JUST, Vol. IV, No. 1, 2016
Trent University
Effects of disorder and charge density wave order
on the local density of states in cuprate
superconductors
Steven K. Ufkes
Abstract
Structural disorder is known to affect the manifestation of charge density waves (CDW) in high-Tc cuprates, but
the precise relationship between CDW order and defects is unclear. A recent NMR experiment on YBa2 Cu3 O6.56
revealed an asymmetric shift in the distribution of the local density of states (LDOS) at the chemical potential
emerging alongside long-range CDW order, thought to be caused by defects[1]. Here we investigate the
relationship between defects and CDW order, and their respective effects on the LDOS. We model a single CuO2
plane using a one-band, tight-binding model. We apply the Hartree-Fock approximation and obtain conduction
electron eigenstates self-consistently. We find that charge modulation occurs along lattice diagonals and exhibits
local dx2 −y2 symmetry. We find that weak disorder results in distinct short and long-range CDW phases at
high and low temperature, respectively. In strongly disordered systems, a gradual increase in CDW order with
decreasing temperature occurs over a large temperature range. We show that the emergence of CDW order is
accompanied by the opening of a gap in the density of states (DOS) at the chemical potential. We find that both
disorder and CDW order individually result in a skewing of the LDOS distribution toward low LDOS, suggesting
that the effect observed in NMR might be due to a combination of these factors.
Keywords
Superconductors — Density of States — Condensed Matter Physics — Computational Physics
Lady Eaton College
1. Introduction
A large part of the condensed matter community is currently
focused on a family of high-temperature superconductors
known as the high-Tc cuprates. Besides superconductivity, a
variety of exotic phases, which compete for expression under
different conditions, have been observed in these materials
[1, 2, 3].
In the recently discovered charge density wave (CDW)
state, which emerges well above the critical temperature for
superconductivity (Tc ) and moderate levels of hole-doping,
conduction electron density is periodically modulated across
the CuO2 planes found in high-Tc cuprates [1, 2, 3]. These
waves are static and exhibit local dx2 −y2 symmetry (henceforth
d symmetry): at Cu sites where charge density is modulated,
conduction electron density tends to shift from adjacent O
atoms in the ±y directions and onto adjacent O atoms in the
±x directions (or vice versa; see Fig. 1)[1, 2, 3].
While this novel electronic phase is interesting in its own
right, furious desire to understand it largely derives from the
fact that the phase is known to compete with superconductivity
[2].
Various experiments have shown that the manifestation
of charge density waves is strongly affected by disorder in
the crystalline structure of these systems [1, 4, 5]. Disorder
can be introduced in practice via chemical doping (e.g. by
addition of O atoms in YBa2 Cu3 O6+x ), which introduces randomly distributed, point-like defects—local variations in the
potential felt by electrons [1, 5, 6]. Moreover, such disorder
is a natural feature in virtually all crystalline systems.
It is well-established that the LDOS near the chemical
potential, N(r, µ), is enhanced in the vicinity of point-like defects [1]. This is relevant, because the LDOS is intimately tied
to the CDW state: spatial modulation of conduction electron
density necessitates spatial modulation of the density of states
which these electron occupy.
Broadly, we seek to understand the relationship between
CDW order, structural disorder, and the distribution of the
LDOS at the chemical potential. Specifically, we are concerned with the effects that randomly-distributed, point-like
defects have on the LDOS. We aim to provide a theoretical
explanation for the findings of a recent nuclear magnetic resonance (NMR) experiment, performed on YBa2 Cu3 O6+x below
the critical temperature for superconductivity [1]. This experiment showed that when superconductivity is suppressed using
a strong magnetic field, a qualitative change in the LDOS
emerges alongside a long-range CDW [1]. The change is
characterized by a spatially inhomogeneous enhancement of
the LDOS on O atoms in the CuO2 planes. In particular, the
distribution of the LDOS becomes asymmetric: a relatively
Effects of disorder and charge density wave order on the local density of states in cuprate superconductors — 2/10
Figure 1. Schematic of a charge density wave in a small section of a CuO2 plane. Conduction electron density (blue) is modulated between Cu
atoms.
small number of O sites have especially large LDOS (see
Fig. 2 and Fig. 3).
