Finite Drude weight for 1D low temperature conductors
Finite Drude weight for 1D low temperature conductors
Dariush Heidarian and Sandro Sorella
Istituto Nazionale di Fisica della Materia (INFM)-Democritos, National Simulation Centre,
and Scuola Internazionale Superiore di Studi Avanzati (SISSA), I-34014 Trieste, Italy
We apply well established finite temperature QuantumMonte Carlo techniques to one dimensional
Bose systems with soft and hardcore constraint, as well as to spinless fermion systems. We give
clear and robust numerical evidence that, as expected, no superfluid density for Bosons or Meissner
fraction for fermions. is possible at any non zero temperature in one dimensional interacting Bose
or fermi lattice models, whereas a finite Drude weight is generally observed in gapless systems, in
partial disagreement to previous expectations.
PACS numbers: 74.25.Fy,71.27.+a,71.10.Fd
I. INTRODUCTION
In the last decades there have been a lot of numerical
and theoretical works to understand the role of strong
correlation in lattice model Hamiltonians.1,2,3,4,5,6,7 Re-
cently this issue has acquired an increasing attention and
remarkable importance, due to the recent advances in the
realization of optical lattices. In these experiments ultra-
cold atoms behave as boson particles trapped on particu-
lar lattice sites, whereas the interaction and the hopping
parameters can be tuned continuously. This important
achievement has opened the possibility to verify directly
the crucial role played by the electron correlation in very
important model Hamiltonians defined on a lattice. An
important example is the realization of a Mott insulating
state in a system with strong on site repulsion8,9. More-
over quite recently the possibility to include the Fermi
statistics in optical lattices appears very promising and
interesting.10
In 1D spinless fermion systems are equivalent to inter-
acting Bose systems with hard-core constraint and are
described by the same low energy theory -the Luttinger
liquid theory-. Indeed this theory holds also for soft-
core bosons, as shown in Ref.(7). Therefore, as far as
the transport properties are concerned one should expect
the same behavior both for fermions and bosons. On the
other hand for lattice models, even in absence of disor-
der, the current does not commute with the Hamiltonian,
implying its possible decay at finite temperature due to
the backscattering processes11. In this case the dynam-
ical current-current correlation function also decays in
time, leading to a current Fourier transform without δ
function at zero energy, namely without a finite Drude
weight within the linear response theory.
Until few decades ago the absence of the Drude weight
was the expected behavior of all interacting metals in lat-
tice models or in real solids at finite temperature. How-
ever a quite clear numerical evidence has been reported
in Ref.12 that current should not decay in integrable 1D
models, namely for Hamiltonians that can be solved by
Bethe ansatz techniques in 1D. These models essentially
possess some hidden conservation law, that was conjec-
tured to forbid the current decay process.12,13 Later sev-
eral groups have reproduced this surprising effect14,15,
with a noticeable exception that a finite Drude weight
at finite temperature was found also for non-integrable
models.15 On the other hand, from purely theoretical
grounds this issue is not settled yet: in Ref.11 it was ar-
gued that backscattering processes can be effective also
at finite temperature and in 1D non integrable models,
whereas in Ref.16, it was proposed that also some par-
ticular non integrable model could provide a conserved
current.
In this work we propose that the general behavior of
1D gapless systems is eventually characterized by a finite
Drude weight at finite temperature, and we have found
no exception in the models that we have studied. This
conclusion is based on a careful and systematic numerical
work on fairly generic one dimensional Bose and Fermi
systems, that all show the same behavior, even though
strong finite size effects are observed in the non integrable
cases.
In the following we investigate the behavior of the
Drude weight in 1D systems in the thermodynamic limit
and finite temperature.
Model and Method : We have studied hardcore and
softcore bosons in a 1D lattice with periodic boundary
conditions. The Hamiltonian studied reads,
iai+1 + h.c.) +
ni(ni − 1)
+V nini+1 +Wnini+2 − µni
The sum is over all lattice sites i, a
i/ai is the boson
creation/annihilation operator at site i, henceforth ni is
the particle number at site i andµ is the chemical poten-
tial. t is the hopping amplitude which is set to one, U is
the on-site repulsion, whereas V and W are the nearest
and the next-nearest neighbor interactions, respectively.
