Manifestation of the Upper Hubbard band in the 2D Hubbard model at low electron density
We consider the 2D Hubbard model in the strong-coupling case (U >> W) and at low electron density (nd² << 1). We find an antibound state as a pole in the two-particle T-matrix. The contribution of this pole in the self-energy reproduces a two-pole structure in the dressed one-particle Gr...
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
2011
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Kagan, M.Yu. Val’kov, V.V. Woelfle, P. 2017-05-31T06:58:39Z 2017-05-31T06:58:39Z 2011 Manifestation of the Upper Hubbard band in the 2D Hubbard model at low electron density / M.Yu. Kagan, V.V. Val’kov, P. Woelfle // Физика низких температур. — 2011. — Т. 37, № 9-10. — С. 1046–1052. — Бібліогр.: 16 назв. — англ. 0132-6414 PACS: 71.10.Fd https://nasplib.isofts.kiev.ua/handle/123456789/118762 We consider the 2D Hubbard model in the strong-coupling case (U >> W) and at low electron density (nd² << 1). We find an antibound state as a pole in the two-particle T-matrix. The contribution of this pole in the self-energy reproduces a two-pole structure in the dressed one-particle Green-function similar to the Hubbard-I approximation. We also discuss briefly the Engelbrecht-Randeria mode which corresponds to the pairing of two holes below the bottom of the band for U >> W and low electron density. Both poles produce nontrivial corrections to Landau Fermi-liquid picture already at low electron density but do not destroy it in 2D. We acknowledge helpful discussions with P.B. Wiegman, D. Vollhardt, P. Fulde, K.I. Kugel, and A.F. Barabanov. This work was supported by RFBR grants №11-02-00798 and 11-02-00741. en Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України Физика низких температур Низкоразмерные структуры Manifestation of the Upper Hubbard band in the 2D Hubbard model at low electron density Article published earlier |
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Manifestation of the Upper Hubbard band in the 2D Hubbard model at low electron density |
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Manifestation of the Upper Hubbard band in the 2D Hubbard model at low electron density Kagan, M.Yu. Val’kov, V.V. Woelfle, P. Низкоразмерные структуры |
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Manifestation of the Upper Hubbard band in the 2D Hubbard model at low electron density |
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Manifestation of the Upper Hubbard band in the 2D Hubbard model at low electron density |
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Manifestation of the Upper Hubbard band in the 2D Hubbard model at low electron density |
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Manifestation of the Upper Hubbard band in the 2D Hubbard model at low electron density |
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manifestation of the upper hubbard band in the 2d hubbard model at low electron density |
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Kagan, M.Yu. Val’kov, V.V. Woelfle, P. |
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Kagan, M.Yu. Val’kov, V.V. Woelfle, P. |
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Низкоразмерные структуры |
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Низкоразмерные структуры |
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2011 |
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English |
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Физика низких температур |
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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We consider the 2D Hubbard model in the strong-coupling case (U >> W) and at low electron density (nd² << 1). We find an antibound state as a pole in the two-particle T-matrix. The contribution of this pole in the self-energy reproduces a two-pole structure in the dressed one-particle Green-function similar to the Hubbard-I approximation. We also discuss briefly the Engelbrecht-Randeria mode which corresponds to the pairing of two holes below the bottom of the band for U >> W and low electron density. Both poles produce nontrivial corrections to Landau Fermi-liquid picture already at low electron density but do not destroy it in 2D.