Figure 2. O17 NMR spectrum (intensity versus frequency) of
YBa2 Cu3 O6.56 at T = 3 K and magnetic field strength H ≈ 28.5 T from
[1]. Oxygen sites O(2) and O(3) lie in CuO2 planes. The NMR
spectrum represents a histogram of Knight shift values, K. The K
values depend on the LDOS. Thus, a plot of number of O sites versus
LDOS should have the same features as the curves above (i.e. an
asymmetric distribution skewed toward low LDOS).[1]
The simultaneous emergence of these two phenomena is
interesting, because CDW order and the LDOS are inextricably linked, yet the nature of their relationship is unclear in this
case. The change in the LDOS is thought not to be a direct
imprint of the CDW state, because the effect of CDW order on
oxygen NMR spectra is already known to be a line-splitting
(different than the effect observed here). Rather, the authors
attribute the change in the LDOS to electron scattering off
point-like defects lying outside the CuO2 planes [1].
In this work, we model a single CuO2 plane using a oneband, tight-binding model. We use the Hartree-Fock approximation and compute electron eigenstates self-consistently.
First, we establish that our model predicts a CDW phase with
primarily local d symmetry emerging at low temperatures.
For weakly disordered systems, the transition to the CDW
phase occurs abruptly below a well-defined temperature; for
Figure 3. Asymmetry of O(2) and O(3) lines, and splitting of O(2)
lines in the O17 NMR spectrum of YBa2 Cu3 O6.56 versus temperature,
from [1]. Oxygen sites O(2) and O(3) lie in CuO2 planes. Splitting of
the O(2) line is a well-established signal of CDW order. Thus, the data
demonstrates that the emergence of CDW order is accompanied by
an asymmetric shift in the NMR spectrum, which can be interpreted
as a similar asymmetric shift in the distribution of the LDOS at the
chemical potential.[1]
strongly disordered systems, the transition to the CDW phase
is gradual, and small CDW order persists at high temperatures.
We show that the emergence of CDW order is accompanied by
the opening of a gap in the spatially averaged density of states,
N(E), at the chemical potential. Second, we show that as
temperature decreases and CDW order emerges, a qualitative
change in the LDOS centred on Cu atoms occurs, similar to
the change observed in the LDOS at O sites described above.
We show that, in the absence of CDW order, the introduction
of strong disorder causes a change in the LDOS distribution
similar to that related to the emergence of CDW order. We
compare changes in the LDOS distribution resulting from
CDW order and disorder, respectively. We show that the
asymmetry of the LDOS distribution predicted by our model
does not vary with CDW order in the well-defined manner
observed in NMR (see Fig. 3)[1].
Explaining features of the LDOS theoretically is important not only by virtue of its intimate relationship with charge
density, but also due to the fact that there are numerous NMR
Effects of disorder and charge density wave order on the local density of states in cuprate superconductors — 3/10
spectroscopy and scanning tunnelling microscopy (STM) experiments probing the LDOS in the high-Tc cuprates.
1.1 Model and method of solution
We model a square section of a single CuO2 plane using a
tight-binding approach, treating the Cu atoms as sites on a
two-dimensional lattice with periodic boundary conditions.
We calculate the conduction electron eigenstates using a oneband model, in which the basis states are centred on Cu atoms.
The Hamiltonian consists of a noninteracting part, Ĥ0 , and an
interacting part, Ĥ1 . The noninteracting part is
Ĥ0 = ∑ ∑ ti j ĉ†iσ ĉ jσ ,
(1)
i, j σ
and the interacting part is
Ĥ1 =
Jbo
∑0 ĉ†iσ ĉiσ ĉ†jσ 0 ĉ jσ 0 .
