# Effective Luttinger parameter and Kane-Fisher effect in quasiperiodic
systems

T.J. Vongkovit  Department of Electrical and Computer Engineering, Princeton
University, Princeton, NJ 08544, USA  Hansveer Singh  Department of Physics,
University of Massachusetts, Amherst, Massachusetts 01033, USA  Romain Vasseur
Department of Physics, University of Massachusetts, Amherst, Massachusetts
01033, USA  Sarang Gopalakrishnan  Department of Electrical and Computer
Engineering, Princeton University, Princeton, NJ 08544, USA

###### Abstract

The ground states of interacting one-dimensional metals are generically
Luttinger liquids. Luttinger liquid theory is usually considered for
translation invariant systems. The Luttinger liquid description remains valid
for weak quasiperiodic modulations; however, as the quasiperiodic modulation
gets increasingly strong, it is increasingly renormalized and eventually
fails, as the system becomes localized. We explore how quasiperiodic
modulation renormalizes the Luttinger parameter characterizing this emergent
Luttinger liquid, using the renormalization of transmission coefficients
across a barrier as a proxy that remains valid for general quasiperiodic
modulation. We find, unexpectedly, that quasiperiodic modulation weakens the
effects of short-range interactions, but enhances those of long-range
interactions. We support the former finding with matrix-product numerics. We
also discuss how interactions affect the localization phase boundary.

The nature of ground-state correlations in interacting disordered systems has
been a question of longstanding interest Imada  _et al._ (1998); Sanchez-
Palencia and Lewenstein (2010); Vojta (2019); Kapitulnik  _et al._ (2019). In
one dimension, this question has been addressed from two complementary
perspectives: perturbatively adding disorder to clean interacting systems
Giamarchi and Schulz (1988), and perturbing the disordered system with weak
interactions using the strong disorder renormalization group (SDRG) Altman
_et al._ (2008). These approaches agree in finding two phases—a localized
insulating phase and a Luttinger-liquid phase—but give contrasting predictions
for the critical behavior between them (see also Refs. Pollet  _et al._ (2009,
2014); Doggen  _et al._ (2017)). In many present-day experiments, the spatial
modulations are quasiperiodic rather than random: both in cold-atom
experiments using incommensurate potentials Roati  _et al._ (2008); Schreiber
_et al._ (2015); Sbroscia  _et al._ (2020); Lüschen  _et al._ (2016, 2018); An
_et al._ (2021) and more recently in Moiré materials Mak and Shan (2022); Fu
_et al._ (2020). Quasiperiodic systems exhibit many of the same phenomena as
random ones—e.g., Anderson localization—but their properties are fundamentally
different because of the deterministic, hyperuniform structure of
quasiperiodic patterns Chandran and Laumann (2017); Crowley  _et al._ (2018a,
b, 2019). Quasiperiodic systems present distinctive theoretical challenges, as
many of the methods used in the random case (such as the replica trick) are no
longer useful.

In the absence of controlled analytical methods, progress on understanding the
ground-state properties of interacting quasiperiodic systems has come from a
combination of numerical Vidal  _et al._ (1999); Roux  _et al._ (2008);
Mastropietro (2015); Naldesi  _et al._ (2016); Cookmeyer  _et al._ (2020);
Agrawal  _et al._ (2020); Schuster  _et al._ (2002); Kraus  _et al._ (2014);
Vu and Das Sarma (2021); Gonçalves  _et al._ (2024); Yoo  _et al._ (2020) and
perturbative approaches. As in the random case, the two complementary
perturbative approaches are to treat interactions nonperturbatively in the
Luttinger-liquid framework, but perturb in disorder Vidal  _et al._ (1999), or
to begin with the scaling theory of the noninteracting critical point, and
perturb in interactions, as in very recent work Gonçalves  _et al._ (2024).
These approaches yield distinct universality classes for the localization
transition: in the former case, the transition is Lorentz-invariant; in the
latter, the Luttinger liquid never emerges and the localization transition
remains in the noninteracting universality class. To reconcile these pictures
it is imperative to understand how Luttinger-liquid behavior emerges (or fails
to emerge) for strong quasiperiodic modulation, and how Luttinger-liquid
properties evolve with quasiperiodicity. We address these questions, focusing
on interacting spinless fermions with finite-range interactions. (The spinful
case was recently explored Chi  _et al._ (2024).)

Increasing the quasiperiodic potential strength causes the single-particle
bands to get increasingly flat Hiramoto and Kohmoto (1989). As flat bands are
associated with enhanced interactions, so one might expect a quasiperiodic
potential to strengthen interaction effects. Surprisingly, this is not the
case in general; instead, the _range_ of the interactions plays a crucial
part. For short-range interactions, the quasiperiodic potential suppresses
interactions and make the system more free-fermion-like Gonçalves  _et al._
(2024). On the other hand, long-range interactions are enhanced by the
quasiperiodic potential. We establish these results by two distinct means.
First, we perform density matrix renormalization group (DMRG) simulations of
one-dimensional quasiperiodic spin chains White (1992); Schollwöck (2011) and
extract the Luttinger parameter, which quantifies interaction strength.
Second, we study of the renormalization of transmission through a barrier,
adapting the treatment of Yue et al. Yue  _et al._ (1994) to the quasiperiodic
case. The renormalization of the transmission coefficient gives us a way of
extracting an effective Luttinger parameter from Hartree-Fock studies.

_Model_.—We will focus on a single-band model of spinless fermions,
interacting with finite-range interactions, governed by the Hamiltonian

$H=\sum_{i}(c^{\dagger}_{i}c_{i+1}+\mathrm{h.c.}+\lambda\cos(\varphi(i+\theta))n_{i})+\sum_{ij}V_{ij}n_{i}n_{j}.$
(1)

where $\varphi$ is the Golden ratio, $c_{i}$ annihilates a fermion on site $i$
and $n_{i}\equiv c^{\dagger}_{i}c_{i}$, and the overall energy scale has been
set to unity. We briefly review the properties of this model in the
noninteracting case $V_{ij}=0$. This is the familiar Aubry-André (AA) model.
When $\lambda<2$ the single-particle eigenstates of this model are delocalized
and ballistic; for $\lambda>2$ all single-particle states are localized.
Unusually, the localization transition occurs at the same value of $\lambda$
for all states in the spectrum. At the critical value $\lambda=2$, not only
are all the eigenstates localized, but the _spectrum_ is highly unusual: all
the single-particle eigenvalues are in a fractal, measure-zero set, so the
bands are anomalously flat Wilkinson (1984); Kohmoto  _et al._ (1987). For
$\lambda<2$ as one approaches the critical point, the spectrum consists of
many increasingly flat minibands. The conventional wisdom is that these flat
minibands will tend to make the system highly susceptible to interactions.

We consider two choices for the interactions $V_{ij}$. First, for our exact
numerical studies, we take the interactions to be nearest-neighbor, i.e.,
$V_{ij}=\Delta\delta_{i,j\pm 1}$. The nearest-neighbor model is appealing for
numerical studies because it can be mapped via a Jordan-Wigner transformation
to a canonical nearest-neighbor spin model, the anisotropic Heisenberg model.
This allows for efficient DMRG simulations, and also allows us to compare our
numerical results in the clean limit with exact results in the literature.
However, as we will see, the range of the interaction plays an important part
in the physics of the model. To allow us to tune this range $d$, we will also
consider interactions that decay with a Gaussian envelope:

$V_{\rm int}(|i-j|)=\frac{v_{\rm int}}{d\sqrt{\pi}}\exp{[-(i-j)^{2}/d^{2}]}.$
(2)

Long-range interactions can be naturally incorporated in our Hartree-Fock
analysis. They are possible Zaletel  _et al._ (2015) but potentially
challenging to treat in DMRG. We note that the Hartree-Fock method has other
advantages, as it allows us to address aspects of the dynamics (and
disentangle distinct physical effects) in a way that would not be possible
with DMRG.

When $\lambda<2$, at a typical filling the Fermi level lies inside a miniband.
One can treat sufficiently weak interactions by linearizing the single-
particle spectrum about the Fermi level and treating the interactions
projected onto a single miniband. This yields a Luttinger liquid, in which the
strength of correlations is set by the Luttinger parameter $K$ (defined
below). Linearizing becomes an increasingly inaccurate approximation as one
approaches $\lambda=2$: any finite interaction strength mixes many minibands.
Nevertheless, we find that the effect of the other minibands manifests itself
as a finite renormalization of the Luttinger parameter, without changing the
asymptotic nature of correlations.

One can understand our main results intuitively by thinking about the behavior
of interactions projected onto a single miniband. The well-defined minibands
at some $\lambda<2$ are essentially the Bloch bands of a periodic approximant
with denominator $q$ to the true incommensurate potential. As $q$ grows, the
minibands split, so the Bloch states of each miniband are made up of
increasingly well separated ($\sim q$) Wannier states. For any finite
interaction range $d$, eventually $q\gg d$, so the interactions projected onto
the miniband are contact-like. A contact interaction has no effect on
identical spinless fermions, so the projected problem becomes noninteracting.
This accounts for the surprising robustness of the flat minibands of the
Aubry-André model to interactions.