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0132-6414 |
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https://nasplib.isofts.kiev.ua/handle/123456789/118762 |
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Manifestation of the Upper Hubbard band in the 2D Hubbard model at low electron density / M.Yu. Kagan, V.V. Val’kov, P. Woelfle // Физика низких температур. — 2011. — Т. 37, № 9-10. — С. 1046–1052. — Бібліогр.: 16 назв. — англ. |
| work_keys_str_mv |
AT kaganmyu manifestationoftheupperhubbardbandinthe2dhubbardmodelatlowelectrondensity AT valkovvv manifestationoftheupperhubbardbandinthe2dhubbardmodelatlowelectrondensity AT woelflep manifestationoftheupperhubbardbandinthe2dhubbardmodelatlowelectrondensity |
| first_indexed |
2025-11-25T15:44:37Z |
| last_indexed |
2025-11-25T15:44:37Z |
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1850517106996543488 |
| fulltext |
© M.Yu. Kagan, V.V. Val’kov, and P. Woelfle, 2011
Low Temperature Physics/Fizika Nizkikh Temperatur, 2011, v. 37, Nos. 9/10, p. 1046–1052
Manifestation of the Upper Hubbard band in the 2D
Hubbard model at low electron density
M.Yu. Kagan
P.L. Kapitza Institute for Physical Problem, 2 ul. Kosygina, Moscow 119334, Russia
E-mail: kagan@kapitza.ras.ru
V.V. Val’kov
Kirensky Institute of Physics, Akademgorodok, 50, bld. 38, Krasnoyarsk 660036, Russia
P. Woelfle
Institute for Theoretical Condensed Matter physics,
Karlsruhe Institute of Technology, Wolfgang-Gaede-Str., 1, Karlsruhe D-76131, Germany
Received May 10, 2011
We consider the 2D Hubbard model in the strong-coupling case (U >> W) and at low electron density (nd2 << 1).
We find an antibound state as a pole in the two-particle T-matrix. The contribution of this pole in the self-energy
reproduces a two-pole structure in the dressed one-particle Green-function similar to the Hubbard-I
approximation. We also discuss briefly the Engelbrecht-Randeria mode which corresponds to the pairing of two
holes below the bottom of the band for U >> W and low electron density. Both poles produce nontrivial
corrections to Landau Fermi-liquid picture already at low electron density but do not destroy it in 2D.
PACS: 71.10.Fd Lattice fermion models (Hubbard model, etc.).
Keywords: Hubbard model, Green-function, low electron density.
Introduction
At low electron density ( 2 1nd << — practically empty
band) and in the strong-coupling case U W>> the
effective interactions in the 2D Hubbard model [1] can be
described in the T-matrix approximation (see Kanamori
[2]). In the low energy sector Fε ≤ ε and in the framework
of this description the 2D Hubbard model becomes
equivalent to a 2D Fermi-gas with quadratic spectrum and
short-range repulsion [3]. Thus it can be characterized by
the 2D gas-parameter of Bloom [4]:
0 2
1 ,
ln (1/ )
f
nd
≈ (1)
where 2 / 2Fn p= π is the electron density in 2D (for both
spin projections, taking into account that / 2n n nσ −σ= =
in the unpolarized case), Fp is the Fermi-momentum, d
is the intersite distance. Accordingly many properties of
the 2D Hubbard model at low electron density, and in
particular the quasiparticle damping near the Fermi-surface
2
2
0~ Im ( , ) ~ lnp F
p
F p
f
ε ε
γ Σ ε
ε ε
p
have Landau Fermi-liquid character (amended with the
specific 2D logarithm) [5], where 2( / 2 )p Fp mε = − ε is
quasi-particle spectrum in the low-energy sector Fε ≤ ε
and 0f is given by (1). Correspondingly the averaging of
Im ( , )pΣ ε p with Fermionic distribution function
( / )F pn Tε produces the familiar result
2( ) ~ Im ( ) ~ lnT T T Tγ Σ in 2D. Accordingly the
quasiparticle residue
1Re1Z
−∂ Σ⎛ ⎞−⎜ ⎟∂ω⎝ ⎠
∼ is nonvanishing
for 0.ω → However, as first mentioned by J. Hubbard [1]
and P.W. Anderson [6], for U W>> the presence of a
band of a finite width produces at high energies an
additional pole in the two-particle T-matrix, well separated
from all other poles, with the energy:
0.Uε >∼ (2)
This pole is usually called the antibound state. Already
in the first iteration of the self-consistent T-matrix
approximation this pole yields a non-trivial contribution to
the self-energy ( , ).Σ ε p As a result the dressed one-particle
Green-function acquires a two-pole structure, very similar
to the Hubbard-I approximation [1].