2 ∑
i, j σ ,σ
(2)
Here, i, j label lattice sites, σ denotes spin, ĉ†iσ , ĉ jσ are fermion
creation and annihilation operators, respectively, and Jbo is a
constant. In the noninteracting part, we consider 1st-, 2nd-,
and 3rd-nearest lattice neighbours. The matrix elements are

t0 i = j




 t1 i, j are nearest neighbours
t2 i, j are 2nd-nearest neighbours
ti j =
(3)


t
i,
j
are
3rd-nearest
neighbours

3


0 otherwise
We use the Hartree-Fock approximation to simplify the interacting part
Ĥ1,HF =
∑ ∑ Pi j ĉ†iσ ĉ jσ .
(4)
hi, ji σ
Here, the first sum is taken over all nearest-neighbour pairs,
denoted hi, ji. The local bond density matrix elements are
defined
Pi j = −Jbo hc†jσ ciσ i,
(5)
and are assumed to be spin-independent. The noninteracting
part of the Hamiltonian accounts for conduction electron kinetic energy; the Coulomb potential due to ionic cores; and
the Coulomb potential due to the other conduction electrons,
which are assumed to create a static potential field. We choose
matrix elements t0 = 0, t1 = 1, t2 = −0.32 and t3 = 0.16, as
in [7]. The interacting part of the Hamiltonian accounts for
the “exchange interaction” which results from consideration
of the exclusion principle. We choose Jbo = 1.8, which sets
the strength of the exchange interaction we are considering.
The choices t1 = 1 and kB = 1 establish energy and temperature scales for our model. In the real system, t1 ∼ 100
meV, so a temperature T = 1 in our model corresponds to
roughly 1000 K.
If sites i, j are nearest neighbours, the matrix element Pi j
represents the “bond density” between them. We use bond
density here as an analogy to the conduction electron density
on O atoms between neighbouring Cu sites—a quantity which
cannot be found directly in our model. We infer features of
charge modulation on O sites using the local bond density
matrix. The temperature-dependence of the effective Hamiltonian is entirely contained in Pi j , so we compare solutions
obtained at different temperatures under the assumption that
any differences between them are related to CDW order.
We define the CDW order parameter, Pd , to be the rootmean-squared value of the d-symmetric component of the
local bond density matrix over all lattice sites (see Appendix
A). The CDW order parameter gives a measure of the degree to
which the system as a whole exhibits CDW order. We consider
similarly defined CDW order parameters for modulation with
local s, px , and py symmetry. Throughout this paper, when
the type of symmetry is not specified in reference to CDW
order, d symmetry is implied.
To simulate doping-induced disorder, we add a random
value in the interval [−w/2, w/2] to each diagonal element
t0 of the noninteracting part of the Hamiltonian, where the
constant w describes the “disorder strength”. When w is small
compared to t1 = 1, we refer to it as weak disorder; when
w is comparable to or larger than t1 , we refer to it as strong
disorder. The diagonal matrix elements t0 represent the energy
of a Cu-centred basis state in isolation. Thus, the effect of
disorder in our model is to shift the basis state energies up or
down in a random fashion.
The matrix elements Pi j are computed self-consistently:
we guess Pi j initially, diagonalize the Hamiltonian (Ĥ0 +
Ĥ1,HF ), use the eigenvalues to make an improved guess for
Pi j , diagonalize the Hamiltonian again, and repeat the process
until the solution converges. We define a converged solution
as one for which
1 (`)
(`−1) 2
Pi j − Pi j
<ε
∑
N i, j
(6)
where N is the number of lattice sites, ` is the iteration number,
and we choose ε = 10−9 .
1.2 Comparison of local density of states distributions
We define the local density of states at lattice position ri and
energy E using a Lorentzian function
N(ri , E) =
|vi,m |2 γ
1
∑
π m (E − Em )2 + γ 2
(7)
where vi,m is the component at lattice site i of the mth electron
eigenstate with energy Em , and we choose the parameter γ =
0.02.
Throughout this paper, we use the term “LDOS distribution” to refer to a plot of the probability density of observing
a particular value of N(r, µ) at any of the lattice sites versus
N(r, µ), where µ is the chemical potential.
