A quantum Monte Carlo study on the superconducting Kosterlitz-Thouless transition of the attractive Hubbard model on a triangular lattice

Abstract

We study the superconducting Kosterlitz-Thouless transition of the attractive Hubbard model on a two-dimensional triangular lattice using auxiliary field quantum Monte Carlo method for system sizes up to $12\times 12$ sites. Combining three methods to analyze the numerical data, we find, for the attractive interaction of $U=-4t$, that the transition temperature stays almost constant within the band filling range of $1.0 < n < 1.4$, while it is found to be much lower in the $n<1$ region. Comment: RevTeX 6 pages

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arXiv:cond-mat/0503465v1 [cond-mat.str-el] 18 Mar 2005
A quantum Monte Carlo study on the superconducting Kosterlitz-Thouless transition
of the attractive Hubbard model on a triangular lattice
Tsuguhito Nakano and Kazuhiko Kuroki
Department of Applied Physics and Chemistry, The University of Electro-Communications, Chofu, Tokyo 182-8585, Japan
(Dated: February 2, 2008)
We study the superconducting Kosterlitz-Thouless transition of the attractive Hubbard model on
a two-dimensional triangular lattice using auxiliary field quantum Monte Carlo method for system
sizes up to 12 × 12 sites. Combining three methods to analyze the numerical data, we find, for the
attractive interaction of U = −4t, that the transition temperature stays almost constant within the
band filling range of 1.0 < n < 1.4, while it is found to be much lower in the n < 1 region.
I. INTRODUCTION
Discovery of superconductivity in layered materials
or quasi-two-dimensional systems in the past several
decades has brought up great interest in the physics
of low dimensional superconductors. Among those are
the high Tc cuprates,1a ruthenate Sr2RuO4,2a cobal-
tate NaxCoO2· y H2O,3MgB2,4a heavy electron sys-
tem CeCoIn5,5and organic conductors such as (BEDT-
TTF)2X (X is an anion).6
Theoretically, it is well known by Mermin-Wagner’s
theorem that no off-diagonal long range order takes place
at finite temperature in purely two-dimensional(2D)
systems.7In the case of pure 2D, superconducting tran-
sition is expected to be of the Kosterlitz-Thouless (KT)
type.8The superconducting KT transition has previ-
ously been studied using finite temperature auxiliary field
quantum Monte Carlo (AFQMC) technique for the at-
tractive Hubbard model, that is, the Hubbard model with
a negative on-site U, on a square lattice9and on a tri-
angular lattice.10Nowadays, a renewed interest for the
KT transition on triangular lattices has arisen because
unconventional superconductivity has been observed in
materials having (anisotropic) triangular lattice structure
such as NaxCoO2· yH2O and (BEDT-TTF)2X.
For the square lattice, it has been shown that the KT
superconducting transition temperature T0 ∼ 0.1t, t is
the hopping integral, for square lattice with n = 0.87 (n is
the band filling) away from the half filling n = 1.0, where
charge density wave (CDW) ordering takes place.9As for
the triangular lattice, it has been concluded that T0takes
its maximum at around half-filling due to the absence
of CDW ordering, while T0 is low near n = 1.4,where
the density of states becomes large due to the van Hove
singularity at n = 1.5.
Since previous studies have been restricted to small
system sizes, here we revisit the problem of the supercon-
ducting KT transition of the attractive Hubbard model
on a triangular lattice using AFQMC technique. Com-
bining three methods of analyzing the numerical data,
we find that the KT transition temperature stays almost
constant within the band filling region of 1 < n < 1.4.Our
results suggest that the estimation of the KT transition
temperature using finite size scaling requires some cau-
tion when the density of states near the Fermi level is
small and the calculation is restricted to small system
sizes.
II. FORMULATION
A.Model
The Hamiltonian of the attractive Hubbard model is
given in standard notation as
H = −t
?
?i,j?,σ
(c†
i,σcj,σ+ c†
j,σci,σ)
+U
?
i
ni,↑ni,↓− µ
?
i,σ
ni,σ, (1)
where c†
ator at site i with spin σ, and ni,σ= c†
notes a pair of nearest neighbors on square or triangular
lattice having periodic boundary condition.The chemical
potential µ controls the band filling n (average number of
electrons per site). The hopping parameter t is taken as
the unit of the energy and is set equal to unity through-
out the study. As for the lattice structure, we mainly
concentrate on the triangular lattice, but we also per-
form calculation on the square lattice at the band filling
of n = 0.87 in order to make comparison with previous
studies and with the triangular lattice. The on-site at-
traction U is fixed at U/t = −4.0 on both lattices, which
also enables us to make comparison with the previous
studies.
i,σ(ci,σ) is a fermion creation (annihilation) oper-
i,σci,σ,< i,j > de-
B.Pairing Correlation Function
Above the superconducting KT transition temperature
T0, the on-site (s-wave) pairing correlation function de-
cays exponentially,
< c†
i,↑c†
i,↓cj,↑cj,↓>∼ exp(−r/ξ). (2)
By contrast, the pairing correlation function exhibits
a power-law decay
< c†
i,↑c†
i,↓cj,↓cj,↑>∼ r−η,(3)

