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arXiv:0809.1423v2 [cond-mat.str-el] 6 Feb 2009
Effects of Strong Correlations and Disorder in d-Wave Sup erconductors
Marcos Rigol,
1, 2
B. Sriram Shastry,
1
and Stephan Haas
3
1
Department of Physics, University of California, Santa Cruz, C alifornia 95064, USA
2
Department of Physics, Georgetown University, Washington, District of Columbia 20057, USA
3
Department of Physics and Astronomy, University of Southern California, Los Angeles, California 90089, USA
(Dated: February 6, 2009)
We use exact diagonalization techniq ues to study the interplay between strong correlations, su-
perconductivity, and disorder in a model system. We study an extension of the t-J model by adding
an infinite-range d-wave superconductivity inducing term and disorder. Our work shows that in t he
clean case t he magnitude of the order parameter is surprisingly small for low-hole filling, thus im-
plying that mean-field theories might be least accurate in that important regime. We demonstrate
that substantial disorder is required to destroy a d-wave superconducting state for low-hole doping.
We provide the first bias free numerical results for th e local density of states of a strongly correlated
d-wave superconducting model, relevant for STM measurements at various fillings and disorders.
PACS numbers: 75.10.Jm,05.50.+q,05.70.-a
The combination of strong correlations and reduced
dimensionality makes the theoretical under standing of
high-T
c
supe rconductivity very difficult and consensus
on its origin has not been reached.
1,2,3
Very recently, ex-
periments using local probes, such as scanning tunneling
sp e c troscopy (STS) and scanning tunneling microscopy
(STM), have shown that the doped cuprates a re highly
inhomogeneous
4
(for a recent review, see Ref. 5). Specific
aspects of high-T
c
supe rconductivity, such a s the robust-
ness of the tunneling spectrum with respect to disorder ,
6
emphasize its contrast with more conventional disorder
sensitive
7
BCS-type superconductivity.
In this work, we probe the effects o f strong cor relations
and disorder in d-wave superconductor s derived fr om a
Mott insulator. We introduce and study a generalized
model derived from the t- J model
8
in which a super c on-
ducting (SC) ground state is argued to be inevitable. We
consider a Hamiltonian H = H
tJ
+ H
d
+ H
random
, with
H
tJ
= −t
X
hi,ji σ
h
˜c
†
iσ
˜c
jσ
+ H.c.
i
+J
X
hi,ji
S
i
· S
j
−
1
4
n
i
n
j
,
(1)
where the sum hi, ji runs over nearest-neighbor sites, σ =
(↑, ↓), a nd standard definitions for the projected creation
and annihilation operators a re employed.
8
The t-J model is microscopically justified from either
the one-band Hubbard model by a large U expansion or
more g e nerally fro m a reduction of the three-band cop-
per oxide model to a single-band model,
9
with a greater
freedom for the parameter ratio J/|t|. While a mean-field
theory (MFT) for the doped t-J model
10,11
gives a d-wave
SC ground state, it is an uncontrolled approximation. In
the MFT, there are often states with other broken sym-
metries in the proximity of the superconductor that can
be missed. Most importantly, in view of the very strong
constraint o f single occupancy in the model, we expect
significant quantum fluctuations, and the MFT cannot
handle these precisely. Therefore it is not clear that the
t-J model does have a SC ground state for the ranges of
parameters studied. In order to precipitate a SC starting
state, we add an attractive ter m
H
d
= −
λ
d
L
L
X
i,j=1
D
†
i
D
j
(2)
where D
i
= (∆
i,i+ˆx
− ∆
i,i+ˆy
), ∆
ij
= ˜c
i↑
˜c
j↓
+ ˜c
j↑
˜c
i↓
, and
L is the number of lattice sites. This is an infinite-range
term of the type that BCS considered in their reduced
Hamiltonian,
12
while building in the d-wave sy mmetry
of SC order. We have also considered imposing an ex-
tended s-wave symmetry, where the results are quali-
tatively quite different and will be repo rted elsewhere.
Within MFT, this model leads to the sa me d-wave state
as found from the t-J model.
10,11
Our model is presum-
ably a superconductor for any λ
d
∼ O(1) in the thermo-
dynamic limit, a nd for sufficiently large λ
d
, for a ny rea-
sonable finite cluster. No tice that we have sidestepped
the issue of the “mechanism” of superconductivity, which
cannot be settled with studies of the kind undertaken
here and focus instead on the natur e of the state so pro-
duced. We argue below that despite the infinite-ranged
nature of H
d
, strong correlations produce a non-mean-
field-like state; this state has an unexpectedly small order
parameter (OP).