Effects of disorder and charge density wave order on the local density of states in cuprate superconductors — 4/10
We measure the asymmetry of the LDOS distributions
using a method similar to that in [1]. First, we find the skewedGaussian-of-best-fit using a least-squares optimization routine.
A skewed Gaussian is defined as
f (x; x0 , a, b, c)
c(x − x0 )
a
−(x−x0 )2 /2b2
√
1 + erf
= √ e
b 2π
2b
(8)
where x0 , a, b, and c are parameters. We define the asymmetry
parameter A of a LDOS distribution in terms of the widths
at half-maximum of the left and right side of the skewedGaussian-of-best-fit, ∆xl and ∆xr
A=
∆xr − ∆xl
∆xr + ∆xl
(9)
A large, positive A describes a LDOS distribution which is
skewed toward low LDOS values; a large, negative A describes
a LDOS distribution which is skewed toward high LDOS
values. See Fig. 7 for examples of skewed Gaussians.
To compare the effects of CDW order and structural disorder on the LDOS distribution, we consider distributions of the
logarithm of the LDOS (i.e. probability density of observing
a particular value of N(r, µ) at any of the lattice sites versus log(N(r, µ))). For each log distribution, we determine
the (standard) Gaussian-of-best-fit. We then compare the
goodness-of-fit (as measured by χ 2 ) of the respective Gaussians between distributions.
2. Results
2.1 Characterization of the charge density wave
state
We first present a typical solution for a system with weak
disorder and low temperature in the CDW state (Fig. 4). An
illustration of the local bond density matrix in real space
(Fig. 4 (a)) reveals a regular pattern of charge modulation. At
a given lattice site, charge density between nearest-neighbours
in the ±y directions tends to shift to the space between nearest
neighbours in the ±x directions (or vice versa). The degree
to which this shift occurs, in turn, oscillates across the lattice
diagonal. To quantify this oscillation, we calculate the degree
of d-symmetric charge modulation, D(ri ), at each site (Eq. 18).
The Fourier transform of D(ri ) across the lattice (Fig. 4(b))
reveals that the degree of d-symmetric charge modulation
oscillates along the lattice diagonal at a well-defined wave
vector Q ≈ ± π2 x̂ ± π2 ŷ.
We examine the local symmetry of charge modulation
(as measured by order parameters defined in the Appendix)
in systems with no disorder, weak disorder and strong disorder, over a range of temperatures (Fig. 5). For all local
symmetries and disorder strengths, CDW order increases with
decreasing temperature. For all disorder strengths, in the low
temperature region where CDW order is maximized, charge
modulation primarily exhibits local d-symmetry. For all local
Figure 4. Typical solution for 24 × 24 lattice with weak disorder
(w = 0.2) at low temperature (T = 0.05) in the CDW phase. (a) Local
bond density matrix illustrated in real space. Red and blue lines
denote bond density above and below the average, respectively; line
thickness denotes the magnitude of the difference between a matrix
element and the average. (b) Squared norm of the Fourier transform
of the d-symmetric component of charge modulation (defined in
Eq. 18), indicating the presence of a charge density wave along the
lattice diagonal (i.e. in the x̂ + ŷ direction).
symmetries and temperatures, CDW order tends to increase
with increasing disorder strength. When disorder is absent
or weak, the transition to the CDW phase occurs abruptly
below a well-defined temperature TCO ≈ 0.11. When disorder
is strong, CDW order increases gradually with decreasing
temperature over a large temperature range—there is no welldefined temperature TCO below which CDW order increases
sharply.
We examine changes in the spatially averaged density
of states, N(E), for a system with strong disorder (Fig. 6).
As temperature decreases and CDW order emerges, a large,
asymmetric gap in the DOS at the chemical potential forms.
The gap deepens as temperature decreases and CDW order
increases.