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where η = 0.25 for T → T0.
calculation, we calculate a summation of the pairing cor-
relation function, Ps, given as
In the actual AFQMC
Ps =
?∆†∆ + ∆∆†?
1
√N
i
(4)
Ơ=
?
c†
i,↑c†
i,↓
(5)
Due to the decaying behavior of the pairing correlation
function mentioned above, Psincreases as N is increased
below T0, while it remains constant above T0.
In the present study, finite temperature AFQMC is
used to calculate Ps.11,12The number of Trotter de-
composition slices L is chosen to satisfy the condition
L ≥ 12β (β = 1/T), so that ∆τ ≤ 0.084/t is satisfied,
where ∆τ = β/L. 3000−30000 Monte Carlo sweeps, de-
pending on the temperature and system size, have been
taken to assure sufficiently small statistical errors. We
have performed calculation on system sizes from 42to
122sites.
In order to obtain T0 from the AFQMC data of Ps,
we use three methods, two of which are based on finite
size scaling, while the other one is a more straightforward
method.
C.Finite size scaling
As shown in eq.3, the pairing correlation decays as
r−ηbelow T0 for a large system size. Hence Ps is pro-
portional to Nx2−ηwith system size N = N2
size system,scaling variable Nx/ξ ,where ξ is the corre-
lation length, becomes more important.Taking this into
account,the scaling hypothesis assumes the following be-
havior for Ps.
x. In a finite
Ps(T,Nx) = N2−η
ξ = exp(A/(T − T0)1/2),
x
f(Nx/ξ)(6)
(7)
where A is a constant and f(x) is a certain scaling func-
tion. Since ξ → ∞ for T → T0, f(x) = f(0) regardless of
the system size N at T = T0. Thus, if we plot PsN−2+η
as functions of T(or β) for different system sizes, they
should coincide regardless of the system size at T = T0.10
Another method to identify T0is to plot PsN−2+η
function of Nx/ξ, where T0and A are chosen so that all
the data points fall on a single curve (namely PsN−2+η
f(Nx/ξ)) regardless of the system size.9
However, as we shall see in the following, these scaling
methods eq.6 turns out to suffer from finite size effects
when the system size is small and/or the density of states
at the Fermi level is small. In fact, in the former method
mentioned above, Psfor small system sizes do not coin-
cide with those for large system sizes even at T = T0. As
for the latter method, PsN−2+η
x
deviate from those for larger system sizes at low temper-
atures.
x
x
as a
x
=
for small system sizes
Due to this problem, here we identify T0as the tem-
perature at which Psfor the largest two system sizes co-
incide in the former method, while in the latter method,
we search the values for T0and A so that as many data
points as possible fall on a single curve, although the
data for small system sizes deviate from the curve at low
temperatures.
D.Extrapolation method
Since the scaling method suffers from finite size ef-
fects on some cases, we have adopted the third method,
namely the extrapolation method.
been adopted for the study of the (real) superconducting
transition temperature for the three dimensional attrac-
tive Hubbard model.13Since Psshould increase sharply
at T = T0 in the thermodynamic limit, it is reason-
able to assume that T0 = limN→∞T(N)
is the inflection point of Ps(T) for the N site system.
T(N)
1
can be obtained by fitting Ps(T) by an appro-
priate function. In this study we use a gaussian form
ANexp[−BN(T−CN)2]+DNas a fitting function, where
AN-DN are fitting parameters. Here, we find that T(N)
scales as 1/N in the range of N considered in the present
study. We may then obtain T0by plotting T(N)
tion of 1/N and linearly extrapolating it to the 1/N → 0
limit.
This method has
1
, where T(N)
1
1
1
as a func-
III.RESULTS
A.Finite size scaling
FIG. 1:
Dashed lines are guides to the eyes.
PsN−1.75
x
plotted against β for the square lattice.
We first present finite size scaling analysis, where we
identify the temperature at which PsN−1.75
tween different system sizes. Before going into the re-
sults for the triangular lattice, we show the result for the
square lattice with n = 0.87, where a scaling analysis has
been performed for system sizes up to 8 × 8 in ref.9. In
x
coincides be-

