Using coherent potential approximation we study zero-temperature Mott transition of the half-filled Hubbard model  in a two-dimensional square lattice with geometrical frustration. It turns out that the geometrical frustration reduces the gap between the Hubbard bands. As a result the metallic phase is stabilized up to a fairly large value of the on-site Coulomb interaction. We found that the critical value $U_C$ for the Mott transition is enhanced by the geometrical frustration. Our results are in good agreement with the ones obtained by the single-site dynamical mean-field theory

Anh, Le Duc

Tuan, Hoang Anh

Vietnam Academy of Science and Technology: Journals Online

Communications in Physics, Vol. 23, No. 1 (2013), pp. 49-55
MOTT TRANSITION OF THE HALF-FILLED HUBBARD MODEL
IN A TWO-DIMENSIONAL FRUSTRATED LATTICE
LE DUC ANH
Department of Physics, Hanoi National University of Education
HOANG ANH TUAN
Institute of Physics, VAST
Abstract. Using coherent potential approximation we study zero-temperature Mott transition of
the half-filled Hubbard model in a two-dimensional square lattice with geometrical frustration. It
turns out that the geometrical frustration reduces the gap between the Hubbard bands. As a result
the metallic phase is stabilized up to a fairly large value of the on-site Coulomb interaction. We
found that the critical value UC for the Mott transition is enhanced by the geometrical frustration.
Our results are in good agreement with the ones obtained by the single-site dynamical mean-field
theory.
I. INTRODUCTION
Strongly correlated electron system with geometrical frustration exhibits a variety
of phenomena and is presently a major topic of great interest in the condensed-matter
community[1, 2]. The competition between strong electronic correlations and the geomet-
rical frustration in metallic spinel compounds may cause some novel phenomena such as
heavy fermion state in LiV2O4[3], superconductivity with relatively high transition temper-
ature of TC = 13.7K in LiT i2O4[4], and so forth. New aspects of the Mott metal-insulator
transitions are also uncovered by the geometrical frustration, which is now one of the cen-
tral issues in the physics of strongly correlated electron systems[5]. Among all possible
microscopic models, the one-band Hubbard model is often used for a study of the interplay
between the geometrical frustration and strong electronic correlations. Previous studies of
such model on triangular and Kagome´ lattices have been performed by various approaches
such as the fluctuation exchange approximation[6], quantum Monte Carlo calculations[7],
coherent potential approximation[8], and cellular dynamical mean field theory[9]. A simi-
lar problem for other frustrated lattices has been also carried out by means of fourth order
perturbative calculation[10], path-integral renormalization group[11], variational Monte
Carlo simulations[12], and a cluster extension of dynamical mean-field theory[13].
In this paper, we investigate the Mott transition of the one-band half-filled Hubbard model
in a two-dimensional square lattice with geometrical frustration. In order to carry out the
calculation we treat the model within the coherent potential approximation (CPA)[14].
Assuming a paramagnetic groundstate, which is valid for the strongly frustrated case, we
show that the frustrated system undergoes a metal-insulator transition at a certain criti-
cal value of the onsite Coulomb repulsion UC . We show that the geometrical frustration
50 LE DUC ANH AND HOANG ANH TUAN
reduces the gap between the Hubbard bands. As a result the metallic phase is stabilized
up to fairly large Hubbard interactions for systems under strong geometrical frustration.
This paper is organized as follows. The Hubbard model and the coherent potential approx-
imation are presented in the next section. In Section 3 we discuss our numerical results.
Finally, the paper is concluded and the directions of future work are in Section 4.
II. MODEL AND FORMALISM
The one-band Hubbard model on the two-dimensional square lattice with nearest
hopping ti,j and on-site Hubbard repulsion U reads
H = −
∑
<i,j>σ
ti,j
(
c+iσcjσ + c
+
jσciσ
)
+ U
∑
i
ni↑ni↓ − µ
∑
i
ni, (1)
where ciσ(c
+
iσ) annihilates (creates) an electron with spin σ at site i, niσ = c
+
iσciσ and
ni = ni↑ + ni↓. The nearest neighbor hopping parameter ti,j, as depicted in Fig. 1, takes
either t or t′. By introducing t′, the so-called crossing hopping parameter, one controls the
geometrical frustration of the frustrated lattice and clarify how the lattice geometry affects
physical properties. The chemical potential µ is chosen such that the average occupancy
is 1 (half-filling).
Fig. 1. Hoppings in a two-dimensional frustrated lattice.
In the alloy-analogue approach the many-body Hamiltonian (1) is replaced by a one-
particle Hamiltonian with disorder which is of the form
H =
∑
i,σ
Eσniσ −
∑
<i,j>σ
ti,j
(
c+iσcjσ + c
+
jσciσ
)
, (2)
where
Eσ =
{
µ with probability 1− n−σ,
µ+ U with probability n−σ.
(3)
The Green function of the Hamiltonian (2) has to be averaged over all possible configura-
tions of the random potential, which can be considered to be due to alloy constituents. The
averaging cannot be performed exactly. Within the CPA, the Green function is determined
by the conditional Green function as follows
Gσ (ω) =
Fσ (ω) (1− n−σ)
1 + Fσ (ω) (Ξσ(ω) + µ)
+
Fσ (ω)n−σ
1 + Fσ (ω) (Ξσ(ω) + µ− U)
. (4)
MOTT TRANSITION OF THE HALF-FILLED HUBBARD MODEL ... 51
Here Fσ (ω) is the local lattice Green function
Fσ (ω) =
∞∫
−∞
ρ0 (ε) dε
ω + iη + µ− Ξσ(ω)− ε
, (5)
where the bare density of states of the frustrated lattice ρ0 (ε) =
1
4π2
π∫
−π
dkx
π∫
π
dkyδ
(
ε− ε~k
)