2.2 Evolution of the local density of states distribution with disorder and CDW order
We examine changes in distribution of the local density of
states at the chemical potential as disorder strength and CDW
order are varied (Fig. 7). When disorder is weak and CDW
order is negligible, the LDOS distribution is approximately
symmetric (Fig. 7(a)). When disorder is strong and CDW
order is negligible, the LDOS distribution becomes skewed
toward low LDOS, with a long “tail” extending to high LDOS
(Fig. 7(b)). When disorder is weak and CDW order is high, the
LDOS distribution is also skewed toward low LDOS, with a
short tail extending to high LDOS (Fig. 7(c)). When disorder
is strong and CDW order is high, the LDOS distribution is
skewed toward low LDOS, with a tail extending toward high
LDOS (Fig. 7(c)).
To distinguish the effects that disorder and CDW order
have on the LDOS distribution, we consider distributions of
the logarithm of the LDOS. We fit the log distributions with
(standard) Gaussians of best fit, and compare the goodness-offit (measured by χ 2 ) between systems (Fig. 8). When disorder
is strong and CDW order is negligible, the log distribution
is well-approximated by a Gaussian function (i.e. the LDOS
Effects of disorder and charge density wave order on the local density of states in cuprate superconductors — 5/10
Figure 5. CDW order parameters for 4 local symmetries as functions of temperature for a 24 × 24 lattice with (a) no disorder; (b) weak disorder,
averaged over 100 disorder configurations; and (c) strong disorder, averaged over 100 disorder configurations. Note that the temperature range
considered in (c) is larger than that in (a,b).
Figure 6. Spatially averaged density of states N(E) for a 24 × 24 lattice with strong disorder (w = 4) at (a) high temperature, with negligible CDW
order; (b) moderate temperature, with moderate CDW order; and (c) low temperature, with high CDW order. The dashed line indicates the
chemical potential. A large gap in the DOS at the chemical potential develops as CDW order emerges. Solutions were averaged over 500
disorder configurations.
distribution is approximately log-normal). When disorder is
weak and CDW order is strong, the log distribution is poorly
approximated by a Gaussian (i.e. the LDOS distribution is not
log-normal).
To quantify the relationship between the LDOS distribution and CDW order, we fit the LDOS distributions with
skewed-Gaussians of best fit, and compare the asymmetry
(Eq. 9) of these curves between systems (Fig. 9). For both
weak and strong disorder, the asymmetry of the LDOS distribution tends to be larger when temperature is low and CDW
order is high. However, there is no well-defined relationship
between CDW order (or temperature) and the asymmetry of
the LDOS distribution. The asymmetry of the LDOS distribution fluctuates unpredictably as temperature decreases and
CDW order increases.
3. Discussion
3.1 Characterization of the charge density wave
state
The solutions obtained in our one-band model for systems
with weak or absent disorder are largely in agreement with
experimental observations of the CDW phase in CuO2 planes
of the high-Tc cuprates. In systems with weak or absent disorder, our model predicts a transition to the CDW phase occuring abruptly below a well-defined transition temperature
TCO ≈ 0.11, corresponding to roughly 100K (Fig. 5). In high-
Tc cuprates with moderate levels of hole-doping, the CDW
phase is known to emerge below 100-150K[8], so our model
predicts a transition temperature of the same order observed
in experiment.
Our model predicts that the charge modulation which
occurs in systems with weak or absent disorder primarily
exhibits local d symmetry, consistent with the wealth of experimental observations in high-Tc cuprates[1, 2, 3]. Experiments
find that charge modulation in the CDW phase occurs primarily at oxygen sites[1, 2, 3]. While the basis states in our
one-band model are centred on Cu sites, preventing direct
observation of charge modulation at O sites, prediction of the
correct local symmetry of charge modulation via the local
bond density matrix, P, suggests that our model nonetheless
contains the physics necessary to describe essential features
of the CDW phase.
For systems with weak or absent disorder, our model
typically finds that charge modulation occurs along lattice
diagonals (i.e. the modulation wave vector Q typically lies in
the Brillouin zone diagonal, as in Fig. 4). While this finding is
consistent with numerous theoretical considerations[3], experiments consistently find that charge modulation occurs along
lattice axes[8] (i.e. along the x̂ or ŷ directions in our scheme).