_DMRG results_.—We begin by extracting the Luttinger parameter directly from
DMRG calculations on the ground state of the quasiperiodically modulated
problem at half-filling. Our primary diagnostic for the Luttinger parameter is
the variance of the charge in a half-system, i.e., the behavior of the
following connected correlation function for a subsystem of length $\ell$:

$F(\ell)=\left\langle\left(\sum_{i=1}^{\ell}n_{i}\right)^{2}\right\rangle-\left\langle\left(\sum_{i=1}^{\ell}n_{i}\right)\right\rangle^{2},$
(3)

where expectation values are taken in the ground state. In a Luttinger liquid
with open boundary conditions in a system of size $L$, one expects
$F(\ell)=\frac{K}{2\pi^{2}}\log(\frac{L}{\pi}\sin(\frac{\pi\ell}{L}))$, where
$K$ is the Luttinger parameter Giamarchi (2003).

Figure 1: Upper panel: Luttinger parameter $K$ extracted from subsystem charge
fluctuations, as a function of the interaction strength $\Delta$, for various
values of the quasiperiodic potential strength $\lambda$. Note that as
$\lambda$ is increased the curves get increasingly flat, at the noninteracting
value $K=1$. Lower panel: dependence of charge fluctuations on subsystem size
for interactions $\Delta=-0.5$ and various values of $\lambda$. DMRG
simulations were done with $L=200$ and a maximum bond dimension of 200. Data
is averaged over 20 phase realizations.

The results for the charge fluctuations are shown in Fig. 1. The charge
fluctuations are logarithmic in $\ell$, as expected for a Luttinger liquid.
Extracting $K$ from the coefficient of the logarithm, we find that $K\to
1$—the noninteracting value—as $\lambda$ is increased at fixed interaction
strength $\Delta$. As one might expect, the onset of the clear logarithmic
scaling regime is pushed out to larger $\ell$ as we increase $\lambda$. In sup
(a) we also consider a separate diagnostic, namely the two-point correlator of
the “spins” related to the fermions of Eq. (1) by a Jordan-Wigner
transformation. This diagnostic is _a priori_ unrelated to the charge
fluctuations, but in a Luttinger liquid, these two observables are tightly
related. Consistent estimates of $K$ from these two approaches are a
consistency check that the system is a Luttinger liquid.

These numerical observations lead to the conclusions that (1) the system is a
Luttinger liquid throughout the delocalized phase of the Aubry-André model,
and (2) the Luttinger parameter is renormalized toward its noninteracting
value as the quasiperiodic potential is increased. These observations are
consistent with the qualitative picture presented in the preceding section.

_Kane-Fisher effect: self-consistent approach_.—The previous section presented
numerically exact calculations for the case of nearest-neighbor interactions.
We now turn to an approximate but versatile approach that sheds light on the
mechanism for Luttinger-parameter suppression, and also lets us extend our
analysis to models with tunable-range interactions. In this section we
introduce the basic idea behind the method for clean systems; the next section
applies the method to quasiperiodic systems.

The basic tool we will use to diagnose correlations is the Kane-Fisher effect
Kane and Fisher (1992)—i.e., the renormalization of tunneling across a
localized impurity due to interactions. At zero temperature and at incident
energies $E$ close to the Fermi energy $E_{F}$, the renormalized transmission
coefficient across the impurity scales as $T(E)\sim|E-E_{F}|^{2(K^{-1}-1)}$.
Thus, tunneling is parametrically suppressed (enhanced) for repulsive
(attractive) interactions. Although the Kane-Fisher effect is normally studied
using bosonization techniques that do not generalize to our setting, there is
an alternative weak-coupling formulation due to Yue et al. Yue  _et al._
(1994) that is more versatile. In this treatment, one begins with the
observation that a static impurity in a Fermi gas creates Friedel oscillations
in the density profile of the surrounding gas. These Friedel oscillations
renormalize the effective scattering shift experienced by electrons farther
away from the impurity—thus further renormalizing the Friedel oscillations,
and so forth. Solving self-consistently for the tunneling across the impurity
one recovers the Kane-Fisher effect.

To explore the effect of quasiperiodic modulations on the Kane-Fisher effect,
we first self-consistently solve for the ground state in the presence of the
impurity, in the Hartree-Fock approximation, and then numerically solve for
tunneling across the impurity. It might seem surprising that a mean-field
approximation like Hartree-Fock can capture a correlation-dominated quantity
like the Luttinger parameter. Intuitively, the reason why this is possible is
that the way the electrons avoid the impurity in the mean-field treatment is
essentially analogous to the way they avoid each other due to correlations. In
any case, we note that the predicted power-law divergences are clearly seen in
the self-consistent Hartree-Fock numerical solution (Fig. 2).

Before turning to our numerical results, we briefly summarize the main results
of Ref. Yue  _et al._ (1994). This work linearizes the dispersion around the
Fermi surface, projecting to states with energies within $D_{0}=v_{F}/d$ of
$E_{F}$. Solving self-consistently for the potential sup (a) gives the result
for the transmission coefficient Yue  _et al._ (1994)

${T}(\epsilon)=\frac{{T}_{0}|\epsilon/D_{0}|^{2\alpha}}{{R}_{0}+{T}_{0}|\epsilon/D_{0}|^{2\alpha}},$
(4)

where $\epsilon=E-E_{F}$ and ${T}_{0}=|t_{0}|^{2}$ and ${R}_{0}=1-{T}_{0}$ are
the bare transmission and reflection coefficients. The dimensionless parameter
$\alpha$ is defined by Yue  _et al._ (1994)

$\displaystyle\alpha=\alpha_{\mathrm{exchange}}-\alpha_{\mathrm{Hartree}}$
$\displaystyle\alpha_{\mathrm{exchange}}=\frac{\tilde{V}(0)}{2\pi
v_{F}},\quad\alpha_{\mathrm{Hartree}}=\frac{\tilde{V}(2k_{F})}{2\pi v_{F}}.$
(5)

Here $\alpha=K^{-1}-1$. In the continuum limit, the exchange and Hartree terms
have opposite effects on the transmission coefficient. In the case of a
repulsive (attractive) interaction, the exchange term decreases (increases)
the transmission coefficient compared to its bare value, while the Hartree
term increases (decreases) it. For zero-range interactions, the exchange and
Hartree terms cancel exactly, as they must because contact interactions have
no effect on spinless fermions.

Figure 2: Data on transmission coefficients in the clean system, computed for
system size $N=2501$ and filling fraction $f=0.1$. Left: Transmission
coefficient as a function of energy measured from the Fermi level, for
$v_{\mathrm{int}}=\pm 0.6$ and without interactions. Middle: same data but on
a log-log plot, showing the nonanalytic behavior of the transmission
coefficient near $E_{F}$. Right: Exponent $\tilde{\alpha}$ (dots) compared
with the continuum analytic prediction $\alpha$ [Eq. (5)], as a function of
interaction strength, for various choices of the interaction range.
Discrepancies at larger interaction strength are expected as lattice
dispersion effects become significant .

We now describe our numerical method for calculating transmission coefficients
on the lattice, focusing first on the case without a quasiperiodic potential.
We will work with Eq. (1) with a finite-range Gaussian interaction Eq. (2). We
perform a mean-field Hartree-Fock decoupling of the interaction term:
$V_{\mathrm{int}}(i,j)n_{i}n_{j}\approx
V_{H}(i)c_{i}^{\dagger}c_{i}+\sum_{i\neq j}V_{\rm
ex}(i,j)c_{i}^{\dagger}c_{j}$, where $V_{H}(i)=\sum_{j\neq i}V_{\rm
int}(|i-j|)\braket{c_{j}^{\dagger}c_{j}}$ and $V_{\rm ex}(i,j)=-V_{\rm
int}(|i-j|)\braket{c_{j}^{\dagger}c_{i}}$.

To compute the transmission coefficient we proceed as follows. First, we solve
the Hartree-Fock problem fully self-consistently for the ground state. Next,
we treat the dynamics of wavepacket transmission under this self-consistent
quadratic Hamiltonian. In the clean case, the transmission coefficient can be
computed using scattering theory. Once the quasiperiodic potential is turned
on, we do not have a simple expression for the asymptotic states. Therefore,
to calculate the transmission coefficient, we construct a wavepacket localized
in real space to one side of the barrier, and sharply localized in energy
space at some energy $E$ displaced by $\epsilon$ from $E_{F}$. We allow the
wavepacket to evolve in time and scatter off the barrier, defining the
transmission coefficient $\tilde{T}(\epsilon)$ as the modulus squared of the
portion of the wavepacket that passes through the barrier to the other side.

Our results for the clean case are summarized in Fig. 2. To minimize lattice
effects we work at a filling $f=0.1$. The left panel shows the transmission
coefficient as a function of energy (measured with reference to $E_{F}$),
showing a dip (peak) for repulsive (attractive) interactions. Note that in the
noninteracting case the transmission coefficient is smooth across $E_{F}$. The
middle panel of Fig. 2 shows that the transmission diverges/vanishes as a
power law, consistent with Eq. (4). From this power-law dependence one can
extract a numerical exponent $\tilde{\alpha}$. This is shown in the right
panel of Fig. 2. In the thermodynamic limit, for weak enough interactions and
low enough filling, we expect that the numerical value $\tilde{\alpha}$ should
approach the continuum analytic value of $\alpha$ (5). Our data are consistent
with this. For stronger interactions, it is no longer quantitatively accurate
to linearize the lattice dispersion around the Fermi surface, so
$\tilde{\alpha}$ deviates from $\alpha$. (Of course, the Hartree-Fock
framework also ceases to be quantitatively accurate.)