Manifestation of the Upper Hubbard band in the 2D Hubbard model at low electron density
Low Temperature Physics/Fizika Nizkikh Temperatur, 2011, v. 37, Nos. 9/10 1047
The theoretical model
We consider the simplest 2D Hubbard model on the
square lattice:
,i j ii i
ij i i
H H N t c c U n n n+
σ σ σ↑ ↓
< >σ σ
′ = −μ = − + −μ∑ ∑ ∑ (3)
where i i in c c+
σ σ σ= is the density operator of electrons on
site i with spin-projection ,σ U is Hubbard repulsion, t
is hopping integral, μ is the chemical potential. The
bandwidth 8W t= on the square lattice. After Fourier-
transforming we get:
p p p p q p qp p
p pp q
H c c U c c c c+ + +
σ σ ′− ↓ + ↑′↑ ↓
′σ
′ = ε +∑ ∑ , (4)
where 2 (cos cos )p x yt p d p dε = − + −μ is the quasiparticle
spectrum of the uncorrelated problem. For low electron
density 1Fp d << we can often use the quadratic
approximation for the spectrum:
2 2
,
2
F
p
p p
m
−
ε = (5)
where 21/ 2m td= is the band-mass; ( / 2) FWμ = − +ε is
chemical potential and
2 (cos cos )p p x yt t p d p d= ε + μ = − + ≈
2
2 2 .
2 2 2
W W ptp d
m
≈ − + = − +
We will mostly consider the physically more transparent
strong-coupling case U W>> at low electron density
2 1.nd <<
T-matrix approximation
We start with the standard definition of the T-matrix
in 2D [4,7]:
2
2
2
2
.
1 ( ) ( )
1
( )(2 )
F p F q p
p q p
UdT
n ndUd
io
σ −σ −
−
=
− ε − ε
−
ω− ε − ε +π∫
p
(6)
The poles of the T-matrix are governed by the condition:
2
2
2
1 ( ) ( )
1 .
( )(2 )
F p F q p
p q p
n ndUd
io
σ −σ −
−
− ε − ε
=
ω− ε − ε +π∫
p (7)
For the antibound state for which ~ Uω we can expand
(7) and get (see also [8]):
2
2
2
1 ( ) ( )
1
(2 )
F p F p qn ndUd σ −σ −− ε − ε
= ×
ωπ∫
p
2
1 ,p p qt t −+ − μ⎡ ⎤
× +⎢ ⎥ω⎣ ⎦
(8)
where ;p ptε = − μ .p q p q q pt− − −ε = − μ = ε
Equivalently we can write:
2 2
21 1 ( ) ( )
(2 )
F p F p q
Ud d n nσ −σ −⎡ ⎤= − ε − ε +⎣ ⎦ω π∫
p
2 2
2 2 1 ( ) ( ) ( 2 )
(2 )
F p F p q p p q
Ud d n n t tσ −σ − −⎡ ⎤+ − ε − ε + − μ⎣ ⎦ω π∫
p
(9)
and use that
2
2 ( ) 0
(2 )
p p q
d t t −+ =
π∫
p
when we integrate
over the Brillouin zone. Thus:
2 2 2 2 2
2 2 2
2 2
2 2
1 1 ( 2 )
2 2 (2 )
( ) ( )
(2 )
F p F p q p p q
Ud nd nd Ud d
d
Ud d n nσ −σ − −
⎛ ⎞
= − − + − μ −⎜ ⎟⎜ ⎟ω ω π⎝ ⎠
⎡ ⎤ ⎡ ⎤− ε + ε ε + ε⎣ ⎦ ⎣ ⎦ω π
∫
∫
p
p
, (10)
where we used that in unpolarized case / 2.n n nσ −σ= =
Note that
/ /2
2 2
2
/ /
1
2 2(2 )
d d
yx
BZ d d
dpdpdd d
π π
−π −π
= =
π ππ∫ ∫ ∫
p
for the integration over the Brillouine zone. Hence:
2
2
2 2
21 (1 )U U Udnd μ
= − − − ×
ω ω ω
2
2 ( ) ( )
(2 )
F p F p q p p q
d n nσ −σ − −⎡ ⎤ ⎡ ⎤× ε + ε ε + ε⎣ ⎦ ⎣ ⎦π∫
p
. (11)
In the third term of (11) the integration is restricted by
Fermi-factors and hence we can use quadratic
approximation for the spectrum 2( / 2 )p Fp mε = − ε . Then
for the third term we get:
____________________________________________________
0 0 02 2 2 2 2
22
0 0
2 cos 2 cos(0) 2
2 2
F F F
D p p p F p F
Ud d p q pq d p q pqN d d d
m m
π π
−ε −ε −ε
⎡ ⎤⎛ ⎞ ⎛ ⎞ϕ + − ϕ ϕ + + ϕ⎢ ⎥− ε ε + ε − ε + ε − ε =⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎢ ⎥π πω ⎝ ⎠ ⎝ ⎠⎣ ⎦
∫ ∫ ∫ ∫ ∫
0 02 2 2 2
2
2 22 2(0) 4 2 (0) 2 2 ,
2 2
F F
D p p p D F F
Ud q Ud qN d d N
m m
−ε −ε
⎡ ⎤ ⎡ ⎤
⎢ ⎥= − ε ε + ε = − − ε + ε⎢ ⎥
⎢ ⎥ω ω ⎢ ⎥⎣ ⎦⎣ ⎦
∫ ∫ (12)
M.Yu. Kagan, V.V. Val’kov, and P. Woelfle
1048 Low Temperature Physics/Fizika Nizkikh Temperatur, 2011, v. 37, Nos. 9/10
where we used that
2 2
2 2( ) ( ) .