While our model weakly suggests charge modulation along
the lattice axes for systems with certain disorder strengths
and chemical potentials, it is unclear why the model typically
Effects of disorder and charge density wave order on the local density of states in cuprate superconductors — 6/10
Figure 7. LDOS distribution for a 24 × 24 lattice under different conditions: (a) high temperature, weak disorder, negligible CDW order; (b) high
temperature, strong disorder, negligible CDW order; (c) low temperature, weak disorder, high CDW order; and (d) low temperature, strong
disorder, high CDW order. Solutions were averaged over 660 disorder configurations. The dashed line indicates the skewed-Gaussian-of-best-fit.
Both disorder and CDW order skew the distribution toward low LDOS.
Figure 8. Distribution of the logarithm of the LDOS for a 24 × 24 lattice under different conditions: (a) low temperature, weak disorder, high CDW
order; and (b) high temperature, strong disorder, negligible CDW order. Solutions were averaged over 660 disorder configurations. (a) When
disorder is weak and CDW order is high, the LDOS distribution is skewed (see Fig. 8(c)), but is poorly approximated by a log-normal distribution.
(b) When disorder is strong and CDW order is negligible, the LDOS distribution approximately log-normal.
predicts diagonal charge modulation.
Our model predicts an increase in CDW order over a large
temperature range with increasing disorder strength (Fig. 5).
When disorder strength is sufficiently weak that an abrupt
Effects of disorder and charge density wave order on the local density of states in cuprate superconductors — 7/10
Figure 9. CDW order and asymmetry of the LDOS distribution as functions of temperature for a 24 × 24 lattice with (a,c) weak disorder and (b,d)
strong disorder. Solutions were averaged over 100 disorder configurations. While the asymmetry of LDOS distribution tends to be higher in the
CDW phase, no clear relationship between CDW order and asymmetry in the LDOS distribution is observed.
transition to the CDW state is discernible (as in Fig. 5(b)), we
can distinguish two regions of temperature over which CDW
order is observed: one at low temperature (e.g. T < 0.11
in Fig. 5(b)), in which a long-range CDW is observed at a
well-defined modulation wave vector Q; and one a higher
temperature, in which short-range charge modulation occurs,
and in which CDW order decreases slowly with increasing
temperature. That such a high-temperature CDW region is
not observed when disorder is absent (as in Fig. 5(a)) suggests
that disorder induces short-range charge modulations. Experiments have firmly established the presence of short-range
charge modulations occuring well above the critical temperature for superconductivity.[1, 2, 3]. More recent experiments
have shown that at low temperatures, when superconductivity
is supressed by applying a strong magnetic field, long-range
charge modulations occur[9]. Our results suggest that the
short-range, higher-temperature CDW phase may be induced
by disorder.
The range of d-symmetric charge modulations is related
to the width of the peak in the Fourier transform of the dsymmetric component of charge modulation (Fig. 4). The
resolution of the Fourier-transform increases with increasing
lattice size. The 24 ×24 system considered in this study yields
a Fourier transform whose resolution is not high enough to permit a careful examination of the charge modulation range via
the Fourier peak width. In future work, larger lattice sizes may
be examined in order to firmly establish the dependences of
charge modulation range on temperature and disorder strength.
The required increase in resolution would come at significant
computational cost, however, given that computation time in
our model is approximately proportional to the cube of the
number of lattice sites (i.e. ∝ N 3 ).
We find that the emergence of CDW order is accompanied by the opening of an asymmetric gap in the DOS at
the chemical potential (Fig. 6). The gap deepens as temperature decreases and CDW order increases. This gap exhibits similar features to the so-called “pseudogap” observed
experimentally[8], in that it is typically asymmetric and the
DOS has a local minimum at the chemical potential. Experimentally, the pseudogap is known to emerge above the temperature at which CDW order emerges, so the pseudogap cannot
be a result of CDW order (at least, not exclusively)[3, 6, 8].
Because we never observe a gap in the DOS at the chemical
potential when CDW order is negligible, we conclude that the
gap we observe cannot be the psuedogap, and must be directly
related to CDW order.