_Kane-Fisher effect in quasiperiodic systems_.— Having checked that our self-
consistent method reproduces the Kane-Fisher effect in clean systems, we turn
to quasiperiodic systems. Quasiperiodic systems have nontrivial spatial
dependence of the self-consistent Hartree and exchange potentials even in the
absence of an impurity. These effects renormalize the band structure and also
shift the localization transition. These renormalization effects were
previously studied in Refs. Mastropietro (2015); An  _et al._ (2021); Štrkalj
_et al._ (2021); Wang  _et al._ (2022), and generally give rise to a mobility
edge Wang  _et al._ (2022); Biddle  _et al._ (2009). Intuitively, attractive
interactions enhance the localization of low-energy states via self-trapping,
while repulsive interactions suppress localization by screening the
quasiperiodic potential. These observations are broadly consistent with our
numerical study of the inverse participation ratio (IPR) as a function of
disorder and energy (Fig. 3).

Figure 3: Localization transition as a function of interactions, for energies
near the Fermi level (marked as a red line), for attractive, noninteracting,
and repulsive cases, with interaction range $d=4$. The inverse participation
ratio (IPR) of a normalized single-particle state with real-space amplitudes
$\psi_{i}$ on a lattice is defined as $\sum\nolimits_{i}|\psi_{i}|^{4}$.
Strictly speaking, localization is diagnosed by how the IPR scales with system
size, but at this large size $N=2001$ a large IPR is a good proxy for
localization. The localization transition clearly drifts to smaller $\lambda$
in the attractive case; the trend in the repulsive case is ambiguous. Data is
averaged over 100 phase realizations.

We now explore the Kane-Fisher effect in the regime where states near the
Fermi energy are clearly delocalized. Following previous studies of impurity
models in quasiperiodic potentials Wu  _et al._ (2019, 2022), we work at a
filling factor $f=0.31$ at which the Fermi level lies well inside a miniband
throughout the delocalized phase. We proceed exactly as above to compute the
transmission coefficient, and extract a power law from its singular behavior
near $E_{F}$. One subtlety that arises in the quasiperiodic case is that the
tunneling depends strongly on the phase of the quasiperiodic potential. In
order to isolate the effect of interactions on the transmission coefficient at
any given combination of $\lambda$ and $\theta$ and to facilitate the
observation of the ways in which attractive and repulsive interactions
renormalize the bare transmission coefficient, we tune the barrier strength to
fix the bare transmission coefficient for the wavepacket at the Fermi surface
to the arbitrary value of $0.5\pm 0.01$.

Our results are shown in Fig. 4. We find that for a short-ranged interaction
with $d=1$, increasing $\lambda$ pushes $\tilde{\alpha}$ toward zero. Long-
ranged interactions have the opposite effect, with $\alpha$ flowing away from
zero as we increase $\lambda$. Past $\lambda\approx 1.7$, the transmission
coefficient no longer behaves as a power-law in $|E-E_{F}|$ in the energy
range we are able to resolve for the available system sizes of size $\lesssim
3000$ sites. This behavior is illustrated in sup (b).

Figure 4: Exponents $\tilde{\alpha}$ characterizing the Kane-Fisher effect in
systems with increasing quasiperiodic potential strength $\lambda$. Data is
computed for system size $N=2501$, filling fraction $f=0.31$, $v_{int}=\pm
0.6$, and averaged over 100 phase realizations. Solid lines: $\tilde{\alpha}$
from repulsive interactions; dashed lines: $\tilde{\alpha}$ from attractive
interactions. Inset: power-law relationship between $\epsilon$ and
$1/\tilde{T}(\epsilon)-1$ for system with $\lambda=1.5$ and interaction range
$d=1$.

To understand the origin of these effects, it is instructive to separate out
the effects due to the Hartree and exchange interaction terms. To do this, we
first solve for the self-consistent potential including both terms. Then to
compute the Hartree (exchange) contribution to the transmission coefficient,
we evaluate wavepacket dynamics in a quadratic Hamiltonian where the exchange
(Hartree) term is artificially set to zero. From this we again extract an
exponent, which we call the Hartree (exchange) exponent (Fig. 5). The exchange
exponent behaves as one would expect, growing stronger with increasing
$\lambda$ as the bands grow flatter. The behavior of the Hartree term is more
subtle. At filling fraction $0.31$, the Bloch states are rather far from
continuum plane waves, so the Hartree term is not well described by simply
taking the Fourier transform of the interaction potential. In the clean limit,
at this filling, we find that the Hartree and exchange terms in fact have the
same sign. Upon increasing $\lambda$, the sign of the Hartree effect flips.
The strength of this term is also non-monotonic with interaction range: for
long-range interactions the term is increasingly suppressed, as one would
expect (a smoothly varying potential has no effect on scattering). However,
for short range potentials, the Hartree term becomes increasingly strong and
negative as $\lambda$ is increased. This is the origin of the suppression of
interaction effects: as remarked in the introduction, the interactions
projected onto the miniband at the Fermi level are increasingly contact-like
as the minibands become sparser, so the Hartree and Fock terms cancel as they
would for contact interactions.

Figure 5: Hartree (exchange) exponents $\tilde{\alpha}$ characterizing Hartree
(exchange) contribution to the Kane-Fisher effect in systems with increasing
quasiperiodic potential strength $\lambda$. Data is computed for system size
$N=2501$, filling fraction $f=0.31$, repulsive interaction $v_{int}=0.6$, and
averaged over 100 phase realizations. Solid lines: Hartree exponent. Dash-dot
lines: Exchange exponent.

_Discussion_.— This work explored how Luttinger-liquid behavior emerges in the
low-energy limit of interacting quasiperiodic systems. We found clear evidence
by two different methods that Luttinger-liquid behavior does emerge, albeit
with a strongly renormalized Luttinger parameter. The Luttinger parameter
exhibits a counterintuitive dependence on the strength of the quasiperiodic
potential $\lambda$: for short-range interactions, as $\lambda$ is tuned
toward the localization transition, the Luttinger parameter approaches that of
the free system, although the bands get very flat. We identified the origin of
this phenomenon as a cancellation between Hartree and Fock terms that becomes
more pronounced at larger $\lambda$. This cancellation occurs because the
characteristic spatial scale of the miniband at the Fermi level grows (with
correlation length $\xi\sim 1/|\lambda_{c}-\lambda|$) with increasing
$\lambda$.

Our conclusions are consistent with (and complementary to) those found by Ref.
Gonçalves  _et al._ (2024) exploring interaction effects at the Aubry-André
critical point: while that work focuses on the critical point itself, we show
how this regime emerges coming from the Luttinger liquid. Ref. Gonçalves  _et
al._ (2024) concluded, based on a scaling argument, that finite-range
interactions are always irrelevant at the Aubry-André critical point. This
would imply that the curves in Fig. 4 are in fact non-monotonic: very close to
the transition, we expect the effective Luttinger parameter to dip again
toward $K=1$. We do not see this numerically, and there are at least two
plausible explanations for why. The first is that the Luttinger parameter
eventually dips back down, but in a parameter regime where we are unable to
see a clean Kane-Fisher effect. A second possibility is that at finite (as
opposed to infinitesimal) interaction strengths, interactions renormalize the
localization problem sufficiently to alter the nature of the Aubry-André
transition (e.g., the fractal structure of energy levels). This second
possibility is less exotic than it sounds. If one considers a two-dimensional
phase diagram with interaction strength and quasiperiodicity as the axes, the
critical behavior found in Ref. Gonçalves  _et al._ (2024) must change to the
behavior found using Luttinger-liquid theory Roux  _et al._ (2008) somewhere
along the phase boundary, and it is natural for this location to depend on
interaction range. Exploring this question in more detail, and developing a
complete theory of this unusual critical phenomenon, is an interesting
challenge for future work.

The framework we have used also naturally extends to two-dimensional systems.
Quasiperiodic systems in higher dimensions exhibit a richer phase diagram at
the noninteracting level, with ballistic, diffusive, and localized phases
Devakul and Huse (2017); Szabó and Schneider (2020). The analog of the Kane-
Fisher effect in two-dimensions is the Altshuler-Aronov suppression of the
density of states Altshuler  _et al._ (1980), which can again be understood in
terms of renormalized Friedel oscillations. It would be interesting to explore
these effects in the quasiperiodic case.

###### Acknowledgements.

We thank David Huse and Jed Pixley for useful conversations. This work was
supported by NSF Grants No. DMR-2103938 (T.J.V., S.G.) and DMR-2104141 (R.V.).

## Supplemental Material

In this document we provide additional numerical results and details on the
following points: (1) Additional DMRG data supporting Luttinger-liquid
behavior, (2) A brief review of the derivation in Ref. Yue  _et al._ (1994) of
the continuum Kane-Fisher effect, and (3) data on how the Kane-Fisher effect
breaks down as one approaches the transition to localization.

##  Appendix A Luttinger-parameter data from correlation functions

In the main text, we showed data extracting the Luttinger parameter from the
dependence of charge variance on subsystem size. Here we present data
extracting the Luttinger parameter from a different observable, namely the
correlation function of the “spin” operator $G_{+-}(x)\equiv\langle
S^{+}(x)S^{-}(0)\rangle$, where
$S^{+}(x)=\exp\left(-i\pi\sum_{y<x}c^{\dagger}(y)c(y)\right)c^{\dagger}(x)$.
In a Luttinger liquid we expect $G_{+-}(x)\sim|x|^{-1/(2K)}$ up to oscillating
terms. This gives us an independent method for extracting the Luttinger
parameter and confirming that the system is a Luttinger liquid (i.e., the
correlations do decay as a power law). The relevant data are shown in Fig. 6.
The data extracted this way are in close agreement with those presented in the
main text, which were extracted from the charge variance.

