#  Understanding Gravity
Some Extra Dimensional Perspectives

$\hbox{V H Satheesh Kumar}^{\star}$ and $\hbox{P K Suresh}^{\dagger}$
School of Physics, University of Hyderabad
Hyderabad 500 046, India.
⋆<EMAIL_ADDRESS>†<EMAIL_ADDRESS>
###### Abstract

Gravity is one of the most inexplicable forces of nature, controlling
everything, from the expansion of the Universe to the ebb and flow of ocean
tides. The search for the laws of motion and gravitation began more than two
thousand years ago but still we do not have the complete picture of it. In
this article, we have outlined how our understanding of gravity is changing
drastically with time and how the previous explanations have shaped the most
recent developments in the field like superstrings and braneworlds.

##  1 Intoduction

Gravity is an immediate fact of everyday experience, but its fundamental
understanding presents some of the deepest theoretical and experimental
challenges in physics today. Gravitational physics is concerned with some of
the most exotic large scale phenomena in the universe. But it is also
concerned with the microscopic quantum structure of space and time and the
unification of all fundamental forces of nature. Gravity is thus important on
both the largest and smallest scales considered in contemporary physics and
remains one of the greatest challenges of twentyfirst century science.

Gravity dominates the large-scale structure of the universe only by default.
Matter arranges itself to cancel electromagnetism, and the strong and weak
forces are intrinsically short range. At a more fundamental level, gravity is
extravagantly feeble. Where does this outlandish disparity come from? Perhaps
the most unusual thing about gravity we know about is that, unlike the other
forces of nature, gravity is intimately related to space and time. Why is it
so different from the other interactions? Why has it not been able to unify it
with the rest? Some such attempts to get quantum gravity have required that we
live in more than four dimensions! If so, why do not we see the other
dimensions? How are these extra dimensions hidden from our world? Is there a
way to detect them? It is the aim of this article as to provide a short
summary of the present status of these extra-dimensional theories of
gravitation. But before graduating to extra dimensional theories, we will have
a look at the well established theories of gravity due to Newton and Einstein.

###  1.1 Newton’s Gravity

The very earliest ideas regarding gravity must have been based on every day
experience. For example, objects fall unless they are supported and climbing a
hill is harder than walking on a level. Aristotle was the first to give some
reasoning for these observed facts. In his view, the whole universe is made up
of four concentric spheres, the innermost being the Earth, then comes Water,
Air and Fire. Since stone is more of ‘Earth type’, it falls down on Earth when
thrown up! Ptolemy extended this view to the heavens and ended up with Geo-
centric theory. Aristotle’s notion of the motion impeded understanding of
gravitation for a long time. Copernicus’s view of the solar system was
important as it allowed sensible consideration of gravitation. Kepler’s laws
of planetary motion were based on the volumes of observational data collected
by Tycho Brahe. Galileo’s understanding of the motion and falling bodies was
through his inclined plane experiments. But till then nobody knew what is the
‘cause’ for such a motion. This set the scene for Isaac Newton’s theory of
gravity which was presented in the Principia[1] in 1687.

Newton started from Galileo’s law of falling objects and applied it to an
unlikely object, the Moon which seems to flout the law of gravity. Newton
realized that the Moon is not immune to gravity and is continuously falling
towards the Earth, but it keeps missing it! Newton thus realized that gravity
was not something special to the Earth, but it also acts in space. This was a
profound and revolutionary idea. According to Aristotle, the laws governing
the heavens were considered to be completely different from the laws of
physics here on Earth. Now, however, if the moon was affected by gravity, then
it made sense that the rest of the Solar System should also be subjected to
gravity. Newton found that he could explain the entire motion of the Solar
System from the planets to the moons to the comets with a single law of
Gravity. Newton’s Universal Law of Gravity states that “all bodies attract all
other bodies, and the strength of the attraction is proportional to the masses
of the two bodies and inversely proportional to the square of the distance
between the bodies”. This is called Universal because it applies to all bodies
in the Universe regardless of their nature. A modern mathematical way of
saying this is,

$F=\dfrac{GMm}{R^{2}}$ (1)

where $G$ is Newton’s Gravitational Constant, $M$ and $m$ are the masses of
the objects and $R$ is the distance between the objects. This law can be
expressed in differential form as Poisson’s Equation,

$\nabla\cdot\nabla\phi=4\pi G\rho$ (2)

where $\phi$ is gravitational potential and $\rho$ is the mass density of the
object.

Despite its power in explaining the planetary orbits in the Solar System,
Newton was unhappy with the lack of a mechanism by which gravity worked. Until
then, all forces were believed to be contact forces \- except the gravity. The
Newtonian concept of “action-at-a-distance” was profoundly disturbing to his
opponents, who attacked his theory as an “occult” and explaining nothing.