While it is clear that the gap in the DOS is related to
CDW order in strongly disordered systems, our results for
weakly disordered systems are unclear. In weakly disordered
systems with or without CDW order, the DOS distribution
consists of sharp peaks separated by large valleys, with an
envelope of shape similar to the DOS distribution for strongly
Effects of disorder and charge density wave order on the local density of states in cuprate superconductors — 8/10
disordered systems (as in Fig. 6). With such a DOS distribution, it is not possible to distinguish the formation of gap.
The DOS for weakly disordered systems could be “smoothed
out” by averaging solutions over boundary conditions (i.e. by
averaging the DOS of sublattices in a superlattice, where a
different phase is introduced to matrix elements linking sites
at each boundary between sublattices). It should be noted
that the disorder strength required to “smooth out” the DOS
without boundary-condition averaging (w ≈ 4, as in Fig. 6) is
unphysically large.
3.2 Evolution of the local density of states distribution with disorder and CDW order
The main result of this work is the qualitative change in the
LDOS distribution associated with disorder and the emergence
of CDW order (Fig. 7). When disorder is weak and CDW
order is negligible, the LDOS distribution is approximately
symmetric (Fig. 7(a)). When strong disorder is introduced at
high temperature, where CDW order is negligible, the LDOS
distribution becomes skewed toward low LDOS (Fig. 7(b)).
For systems with weak disorder in the CDW phase, the LDOS
distribution is again skewed toward low LDOS (Fig. 7(c)).
That disorder and CDW order appear to individually skew
the LDOS distribution toward low LDOS suggests that the
asymmetry in the LDOS distribution observed in NMR[1]
might be due to a combination of disorder and CDW order,
rather than disorder alone as the authors suggested.
While both disorder and CDW order skew the LDOS distribution toward low LDOS, our results suggests that they do
so in slightly different manners. To quantify this difference we
examine distributions of the logarithm of the LDOS (Fig. 8).
We find that when disorder is strong and CDW order is negligible, the LDOS distribution is approximately log-normal. In
contrast, when disorder is weak and CDW order is high, the
LDOS distribution is poorly approximated by a log-normal
distribution. A well-established feature of Anderson localization, in which electron wavefunctions become localized due to
strong disorder, is a log-normal distribution of the LDOS[10].
Thus, the skewing of the LDOS distribution in the presence
of strong disorder in our model may be due to localization
of the electron wave functions. Whether such localization
occurs as a result of disorder could be investigated easily in
our model by examining the spatial extent of the individual
electron eigenstates.
While our model suggests that the LDOS distribution becomes skewed in a manner similar to that observed in NMR[1],
we do not observe a simple relationship between the asymmetry of the LDOS distribution and CDW order as presented
in Fig. 3. While we find that the asymmetry of the LDOS
distribution tends to be higher in temperature regions where
CDW order is high (Fig. 9), the asymmetry of the LDOS
distribution fluctuates considerably in regions of both high
and low CDW order. It should also be noted that we do not
observe a line-splitting in the LDOS distribution, which is a
characteristic of the CDW phase[1].
There are numerous factors which may contribute to the
poorly defined relationship between CDW order and the asymmetry of the LDOS distribution in our model (Fig. 9). Most
importantly, the asymmetry observed in NMR[1] was that of
the LDOS distribution at oxygen sites in the CuO2 planes.
Moreover, the dominant feature of the CDW state observed
in high-Tc cuprates is charge modulation at O sites[1, 2, 3].
The basis states in our model are centred Cu sites, so the distributions considered here are those of the LDOS centred on
Cu sites. While changes in the LDOS at O sites in the real
system may well be reflected in the LDOS centred on Cu sites
in our model, it is likely that any such changes would be obscured by our restriction to Cu-centred basis states. In future
work, we may consider a three-band model with some basis
states centred on O sites, which would permit examination
of changes in the LDOS (and charge modulation in general)
directly at O sites.