Figure 6: Data on the correlation function $G_{+-}(r)$ with distance $r$
extracted using DMRG on a system of size $100$. Left: variation with
interaction strength $\Delta$ at fixed $\lambda=1$. Center: variation with
$\lambda$ at fixed $\Delta=-0.5$. Right: Luttinger parameter $K$ extracted
from these power laws. These results for the Luttinger parameter are
consistent with those presented in the main text, where they were extracted
from the charge variance. DMRG simulations were done with $L=200$ and a
maximum bond dimension of 200. Data is averaged over 20 phase realizations.

##  Appendix B Review of the continuum argument in Ref. Yue  _et al._ (1994)

In this section, we review the scattering of spinless interacting electrons
from a barrier $U(x)$ localized at the origin of a continuous, infinite one-
dimensional system. In the absence of interactions, the electronic
eigenfunctions are

$\phi_{k}(x)=\frac{1}{\sqrt{2\pi}}\left[\begin{cases}e^{ikx}+r_{0}e^{-ikx}&x<-a\\\
t_{0}e^{ikx}&a<x\\\ \end{cases}\right],$ (6)
$\phi_{-k}(x)=\frac{1}{\sqrt{2\pi}}\left[\begin{cases}t_{0}^{\prime}e^{-ikx}&x<-a\\\
e^{-ikx}+r_{0}^{\prime}e^{ikx}&a<x\\\ \end{cases}\right].$ (7)

$a$ denotes the spatial extent of the barrier. In the presence of an
interaction $V(|x-y|)$, one can use Hartree-Fock theory to describe the way in
which electrons are affected by the other electrons in the Fermi sea. In this
theory, each electron is subject to a Hartree potential $V_{H}(x)$ and an
exchange potential $V_{ex}(x,y)$, defined by

$V_{H}(x)=\int dyV(|x-y|)n(y),$ (8)
$V_{ex}(x,y)=-V(|x-y|)\sum_{|q|<k_{F}}\phi_{q}^{*}(y)\phi_{q}(x),$ (9)

where $n(y)=\sum_{|q|<k_{F}}\phi_{q}^{*}(y)\phi_{q}(y)$.

The Hartree-Fock equations determining the single-electron wavefunctions are

$\begin{split}[-\frac{1}{2m}\frac{d^{2}}{dx^{2}}&+U(x)+V_{H}(x)]\phi_{k}(x)\\\
&+\int dyV_{ex}(x,y)\phi_{k}(y)=E\phi_{k}(x)\end{split}$ (10)

Naively, one can calculate modified transmission amplitudes $t_{k}$ to leading
order by obtaining the Hartree and exchange potentials in the first Born
approximation, substituting them into Eq. 10, using the method of Green’s
functions to obtain the asymptotic form of the single-electron wavefunctions,
and reading off the coefficient $t_{k}$:

$\phi_{k}(x)=\frac{1}{\sqrt{2\pi}}t_{k}e^{ikx},\quad\quad x\rightarrow\infty.$
(11)

However, corrections to $t_{0}$ due to the Hartree and exchange potentials at
first order in perturbation theory contain logarithmic divergences as
$k\rightarrow k_{F}$ Yue  _et al._ (1994). In fact, these divergences are
present at any finite order in perturbation theory. One way to proceed is to
sum the most divergent terms at all orders in perturbation theory, which can
be accomplished via renormalization in energy space. Performing this
renormalization via a poor man’s scaling approach yields the results quoted in
the main text.

##  Appendix C Breakdown of Kane-Fisher effect at large $\lambda$

In the main text we presented data on the numerical exponent $\tilde{\alpha}$
characterizing the Kane-Fisher effect, in the regime of quasiperiodic
potential $\lambda$ for which this was well-defined. In Fig. 7 we provide an
example of how this structure breaks down as we increase $\lambda$ keeping the
other parameters fixed.

Figure 7: Behavior of transmission amplitudes near the localization
transition. Left: behavior of the transmission amplitude near the Fermi energy
for various $\lambda$ at $v_{\mathrm{int}}=0.6$. Right: same data but on a
log-log plot, showing the breakdown of the Kane-Fisher power law (except
presumably at inaccessibly small energies).