From the period immediately following Newton’s discovery of his Universal Law
of Gravitation, to about the turn of the nineteenth century, the theory of
gravitation stayed essentially unchanged. More sophisticated mathematical
tools for understanding the interplay of the planets were developed, but the
underlying theory remained stable. The excitement during this period mainly
came from the systematic application of the theory of gravity to the heavens.
For example, Halley’s prediction of the return of the comet that now bears his
name, discovery of the Neptune by John Adams and Urbain Leverrier, William
Hershel’s observations of binary stars and the calculation of the mass of
stars, James Maxwell’s explanation of Rings of the Saturn. Of course, other
advances were made, among the most important were the experiments of
Cavendish, who directly demonstrated the gravitational force between two
objects in the laboratory.

###  1.2 Einstein’ Gravity

The twentieth century was a time of tremendous progress in physical science.
For the understanding of gravity, the century began with two puzzles. The
first of these puzzles concerned the orbit of the planet Mercury. The second
puzzle was related to a series of experiments performed by the Hungarian
physicist Roland Eötvös at the end of the nineteenth century. Eötvös was
intrigued by a curious fact about Newton’s laws of gravity and motion. His
experiments showed that the gravitational mass was the same as the inertial
mass to at least a few parts in a hundred million.

Einstein’s theory of General Relativity[2], published in 1915, is our most
detailed mathematical theory for how gravity works. The foundation stone for
the general relativity is the equivalence principle, which assumes equivalence
between the inertial mass with the gravitational mass. This implies ‘the weak
equivalence principle’, i.e., the effects of gravitation can be transformed
away locally by using suitably accelerated frames of reference. This can be
generalized to ‘the strong equivalence principle’, which allows us to study
gravitational interaction by studying only the geometry of the space-time. The
modern approach to gravity as the geometry of curved spacetime is based on
this theory.

To understand the geometry of spacetime, consider the distance between two
spacetime points in any inertial frame,

$ds^{2}=c^{2}dt^{2}-dx^{2}-dy^{2}-dz^{2}.$ (3)

But if these two points are not connected by a ‘straight’ line, the distance
can be given by a more general form,

$ds^{2}=g_{\mu\nu}dx^{\mu}dx^{\nu},$ (4)

where sum over repeated indices is implied. The indices $\mu,\nu=0,1,2,3$ run
over four spacetime coordinates. The coefficient $g_{\mu\nu}$ is a function of
the spacetime coordinate $x^{\mu}$. This is called the metric and it specifies
the geometry of the spacetime. To study the geometry of any spacetime for an
understanding of the theory of gravity, it is enough to study the metric
$g_{\mu\nu}$.

It follows then from the Principle of Equivalence that the equations which
govern gravitational fields of arbitrary strength must take the form,

$G_{\mu\nu}=-8\pi GT_{\mu\nu}$ (5)

where $G_{\mu\nu}$ is called the Einstein tensor which has the geometrical
information about the spacetime, $G$ is the Newton’s gravitational constant
and $T_{\mu\nu}$ is the energy-momentum tensor of the matter present. Einstein
tensor is given by,

$G_{\mu\nu}=R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R$ (6)

where $R_{\mu\nu}$ is the Ricci curvature tensor and $R$ is Ricci scalar.

In general relativity, one performs calculations to compute the evolution and
structure of an entire universe at a time. A typical way of working is to
propose some particular collection of energy and matter in the universe, to
provide the $T_{\mu\nu}$. Given a particular $T_{\mu\nu}$, the Einstein
equation turns into a system of second order nonlinear differential equations
whose solutions give us the metric of spacetime $g_{\mu\nu}$, which holds all
the information about the structure and evolution of a universe with that
given $T_{\mu\nu}$.

General Relativity is perhaps the most beautiful physical theory and one of
the crowning glories of modern physics. It is powerful, pleasing to the
aesthetic sense and well-tested. General Relativity has survived many
different tests, and it has made many predictions which have been confirmed.
Currently, physicists are exploring two of its most fundamental predictions:
The first is the detection of gravitational waves (LISA and LIGO); the second
is the phenomenon called ‘frame-dragging’ that exists around rotating massive
objects (Gravity Probe-B). Other tests focus on the laboratory-scale
measurements of the force of gravity to look for signs of ‘extra dimensions’.
Data from any of these experimental studies will greatly improve our
understanding of gravity, and will show us how to go beyond the mathematics of
General Relativity to create an even-better theory.

The unification of quantum theory and general relativity has been a major
problem in physics ever since these theories were proposed. The problem is
that since all fields carrying energy are affected by gravity, gravity
contributes to its own source. When trying to do calculations on the energy
scale where gravity is usually thought to be similar in strength as the other
forces, the graviton self coupling causes fluctuations which introduces
infinities in the calculations. This has lead many theories to accomadate the
idea of extra dimensions to get the quantum gravity. A recent non-technical
review of extra dimensional theories can be found in [3]. One of the early
possibilities for such a unification of the then known interactions i.e.,
gravity and electromagnetism, was suggested by Kaluza[4] and Klein[5]. But
historically, it was Gunnar Nordström[6] who brought the idea of extra spacial
dimension into physics.