(2 ) (2 )
F p q p F p p q
d dn n−σ − −σ +ε ε = ε ε
π π∫ ∫
p p
In (12) 2 (0) / 2DN m= π is the density of states in 2D for
the quadratic spectrum. Hence:
2 2
2
2 2
21 (1 ) 2 .
2
F
F
mU U Ud qnd
m
⎡ ⎤εμ
= − − − − ε +⎢ ⎥
ω πω ω ⎢ ⎥⎣ ⎦
(13)
Having in mind that 2( / 2 ) ( / 4 ) / 2F Fm p nε π = π = we get
2 2
2
2 2
21 (1 ) 2 .
2
F
U U Und qnd
m
⎛ ⎞μ
= − − − − ε +⎜ ⎟⎜ ⎟ω ω ω ⎝ ⎠
(14)
Accordingly for the antibound state:
2 2
2
2 2
2(1 ) 2 .
(1 ) 2 (1 )
ab F
U Und qU nd
mU nd U nd
⎛ ⎞μ
ω ≈ − − + ε −⎜ ⎟⎜ ⎟− − ⎝ ⎠
(15)
Or respectively
2 2
2
2 2
2(1 )
21 1
ab F
nd qU nd
mnd nd
⎛ ⎞μ
ω ≈ − − + ε − =⎜ ⎟⎜ ⎟− − ⎝ ⎠
2 2 2
2
2 2
2(1 ) 2
21 1
F
nd nd qU nd
mnd nd
⎛ ⎞μ
= − − μ − + ε − =⎜ ⎟⎜ ⎟− − ⎝ ⎠
2 2 2
2
2 2(1 ) 2 ( 2 ) .
21 1
F
nd nd qU nd
mnd nd
= − − μ + ε − μ −
− −
(16)
By analogy with attractive-U Hubbard model [9] we
can introduce “bosonic” chemical potential:
2 ,B bEμ = μ − (17)
where
2
2 2 2
2(1 ) ( 2 ) (1 )
1
b F
ndE U nd U nd nd W
nd
= − + ε − μ ≈ − +
−
(18)
is a “binding” energy of antibound pair and 2 *( / 4 )q m−
for the spectrum, where the effective mass reads:
2
*
2
(1 )
2
ndm m m
nd
−
= >> for 2 1.nd << (19)
Then we can represent:
2 2
2 ,
4 4
ab b B
q qE
m m∗ ∗
ω = − μ − = − −μ (20)
which is quite nice. The spectrum (20) closely resembles
the pole of the attractive-U Hubbard model for b FE > ε
[9]. The important difference is, however, in the relative
sign between 2μ and .bE In the attractive-U Hubbard
model 2B bEμ = μ + and the real pairs are created below
the bottom of the band. Thus (| | /2)bEμ ≈ − and 0Bμ →
at low temperatures. In the repulsive-U Hubbard model for
low electron density 2 1:nd << ( / 2) FWμ ≈ − +ε for low
temperatures. Only in the case of half-filled band 2 1nd =
(one electron per site) the chemical potential / 2Uμ ≈
“jumps” in the middle of the Mott–Hubbard gap
.MH UΔ = The situation resembles that for a
semiconductor: the chemical potential for 2 1nd = lies in
the middle of the forbidden gap. Another important
difference is connected with the hole-like dispersion in
(20) that is with the sign “–” in front of 2 / 4 .q m∗
The T-matrix close to the pole reads [9]:
2
*
( , ) .