## References

  * Imada  _et al._ (1998) Masatoshi Imada, Atsushi Fujimori, and Yoshinori Tokura, “Metal-insulator transitions,” Rev. Mod. Phys. 70, 1039–1263 (1998).
  * Sanchez-Palencia and Lewenstein (2010) Laurent Sanchez-Palencia and Maciej Lewenstein, “Disordered quantum gases under control,” Nature Physics 6, 87–95 (2010).
  * Vojta (2019) Thomas Vojta, “Disorder in quantum many-body systems,” Annual Review of Condensed Matter Physics 10, 233–252 (2019).
  * Kapitulnik  _et al._ (2019) Aharon Kapitulnik, Steven A. Kivelson, and Boris Spivak, “Colloquium: Anomalous metals: Failed superconductors,” Rev. Mod. Phys. 91, 011002 (2019).
  * Giamarchi and Schulz (1988) T. Giamarchi and H. J. Schulz, “Anderson localization and interactions in one-dimensional metals,” Phys. Rev. B 37, 325–340 (1988).
  * Altman  _et al._ (2008) Ehud Altman, Yariv Kafri, Anatoli Polkovnikov, and Gil Refael, “Insulating phases and superfluid-insulator transition of disordered boson chains,” Phys. Rev. Lett. 100, 170402 (2008).
  * Pollet  _et al._ (2009) L. Pollet, N. V. Prokof’ev, B. V. Svistunov, and M. Troyer, “Absence of a direct superfluid to mott insulator transition in disordered bose systems,” Phys. Rev. Lett. 103, 140402 (2009).
  * Pollet  _et al._ (2014) Lode Pollet, Nikolay V. Prokof’ev, and Boris V. Svistunov, “Asymptotically exact scenario of strong-disorder criticality in one-dimensional superfluids,” Phys. Rev. B 89, 054204 (2014).
  * Doggen  _et al._ (2017) Elmer V. H. Doggen, Gabriel Lemarié, Sylvain Capponi, and Nicolas Laflorencie, “Weak- versus strong-disorder superfluid—bose glass transition in one dimension,” Phys. Rev. B 96, 180202 (2017).
  * Roati  _et al._ (2008) Giacomo Roati, Chiara D’Errico, Leonardo Fallani, Marco Fattori, Chiara Fort, Matteo Zaccanti, Giovanni Modugno, Michele Modugno, and Massimo Inguscio, “Anderson localization of a non-interacting bose–einstein condensate,” Nature 453, 895–898 (2008).
  * Schreiber  _et al._ (2015) Michael Schreiber, Sean S. Hodgman, Pranjal Bordia, Henrik P. Lüschen, Mark H. Fischer, Ronen Vosk, Ehud Altman, Ulrich Schneider, and Immanuel Bloch, “Observation of many-body localization of interacting fermions in a quasirandom optical lattice,” Science 349, 842–845 (2015), arXiv:1501.05661 .
  * Sbroscia  _et al._ (2020) Matteo Sbroscia, Konrad Viebahn, Edward Carter, Jr-Chiun Yu, Alexander Gaunt, and Ulrich Schneider, “Observing localisation in a 2D quasicrystalline optical lattice,”  (2020), arXiv:2001.10912 .
  * Lüschen  _et al._ (2016) H. P. Lüschen, P. Bordia, S. Scherg, F. Alet, E. Altman, U. Schneider, and I. Bloch, “Evidence for Griffiths-Type Dynamics near the Many-Body Localization Transition in Quasi-Periodic Systems,” ArXiv e-prints (2016), arXiv:1612.07173 [cond-mat.quant-gas] .
  * Lüschen  _et al._ (2018) Henrik P. Lüschen, Sebastian Scherg, Thomas Kohlert, Michael Schreiber, Pranjal Bordia, Xiao Li, S. Das Sarma, and Immanuel Bloch, “Single-particle mobility edge in a one-dimensional quasiperiodic optical lattice,” Phys. Rev. Lett. 120, 160404 (2018).
  * An  _et al._ (2021) Fangzhao Alex An, Karmela Padavić, Eric J. Meier, Suraj Hegde, Sriram Ganeshan, J. H. Pixley, Smitha Vishveshwara, and Bryce Gadway, “Interactions and mobility edges: Observing the generalized aubry-andré model,” Phys. Rev. Lett. 126, 040603 (2021).
  * Mak and Shan (2022) Kin Fai Mak and Jie Shan, “Semiconductor moiré materials,” Nature Nanotechnology 17, 686–695 (2022).
  * Fu  _et al._ (2020) Yixing Fu, Elio J König, Justin H Wilson, Yang-Zhi Chou, and Jedediah H Pixley, “Magic-angle semimetals,” npj Quantum Materials 5, 71 (2020).
  * Chandran and Laumann (2017) A. Chandran and C. R. Laumann, “Localization and symmetry breaking in the quantum quasiperiodic ising glass,” Phys. Rev. X 7, 031061 (2017).
  * Crowley  _et al._ (2018a) P. J.D. Crowley, A. Chandran, and C. R. Laumann, “Quasiperiodic Quantum Ising Transitions in 1D,” Physical Review Letters 120, 175702 (2018a).
  * Crowley  _et al._ (2018b) P. J. D. Crowley, A. Chandran, and C. R. Laumann, “Critical behavior of the quasi-periodic quantum Ising chain,” arXiv:1812.01660 [cond-mat] (2018b), arXiv:1812.01660 .
  * Crowley  _et al._ (2019) P. J. D. Crowley, C. R. Laumann, and S. Gopalakrishnan, “Quantum criticality in ising chains with random hyperuniform couplings,” Phys. Rev. B 100, 134206 (2019).
  * Vidal  _et al._ (1999) Julien Vidal, Dominique Mouhanna, and Thierry Giamarchi, “Correlated fermions in a one-dimensional quasiperiodic potential,” Physical review letters 83, 3908 (1999).
  * Roux  _et al._ (2008) Guillaume Roux, T Barthel, IP McCulloch, Corinna Kollath, U Schollwöck, and Thierry Giamarchi, “Quasiperiodic bose-hubbard model and localization in one-dimensional cold atomic gases,” Physical Review A 78, 023628 (2008).
  * Mastropietro (2015) Vieri Mastropietro, “Localization of interacting fermions in the aubry-andré model,” Physical review letters 115, 180401 (2015).
  * Naldesi  _et al._ (2016) Piero Naldesi, Elisa Ercolessi, and Tommaso Roscilde, “Detecting a many-body mobility edge with quantum quenches,” SciPost Physics 1, 010 (2016).
  * Cookmeyer  _et al._ (2020) Tessa Cookmeyer, Johannes Motruk, and Joel E Moore, “Critical properties of the ground-state localization-delocalization transition in the many-particle aubry-andré model,” Physical Review B 101, 174203 (2020).
  * Agrawal  _et al._ (2020) Utkarsh Agrawal, Sarang Gopalakrishnan, and Romain Vasseur, “Universality and Quantum Criticality in Quasiperiodic Spin Chains,” Nature Communications 2020 11:1 11, 1–7 (2020), arXiv:1908.02774 .
  * Schuster  _et al._ (2002) Cosima Schuster, Rudolf A. Römer, and Michael Schreiber, “Interacting particles at a metal-insulator transition,” Phys. Rev. B 65, 115114 (2002).
  * Kraus  _et al._ (2014) Yaacov E. Kraus, Oded Zilberberg, and Richard Berkovits, “Enhanced compressibility due to repulsive interaction in the harper model,” Phys. Rev. B 89, 161106 (2014).
  * Vu and Das Sarma (2021) DinhDuy Vu and S. Das Sarma, “Moiré versus mott: Incommensuration and interaction in one-dimensional bichromatic lattices,” Phys. Rev. Lett. 126, 036803 (2021).
  * Gonçalves  _et al._ (2024) Miguel Gonçalves, J. H. Pixley, B. Amorim, Eduardo V. Castro, and Pedro Ribeiro, “Short-range interactions are irrelevant at the quasiperiodicity-driven luttinger liquid to anderson glass transition,” Phys. Rev. B 109, 014211 (2024).
  * Yoo  _et al._ (2020) Yongchan Yoo, Junhyun Lee, and Brian Swingle, “Nonequilibrium steady state phases of the interacting aubry-andré-harper model,” Phys. Rev. B 102, 195142 (2020).
  * Chi  _et al._ (2024) Runze Chi, Josephine J Yu, Chaitanya Murthy, and T Xiang, “Luttinger liquid phase in the aubry-andr$\backslash$’e hubbard chain,” arXiv preprint arXiv:2401.08980 (2024).
  * Hiramoto and Kohmoto (1989) Hisashi Hiramoto and Mahito Kohmoto, “Scaling analysis of quasiperiodic systems: Generalized harper model,” Phys. Rev. B 40, 8225–8234 (1989).
  * White (1992) Steven R. White, “Density matrix formulation for quantum renormalization groups,” Phys. Rev. Lett. 69, 2863–2866 (1992).
  * Schollwöck (2011) Ulrich Schollwöck, “The density-matrix renormalization group in the age of matrix product states,” Annals of Physics 326, 96–192 (2011), january 2011 Special Issue.
  * Yue  _et al._ (1994) Dongxiao Yue, L. I. Glazman, and K. A. Matveev, “Conduction of a weakly interacting one-dimensional electron gas through a single barrier,” Phys. Rev. B 49, 1966–1975 (1994).
  * Wilkinson (1984) M Wilkinson, “Critical properties of electron eigenstates in incommensurate systems,” Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences 391, 305–350 (1984).
  * Kohmoto  _et al._ (1987) Mahito Kohmoto, Bill Sutherland, and Chao Tang, “Critical wave functions and a cantor-set spectrum of a one-dimensional quasicrystal model,” Physical Review B 35, 1020 (1987).
  * Zaletel  _et al._ (2015) Michael P. Zaletel, Roger S. K. Mong, Christoph Karrasch, Joel E. Moore, and Frank Pollmann, “Time-evolving a matrix product state with long-ranged interactions,” Phys. Rev. B 91, 165112 (2015).
  * Giamarchi (2003) Thierry Giamarchi,  _Quantum physics in one dimension_ , Vol. 121 (Clarendon press, 2003).
  * sup (a) See Supplemental Material at [url] for numerical details and additional plots (a).
  * Kane and Fisher (1992) C. L. Kane and Matthew P. A. Fisher, “Transport in a one-channel luttinger liquid,” Phys. Rev. Lett. 68, 1220–1223 (1992).
  * Štrkalj  _et al._ (2021) Antonio Štrkalj, Elmer V. H. Doggen, Igor V. Gornyi, and Oded Zilberberg, “Many-body localization in the interpolating aubry-andré-fibonacci model,” Phys. Rev. Res. 3, 033257 (2021).
  * Wang  _et al._ (2022) Yunfei Wang, Jia-Hui Zhang, Yuqing Li, Jizhou Wu, Wenliang Liu, Feng Mei, Ying Hu, Liantuan Xiao, Jie Ma, Cheng Chin, and Suotang Jia, “Observation of interaction-induced mobility edge in an atomic aubry-andré wire,” Phys. Rev. Lett. 129, 103401 (2022).
  * Biddle  _et al._ (2009) J. Biddle, B. Wang, D. J. Priour, and S. Das Sarma, “Localization in one-dimensional incommensurate lattices beyond the aubry-andré model,” Phys. Rev. A 80, 021603 (2009).
  * Wu  _et al._ (2019) Ang-Kun Wu, Sarang Gopalakrishnan, and J. H. Pixley, “Fractal x-ray edge problem at the critical point of the aubry-andré model,” Phys. Rev. B 100, 165116 (2019).
  * Wu  _et al._ (2022) Ang-Kun Wu, Daniel Bauernfeind, Xiaodong Cao, Sarang Gopalakrishnan, Kevin Ingersent, and J. H. Pixley, “Aubry-andré anderson model: Magnetic impurities coupled to a fractal spectrum,” Phys. Rev. B 106, 165123 (2022).
  * sup (b) (b), see Supplemental Material.
  * Devakul and Huse (2017) Trithep Devakul and David A Huse, “Anderson localization transitions with and without random potentials,” Physical Review B 96, 214201 (2017).
  * Szabó and Schneider (2020) Attila Szabó and Ulrich Schneider, “Mixed spectra and partially extended states in a two-dimensional quasiperiodic model,” Phys. Rev. B 101, 014205 (2020).
  * Altshuler  _et al._ (1980) B. L. Altshuler, A. G. Aronov, and P. A. Lee, “Interaction effects in disordered fermi systems in two dimensions,” Phys. Rev. Lett. 44, 1288–1291 (1980).


Also, using the results from Appendix <ref>, we can deduce that the common variance in the two groups is

\begin{multline*}
V\left( {t_0 |k = 1} \right) = \mathbb{E}\left( {t_0^2 |k = 1} \right) - \left[ {\mathbb{E}\left( {t_0 |k = 1} \right)} \right]^2  \\
= \frac{{V\left( {t_0 } \right)\left[ {p_e  + \rho _{\operatorname{e} ,t_0 }^2 (1 - 2p_e )} \right]}}
{{p_e }} - \rho _{\operatorname{e} ,t_0 }^2 \frac{{(1 - p_e )}}
{{p_e }}V\left( {t_0 } \right) = V\left( {t_0 } \right)\left( {1 - \rho _{\operatorname{e} ,t_0 }^2 } \right).
\end{multline*}

Therefore, we can write

\begin{multline*}
f(t_{0i} |k_i  = 1) = \\
\frac{1}
{{\sqrt {2\pi V\left( {t_0 } \right)\left( {1 - \rho _{\operatorname{e} ,t_0 }^2 } \right)} }}\exp \left[ {\frac{{ - 1}}
{{2V\left( {t_0 } \right)\left( {1 - \rho _{\operatorname{e} ,t_0 }^2 } \right)}}\left( {t_{0i}  - \rho _{\operatorname{e} ,t_0 } \sqrt {\frac{{(1 - p_e )}}
{{p_e }}} \sqrt {V\left( {t_0 } \right)} } \right)^2 } \right]
\end{multline*}

\begin{multline*}
f(t_{0i} |k_i  = 0) = \\
\frac{1}
{{\sqrt {2\pi V\left( {t_0 } \right)\left( {1 - \rho _{\operatorname{e} ,t_0 }^2 } \right)} }}\exp \left[ {\frac{{ - 1}}
{{2V\left( {t_0 } \right)\left( {1 - \rho _{\operatorname{e} ,t_0 }^2 } \right)}}\left( {t_{0i}  + \rho _{\operatorname{e} ,t_0 } \sqrt {\frac{{p_e }}
{{(1 - p_e )}}} \sqrt {V\left( {t_0 } \right)} } \right)^2 } \right].
\end{multline*}