Finally, we consider a quenched r andom disor der term
of the form
H
random
=
X
i
ε
i
n
i
(3)
where the ε
i
’s are taken randomly from a uniform distri-
bution between [−Γ, Γ]. The full Hamiltonian Eqs. (1)-
(3) thus describes an inhomogeneous strongly corr e lated
supe rconductor. In our study, we use numerical diago-
nalization of clusters with 18 and 20 sites. The dimension
of the largest Hilbert s pace diagonalized here is ∼ 10
8
.
In our model, we are interested in understanding how
the evolution into the SC state occurs, as λ
d
is turned
on. Towards this end, we show in Fig. 1 the derivative
of the energy (main panels) and the energy itself (in-
sets) as λ
d
is increased, for different fillings of the 18 and
2
-3
-2.5
-2
-1.5
-1
-0.5
0
dE/dλ
d
L=20, N=4
L=20, N=6
0 0.2 0.4
0.6
0.8 1
λ
d
-4
-3.5
-3
-2.5
-2
-1.5
-1
dE/dλ
d
L=18, N=14
L=18, N=16
L=20, N=16
L=20, N=18
0 0.2 0.4
0.6
0.8 1
-20
-18
-16
-14
-12
-10
E
0 0.2 0.4
0.6
0.8 1
-16
-14
-12
-10
E
-3
-2.5
-2
-1.5
-1
-0.5
0
dE/dλ
d
L=20, N=8
L=20, N=10
(a)
(b)
FIG. 1: (color online). Energy (insets) and its derivative
(main panels) vs λ
d
. (a) Low density of particles (less corre-
lated) and (b) low density of holes (strongly correlated). N
is the number of electrons, and J = 0.3. All energies are in
units of t.
20 site clusters. For low fillings of electrons [Fig. 1(a)],
we find that for certain cases of the numbe r of particles,
level crossings occur, as signaled by a jump in the en-
ergy derivative, indicating a change of the symmetry of
the ground sta te.
13
A similar jump is seen for all fi llings
between N = 4 and N = 10 in the 18-site cluster [not
shown in Fig. 1(a)]. On the other hand, we find that for
low-hole fillings in the 18 and 20 site clusters, the energy
derivative is continuous with λ
d
, suggesting a particu-
lar compatibility b e tween the d-wave order and the t-J
model. It is interesting that there is evidence of this com-
patibility from high-temperature expansions
14
and exact
diagonaliza tion studies,
15
which are also unbiased such
as the present one.
One of our diagnostic tools for studying the nature
of the SC state is the d-wave pair density matrix P
ij
=
hD
†
i
D
j
i.
16
Its largest (Λ
1
) and next largest (Λ
2
) eigenval-
ues are computed and their ratio R(≥ 1) is monitored.
This ratio is a n effective prob e of the order, both for
clean and disordered sup erconductors. Tak ing the ratio
eliminates uninteresting normalization effects re lated to
the change in the particle density, etc. This procedure,
for example, eliminates the ex pected diminishing of all
the eigenva lues of P
ij
as the hole doping decreases (due
to Gutzwiller cor relations). It is thus constructed as a
pure number. For a SC ground state, it is expected to
scale for large L like R ∼ Ψ
2
L + Φ , where Ψ is the di-
mensionless O(1) OP (Ref. 16) and Φ ∼ O(1) represents
the depletion (i.e., spillover) from the condensate. This
0
10
20
30
R=Λ
1
/Λ
2
L=20, N=4
L=20, N=6
0 0.2 0.4
0.6
0.8 1
λ
d
1
1.2
1.4
1.6
1.8
2
R=Λ
1
/Λ
2
L=18, N=14
L=18, N=16
0 1 2 3 4
1
10
L=20, N=6 (Short)
L=20, N=6 (Long)
0 1 2 3 4
1
1.5
2
L=20, N=18 (Short)
L=20, N=18 (Long)
0
10
20
30
L=20, N=8
L=20, N=10
0 0.2 0.4
0.6
0.8 1
λ
d
1
1.2
1.4
1.6
1.8
2
R=Λ
1
/Λ
2
L=20, N=16
L=20, N=18
(a)
(b)
FIG. 2: (color online). Ratio R = Λ
1
/Λ
2
with increasing
λ
d
.(a) Low density of particles (less correlated) and (b) low
density of holes (strongly correlated). In the insets, we com-
pare the infinite-range model Eq . (2)] with a short-range ver-
sion of Eq. (2) [taking i = j and dropping the L in the de-
nominator] up to larger values of λ
d
. N is th e number of
electrons, and J = 0.3.
depletion occurs due to repulsive interactions, i.e., strong
correla tio ns. In the parallel case of a Bose system with
N
b
bosons,
17
we expect Λ
1
∼ O(N
b
) while Λ
2
∼ O(1).