4
B
UT
q io
m
ω
ω ≈
ω+ +μ +
q (21)
Imaginary part of the self-energy
In the first iteration to the self-consistent T-matrix
approximation (see [10,11]):
____________________________________________________
2
2Im ( , ) Im ( , ) ( ) ( )
(2 )
p F p B p
d T n n⎡ ⎤Σ ω = ω+ε + ε + ε +ω =⎣ ⎦π∫
pk p k
2 2 2
2 * *
( ) ( )( ) ( ) ( ) ,
(2 ) 4 4
p p B F p B B
d U n n
m m
⎡ ⎤⎛ ⎞+ +
= π ω+ ε δ ω+ ε +μ + ε + − −μ⎢ ⎥⎜ ⎟⎜ ⎟π ⎢ ⎥⎝ ⎠⎣ ⎦
∫
p p k p k
(22)
_______________________________________________
where ( )F pn ε is fermionic distribution function,
2 *[ (( ) / 4 ) ]B Bn m− + −μp k is bosonic distribution
function. Having in mind that 2 ~B bE Uμ = μ − − we get
for U T>> [11]:
2
*
2
* ( )
4
( ) 1 0
4
e e 1
B
B B
m T T
n
m + μ− −
⎛ ⎞+
− −μ = →⎜ ⎟⎜ ⎟
⎝ ⎠
−
p k
p k .
Manifestation of the Upper Hubbard band in the 2D Hubbard model at low electron density
Low Temperature Physics/Fizika Nizkikh Temperatur, 2011, v. 37, Nos. 9/10 1049
Thus:
2 2
2 *
( )Im ( , ) (
(2 ) 4
B p
dU
m
⎡ ⎤+
Σ ω = π − −μ δ ω+ ε +⎢ ⎥
π ⎢ ⎥⎣ ⎦
∫
p p kk
2
*
( ) ) ( )
4
B F pn
m
+
+μ + ε
p k . (23)
Here again we have the important difference with
attractive-U Hubbard model where at low temperatures
0:T → 2 Bn n= while 0.Fn = In repulsive-U Hubbard
model we have vice-versa 0Bn = and Fn n= for 0.T →
Having in mind that * / 1m m >> for 2 1nd << we can
neglect 2 *( ) / 4m+p k in (23). Thus we get [11]:
0
2Im ( , ) (0) ( )
F
D B p p BN U d
−ε
Σ ω = −π μ ε δ ω+ ε + μ =∫k
[ ]2 (0) ( ) ( ) .D B B B FN U= −π μ θ ω+ μ − θ ω+ μ − ε (24)
Real part of the self energy
Correspondingly for the real part of the self-
energy[10,11]:
Re ( , )Σ ω =k
2
2Re ( , ) ( ) ( )
(2 )
p B p F p
dT n n⎡ ⎤= ω+ε + ω+ε + ε⎣ ⎦ π∫
pp k (25)
and again neglecting ( )B pn ω+ ε for U T>> we get:
Re ( , )Σ ω =k
2 2
*
(0) ( )
( )
4
p
D F p p
p B
UN n d
m
ω+ ε
= ε ε
+
ω+ ε +μ +
∫ p k
. (26)
For * / 1m m >> : 2 *( ) / 4m+p k is small and thus:
0
2Re ( , ) (0)
F
p
D p
p B
UN d
−ε
ω+ ε
Σ ω = ε =
ω+ ε + μ∫k
0
2 (0)
F
p
D F B
p B
d
UN
−ε
⎡ ⎤ε
⎢ ⎥= ε −μ =
⎢ ⎥ω+ ε +μ
⎣ ⎦
∫
2 (0) ln B
D F B
B F
UN
⎡ ⎤ω+ μ
= ε −μ⎢ ⎥
ω+ μ − ε⎢ ⎥⎣ ⎦
. (27)
Assuming that B Fω+ μ > ε and expanding the logarithm
in the second term we get:
2
2Re ( , ) (0)
2
F
D
B B
ndUN U
ε ω ω
Σ ω = =
ω+μ ω+μ
k . (28)
Thus the pole of the dressed one-particle Green-function
[12] 1 1
0( , ) ( , ) ( , )G G− −ω = ω −Σ ωk k k reads:
2
0.