We can write this conditional distribution as

\begin{multline*}
f(t_{0i} |k_i ) = \frac{1}
{{\sqrt {2\pi V\left( {t_0 } \right)\left( {1 - \rho _{\operatorname{e} ,t_0 }^2 } \right)} }} \\
\exp \left[ {\frac{{ - 1}}
{{2V\left( {t_0 } \right)\left( {1 - \rho _{\operatorname{e} ,t_0 }^2 } \right)}}\left( {t_{0i}  - \rho _{\operatorname{e} ,t_0 } \sqrt {V\left( {t_0 } \right)} ( - 1)^{1 - k_i } (1 - p_e )^{k_i  - \frac{1}
{2}} p_e ^{\frac{1}
{2} - k_i } } \right)^2 } \right].
\end{multline*}

Then, the joint distribution is

\begin{multline*}
f(t_{0i} ,k_i )$ $ = \frac{{p_e^{k_i } (1 - p_e )^{1 - k_i } }}
{{\sqrt {2\pi V\left( {t_0 } \right)\left( {1 - \rho _{\operatorname{e} ,t_0 }^2 } \right)} }} \\
\exp \left[ {\frac{{ - 1}}
{{2V\left( {t_0 } \right)\left( {1 - \rho _{\operatorname{e} ,t_0 }^2 } \right)}}\left( {t_{0i}  - \rho _{\operatorname{e} ,t_0 } \sqrt {V\left( {t_0 } \right)} ( - 1)^{1 - k_i } (1 - p_e )^{k_i  - \frac{1}
{2}} p_e ^{\frac{1}
{2} - k_i } } \right)^2 } \right].
\end{multline*}

The marginal distribution of $t_{0i} $ is a mixture of two normals,

f(t_{0i} ) = (1 - p_e )f(t_{0i} |k_i  = 0) + p_e f(t_{0i} |k_i  = 1).

Then, we can derive

\mathbb{E}\left[ {a(t_{0i} )} \right] = \int {a(t_{0i} )f(t_{0i} )}  = (1 - p_e )\int {a(t_{0i} )f(t_{0i} |k_i  = 0)}  + p_e \int {a(t_{0i} )f(t_{0i} |k_i  = 1)},

and equivalently for $\mathbb{E}\left[ {c(t_{0i} )} \right]$ and $\mathbb{E}\left[ {d(t_{0i} )} \right]$. We can also derive

\begin{multline*}
\mathbb{E}\left[ {k_i a(t_{0i} )} \right] = \int {\int {k_i a(t_{0i} )f(t_{0i} ,k_i )dk_i dt_{0i} } }  = \sum\limits_{k_i  = 0,1} {\int {k_i a(t_{0i} )f(t_{0i} ,k_i )dt_{0i} } }  = \\
\int {a(t_{0i} )f(t_{0i} ,1)dt_{0i} }  =  \int {a(t_{0i} )f(t_{0i} ,1)dt_{0i} }  = \\
\int a(t_{0i} )\frac{{p_e }}
{{\sqrt {2\pi V\left( {t_0 } \right)\left( {1 - \rho _{\operatorname{e} ,t_0 }^2 } \right)} }} \\
\exp \left[ {\frac{{ - 1}}
{{2V\left( {t_0 } \right)\left( {1 - \rho _{\operatorname{e} ,t_0 }^2 } \right)}}\left( {t_{0i}  - \rho _{\operatorname{e} ,t_0 } \sqrt {V\left( {t_0 } \right)} (1 - p_e )^{\frac{1}
{2}} p_e ^{ - \frac{1}
{2}} } \right)^2 } \right]dt_{0i}   \\
= p_e \int {a(t_{0i} )f(t_{0i} |k = 1)dt_{0i} },
\end{multline*}

and equivalently for $\mathbb{E}\left[ {k_i c(t_{0i} )} \right]$ and $\mathbb{E}\left[ {k_i d(t_{0i} )} \right]$.
Using the expressions derived here, our program computes the expectations numerically to obtain $\mathbb{E}\left[ {{\mathbf{X}}_i ^\prime  {\mathbf{\Sigma }}_i^{ - 1} {\mathbf{X}}_i } \right]$, then it inverts this matrix and extracts the [4,4] component, which will be $Var\left( {\hat \gamma _3 } \right)$. For CMD, the procedure is exactly the same but using the matrix

{\mathbf{W}}_i  = \left( {\begin{array}{*{20}c}
1 & 0 & {k_i }  \\
0 & 1 & 0  \\
\end{array} } \right).

§ PROOF THAT $R_{OPT} $ IS THE SAME FOR BOTH THE COST CONSTRAINT AND THE POWER CONSTRAINT, AND REDUCES TO THE SOLUTION TO THE UNCONSTRAINED PROBLEM (<REF>), BUT $N_{OPT} $ DEPENDS UPON THE CONSTRAINT

The power optimization problem is

\mathop {Max}\limits_r \;\Phi \left[ {\frac{{\sqrt N \left| {\left( {{\mathbf{c'B}}} \right)_{H_A } } \right|}}
{{\sqrt {{\mathbf{c'\Sigma }}_{\rm B} (r){\mathbf{c}}} }} - z_{1 - \alpha /2} } \right]{\text{ subject to }}COST = Nc_1  + \frac{{Nrc_1 }}
{\kappa }.

The cost constraint,

N = \frac{{\kappa COST}}
{{c_1 (\kappa  + r)}},

can be plugged in the optimization function to obtain the unconstrained problem

\mathop {Max}\limits_r \;\Phi \left[ {\frac{{\sqrt {\frac{{\kappa COST}}
{{c_1 (\kappa  + r)}}} \left| {\left( {{\mathbf{c'B}}} \right)_{H_A } } \right|}}
{{\sqrt {{\mathbf{c'\Sigma }}_{\rm B} (r){\mathbf{c}}} }} - z_{1 - \alpha /2} } \right].

Since $\Phi $ is a monotone function, this is equivalent to

\mathop {Max}\limits_r \;\frac{{\sqrt {\frac{{\kappa COST}}
{{c_1 (\kappa  + r)}}} \left| {\left( {{\mathbf{c'B}}} \right)_{H_A } } \right|}}
{{\sqrt {{\mathbf{c'\Sigma }}_{\rm B} (r){\mathbf{c}}} }} - z_{1 - \alpha /2}.

Removing positive constant terms with respect to $r$, it is equivalent to

\mathop {Max}\limits_r \;\frac{1}
{{(\kappa  + r){\mathbf{c'\Sigma }}_{\rm B} (r){\mathbf{c}}}},

which is in turn equivalent to $\mathop {Min}\limits_r \;(\kappa  + r){\mathbf{c'\Sigma }}_{\rm B} (r){\mathbf{c}}$. Once $r_{opt} $ is found solving this minimization problem, $N_{opt} $ would be

N_{opt}  = \frac{{\kappa \,COST}}
{{c_1 \left( {\kappa  + r_{opt} } \right)}}.

The cost optimization problem is

\mathop {Min}\limits_r \;Nc_1  + \frac{{Nrc_1 }}
{\kappa }{\text{ subject to }}\Phi \left[ {\frac{{\sqrt N \left| {\left( {{\mathbf{c'B}}} \right)_{H_A } } \right|}}
{{\sqrt {{\mathbf{c'\Sigma }}_{\rm B} (r){\mathbf{c}}} }} - z_{1 - \alpha /2} } \right] = \pi.

Noting that

Nc_1  + \frac{{Nrc_1 }}
{\kappa } = Nc_1 \left( {\frac{{\kappa  + r}}
{\kappa }} \right)

and that from the power constraint

N = \frac{{\left( {{\mathbf{c'}}\,{\mathbf{\Sigma }}_{\rm B} (r)\,{\mathbf{c}}} \right)\left( {z_{1 - \alpha /2}  + z_\pi  } \right)^2 }}
{{\left( {\left( {{\mathbf{c'{\rm B}}}} \right)_{H_A } } \right)^2 }},

this is equivalent to the unconstrained problem

\mathop {Min}\limits_r \;\frac{{({\mathbf{c'}}\,{\mathbf{\Sigma }}_{\rm B} (r)\,{\mathbf{c}})\left( {z_{1 - \alpha /2}  + z_\pi  } \right)^2 }}
{{\left( {\left( {{\mathbf{c'{\rm B}}}} \right)_{H_A } } \right)^2 }}c_1 \left( {\frac{{\kappa  + r}}
{\kappa }} \right).

Removing positive constant terms with respect to $r$, the problem becomes $\mathop {Min}\limits_r \;\left( {\kappa  + r} \right)({\mathbf{c'}}\,{\mathbf{\Sigma }}_{\rm B} (r)\,{\mathbf{c}})$, which is equivalent to the minimization problem obtained before. Thus, given $\kappa $, ${\mathbf{c}}$ and ${\mathbf{\Sigma }}_{\rm B} (r)$, the same $r_{opt} $ maximizes power and minimizes cost. For the cost problem, once $r_{opt} $ is found solving the minimization problem,

N_{opt}  = \frac{{\left( {{\mathbf{c'\Sigma }}_{\rm B} (r_{opt} ){\mathbf{c}}} \right)\,\left( {z_\pi   + z_{1 - \alpha /2} } \right)^2 }}
{{({\mathbf{c'{\rm B}}}_{H_A } )^2 }}.