##  2 Kaluza-Klein Theory

An early proposal to unite general relativity and classical electrodynamics
was given by Theodor Kaluza[4] in 1921. He showed that the gravitational and
electromagnetic fields stem from a single universal tensor and such an
intimate combination of the two interactions is possible in principle, with
the introduction of an additional spacial dimension. Although our rich
physical experience obtained so far provides little suggestion of such a new
spacial dimension, we are certainly free to consider our world to be four
dimensional spacetime of the bigger five dimensional spacetime. In this
scenario, one has to take into account the fact that we are only aware of the
spacetime variation of state-quantities, by making their derivatives with
respect to the new parameter vanish or by considering them to be small as they
are of higher order. This assumption is known as the cylindrical condition.

The five dimensional line element is given by

$d\hat{s}^{2}=\hat{g}_{\hat{\mu}\hat{\nu}}(x^{\mu},y)d\hat{x}^{\hat{\mu}}d\hat{x}^{\hat{\nu}}$
(7)

with $y$ as the additional spatial coordinate. The five dimensional metric can
be expressed as,

$\hat{g}_{\hat{\mu}\hat{\nu}}=\left(\begin{array}[]{cc}g_{\mu\nu}&g_{\mu 5}\\\
g_{5\nu}&g_{55}\\\ \end{array}\right)$ (8)

where all unhatted quantities are four-dimensional and all hatted quantities
are five-dimensional.

Once we have a spacetime metric, like in standard general relativity we can
construct the Christoffel symbols $\Gamma^{\mu}_{\nu\rho}$, the Riemann-
Christoffel curvature tensor $R^{\mu}_{\nu\rho\sigma}$, the Ricci tensor
$R_{\mu\nu}$, the curvature invariant $R$ and then the field equations. This
approach gave a striking result, the fifteen higher-dimensional field
equations naturally broke into a set of ten formulae governing a tensor field
representing gravity, four describing a vector field representing
electromagnetism, and one wave equation for a scalar field. Furthermore, if
the scalar field was constant, the vector field equations were just Maxwell’s
equations in vacuo, and the tensor field equations were the 4-dimensional
Einstein field equations sourced by an electromagnetic field.

In one fell swoop, Kaluza had written down a single covariant field theory in
five dimensions that yielded the four dimensional theories of general
relativity and electromagnetism! But many problems plagued Kaluza’s theory.
Not the least of which was the nature of the fifth dimension. There was no
explanation given for Kaluza’s ad hoc assumption, the cylindrical condition.

In 1926, Oscar Klein[5] provided an explanation for Kaluza’s fifth dimension
by proposing it to have a circular topology so that the coordinate $y$ is
periodic i.e., $0\leq y\leq 2\pi R,$ where $R$ is the radius of the circle
$S^{1}$. Thus the global space has topology $R^{4}\times S^{1}$. So Klein
suggested that there is a little circle at each point in four-dimensional
spacetime. This is the basic idea of Kaluza-Klein compactification. Although
there are four space dimensions, one of the space dimensions is compact with a
small radius. As a result, in all experiments we could see effects of only
$3+1$ dimensions. Thus Klein made the Kaluza’s fifth dimension less artificial
by suggesting plausible physical basis for it in compactification of the fifth
dimension. The theory of gravity on a compact space-time is called Kaluza-
Klein theory. The detailed pedogoical account of this is given in [7] and a
very good reviews can be found in [8].

We introduce the following notations,

$\displaystyle\hat{g}_{55}$ $\displaystyle=$ $\displaystyle\phi$ (9)
$\displaystyle\hat{g}_{5\mu}$ $\displaystyle=$ $\displaystyle\kappa\phi
A_{\mu}$ (10) $\displaystyle\hat{g}_{\mu\nu}$ $\displaystyle=$ $\displaystyle
g_{\mu\nu}+\kappa^{2}\phi A_{\mu}A_{\nu}.$ (11)

Hereby the quantities $\hat{g}_{\hat{\mu}\hat{\nu}}$ are redused to known
quantities. Now, the new metric can be written as

$\hat{g}_{\hat{\mu}\hat{\nu}}=\phi^{-1/3}\left(\begin{array}[]{cc}g_{\mu\nu}+\kappa^{2}\phi
A_{\mu}A_{\nu}&\kappa\phi A_{\mu}\\\ \kappa\phi A_{\nu}&\phi\\\
\end{array}\right),$ (12)

where the field $\phi$ appears as a scaling parameter in the fifth dimension
and is called the dilaton field. The fields $g_{\mu\nu}(x,y),A_{\mu}(x,y)$ and
$\phi(x,y)$ transform respectively as a tensor, a vector and a scalar under
four-dimensional general coordinate transformations.