2k
B
ndU ω
ω−ε − =
ω+μ
(29)
Correspondingly:
2
2
2 22 2
0;
2
2 2 0.
2 2
B k k B
B k B k
k B
Und
Und Und
⎛ ⎞
ω + μ − ε − ω− ε μ =⎜ ⎟⎜ ⎟
⎝ ⎠
⎛ ⎞ ⎛ ⎞
μ − ε − μ − ε −⎜ ⎟ ⎜ ⎟
⎜ ⎟ ⎜ ⎟ω+ − − ε μ =
⎜ ⎟ ⎜ ⎟
⎜ ⎟ ⎜ ⎟
⎝ ⎠ ⎝ ⎠
As a result:
22 2
1,2
2 2 .
2 2
B k B k
k B
Und Und⎛ ⎞
μ − ε − μ − ε −⎜ ⎟
⎜ ⎟ω = − ± + ε μ
⎜ ⎟
⎜ ⎟
⎝ ⎠
(30)
Having in mind that ~B Uμ − we can expand the square
root in (30). Then:
2
1,2
2
2
B k
Und
μ − ε −
ω = − ±
2
2
2 .
2
2
B k k B
B k
Und
Und
⎛ ⎞
⎜ ⎟μ − ε −⎜ ⎟ε μ
± +⎜ ⎟
⎜ ⎟μ − ε −⎜ ⎟
⎝ ⎠
(31)
We know that 0Bμ < and
2
;
2B k
Und⎧ ⎫⎪ ⎪μ >> ε⎨ ⎬
⎪ ⎪⎩ ⎭
.
That is why
2 2
2 2
2 2
B k B k
Und Und
μ − ε − μ − ε −
= −
and hence:
2
1,2
2
2
B k
Und
μ − ε −
ω = − ∓
2
2
2
2
2
B k k B
B k
Und
Und
⎛ ⎞
μ − ε −⎜ ⎟ε μ⎜ ⎟+
⎜ ⎟
μ − ε −⎜ ⎟
⎝ ⎠
∓ . (32)
M.Yu. Kagan, V.V. Val’kov, and P. Woelfle
1050 Low Temperature Physics/Fizika Nizkikh Temperatur, 2011, v. 37, Nos. 9/10
Finally:
2
1 2
2 2
2
2
.
2
k B
B k
B k
k B
B k
Und
Und
Und
⎧ ⎛ ⎞ ε μ
ω = − μ − ε − −⎪ ⎜ ⎟⎜ ⎟⎪ ⎝ ⎠ μ − ε −⎪
⎨
ε μ⎪ω =⎪
⎪ μ − ε −
⎩
(33)
The dressed Green-function ( , )G ω k reads:
1 2 1 2 1 2
1 1( , )
( )( )
B BG
⎡ ⎤ω+μ ω+μ
ω = = − =⎢ ⎥ω−ω ω−ω ω −ω ω−ω ω−ω⎣ ⎦
k
1 2
1 2 1 1 2 2 1 2
1 1 1B B⎛ ⎞ ⎛ ⎞ω +μ ω +μ
= − + −⎜ ⎟ ⎜ ⎟ω −ω ω−ω ω −ω ω−ω ω −ω⎝ ⎠ ⎝ ⎠
1 2
1 2 1 2 1 1 2 2
1 1 1 .B B⎛ ⎞ ⎛ ⎞ω +μ ω +μ
− = −⎜ ⎟ ⎜ ⎟ω −ω ω −ω ω−ω ω −ω ω−ω⎝ ⎠ ⎝ ⎠
(34)
Let us check the poles structure:
2
1 2
2 2
2
2
.
2
k B
B k
B k
k B
B k
Und
Und
Und
⎧ ⎛ ⎞ ε μ
ω−ω = ω+ μ − ε − +⎪ ⎜ ⎟⎜ ⎟⎪ ⎝ ⎠ μ − ε −⎪
⎨
ε μ⎪ω−ω = ω−⎪
⎪ μ − ε −
⎩
(35)
But 22 (1 )B b bE E U ndμ = μ− ≈ − ≈ − − and
2 2
1 .