§ OPTIMAL $R$ UNDER LDD AND FIXED $S$, FOR CS

The optimal $r$ solves $\mathop {Min}\limits_r \;(\kappa  + r){\mathbf{c'\Sigma }}_{\rm B} {\mathbf{c}}$ (Appendix <ref>). Plugging in the appropriate value for ${\mathbf{c'\Sigma }}_{\rm B} {\mathbf{c}}$ from Table <ref>, the problem under LDD, CS and fixed $s$ is

\mathop {Min}\limits_r \;(\kappa  + r)\frac{{12\sigma ^2 (1 - \rho )}}
{{p_e (1 - p_e )s^2 r(r + 1)(r + 2)}}.

Removing positive constant terms with respect to $r$, this problem becomes

\mathop {Min}\limits_r \quad F(r) = \frac{{(\kappa  + r)}}
{{r\;(r + 1)(r + 2)}}.

\frac{{\partial F}}
{{\partial r}} = \frac{{ - 2\kappa  - 6\kappa r - 3r^2  - 3\kappa r^2  - 2r^3 }}
{{r^2 (r + 1)^2 (r + 2)^2 }} < 0\;\forall \kappa,

$F(r)$ decreases as $r$ increases, and $r_{opt}  \to \infty $ subject to the cost constraint.

§ $R_{OPT} $ UNDER LDD, RS AND FIXED $S$

The optimal $r$ solves $\mathop {Min}\limits_r \;(\kappa  + r){\mathbf{c'\Sigma }}_{\rm B} {\mathbf{c}}$ (Appendix <ref>). Plugging in the appropriate value for ${\mathbf{c'\Sigma }}_{\rm B} {\mathbf{c}}$ from Table <ref>, the problem under LDD, RS and fixed $s$ is

\mathop {Min}\limits_r \;(\kappa  + r)\left( {\frac{{12\sigma ^2 (1 - \rho _{t_0 } )}}
{{s^2 p_e (1 - p_e )}}} \right)\left( {\frac{1}
{{r(r + 1)(r + 2)}} + \left( {\frac{{\rho _{b_1 ,s,\tilde r} }}
{{1 - \rho _{b_1 ,s,\tilde r} }}} \right)\frac{1}
{{\tilde r(\tilde r + 1)(\tilde r + 2)}}} \right).

Removing positive constant terms with respect to $r$, this problem becomes

\mathop {Min}\limits_r \quad G(r) = (\kappa  + r)\left( {\frac{1}
{{\,r(r + 1)(r + 2)}} + \left( {\frac{{\rho _{b_1 ,s,\tilde r} }}
{{1 - \rho _{b_1 ,s,\tilde r} }}} \right)\frac{1}
{{\tilde r(\tilde r + 1)(\tilde r + 2)}}} \right).

The solution, $r_{opt} $, solves

\begin{multline*}
\frac{{\partial G}}
{{\partial r}} = \left( {\frac{{\rho _{b_1 ,s,\tilde r} }}
{{1 - \rho _{b_1 ,s,\tilde r} }}} \right)\frac{1}
{{\tilde r(\tilde r + 1)(\tilde r + 2)}} + \frac{{ - 2\kappa  - 6\kappa r - 3r^2  - 3\kappa r^2  - 2r^3 }}
{{r^2 (r + 1)^2 (r + 2)^2 }} \\
= \left( {\frac{{\rho _{b_1 ,s,\tilde r} }}
{{1 - \rho _{b_1 ,s,\tilde r} }}} \right)\frac{1}
{{\tilde r(\tilde r + 1)(\tilde r + 2)}} + \frac{{\partial F}}
{{\partial r}} = 0,
\end{multline*}

where $\frac{{\partial F}} {{\partial r}}$ is the derivative of the objective function $F(r)$ for the analogous problem under compound symmetry (Appendix <ref>). We showed in Appendix <ref> that $\frac{{\partial F}}{{\partial r}}$ is always negative, and since

\frac{{\partial ^2 F}}
{{\partial r^2 }} = \frac{{2\left( {4\kappa  + 18\kappa r + 33\kappa r^2  + 7r^3  + 24\kappa r^3  + 9r^4  + 6\kappa r^4  + 3r^5 } \right)}}
{{r^3 \left( {r + 1} \right)^3 (r + 2)^3 }} \geqslant 0,

$\frac{{\partial F}} {{\partial r}}$ is also an increasing function of $r$. In addition, $\mathop {\lim }\limits_{r \to \infty } \frac{{\partial F}}
{{\partial r}} = 0^ - $. Since $\frac{{\partial G}} {{\partial r}}$ is $\frac{{\partial F}} {{\partial r}}$ plus a constant, $\frac{{\partial G}}
{{\partial r}}$ will equal 0 at some interior point of $r$ between 1 and $\infty $. Since

\frac{{\partial ^2 G}}
{{\partial r^2 }} = \frac{{2\left( {4\kappa  + 18\kappa r + 33\kappa r^2  + 7r^3  + 24\kappa r^3  + 9r^4  + 6\kappa r^4  + 3r^5 } \right)}}
{{r^3 \left( {r + 1} \right)^3 (r + 2)^3 }} \geqslant 0

for all $r > 0$, $G(r)$ is convex and the point that solves $\frac{{\partial G}}
{{\partial r}} = 0$ is a global minimum and therefore it is $r_{opt} $.

\frac{{\partial G}}
{{\partial r}} = 0 \Leftrightarrow \kappa  = \frac{{r_{opt} ^2 \left( { - (3 + 2r_{opt} )\tilde r(\tilde r + 1)(\tilde r + 2) + (r_{opt}  + 1)^2 (r_{opt}  + 2)^2 \frac{{\rho _{b_1 ,s,\tilde r} }}
{{1 - \rho _{b_1 ,s,\tilde r} }}} \right)}}
{{\,\left( {2 + 6r_{opt}  + 3r_{opt} ^2 } \right)\tilde r(\tilde r + 1)(\tilde r + 2)}}.

Figure <ref> of the paper shows $r_{opt} $ for several values of $\kappa $ and $\rho _{b_1 ,s,\tilde r} $.

§ $\LEFT( {N_{OPT} ,R_{OPT} } \RIGHT)$ UNDER LDD, CS AND FIXED $\TAU $

As shown in Appendix <ref>, the optimal $r$ solves $\mathop {Min}\limits_r \;(\kappa  + r){\mathbf{c'\Sigma }}_{\rm B} {\mathbf{c}}$  Plugging in the appropriate value for ${\mathbf{c'\Sigma }}_{\rm B} {\mathbf{c}}$ from Table <ref>, the problem under LDD, CS and fixed $\tau $ is

\mathop {Min}\limits_r \;(\kappa  + r)\frac{{12\sigma ^2 (1 - \rho )r}}
{{p_e (1 - p_e )\tau ^2 (r + 1)(r + 2)}}.

Removing positive constant terms with respect to $r$, this problem becomes

\mathop {Min}\limits_r \quad H(r) = \frac{{(\kappa  + r)\,r}}
{{(r + 1)(r + 2)}}.

Taking derivatives with respect to $r$, $r_{opt} $ solves

\frac{{\partial H}}
{{\partial r}} = \frac{{(3 - \kappa )r^2  + 4r + 2\kappa }}
{{\left( {r + 1} \right)^2 (r + 2)^2 }} = 0.

For $\kappa  < 3$ the derivative is positive. Therefore, when $\kappa  < 3
$ $H(r)$ increases with r and, consequently, the minimum is at $r =1$. If $\kappa  > 3
$, the derivative equals 0 at

r = \frac{{2 \pm \sqrt 2 \sqrt {2 - 3\kappa  + \kappa ^2 } }}
{{\kappa  - 3}},

which gives a positive solution only at

r = \frac{{2 + \sqrt 2 \sqrt {2 - 3\kappa  + \kappa ^2 } }}
{{\kappa  - 3}}.

Now, we need to check whether at this point there is a maximum or a minimum of $H(r)
$. The second derivative of $H(r)$ is

\frac{{\partial ^2 H}}
{{\partial r^2 }} = \frac{{2\left( {4 - 6\kappa  - 6\kappa r - 6r^2  - 3r^3  + \kappa r^3 } \right)}}
{{(r + 1)^3 (r + 2)^3 }}.

We evaluated the second derivative at the point

r = \frac{{2 + \sqrt 2 \sqrt {2 - 3\kappa  + \kappa ^2 } }}
{{\kappa  - 3}}

with Mathematica <cit.> and obtained

\left( {24 + \frac{{3\sqrt 2 }}
{{\sqrt {(\kappa  - 2)(\kappa  - 1)} }}} \right)\kappa  - 17\sqrt 2 \sqrt {(\kappa  - 2)(\kappa  - 1)}  - \frac{{7\sqrt 2 }}
{{\sqrt {(\kappa  - 2)(\kappa  - 1)} }} - 40.