The Einstein-Hilbert action for five dimensional gravity can be written as,

${\hat{S}}=\frac{1}{2\hat{k}^{2}}\int d^{5}\hat{x}\sqrt{-\hat{g}}\hat{R}$ (13)

where $\hat{k}$ is the five dimensional coupling constant and $\hat{R}$ is the
five dimensional curvature invariant. We can get the field equations of
gravity and electromagnetism from the above action by variational principle.

As Klein suggested, the extra dimension has become compact and satisfies the
boundary condition

$y=y+2\pi R,$ (14)

all the fields are periodic in $y$ and may be expanded in a Fourier series,

$\displaystyle g_{\mu\nu}(x,y)$ $\displaystyle=$
$\displaystyle\sum_{n=-\infty}^{+\infty}g_{\mu\nu n}(x)e^{in\cdot y/R}$ (15)
$\displaystyle A_{\mu}(x,y)$ $\displaystyle=$
$\displaystyle\sum_{n=-\infty}^{+\infty}A_{\mu n}(x)e^{in\cdot y/R}$ (16)
$\displaystyle\phi(x,y)$ $\displaystyle=$
$\displaystyle\sum_{n=-\infty}^{+\infty}\phi_{n}(x)e^{in\cdot y/R}$ (17)

The equations of motion corresponding to the above action are,

$\displaystyle(\partial^{\mu}\partial_{\mu}-\partial^{y}\partial_{y})g_{\mu\nu}(x,y)$
$\displaystyle=$
$\displaystyle(\partial^{\mu}\partial_{\mu}+\dfrac{n^{2}}{R^{2}})g_{\mu\nu
n}(x)=0$ (18)
$\displaystyle(\partial^{\mu}\partial_{\mu}-\partial^{y}\partial_{y})A_{\mu}(x,y)$
$\displaystyle=$
$\displaystyle(\partial^{\mu}\partial_{\mu}+\dfrac{n^{2}}{R^{2}})A_{\mu
n}(x)=0$ (19)
$\displaystyle(\partial^{\mu}\partial_{\mu}-\partial^{y}\partial_{y})\phi(x,y)$
$\displaystyle=$
$\displaystyle(\partial^{\mu}\partial_{\mu}+\dfrac{n^{2}}{R^{2}})\phi_{n}(x)\;\;=0$
(20)

Comparing these with the standard Klein-Gordon equation, we can say that only
the zero modes $(n=0)$ will be massless and observable at our present energy
and all the excited states, called as Kaluza-Klein states, will have masses

$m_{n}\sim\dfrac{|n|}{R}$ (21)

as well as charge

$q_{n}=\sqrt{2}\kappa\dfrac{n}{R}$ (22)

as shown by Salam and Strathdee[9], where $n$ is the mode of excitation. So,
in four dimensions we shall see all these excited states with mass or momentum
$\sim O(n/R)$. Since we want to unify the electromagnetic interactions with
gravity, the natural radius of compactification will be the Planck length,

$R=\dfrac{1}{M_{p}}$ (23)

where the Planck mass $M_{p}\sim 10^{18}GeV.$

Since the Kaluza-Klein metric is a $5\times 5$ symmetric tensor, it has 15
independent components. However, because of various gauge fixings we will have
only 5 independent degrees of freedom. Whereas in four dimensions we have only
2 degrees of freedom for a massless graviton. This implies that from four
dimensional point of view a higher dimensional graviton will contain particles
other than just ordinary four dimensional graviton. The zero-mode of five
dimensional graviton contains a four dimensional massless graviton with 2
physical degrees of freedom; a four dimensional massless gauge boson with 2
physical degrees of freedom and a real scalar with 1 physical degree of
freedom. Whereas the non-zero mode of five dimensional graviton is massive and
has 5 physical degrees of freedom.

Kaluza and Klein’s fivedimensional version general relativity although flawed,
is an example of such an attempt to unite the forces of nature under one
theory. It led to glaring contradictions with experimental data. But some
physicists felt that it was on the right track, that it in fact did not
incorporate enough extra dimensions! This led to modified versions of Kaluza-
Klein theories incorporating numerous and extremely small extra dimensions.
The three main different approaches to higher dimensional unification are

  1. 1.

Compactified Approach

In this scenario extra dimensions are forbidden for us to experience as they
are compactified and are unobservable on presently accessible energy scales.
This approach has been successful in many ways and is the dominant paradigm in
the higher dimensional unification. This has lead to new theories like
11-dimensional supergravity, 10-dimensional superstring theory, the latest
11-dimensional M-theory and Braneworld theory.

  2. 2.