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FIG. 2:
lar lattice with several band filling.
0.8(a),1.0(b),1.2(c),1.4(d).
A plot similar to Fig.1 but for triangu-
Band filling n is
Fig.1, PsN−1.75
sizes from 4 × 4 to 12 × 12. We can see that the results
for the largest two system sizes, 10 × 10 and 12 × 12
merges at around β = 7. The results for smaller sys-
tems do not cross or merge with each other within the
present temperature range, but if we assume that the re-
sults for small systems are strongly affected by finite size
effects, we may adopt T0/t ≃ 1/7 ≃ 0.13 for this band
filling, which is roughly consistent with what has been
concluded (T0/t ≃ 0.1) in ref.9.
We now move on to the results for the triangular lat-
tice. In Fig.2, PsN−1.75
x
is plotted as functions of β. For
n = 1.4, the results for the largest two systems, 10 × 10
and 12 × 12 merges at around β = 6. The results for
smaller systems do not cross or merge with each other
within the present temperature range as in the case of
the square lattice, but if we again assume that the re-
sults for small systems are strongly affected by finite size
effects, we may adopt T0/t ≃ 1/6 = 0.17 for this band
filling. For n = 1.2, the results for the largest three sys-
tem sizes cross at β ≃ 5.3. Again assuming that results
for smaller system sizes are affected by finite size effects,
we may take T0/t ≃ 0.19 for this band filling. As for
n = 1.0, although the results for 12×12 and 8×8 crosses
at around β = 6, those of the largest two sizes do not
cross with each other. Therefore, T0/t cannot be esti-
mated from these results at this band filling. The situa-
tion is even worse in the case of n = 0.8, where PsN−1.75
of the largest two system sizes do not even come close at
x
is plotted as functions of β for system
x
low temperatures. Here again we cannot evaluate T0/t
for this band filling from these results.
B.Extrapolation method
FIG. 3: The AFQMC data of pairing correlation function Ps
for square lattice with n = 0.87. Dashed curves are the fitting
results.
FIG. 4: TN
n = 0.87. The dashed line is a least squares fit.
1 plotted against 1/N for the square lattice with
We now evaluate T0using an alternative method, that
is, the extrapolation method. Here again, we first show
the result for the square lattice with n = 0.87. In Fig.3,
the raw Ps data is plotted as functions of temperature
for each system size. T(N)
1
extracted from these data are
plotted against 1/N in Fig.4. The results for N = 42
turn out to be too much affected by finite size effects, so
we have omitted these data in the extrapolating process.
T0obtained by extrapolating the results to 1/N → 0 is
T0/t ≃ 0.15, which is a little bit higher than, but in fair
agreement with the value obtained by the scaling method.
We move on to the triangular lattice. In Fig.5, the
raw Ps data as well as the fitting curves are plotted as
functions of T for each system size and band filling. (We
do not show the fitting curves for n = 0.8 since T(N)
turns out to be negative.) T(N)
1
are plotted against 1/N for each band filling in Fig.6. T0
obtained by extrapolation is 0.17, 0.18, 0.17 for n = 1.4,
n = 1.2, n = 1.0, respectively. For n = 1.4 and n = 1.2,
T0obtained using the scaling method coincides fairly well
1
obtained fromthese data