This expression can be proven to be negative for all $\kappa  > 3$. Therefore, $H(r)$ has a maximum at

r = \frac{{2 + \sqrt 2 \sqrt {2 - 3\kappa  + \kappa ^2 } }}
{{\kappa  - 3}},

while we were looking for a minimum. Since this is the only local maximum or minimum of $H(r)$, the global minimum of $H(r)$ will be at $r = 1$ or at $r = \infty $ . The global minimum will be at $r = \infty$ if we can find a value of r such that

H(r) < H(1) = \frac{{(1 + \kappa )}}

With a little bit of algebra , we get

H(r) = \frac{{r(r + \kappa )}}
{{(r + 1)(r + 2)}} < \frac{{(1 + \kappa )}}
{6} \Leftrightarrow r^2 (\kappa  - 5) + r( - 3\kappa  + 3) + 2(\kappa  + 1) > 0,

which has roots at $r = 1$ and ${\text{ }}r = \frac{{2(\kappa  + 1)}}
{{\kappa  - 5}}$. If $\kappa  < 5$, then ${\text{ }}r = \frac{{2(\kappa  + 1)}}
{{\kappa  - 5}} < 0$, outside of its valid range. The global minimum is then  $r_{opt}  = 1$,

N_{opt}  = \frac{{\kappa COST}}
{{c_1 (\kappa  + 1)}}

N_{opt}  = \frac{{2\sigma ^2 (1 - \rho )\,\left( {z_\pi   + z_{1 - \alpha /2} } \right)^2 }}
{{\tau ^2 p_e (1 - p_e )\gamma _3^2 }}

for the power maximization or cost minimization problems, respectively. If $\kappa  > 5$, then $r = \frac{{2(\kappa  + 1)}}{{\kappa  - 5}} > 1$ within the range, so taking $r$as large as possible subject to the cost constraint,

r_{opt}  = \kappa \left( {\frac{{COST}}
{{c_1 }} - 1} \right)

and $N_{opt}  = 1$.  Under a power constraint, one would find the smallest $r$ that satisfies the power constraint and set $N_{opt}  = 1$. In reality, the investigator will set $r$ as large as is feasible, and then find $N_{opt} $ to satisfy the cost or power constraint.

§ $\LEFT( {N_{OPT} ,R_{OPT} } \RIGHT)$ UNDER LDD, RS AND FIXED $\TAU $

The optimal $r$ solves $\mathop {Min}\limits_r \;(\kappa  + r){\mathbf{c'\Sigma }}_{\rm B} {\mathbf{c}}$ (Appendix <ref>). Plugging in the appropriate value for ${\mathbf{c'\Sigma }}_{\rm B} {\mathbf{c}}$ from Table<ref>, the problem under LDD, RS and fixed $\tau $ is

\mathop {Min}\limits_r \;(\kappa  + r)\left( {\frac{{12\sigma ^2 (1 - \rho _{t_0 } )}}
{{s^2 p_e (1 - p_e )}}} \right)\left( {\frac{r}
{{(r + 1)(r + 2)}} + \left( {\frac{{\rho _{b_1 ,\tau ,\tilde r} }}
{{1 - \rho _{b_1 ,\tau ,\tilde r} }}} \right)\frac{{\tilde r}}
{{(\tilde r + 1)(\tilde r + 2)}}} \right).

Removing positive constant terms with respect to $r$, this problem becomes

\mathop {Min}\limits_r \quad I(r) = (\kappa  + r)\left( {\frac{r}
{{\,(r + 1)(r + 2)}} + \left( {\frac{{\rho _{b_1 ,\tau ,\tilde r} }}
{{1 - \rho _{b_1 ,\tau ,\tilde r} }}} \right)\frac{{\tilde r}}
{{\,(\tilde r + 1)(\tilde r + 2)}}} \right).

Taking derivatives with respect to r, $r_{opt} $ solves

\begin{multline*}
\frac{{\partial I}}
{{\partial r}} = \left( {\frac{{\rho _{b_1 ,\tau ,\tilde r} }}
{{1 - \rho _{b_1 ,\tau ,\tilde r} }}} \right)\frac{{\tilde r}}
{{\,(\tilde r + 1)(\tilde r + 2)}} + \frac{{(3 - \kappa )r^2  + 4r + 2\kappa }}
{{\left( {r + 1} \right)^2 (r + 2)^2 }} = \\
\left( {\frac{{\rho _{b_1 ,\tau ,\tilde r} }}
{{1 - \rho _{b_1 ,\tau ,\tilde r} }}} \right)\frac{{\tilde r}}
{{\,(\tilde r + 1)(\tilde r + 2)}} + \frac{{\partial H}}
{{\partial r}} = 0,
\end{multline*}

where $\frac{{\partial H}} {{\partial r}}$ is the derivative of the objective function $H(r)$ for the analogous problem under CS, given in Appendix <ref>. There we showed that if $\kappa  < 3$ then $\frac{{\partial H}} {{\partial r}}$ was strictly positive for all $r$, and therefore so is $\frac{{\partial I}} {{\partial r}}$. Thus, if $\kappa  < 3$, $I(r)$ is minimized at $r_{opt}  = 1$. For $\kappa  > 3$, we know that $\frac{{\partial H}} {{\partial r}}$ is continuous, has only one root in the range of interest and it can be shown that $\mathop {\lim }\limits_{r \to \infty } \frac{{\partial H}} {{\partial r}} = 0^ -  $ and $\frac{{\partial H(1)}}
{{\partial r}} = \frac{{7 + \kappa }} {{36}}$. It can also be shown with Mathematica <cit.> that $\frac{{\partial ^2 H}} {{\partial r^2 }}$ has only one real root, $r*$. Therefore, $\frac{{\partial H}} {{\partial r}}$ is positive at $r = 1$, it crosses 0 at the root

r = \frac{{2 + \sqrt 2 \sqrt {2 - 3\kappa  + \kappa ^2 } }}
{{\kappa  - 3}},

as shown in Appendix <ref>, it has a minimum at the only root of  $\frac{{\partial ^2 H}}
{{\partial r^2 }}$ and it increases again towards zero, where it reaches an asymptote. Because of the form of $\frac{{\partial I}} {{\partial r}}$, it will have a similar shape, since it is equal to $\frac{{\partial H}} {{\partial r}}$ but moved upwards by a factor of

\left( {\frac{{\rho _{b_1 ,\tau ,\tilde r} }}
{{1 - \rho _{b_1 ,\tau ,\tilde r} }}} \right)\frac{{\tilde r}}
{{\,(\tilde r + 1)(\tilde r + 2)}}.

Therefore, $\frac{{\partial I}}{{\partial r}}$ will have zero roots if

\left( {\frac{{\rho _{b_1 ,\tau ,\tilde r} }}
{{1 - \rho _{b_1 ,\tau ,\tilde r} }}} \right)\frac{{\tilde r}}
{{\,(\tilde r + 1)(\tilde r + 2)}} > \frac{{\partial H(r*)}}
{{\partial r}},

or two roots otherwise. In the first case, when $\frac{{\partial I}} {{\partial r}}$ has zero roots, $\frac{{\partial I}} {{\partial r}}$ is always positive and therefore $I(r)$ increases as $r$ increases and the minimum of $I(r)$ is at $r_{opt}  = 1$. In the second case, $\frac{{\partial I}} {{\partial r}}$ has two roots, which solve

\kappa  = \frac{{r(4 + 3r)(\tilde r + 1)(\tilde r + 2) + \tilde r\left( {r + 1} \right)^2 (r + 2)^2 \left( {\frac{{\rho _{b_1 ,\tau ,\tilde r} }}
{{1 - \rho _{b_1 ,\tau ,\tilde r} }}} \right)}}
{{\,(r^2  - 2)(\tilde r + 1)(\tilde r + 2)}}.

Also, $\frac{{\partial ^2 H}}
{{\partial r^2 }} = \frac{{\partial ^2 I}}
{{\partial r^2 }}$, and $\frac{{\partial ^2 H}}
{{\partial r^2 }}$ is continuous and it has only one root at $r*$.  $\frac{{\partial ^2 H}}
{{\partial r^2 }}$ is negative for $r < r*$ and positive for $r > r*$. Since $r*$ lies between the first and second roots of $\frac{{\partial I}} {{\partial r}}$, it can be concluded that the first root is a maximum of $I(r)$ and the second root is a minimum of $I(r)$. The function $I(r)$ has, therefore, two local minima, one at $r = 1$  and the other at the second root of $\frac{{\partial I}} {{\partial r}}$. To find out when the second root is the global minimum of $I(r)$ we need to solve $I(1) > I(r)
$, where

I(1) = (\kappa  + 1)\left( {\frac{1}
{6} + \left( {\frac{{\rho _{b_1 ,\tau ,\tilde r} }}
{{1 - \rho _{b_1 ,\tau ,\tilde r} }}} \right)\frac{{\tilde r}}
{{\,(\tilde r + 1)(\tilde r + 2)}}} \right)

I(r) = (\kappa  + r)\left( {\frac{r}
{{\,(r + 1)(r + 2)}} + \left( {\frac{{\rho _{b_1 ,\tau ,\tilde r} }}
{{1 - \rho _{b_1 ,\tau ,\tilde r} }}} \right)\frac{{\tilde r}}
{{\,(\tilde r + 1)(\tilde r + 2)}}} \right).

Provided $r > 2$, this is equivalent to

\rho _{b_1 ,\tau ,\tilde r}  < \frac{{\left[ { - 2(\kappa  + 1) + (\kappa  - 5)r} \right](\tilde r + 1)(\tilde r + 2)}}
{{6\tilde r(r + 1)(r + 2) + \left[ { - 2(\kappa  + 1) + (\kappa  - 5)r} \right](\tilde r + 1)(\tilde r + 2)}}.

The condition only makes sense if $ - 2(\kappa  + 1) + (\kappa  - 5)r > 0$, which is equivalent to the conditions $\kappa  > 5$ and $r > \frac{{2\left( {\kappa  - 1} \right)}} {{\kappa  - 5}}$. Figure <ref> of the paper shows this region for different values of $\kappa $and $\rho _{b_1 ,\tau ,\tilde r} $, together with a line for the optimal value.

§ DEMONSTRATION OF THE PROGRAM USE

This is the output of the program for the computation of the optimal combination of $(N, r)$ that minimizes the total cost of the study subject to achieving a fixed power under LDD and RS. For other examples and a user's guide, go to <http://www.hsph.harvard.edu/faculty/spiegelman/optitxs.html>.