Projective Approach

Projective theories were designed to emulate the successes of Kaluza-Klein
theory without epistemological burden of a real fifth dimension. In this way
of unification, the extra dimensions are treated as mathematical artifacts of
a more complicated theory. The fifth dimension is absorbed into ordinary four
dimensional spacetime by replacing the classical tensors of general relativity
with projective ones, which in turn alters the geometrical foundation of
general relativity itself.

  3. 3.

Noncompactified Approach

This theory prefers to stay with idea that the new coordinates are physical.
Following Minkowski’s example, one can imagine coordinates of other kinds,
scaled by appropriate dimension transporting parameters to give them units of
length. In this approach the extra dimensions may not necessarily be
spacelike. This takes the observable quantity such as rest mas as the extra
dimension.

Here we discuss only the compactified approaches and the interested readers
can refer to [10] for the detailed review and the comparative study of these
three approaches.

##  3 String Theory

After the discovery of nuclear interactions, physicists discovered that it no
longer seemed that the Kaluza-Klein theory with one extra dimension was a
viable candidate to include all the gauge interactions. The electromagnetic
interaction could be accomodadted with only one extra dimension. But the
strong, weak and electromagnetic interactions, i.e. the $SU(3)_{c}\times
SU(2)_{L}\times U(1)_{Y}$ gauge theory require more degrees of freedom than a
5-metric could offer. However, the way in which to address the additional
requirements of modern physics is not hard to imagine, one merely has to
further increase the dimensionality of theory until all of the desired gauge
bosons are accounted for. Then how many dimensions do we need to unify modern
particle physics with gravity via the Kaluza-Klein mechanism?

The answer comes from N=8 supersymmetry which contains spin-2 particle. When
N=8 supersymmetry is coupled with general relativity, one has 11 dimensional
supergravity theory. But it was realized that it is not possible to get all
the gauge interactions and the required fermion contents of the standard model
from this theory. Then there were attempts to consider 11 dimensional theories
with gauge groups. Of course, the main motivation of obtaining all gauge
interactions and gravity from one Einstein-Hilbert action at 11-dimensions
would be lost, but still this became an important study for sometime. In this
construction the main problem was due to new inconsistency, the anomaly.

This problem was tackled with String theory. A good intoduction to the subject
is given in [11]. Briefly, the origin of string theory was the discovery by
Veneziano [12] and Virasoro [13] of simple formulas as a model for describing
the scattering of hadrons. These formulae revealed a rather novel mathematical
structure which was soon interpreted by the physical picture based on the
relativistic dynamics of strings by Nambu, Nielsen and Susskind [14]. This
string interpretation of ’dual resonance model’ of hadronic physics was not
influential in the development of the subject until the appearance of the 1973
paper by Goddard et al [15]. It explained in detail how the string action
could be quantized in light-cone gauge. Interestingly, among the massles
string states, there is one that has spin two. In 1974, it was shown by Scherk
and Schwarz [16] and independently by Yoneya [17] that this particle interacts
like graviton, so the theory actually includes general relativity. This lead
them to propose that string theory should be used for unification rather than
for hadrons. This implied, in particular that the string length scale should
be comparable to the plank length, rather than the size of hadrons i.e.,
$10^{-15}m$, as it was previously assumed. All this made string theory a
potential candidate to be a theory of quantum gravity.

String theory replaces all elementary point-particles that form matter and its
interactions with a single extended object of vanishing length. Thus every
known elementary particle, such as the electron, quark, photon or neutrino
corresponds to a particular vibration mode of the string. The diversity of
these particles is due to the different properties of the corresponding string
vibrations. In fact the laws of quantum mechanics tell us that a single
elementary string has infinite number of vibrational states. Since each such
vibrational state behaves as a particular type of elementary particle, string
theory seems to contain infinite types of elementary particles. This would be
in contradiction with what we observe in nature were it not for the fact that
most of these elementary particles in string theory turn out to be very heavy,
and not observable in present experiments. Thus there is no immediate conflict
between what string theory predicts and what we observe in actual experiments.
On the other hand these additional heavy elementary particles are absolutely
essential for getting finite answers in string theory.

The possible advantage of string theory is that the anomalies faced by
Supergravity are fixed naturally by the extended nature of strings. The analog
of a Feynman diagram in string theory is a two-dimensional smooth surface, and
the loop integrals over such a smooth surface lack the zero-distance, infinite
momentum problems of the integrals over particle loops. In string theory
infinite momentum does not even mean zero distance, because for strings, the
relationship between distance and momentum is roughly like

$\bigtriangleup L\sim\dfrac{\hbar}{p}+\alpha^{\prime}\dfrac{p}{\hbar}$ (24)

The parameter $\alpha^{\prime}$ is related to the string tension, the
fundamental parameter of string theory, by the relation

$T_{string}=\dfrac{1}{2\pi\alpha^{\prime}}$ (25)

The above relation implies that a minimum observable length for a quantum
string theory is

$L_{min}\sim 2\sqrt{\alpha^{\prime}}$ (26)

Thus zero-distance behavior which is so problematic in quantum field theory
becomes irrelevant in string theories, and this makes string theory very
attractive as a theory of quantum gravity.