In Fig. 2, we show R = Λ
1
/Λ
2
for different fillings of
the 18 and 20 lattices as a function of λ
d
. By comparing
the low electron filling [Fig. 2(a)] to the low hole doping
case [Fig. 2(b)], one can gauge the effects of correlations
for overdoped (a), and optimal or underdoped cuprates
(b). For the lowe st electron densities (N = 4), R reaches
very large values (R > 30) and decreases as the density
is increased, up to a round R ∼ 6 for ten electrons. We
notice that sometimes for low electron filling one needs
λ
d
to exceed a critical value before R starts increasing ,
e.g., N = 6 in Fig. 2(a). This is a s ignature of a quantum
phase transition into a SC state, and oc c urs in many but
not all instances. The transition point coincides with the
jump seen in the derivative of the energy in Fig. 1. Taken
together, these confirm that the new ground state has a
different symmetry than the ground state of the plain
vanilla t- J model.
On the opposite end, for low-hole doping [two and four
holes in 18 and 20 sites in Fig. 2(b)], one can see that R
increases continuously with λ
d
, i.e., no a brupt transition
occurs. Figure 2(b) also shows that in that regime R in-
creases very slowly with λ
d
and does not exceed R = 2 for
λ
d
≤ 1. We can interpret this as a small value of Ψ and
a large value of Φ as defined above, implying a large de-
3
0
10
20
30
40
L=20, N=4
L=20, N=6
0
0.5
1
1.5
2
Γ
1
1.2
1.4
1.6
1.8
2
R=Λ
1
/Λ
2
L=18, N=14
L=18, N=16
0.5
1
1.5
2
Γ
0.5
1
1.5
2
2.5
3
Λ
1
, Λ
2
0.5
1
1.5
2
Γ
1.5
2
2.5
Λ
1
, Λ
2
0
10
20
30
40
R=Λ
1
/Λ
2
L=20, N=8
L=20, N=10
0
0.5
1
1.5
2
Γ
1
1.2
1.4
1.6
1.8
2
R=Λ
1
/Λ
2
L=20, N=16
L=20, N=18
(a)
(b)
FIG. 3: (color online). Ratio R = Λ
1
/Λ
2
with increasing
disorder (Γ). (a) Low density of particles (less correlated)
and (b) low density of holes (strongly correlated). In the
insets, we show the separate values of Λ
1
(upper plots) and
Λ
2
(lower plots), corresponding to the results shown in the
main panels. Here J = 0.3 and λ
d
= 1. These results were
obtained averaging over ten different disorder realizations.
pletion of the condensate. This shows, that even in our
infinite-range model, the SC OP at low doping is very
strongly depleted. However by no means should we un-
derstand that superconductivity is weaker in that regime.
From studies of pur e ly bosonic systems, it is known that
due to strong corr elations, the superfluid (SC) fraction
(i.e., ρ
s
) can be much larger than the condensate frac-
tion (i.e., the OP Ψ) (see, e.g., Ref. 18). Correlations
also tend to make superfluidity (superconductivity) more
stable against perturbations.
We now compare our results for the infinite-range
model of Eq. (2) to those produced by the more s tan-
dard short-range case [Eq. (2), for i = j and no normal-
ization by L in the denominator ] used in the literature
dealing with the Hubbard model. The inse ts in Fig. 2
show that while in the infinite-range model R saturates
with increasing λ
d
, in the short-range model R attains a
maximum value for λ
d
∼ 1 and then decreases towards
unity. The latter occurs because for λ
d
≫ 1 one produces
localized pairs, i.e., there is no long-range coherence.
Turning to diso rder, in the ma in panel in Fig. 3, we
show how R evolves with increasing diso rder for λ
d
= 1.
(Those results were obtained averaging over ten different
disorder realizations.) Here one can see that disorder
produces a very large reduction of Λ
1
/Λ
2
for low electron
filling, i.e., disorder ha s a very large impact on the SC OP.
For the case of low-hole density, the relative reduction of
Λ
1
/Λ
2
is much smaller, i.e., as λ
d
had a small effect in
increasing R, so is Γ having a smaller effect in reducing
it.
From the main panels in Fig. 3 we see that the effect
of disorder in the SC state is always to make R decrease.
It is of co nsiderable interests to unders tand what hap-
pens to Λ
1
and Λ
2
separately as Γ is increased. Results
for these quantities are presented in the insets in Fig.