For hardcore bosons in the U → ∞ limit the Hamilto-
nian can be mapped onto an S = 1/2 spin system with
Szi = ni − 1/2 and S
i = a
i . In this work we present
our results for the half filled case of hardcore and soft-
core models. Most of our results have been obtained by
Quantum Monte Carlo (QMC), using the stochastic se-
ries expansion (SSE)5,17 with the directed loop update18.
http://arxiv.org/abs/0704.0406v1
Superfluid density ρs (or spin stiffness in the equiva-
lent spin model), is defined as the second derivative of
the free energy with respect to a twist in the boundary
conditions. In order to compute this quantity by QMC,
it is convenient to apply linear response theory, relating
this quantity to the current current response function
Λ(q, iωn) =
dτ exp(iωnτ)〈J(q, τ)J(−q, 0)〉/N , where
J is the current operator and ωn is Matsubara frequency.
Then the following expression for the superfluid density
is obtained:
ρs = 〈−K〉 − Λ(q = 0; iωn = 0) =
〈W 2〉
where 〈K〉 is the average kinetic energy per site, ωn =
2πn/β are the Matsubara frequencies andW is the wind-
ing number. Similarly the Drude weight is obtained with
the same expression but with a different order in the limit
ω → 0 and q → 0, namely15,19,20
D = 〈−K〉 − ReΛ(q = 0, ω → 0). (3)
In SSE one can obtain Λ very accurately in terms of
Matsubara frequencies. Therefore analytic continuation
of the data is required. In order to avoid difficulties of
extrapolation to iωn → 0 at large temperatures, we have
worked at relatively low temperatures (β ≥ 10).
In principle, due to the different order of lim-
its, the Drude weight and the superfluid density may
be different when the following quantity remains fi-
nite in the thermodynamic limit15: D − ρs =
En=Em
β exp(−βEn)|〈ψn|J |ψm〉|
2/L, where, J is the
current operator, while En and |ψn〉 are the n
th eigen-
value and eigenstate of the many body system, respec-
tively.
The current operator can be written as J(q = 0) =
) where H+
al+1 and b is the bond
index, corresponding to the site index l. The ensemble
average of product of two local operatorsHσ1
andHσ2
(τ)Hσ1
(0)〉 =
n,m=0
(τ − β)n(−τm)
〈ψk|H
HmHσ1
|ψk〉 (4)
where τ is the imaginary time, Z is partition function
and the summation over n and m comes from Taylor-
expansion of e(−β+τ)H and e−τH . Following Ref.17 the
relation (4) can be simplified to
(β − τ)ns−m−2τm
(ns − 1)!
(ns −m− 2)!m!
N b1b2,σ1σ2m
where ns is the length of sequence of the local operators
and it changes in each QMC sampling. N b1b2,σ1σ2m is the
number of times that two operatorsHσ1
and Hσ2
appear
in this sequence with distance of m local operators, and
〈...〉W indicates an arithmetic average using configura-
tions with relative weight W . In this work we introduce
an efficient way to sample by SSE the current-current
response function. To this end, we multiply expression
(5) by eiωnτ and integrate over the imaginary time τ , we
obtain:
1F1(m+ 1, ns; 2iπn)N
b1b2σ1σ2
where
1F1(m+ 1, n; z) =
(n− 1)!
(n−m− 2)!m!
dx exp(zx)xm(1− x)n−m−2 (7)
is the confluent hypergeometric function.
Therefore, the current-current correlation acquires
contributions determined by length of operator string
ns. All these contributions are stochastically sampled
in an efficient way, and in each statistical measurement
the correlation function Λ(q = 0, iωn) has the following
estimator:
σ1,σ2=±
1F 1(m+ 1, ns; 2iπn)N
m (8)
where Nσ1σ2m =
b1,b2
N b1b2,σ1σ2m .