If string theory is a theory of quantum gravity, then this minimum length
scale should be at least the size of the Planck length, which is the length
scale made by the combination of Newton’s constant, the speed of light and
Planck’s constant

$L_{p}=\sqrt{\dfrac{\hbar G_{N}}{c^{3}}}=1.6\times 10^{-35}m$ (27)

All was well, but this was only consistent if the dimension of spacetime is 26
and had only gauge bosons in it. Moreover these bosonic string theories are
all unstable because the lowest excitation mode, or the ground state, is a
tachyon. Adding fermions to string theory introduces a new set of negative
norm states or ghosts. String theorists learned that all of these bad ghost
states decouple from the spectrum when two conditions are satisfied: the
number of spacetime dimensions is 10, and theory is supersymmetric, so that
there are equal numbers of bosons and fermions in the spectrum. The resulting
consistent string theories are called Superstring theories and they do not
suffer from the tachyon problem that plagues bosonic string theories.

A very nice feature of such superstring theories is that, in 10-dimensions the
gauge and gravitational anomalies cancel for $E_{8}\times E_{8}$ group and the
$SO(32)$ group. It was then found that when the extra six-dimensional space is
compactified, the four-dimensional world contains all the required fermions
and the standard model gauge groups. Supersymmetry could remain unbroken till
the electroweak scale to take care of the gauge hierarchy problem. This the
appears to be the unified theory of all know interactions. At that time
(1984-85), string theorists believed there were five distinct superstring
theories. They differ by very general properties of the strings:

  * •

In the first case (Type I) the strings are unoriented and insulating and can
have boundaries in which case they carry electric charges on their boundaries.

  * •

In two theories (the Type IIA and Type IIB) the strings are closed and
oriented and are electrical insulators.

  * •

In two theories(the heterotic superstrings with gauge group SO(32) and
$\mathbf{E_{8}\times E_{8}}$) the strings are closed, oriented and
sperconducting.

But now it is known that this naive picture was wrong, and that the the five
superstring theories are connected to one another as if they are each a
special case of some more fundamental theory. In the mid-nineties it was
learned that various string theories floating around were actually related by
duality transformations known as T-duality and S-duality. T-duality is a
symmetry of string theory, relating type IIA and type IIB string theory, and
the two heterotic string theories. S-duality relates Type I string theory to
the heterotic SO(32) theory. Using various known dualities between different
compactification of different string theories one can now argue that all five
string theories are different ways of describing a single theory. These ideas
have collectively become known as M-theory, where M is for membrane, matrix,
or mystery, depending on your point of view!

In string theory, we assume that the particles we see around us are actually
like strings. Since the entire string propagates with time, we have to apply
boundary conditions to the end points for consistency. This lead us to either
open or closed strings, which have different boundary conditions. When this
theory was extended to a membrane (or brane for short), one has to apply
boundary conditions to its boundary surfaces. This can then be extended to
higher $n$-dimensional branes. In general, branes are static classical
solutions in string theories. A $p$-brane denotes a static configuration which
extends along $p$-spatial directions and is localized in all other directions.
A $p$-brane is described by a $(p+1)$-dimensional gauge field theory. Strings
are equivalent to 1-branes, membranes are 2- branes and particles are
0-branes.

A special class of p-branes in string theory are called D-branes. Roughly
speaking, a D-brane is a p-brane where the ends of open strings are localized
on the brane. D-branes were discovered by investigating T-duality for open
strings. Open strings don’t have winding modes around compact dimensions, so
one might think that open strings behave like particles in the presence of
circular dimensions.

Although these theories now appear to be far from any experiments, it is now
established that these theories have the prospect of becoming theory of
everything. The scale at which this theory is operational is close to the
Planck scale. This makes it experimentally non-viable for a very long time, or
probably at any time!

##  4 Braneworld Models

The large separation between the weak scale $(10^{3}Ge)$ and the traditional
scale of quantum gravity, the Planck scale $(10^{19}GeV)$ is one of the most
puzzling aspects of nature. This is known as the hierarchy problem. One
theoretical means of solving this problem is to introduce supersymmetry.
Alternatively one may hope to address the hierarchy by exploiting the geometry
of spacetime. An extremely popular theory which cures the hierarchy problem by
changing the geometry of spacetime with extra space dimensions is the so-
called braneworld scenario.