3. There one can see that Λ
1
behaves qualitatively very
differently between low-electron fillings and low-hole fill-
ings. In the first case Λ
1
exhibits a very large reduction,
which points towards the destruction of superc onductiv-
ity. On the other hand, for low-hole doping , Λ
1
is almost
unaffected by the increase of disorder and can even be
enhanced, as shown for 14 particles in 18 sites. Unex-
pectedly, the reduction of R in this case is related to an
increase of Λ
2
. This increases points towards a slower
decay of P
ij
when disorder is increased. This suggests
the possibility of an emerging algebraic long- range order,
producing a different signature in the density matrix than
the case of standard LRO. For example, in the 2D XY
model below T
c
, or in the 1D Heisenberg antiferromag-
netic ground state, there is no true LRO, but several of
the largest density matrix eigenvalues scale as L
η
, with
η < 1. The system sizes we trea t here are too small
to make definitive statements. However, it is interesting
to note that for low-hole doping the behavior is quali-
tatively different from the low electron filling, in which
the largest eigenvalue ex hibits a large decrease with in-
creasing disorder. An analysis of the data for the s-wave
supe rconductor studied in Ref. 19 exhibits e xactly the
latter behavior, in contrast to the one we se e for the SC
t-J model in the low-hole doping regime. Our results
therefore suggest unusual power-law type superconduc-
tivity in the presence of disorder close to half filling.
In order to make connection with experimentally mea -
surable STM curves, we show in Fig. 4 the local density
of states of the L = 20 site cluster for two different fillings
in the prese nce of dis order and λ
d
. Figures 4(a) and 4(d)
correspond to fillings where the ground state of the plain
t-J model (λ
d
= 0) is a diabatically connected to the SC
ground state at finite λ
d
. In the presence of disorder, the
density of states is similar to the one reported pre viously
for the translational invariant t-J model with L = 16
sites.
20
These curves display a striking asymmetry be-
tween adding a particle and taking out a particle, and
the evolution of this asymmetry with doping is similar to
that of the clean t-J model.
Adding the SC ter m (λ
d
> 0) to the disorder ed sys-
tem opens a gap. This can be clear ly seen in Figs. 4(b)
and 4(e). Our system sizes are too small to see the V
shape expected for a d-wave superconductor, i.e., we see
a real gap. As disorder is incr e ased, Figs. 4(c) and 4(f)
show the reduction of the gap. From the results shown in
Figs. 4, we see that the SC gap c loses only for a substan-
tial disorder (Γ & 2λ
d
). Our calculations therefore a lso
sheds light on this aspec t of the STM spectra, namely,
the robustness against disorder.
4
0
0.1
0.2
0.3
N(ω)
0
0.1
0.2
0.3
N(ω)
-8 -6 -4 -2 0 2 4 6 8
ω
0
0.1
0.2
N(ω)
0
0.1
0.2
N(ω)
0
0.1
0.2
0.3
N(ω)
-10 -8 -6 -4 -2 0 2 4 6
ω
0
0.1
0.2
N(ω)
(a)
(b)
(d)
(e)
L=20, N=8
L=20, N=16
(c)
(f)
Γ=1.0, λ
d
=0.0
Γ=1.0, λ
d
=1.0
Γ=2.0, λ
d
=1.0
FIG. 4: (color online). Averaged density of states [N(ω)]
for two different fillings (N = 8 and N = 16) of the 20 site
cluster. We show results for: Γ = 1, λ
d
= 0 (a) and (d);
Γ = 1, λ
d
= 1 (b) and (e); and Γ = 2, λ
d
= 1 (c) and (f).
N(ω) was computed as the average over different lattice sites
and over two different disorder realizations.
In conclusion, we have presented and studied a vari-
ant of the t-J model, with an infinite-range d-wave su-
perconducting term. We have shown how the energy, its
derivative, and the d-wave superconducting order param-
eter evolve with increas ing the strength of the supercon-
ducting term. In addition to discontinuities in all the
above quantities for low electron densities, we find a se-
vere reduction of the magnitude of the order parameter
at low-hole filling. This is a signature of strong quan-
tum fluctuations near the Mott insulator. In relation
to current STM experiments, we find that superconduc-
tivity s urvives considerable disorder close to half filling.
The local density-of-states curves yield bias free (i.e., non
variational) results for a strongly co rrelated d-wave su-
perconductor in the presence of disorder and provide a
picture of the large energy scale structure of this imp or-
tant object.
Acknowledgments
We acknowledge support from NSF under Contract
NO. DMR-0706128 and DOE-BES under Contract NO.
DE-FG02-06ER46319. We thank M. A. P. Fisher, G.
H. Gweon, A. Pasupathy, M. Randeria, J. A. Riera, and
R. T. Scalettar for helpful discussions. Computational
facilities were provided by HPCC-USC center.
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state are the same; broken symmetry emerges only in the
thermodynamic limit. We therefore generically expect con-
tinuous dE/dλ
d
curves, rather than discontinuous curves
that level crossings imply. For d-wave symmetry, we found
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