2 2B
Und ndU
⎛ ⎞
μ − ≈ − −⎜ ⎟⎜ ⎟
⎝ ⎠
Of course
2
2B k
Und
μ − >> ε . Hence:
____________________________________________________
2 22 2 2 2
1 22
(1 ) (1 )
1 1 1
2 2 2 2
11
22
k k
k k k
U nd ndnd nd nd ndU U U
ndndU
⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ε − ε −
ω−ω = ω− ε − − − = ω− ε − − + = ω− − + ε − =⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎛ ⎞⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠−− −⎜ ⎟⎜ ⎟
⎝ ⎠
2 22 2 2
2 22
(1 ) (1 )
1 ; 1 .
2 2 2
1( ) 1
22
k k
k k
U nd ndnd nd ndU
ndndU
⎛ ⎞ ⎛ ⎞ε − ε −
= ω− − − ε ω−ω = ω+ = ω− ≈ ω− ε −⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎛ ⎞⎝ ⎠ ⎝ ⎠−− −⎜ ⎟⎜ ⎟
⎝ ⎠
(36)
In the same time the first term in (34) yields:
2 2
1
2 2 21 1 2 1
2
2
2 2
221 1 1
1
2 21 1
1 1
2 2 2
1 (1 )
21 1 11 .
2( )
11
22
k B
B
k k
nd ndU
nd nd ndU
ndU U nd
nd nd
ndndU
⎛ ⎞
− + ε +μ⎜ ⎟⎜ ⎟ω +μ ⎝ ⎠= ≈
ω−ω ω −ω ω−ω ⎛ ⎞ ⎛ ⎞
− + ε − ε −⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟
⎝ ⎠ ⎝ ⎠
⎛ ⎞ ⎛ ⎞− − −⎜ ⎟⎜ ⎟ ⎜ ⎟−⎝ ⎠ ⎜ ⎟≈ = − ≈
ω−ω ω−ω ω−ω⎜ ⎟⎛ ⎞
−−⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎝ ⎠⎝ ⎠
(37)
_______________________________________________
The second term in (34) reads:
2
2
2
22 1 2 2
1 (1 )
21 1
1
2
k
B
nd U nd
ndU
⎛ ⎞
ε − − −⎜ ⎟⎜ ⎟ω +μ ⎝ ⎠≈ ≈
ω−ω ω −ω ω−ω ⎛ ⎞
−⎜ ⎟⎜ ⎟
⎝ ⎠
2 2
22 2
1 (1 ) 1 1 .
2
1
2
U nd nd
ndU
⎛ ⎞−
≈ − ≈ − −⎜ ⎟⎜ ⎟ω−ω ω−ω⎛ ⎞ ⎝ ⎠−⎜ ⎟⎜ ⎟
⎝ ⎠
(38)
Thus
22
2 2 2
1
22( , )
1 1
2 2 2k k
ndnd
G
nd nd ndU
⎡ ⎤⎛ ⎞
−⎢ ⎥⎜ ⎟⎜ ⎟⎢ ⎥⎝ ⎠ω ≈ +⎢ ⎥⎛ ⎞ ⎛ ⎞⎢ ⎥ω− − − ε ω− ε −⎜ ⎟ ⎜ ⎟⎢ ⎥⎜ ⎟ ⎜ ⎟
⎝ ⎠ ⎝ ⎠⎣ ⎦
k
(39)
and we completely recover the Hubbard-I approximation
[1,13]. The first pole in (39) corresponds to the Upper
Hubbard band (UHB). Thus 2 / 2UHBZ nd= . The second
Manifestation of the Upper Hubbard band in the 2D Hubbard model at low electron density
Low Temperature Physics/Fizika Nizkikh Temperatur, 2011, v. 37, Nos. 9/10 1051
pole corresponds to the lower Hubbard band (LHB):
2
1
2LHB
ndZ
⎛ ⎞
= −⎜ ⎟⎜ ⎟
⎝ ⎠
. Of course, 1.UHB LHBZ Z+ = We can
rewrite ( , )G ω k as:
( , ) LHB
k LHB
Z
G
Z io
ω = +
ω− ε +
k
2
.