A second potential factor in the poorly defined relationship between CDW order and the asymmetry of the LDOS
distribution is that the skewed-Gaussians-of-best-fit do not
approximate the LDOS distributions well in all circumstances
(e.g. Fig. 7 (c,d)). However, the left and right peak widths of
the skewed-Gaussians do appear to reflect the left and right
peak widths of the LDOS distributions which they fit, in most
circumstances. Thus, it is unlikely that this poorness-of-fit is
large factor in the poor relationship between asymmetry of
the LDOS distribution and CDW order in our model.
3.3 Finite-size effects and CDW commensuration
There are various issues associated with the smallness of the
lattice and the use of periodic boundary conditions in our
model. If the lattice size in one direction is not approximately
divisible by the period of charge modulation in that direction
(i.e. if the CDW is incommensurate with the lattice), the CDW
will destructively interfere with itself. Such destructive selfinterference reduces CDW order in the system, and would not
occur in the bulk in the real system. Because the period of
charge modulation partly depends on the mean occupation
of the electron eigenstates (i.e. the “filling” of eigenstates),
we mitigate such destructive self-interference by tuning the
filling such that the CDW is commensurate with the lattice. It
is feasible that the period of charge modulation also depends
on disorder strength. Thus, when two systems with equal
filling but different disorder strengths are compared, it is
unclear whether differences in CDW order is partly due to
differences in (unphysical) destructive self-interference of the
CDW between systems.
In a similar vein, because we use periodic boundary conditions, individual electron eigenstates intefere destructively
with themselves. This destructive interference is more pronounced at certain wave vectors (and energies) than others.
The result is that the DOS is diminished at certain energies.
This effect is apparent in the high-energy region of the density
of states for strongly disordered systems (Fig. 6). The effect
diminishes with increasing disorder strength, which is why
Effects of disorder and charge density wave order on the local density of states in cuprate superconductors — 9/10
the weakly disordered DOS distribution gives a series of sharp
peaks separated by large valleys—the valleys correspond to
energy regions over which a high degree of destructive selfinterference occurs.
Appendix: Form Factors and CDW Order
Parameters
We decompose the local bond density matrix into “formfactors”, which describe, as a function of position, the degree
of s, px , py or d-symmetric charge modulation, respectively.
Assume the lattice sites are separated by unit length in the x
and y directions. Let ri be the position of the ith lattice site.
We treat the matrix element Pi j as a function of lattice site
positions ri , r j , and decompose it as
magnitudes of the form-factor functions
!1/2
1 N
2
Ps =
∑ |S(ri )|
N j=1
!1/2
1 N
2
Ppx =
∑ |Px (ri )|
N j=1
!1/2
1 N
2
Ppy =
∑ |Py (ri )|
N j=1
!1/2
1 N
2
Pd =
∑ |D(ri )|
N j=1
(19)
(20)
(21)
(22)
Throughout this paper, when the type of symmetry is not
specified in reference to CDW order, d symmetry is implied.
Pi j = S(ri )As (r j − ri ) + Px (ri )A px (r j − ri )
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where
As (r j − ri ) =
A px (r j − ri ) =
A py (r j − ri ) =





1/2 r j − ri = ±x̂, ±ŷ
0
otherwise
√
1/ √
2
r j − ri = x̂
−1/ 2 r j − ri = −x̂
0
otherwise
√
2
r j − ri = ŷ
1/ √
−1/ 2 r j − ri = −ŷ
0
otherwise


r j − ri = ±x̂
 1/2
−1/2 r j − ri = ±ŷ
Ad (r j − ri ) =

0
otherwise
(11)
(12)
(13)
N
S(ri ) =
∑ As (r j − ri )Pi j
(15)
j=1
N
Px (ri ) =
∑ A px (r j − ri )Pi j
(16)
j=1
N
Py (ri ) =
∑ A py (r j − ri )Pi j
(17)
j=1
N
D(ri ) =
∑ Ad (r j − ri )Pi j
(18)
j=1
where N is the number of lattice sites.
We define CDW order parameters, which describe the
degree of s, px , py and d-symmetric CDW order in the system,
as the root-mean-squared values over all lattice sites of the
Effects of disorder and charge density wave order on the local density of states in cuprate superconductors — 10/10
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