This phenomenological model has been motivated by the work of Horava and
Witten [18], who found a certain 11-dimensional string theory scenario where
the fields of the standard model are confined to a 10-dimensional
hypersurface, or brane. In this picture, the non-gravitational degrees of
freedom are represented by strings whose endpoints reside on the brane and on
the other hand, gravitational degrees of freedom in string theory are carried
by closed strings, which cannot be tied-down to any lower-dimensional object.
Hence, the main feature of this model is that the standard model particles are
localized on a three dimensional space called the brane, while gravity can
propagate in $4+n$ dimensions called the bulk. It is usually assumed that all
$n$ dimensions are transverse to the brane and have a common size $R$.
However, the brane can also have smaller extra dimensions associated with it,
of size $r\ll R$ leading to effects similar to a small finite thickness.

The three main features of braneworld models are

  1. 1.

Localization of standard model particles on the brane: A first particle
physics application of this idea was put forwarded by Rubakov and Shaposhnikov
[19] and independently by Akama [20].

  2. 2.

Localization of gauge fields on the brane: A mechanism for gauge field
localization within the field theory context was proposed by Dvali and Shifman
[21]. Localization of gauge fields is a rather natural property of D-branes in
closed string theories [22].

  3. 3.

Obtaining four-dimensional gravity on the brane: All the existing braneworld
models obtain the laws of (3+1) dimensional gravity on the brane as their low
energy approximation.

The size and geometry of the bulk, as well as the types of particles which are
allowed to propagate in the bulk and on the brane, vary between different
models. Some important braneworld models are discussed briefly here in the
order of their appearance. Some what detailed discussion is given in the
reference [23].

###  4.1 Braneworlds with Compact Extra Dimensions

Here to obtain (3+1) dimensional gravity on the brane the idea of KK
compactification is combined with braneworld idea. This was proposed in 1998
by Arkani-Hamed, Dimopoulos and Dvali[24] along with Antoniadis [25] . The
additional dimensions are compact may be as large as as millimeter! As one of
its attractive features, the model can explain the weakness of gravity
relative to the other fundamental forces of nature. In the brane picture, the
other three SM interactions are localized on the brane, but gravity has no
such constraint and “leaks” into the bulk. As a consequence, the force of
gravity should appear significantly stronger on small say, sub-millimeter
scales, where less gravitational force has “leaked”. This opens up new
possibilities to solve the Higgs mass hierarchy problem and gives rise to new
predictions that can be tested in accelerator, astrophysical and table-top
experiments.

The action for gravity in (4+n) dimensions is give by,

$S_{4+n}=\dfrac{M_{\ast}^{2+n}}{2}\int d^{4}x\int_{0}^{2\pi
R}d^{n}y\sqrt{G}{R_{4+n}}+\int d^{4}x\sqrt{g}(T+L_{SM})$ (28)

where $M_{\ast}\sim(1-10)TeV,$ $g(x)=G(x,y=0)$ and $T+\langle
L_{SM}\rangle=0.$ The low effective four dimensional action for a zero mode
takes the form,

$S=\dfrac{M_{\ast}^{2+n}{2\pi R}^{n}}{2}\int
d^{4}x\sqrt{{g_{zm}}}{R_{zm}}+\int d^{4}x\sqrt{g}(T+L_{SM}).$ (29)

Comparing it with standard four dimensional pure gravity action we get,

$M_{pl}^{2}=M_{\ast}^{2+n}(2\pi R)^{n}.$ (30)

Postulating that new quantum gravity scale is at a few TeV, we find the size
of the extra dimensions to be,

$R=10^{-17+30/n}cm.$ (31)

For one extra dimension, $n=1$, the size of extra dimension would be $R\sim
10^{13}cm$. This is excluded since it would have modified gravity in solar
system scale. For $n=2$ we get $R\sim 10^{-2}cm$, which is interesting since
it predicts modification of four dimensional laws of gravity at submillimeter
scale.

Two static sources on the brane interact with the following non-relativistic
gravitational potential

$V(r)=-G_{N}m_{1}m_{2}\sum_{n=-\infty}^{n=+\infty}|\Psi_{n}(y=0)|^{2}\dfrac{e^{-m_{n}r}}{r},$
(32)

where $\Psi_{n}(y=0)$ denotes the wave function of $n^{th}$ KK mode at a
position of the brane and $m_{n}=|n|/R$. If $r\gg L$ from the above expression
we get

$V(r)=\dfrac{-G_{N}m_{1}m_{2}}{r}.$ (33)

This recovers the conventional four dimensional law of Newtonian dynamics. In
the limit $r\ll L$ we get,

$V(r)=\dfrac{-m_{1}m_{2}}{M_{\ast}^{2+n}r^{1+n}}.$ (34)

This is the law of (4+n) dimensional gravitational interactions. Therefore,
the laws of gravity are modified at distances of order R.