and the bare one electron dispersion[13] ε~k = −2t (cos kx + cos ky)− 2t
′cos (kx + ky).
The self-consistent condition of the CPA requires that the conditional Green function must
coincide with the local Green function of the original lattice, i.e.,
Gσ (ω) = Fσ (ω) . (6)
So far, for determining the Green function we have obtained a closed system of equations,
which can be solved numerically by iterations[15, 16].
III. NUMERICAL RESULTS AND DISCUSSION
We solve numerically the self-consistent equations (4) – (6) to determine the self-
energy and the Green function by simple iterations[15, 16]. The algorithm is summarized
as follows. Begin with an initial self-energy guess, one obtains the local Green function
from Eq. (5). Substituting the self-energy and the local Green function were calculated in
the previous step to Eq. (4) one calculates the conditional Green function. Finally, a new
self energy is determined by
Ξ′σ(ω) = Ξσ(ω) +
1
Fσ (ω)
−
1
Gσ (ω)
, (7)
which is equivalent to Eq. (6). This procedure is iterated until convergence is reached.
Normally, a relative error for the Green function of less than 10−10 is achieved after
few hundreds of iterations. However, the number of iterations for such relative error
will be of the order of thousands when one reaches the Mott transition. Throughout
this work, for simplicity we assume a paramagnetic solution for the groundstate. Note
that for highly frustrated systems at half-filling, this assumption is valid since the an-
tiferromagnetic order is expected to be destroyed by frustration[13, 17]. Hereafter, we
take t = 1 as the energy unit, total band-filling 1, zero temperature, and η = 0.01 in
numerical calculations. Now we turn to present our numerical results for all the unfrus-
trated model t′ = 0, moderately frustrated model t′ = 0.5, and strongly frustrated one
t′ = 1. Fig. 2 shows the non-interacting Green function for various values of the crossing
hopping parameter t′. It is clear that the geometrical frustration, due to the presence
of the crossing hopping parameter t′, makes the non-interacting DOS being asymmet-
ric. The absence of particle-hole symmetry suggests that a system magnetic long-range
order should be suppressed, thus, the ground state is likely to be paramagnetic. Fur-
thermore, the absence of particlehole symmetry also implies that for strongly frustrated
systems the weights of the Hubbard bands may be different. The latter signifies that
the chemical potential for large U at half-filling may lie at one of these Hubbard bands
52 LE DUC ANH AND HOANG ANH TUAN
or that the Mott insulator may not be formed. However, it is not the case though.
 ω
-0.5
0.0
0.5
1.0
R
e 
G
( ω
)
t’ = 0.00
t’ = 0.25
t’ = 0.50
t’ = 0.75
t’ = 1.00
-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7
-1.2
-0.8
-0.4
0.0
Im
 G
( ω
)
Fig. 2. Non-interacting Green function for various values of the crossing hopping
parameter t′.
0
0.05
0.1
0.15
0.2
0.25
0.3
A
(ω
)
0
0.05
0.1
0.15
0.2
0.25
0.3
0
0.05
0.1
0.15
0.2
0.25
A
(ω
)
0
0.05
0.1
0.15
0.2
0.25
0
0.05
0.1
0.15
0.2
A
(ω
)
0
0.05
0.1
0.15
0.2
-8 -6 -4 -2 0 2 4 6 8
 ω
0
0.05
0.1
0.15
0.2
A
(ω
)
-8 -6 -4 -2 0 2 4 6 8
 ω
0
0.05
0.1
0.15
0.2
U = 1, t’ = 0.5
U = 2, t’ = 0.5
U = 4, t’ = 0.5
U = 2, t’ = 1
U = 1, t’ = 1
U = 4, t’ = 1
U = 6, t’ = 1U = 6, t’ = 0.5
Fig. 3. Spectral functions for the unfrustrated t = 0′ model (dashed lines), mod-
erately frustrated model t′ = 0.5 (solid lines on the left panel), and strongly
frustrated one t′ = 1 (solid lines on the right panel).
From Fig. 3, spectral functions for various values of the onsite Coulomb repulsion U and
the crossing hopping parameter t′, it turns out that the weights of the Hubbard bands
MOTT TRANSITION OF THE HALF-FILLED HUBBARD MODEL ... 53
are still the same, implying that for large value of U the chemical potential lies in between
the two Hubbard bands and the system is Mott insulator. Furthermore, it is noticable that
the metallic region is extended as the geometrical frustration reduces the gap between the
Hubbard bands. The reduction of the gap implies that the critical correlation-driven metal-
insulator transition Uc of the frustrated model is larger that that of the unfrustrated one.
-0.2
-0.1
0
Σ(
ω)
-0.2
-0.1
0
0.1
0.2
-0.6
-0.3
0
Σ(
ω)
-0.3
0
0.3
0.6
-12
-9
-6
-3
0
Σ(
ω)
-3
-1.5
0
1.5
3
-8 -6 -4 -2 0 2 4 6 8
 ω
-60
-30
0
Σ(
ω)
-8 -6 -4 -2 0 2 4 6 8
 ω
-60
-30
0
30
60
U = 1, t’ = 1
U = 2, t’ = 1
U = 4, t’ = 1
U = 2, t’ = 1
U = 1, t’ = 1
U = 4, t’ = 1
U = 6, t’ = 1U = 6, t’=1
Real partImaginary part
Fig. 4. The imaginary and real part of the self-energy as a function frequency for
t′ = 1 and different values of U . The dashed (solid) lines are of the unfrustrated
(strongly frustrated) model, respectively.
Fig. 4 shows the imaginary and real part of the self-energy for the strongly frustrated
model t′ = 1 with different values of U as chosen in Fig. 3. At Fermi energy, in the
insulating regime U > UC , the imaginary part of the self-energy has a sharp peak whose
weight is roughly independent of U . The influence of the geometrical frustration on the
critical value UC is presented in Fig. 5. It turns out that when t
′ is increased, UC is
getting larger and finally it levels off at UC/W (t
′) ≈ 1.1 for highly frustrated system. Our
prediction UC/W (t
′) ≈ 1.1 when t′ ≈ 1 is in good agreement with the single-site DMFT
results [18, 19]. However, that UC calculated within cluster DMFT[13] is approximately
twice times larger than that we obtained here. This is not surprised since the importance
features of the non-local correlations are not taken into account within single-site theories.
54 LE DUC ANH AND HOANG ANH TUAN
0.0 0.2 0.4 0.6 0.8 1.0
4.0
4.2
4.4
4.6
4.8
5.0
 