Discussion: At zero temperature, for non degenerate
ground state, the Drude weight and the superfluidity are
the same. In a 1D system at any finite temperature ρs is
expected to be zero in the thermodynamic limit, whereas
the Drude weight can be non-zero. For hardcore and soft-
core bosons in a 1D lattice, a systematic size scaling of
the superfluid density ρs clearly shows that this quantity
vanishes in the thermodynamic limit and for any finite
temperature (see figures 1 and 2). Further, we find that,
for a fixed set of parameters and at half filling, all super-
fluidity data versus 1/L collapse to one curve whenever
the x-axis is appropriately scaled with the temperature
T (see figures 1 and 2). This analysis suggests the scal-
ing form ρs(β, L) ≡ ρs(β/L). If one takes the order of
limit T → 0 after L → ∞, superfluidity remains zero
even at zero temperature. Notice that by taking first the
limit T → 0 and then L → ∞ superfluidity has a finite
value for the gapless phase, but this is not a signature of
superfluidity, rather the occurrence of a finite zero tem-
perature Drude weight. Though in 1D is not possible to
have a finite superfluid density at any non zero tempera-
ture, several authors have identified the finite zero tem-
perature Drude weight with the superfluid density for a
superfluid with vanishing critical temperature. We be-
lieve that this identification is a bit confusing and there-
fore we prefer to think about absence of superfluidity and
superconductivity in 1D systems, as commonly reported
in the textbooks.
Fig. 3 shows the current-current correlation versus ωn
in the metallic and insulating phases of an integrable
FIG. 1: (color online) Superfluid stiffness for an integrable (a)
and a non-integrable (b) model versus β/L. The system size
L is ranging from 50 to 1200.
FIG. 2: (color online) Superfluidity of the soft-core bosons
versus scaled system size at half filling, the on-site interaction
is U = 4
model (W = 0, U = ∞). The zero-frequency value is
the superfluid density ρs and the limit ωn → 0 gives the
Drude weight D. For W = 0 at zero temperature, there
exists a critical value Vc/t = 2 below which the Drude
weight is finite. In the first case (a) shown in Fig.(3)
with V/t = 2 the Drude weight has a finite value at any
finite temperature, which is consistent with the previous
works12. In the insulating phase (case b) with V/t = 3,
the superfluid density coincides with the Drude weight
and they both tend to zero as the system size increases.
In a non-integrable model such as hard-core bosons
with nearest and next nearest neighborer interactions
earlier works have suggested zero Drude weight as system
size increases. With SSE we can go to very large system
sizes and low temperatures and check the scaling depen-
dence of the Drude weight. In Fig. 4 we have plotted
current-current correlation versus Matsubara frequency
for different L, and a fixed temperature T = 1/100. As
shown in the same Figure (4) we have also found a finite
Drude weight at finite T in the celebrated Bose-Hubbard
model with softcore constraint and in several other mod-
FIG. 3: (color online) (a) Current-current correlation for an
integrable model in the metallic phase. The zero frequency
data shows superfluidity while the extrapolation to n → 0 is
the Drude weight. D remains finite with increasing L while
ρs vanishes. (b) In the insulating phase D and ρs have the
same value and both tend to zero by increasing L.
FIG. 4: (color online) Response function vs. n for (a) hard-
core bosons with V/t = 1.5, W/t = 1, T/t = 1/100 and (b)
Bose-Hubbard model with softcore constraint and U/t = 2,
µ/t = −0.4, T/t = 1/25. The system sizes ranges from
L = 100 to L = 800.
els (not shown). Although some evidence that few par-
ticular non integrable models could have a finite Drude
weight at finite temperature have been reported before,
here we have found a very convincing evidence that this
behavior should be generic for 1D gapless system regard-
less from their integrability. We have supported this
statement by state of the art numerical calculations ob-
tained for very large system sizes and low temperature so
that all possible extrapolations are perfectly under con-
trol.
In conclusion it turns out that, at low energy, all gap-
less lattice models studied scale to the Luttinger liquid
fixed point where the backscattering is a marginally irrel-
evant coupling and the current is therefore conserved at
the fixed point. This is therefore a peculiar and generic
feature of 1D. Indeed in 2D systems, such as hardcore
bosons with n.n. repulsion in a square and triangular
lattice, we found no difference between ρs and D.
Acknowledgments
We thank M. Troyer for useful discussions. This work
is partially supported by COFIN-2005 and CNR.
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