна та розв’язано його у однопетлевому наближеннi для двовимiрного випадку та пе-
реносу мiж найближчими сусiдами. При пониженнi електронної концентрацiї крiм чотирьох зон, що
iснують при половинному заповненнi, з’являється вузька зона поблизу рiвня Фермi. Дисперсiя цiєї
нової зони, її ширина та змiна при змiнi концентрацiї є подiбними до таких же величин для випадку
спiн-поляронної зони в t-J моделi.
Ключовi слова: модель Хаббарда, дiаграмна технiка, енергетичний спектр
PACS: 71.10.Fd, 71.10.-w
544
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