1
2
UHB
UHB k
Z
ndU Z io
+
⎛ ⎞
ω− − − ε +⎜ ⎟⎜ ⎟
⎝ ⎠
(40)
Note that the second iteration to the self-consistent
T-matrix approximation does not change the gross
features of (40). Thus the antibound state yields
nontrivial corrections to Landau Fermi-liquid picture
already at low electron density, but does not destroy it
in 2D. The simplest Hartree-Fock contribution to the
thermodynamic potential Ω from the upper Hubbard band
2
0 2( , ) ( , )
2(2 )
d dG ω
ΔΩ Σ ω ω
ππ∫
pp p∼ with 0( , )G ω p and
( , )Σ ω p given by (28), (29) yields 3~ .UHB FZ n nΔΩ ε∼
Engelbrecht–Randeria mode
For the sake of completeness let us discuss briefly the
Engelbrecht–Randeria mode[14] which also corresponds to
the pole of the T-matrix for U W>> and 2 1.nd <<
According to [14] it has a spectrum for 2 Fq p< :
2
0
1exp .
2
q
ER q
Ff
ω⎧ ⎫
ω ≈ ω − −⎨ ⎬
ε⎩ ⎭
(41)
Note that while antibound state exists also in 3D physics,
the Engelbrecht–Randeria mode is specific for 2D
Hubbard model.
In (41)
2
2
4q F
q
m
ω = − ε and 2
0
1exp nd
f
⎧ ⎫
− =⎨ ⎬
⎩ ⎭
in
agreement with (1). Note that for 0:q =
22 2 0ER F F ndω = − ε − ε < . (42)
The collective character of Engelbrecht–Randeria mode
is connected with the fact that in the absence of fermionic
background (for 0Fε = ) 0ERω = in (42). Moreover
2 .ER Fω < − ε Hence this mode lies below the bottom of
the band and corresponds to the binding of two holes
(Recall that the antibound state lies above the upper edge
of the band).
In terms of the “bosonic” chemical potential :Bμ
2
,
4ER B
q
m
ω ≈ −μ (43)
where in terms of ( / 2) ,FWμ ≈ − +ε 2 | |B bEμ = μ+ and
the binding energy 2| | 2b FE W nd≈ + ε .
Conclusion and acknowledgements
We considered the excitation spectrum of the Hubbard
model at low electron density, where a small parameter
(gas parameter) allows a controlled expansion. On the level
of the first iteration to the self-consistent T-matrix
approximation we found the contribution of the T-matrix
pole corresponding to the antibound state to the self-energy
.Σ As a result we got a two-pole structure of the dressed
one-particle Green-function which closely resembles the
Hubbard-I approximation.
It would be interesting to find the possible contribution
of the Upper Hubbard band to the ground-state energy or
compressibility and to build the bridge between the
Galitskii–Bloom Fermi-gas expansion for the ground-state
energy (or compressibility) and the Gutzwiller type of
expansion for the partially filled band [15] when the
electron density is increased.
For the sake of completeness we also analyzed the
Engelbrecht-Randeria mode which corresponds to the
pairing of two holes below the bottom of the band.
According to [14] this mode, when keeping the full q-de-
pendence for 0 2 ,Fq p≤ < gives nonanalytic corrections
5/2~ ω to the imaginary part of the self-energy Im ( )Σ ω
in 2D. It also contributes to the thermodynamics at 0T =
in the same order in density as the contribution of the
antibound state: 2 3~ ~ 0F n nd nΔΩ ε ⋅ > — amounting to
an increase of the thermodynamic potential Ω [14]. Thus
the Engelbrecht–Randeria mode as well as the Hubbard-
Anderson mode corresponding to the antibound state yield
interesting corrections to the Landau Fermi-liquid picture
in 2D already at low electron density, but do not destroy it
completely in contrast to the 1D-case, where we have the
Luttinger liquid state and a vanishing quasiparticle residue
0Z → for 0ω→ [16].
We acknowledge helpful discussions with
P.B. Wiegman, D. Vollhardt, P. Fulde, K.I. Kugel, and
A.F. Barabanov.
This work was supported by RFBR grants №11-02-
00798 and 11-02-00741.
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