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FIG. 5: A plot similar to Fig.3 but for the triangular lattice
with n = 0.8(a), 1.0(b), 1.2(c), 1.4(d). Dashed lines are fitting
results.
FIG. 6: A plot similar to Fig.4 but for the triangular lattice
with n = 1.0, 1.2, 1.4.
with the values obtained here. For n = 0.8, Psturns out
to decrease upon increasing the system size from 8×8 to
12×12, which seems to indicate that there is no symptom
of KT transition within the temperature range (> 0.1t)
investigated in the present study, so T0, if any, should
be lower than 0.1t. In Fig.7, T0obtained by the present
method is plotted against the band filling.
C.Justification by scaling
The KT transition temperature determined by the
above method can be further justified by checking
whether Ps actually scales as Ps = N1.75
x
f(Nx/ξ) with
FIG. 7:
temperature T0.
eyes.
The band filling dependence of the KT transition
Solid and dashed lines are guides to the
FIG. 8:
lattice.
PsN−1.75
x
plotted against Nx/ξ for the triangular
ξ = exp(A/(T − T0)1/2)). In Fig.8, we plot PsN−1.75
functions of Nx/ξ for all the system sizes. We find for
n = 1.0, 1.2, and 1.4 that the numerical results for 8×8,
10 × 10 and 12 × 12 fall on a single curve by adopting
the T0obtained above and choosing appropriate values of
A. The results for smaller systems also fall on the same
curves at high temperatures (namely for large Nx/ξ), but
at low temperatures they start to deviate due to finite
size effects. As for n = 0.8, if A = 0.3 and T0 = 0.05
are chosen, the results for 12 × 12 and 10 × 10 fall on a
similar curve. These results further justify the values of
T0obtained in the preceding sections. Similar results are
also obtained for the square lattice as seen in Fig.9.
x
as

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FIG. 9:
n = 0.87.
A plot similar to Fig.8 for the square lattice with
IV.DISCUSSION
A. Comparison of the methods for determining T0
Our results show that the scaling method using the
crossing point of PsN−1.75
x
is strongly affected by finite
size effects, and that it is difficult to know from the begin-
ning the system size necessary to obtain an accurate T0.
The present analysis suggests that this method seems to
work well at band fillings where the density of states at
EF is relatively large, namely at n = 1.2 and n = 1.4 in
the present case. There, the agreement with the results
of the extrapolation method is also good. This may be
because the discreteness of the energy levels due to the
finite system size is small when the density of states is
large.
B. Correlation between T0 and the density of states
In order to look into a possible correlation between T0
and the density of states at the Fermi level, we compare
the present results of T0 with the transition tempera-
ture obtained by mean field approximation (Fig.11). Al-
though the values of TMF
c
than T0, we find a similar band filling dependence,
namely, TMF
c
is almost constant for 1 < n < 1.5, and
smaller for n < 1. The origin of this similarity is not
clear, but this does seem to suggest that T0 is roughly
correlated with the density of states near the Fermi level
at least for the value of U(= −4t) adopted in the present
study. It would be an interesting future study to ana-
lyze this point from the viewpoint of the study by Timm
et al, where the relation between the mean field Tcand
the KT superconducting transition temperature in the
repulsive Hubbard model has been investigated by cal-
culating the superfluid density as a function of temper-
ature and combining it with the Berezinskii-Kosterlitz-
Thouless theory.14
T0for 1 ≤ n ≤ 1.4 is still larger than T0for the square
lattice at n = 0.87 despite the fact that the density of
themselves are much larger
FIG. 10:
gular lattices on the tight binding model.
The density of states of the square and the trian-
0
0.2
0.4
0.6
0.8
0 0.51 1.52
FIG. 11:
triangular lattice obtained by applying BCS mean field ap-
proximation.
The band filling dependence of the TMF
c
for the
states near the Fermi level for the square lattice with
n = 0.87 is larger than for the triangular lattice with
n = 1.0 (see Fig.10).
If T0 is indeed positively correlated with the density
of states near the Fermi level, this “inversion” of T0be-
tween the square and the triangular lattice may be due to
the absence of charge density wave ordering in the latter
lattice due to frustration, as discussed previously.10
V.CONCLUSIONS
In the present study, we have investigated the su-
perconducting KT transition of the attractive Hubbard
model on a two-dimensional triangular lattice using aux-
iliary field quantum Monte Carlo method for system sizes
up to 12×12 sites. Combining three methods to analyze
the numerical data for the pairing correlation function,
we find that the transition temperature stays almost con-
stant within the band filling range of 1.0 ≤ n ≤ 1.4, while
it is found to be much lower in the n < 1 region. Among
the three methods, the extrapolation method is found to
work well regardless of the band filling, while the methods
relying on finite size scaling require some caution when
the density of states near the Fermi level is small and the
calculation is restricted to small system sizes.