UC
t'
1.00
1.02
1.04
1.06
1.08
1.10
UC/W(t') 
Fig. 5. The critical value UC and UC/W (t
′) are shown as a function of the crossing
hopping parameter t′, whereW (t′) is the bare bandwidth of the system. Note that
t is chosen to be 1 as the energy unit.
IV. CONCLUSIONS
We have applied the coherent potential approximation to study Mott transition of
the half-filled Hubbard model in the two-dimensional square lattice with geometrical frus-
tration. The system has been analyzed for a wide range of the Hubbard on-site Coulomb
repulsion U and the crossing hopping integral t′. It shows that the geometrical frustration
reduces the gap between the Hubbard bands. As a result the metallic phase is stabilized
up to fairly large Hubbard interactions under the strong geometrical frustration. Our
results are in good agreement with the ones obtained by single-site dynamical mean-field
theory. However, our UC is approximately half of that obtained by cluster DMFT. This
suggests a cluster extention theory of the CPA is needed to get better satisfactory with
cluster DMFT results. We leave this problem for future study.
ACKNOWLEDGMENTS
This research is funded by Vietnam National Foudation for Science and Technology
Development (NAFOSTED) under grant number 103.02-2011.05.
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Received 27 April 2012.


Mott Transition of the Half-filled Hubbard Model in a Two-dimensional Frustrated Lattice

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Mott Transition of the Half-filled Hubbard Model in a Two-dimensional Frustrated Lattice

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