###  4.2 Braneworlds with Wrapped Extra Dimensions

This phenomenon of localizing gravity was discovered by Randall and Sundrum
[26] in 1999. RS brane-worlds do not rely on compactification to localize
gravity on the brane, but on the curvature of the bulk, sometimes called
“warped compactification”. What prevents gravity from leaking into the extra
dimension at low energies is a negative bulk cosmological constant. There are
two popular models. The first, called RS1, has a finite size for the extra
dimension with two branes, one at each end. The second, RS2, is similar to the
first, but one brane has been placed infinitely far away, so that there is
only one brane left in the model. They also used their model to explain the
hierarchy problem [27] in particle physics.

For simplicity, we consider RSII model which has a single brane embedded in
five dimensional bulk with negative cosmological constant. The action for this
model is given by,

$S_{5}=\dfrac{M_{\ast}^{3}}{2}\int
d^{4}x\int_{-\infty}^{+\infty}dy\sqrt{G}(R_{5}-2\Lambda)+\int
d^{4}x\sqrt{g}(T+L_{SM}),$ (35)

where $\Lambda$ denotes the negative cosmological constant and $T$ is the
brane tension. The equation of motion derived from this action is given by,

$M_{\ast}\sqrt{G}(R_{AB}-\frac{1}{2}G_{AB}R)=-M_{\ast}^{3}\Lambda\sqrt{G}G_{AB}+T\sqrt{g}g_{\mu\nu}\delta_{A}^{\mu}\delta_{B}^{\nu}\delta(y).$
(36)

In this convention the brane is located in extra space at $y=0$. The above
equations have a solution in four dimensional world volume as

$ds^{2}=e^{-|y|/R}\eta_{\mu\nu}dx^{\mu}dx^{\nu}+dy^{2}.$ (37)

It is important to emphasize that the five dimensional action is integrable
with respect to $y$ for the zero mode. That is,

$\dfrac{M_{\ast}^{3}}{2}\int
d^{4}x\int_{-\infty}^{+\infty}dy\sqrt{G}R_{5}\longrightarrow\dfrac{M_{\ast}^{3}(2R)}{2}\int
d^{4}x\sqrt{g}R.$ (38)

The result of this integration is a conventional four dimensional action.
Hence we find the relation between four dimensional Plank mass and $M_{\ast}$,

$M_{pl}^{2}=M_{\ast}^{3}(2R)$ (39)

This looks similar to that in ADD model with one extra dimension. The
similarity is due to the fact that the effective size of the extra dimension
that is felt by the zero-mode graviton is finite and is of the order of $R$ in
both the models.

Besides the zero-mode there are an infinite number of KK modes. Since the
extra dimension is not compactified the KK modes have no mass gap. In the zero
mode approximation these states are neglected. However at distances smaller
than the size of the extra dimension, the effects of these modes become
important.

The static potential between tow sources on the brane is given by

$V(r)=\dfrac{-G_{N}m_{1}m_{2}}{r}(1+\dfrac{(2R)^{2}}{r^{2}}).$ (40)

The first term is the conventional four dimensional law of Newtonian dynamics
whereas the second term is due to exchange of KK modes which becomes dominant
when $r\lesssim R.$

###  4.3 Braneworlds with Infinite Volume Extra Dimensions

This mechanism of obtaining (3+1) gravity on the brane is different from the
earlier two as it allows the volume of the extra dimension to be infinite.
This model was proposed in 2000 by G.R.Dvali, G.Gabadadze and M.Porrati [28].
In the first model the four dimensional gravity could be reprodused at large
distances due to finite volume of extra space. This is usually done by
compactifying the extra space. Alternatively this done by warping the extra
dimensions in the second model where still the volume of extra space is
finite. But in this scenario the size of the extra dimensions does not need to
be stabilized since the extra dimensions are neither compactified nor wrapped
because of the presence of infinite-volume extra dimensions and hence gravity
is modified at large distances. This gives rise to new solutions for late-time
cosmology and acceleration of the universe which comes from type Ia supernovae
observations. This can also explain dark energy problem and Cosmic Microwave
Background.

The action in five dimensions with one infinite volume extra dimension is give
by,

$S_{5}=\dfrac{M_{\ast}^{3}}{2}\int
d^{4}x\int_{-\infty}^{+\infty}dy\sqrt{G}R_{5}+\int
d^{4}x\sqrt{g}(\frac{M_{pl}^{2}}{2}T+L_{SM})$ (41)

To study the gravity described by this model, we introduce the quantity

$r_{c}\sim M_{pl}^{2}/M_{\ast}^{3}.$ (42)

When $r_{c}\rightarrow\infty$ the four dimensional term dominates but in the
opposite limit $r_{c}\rightarrow 0,$ the five dimensional term dominates.
Therefore we expect that for $r\ll r_{c}$ to recover the four dimensional laws
on the brane while for $r\gg r_{c}$ five dimensional laws.

The static gravitational potential between the sources in the four dimensional
world volume of the brane is given by,

$V(r)=-\frac{1}{8\pi^{2}M_{pl}^{2}}\frac{1}{r}\\{sin(\frac{r}{r_{c}})Ci(\frac{r}{r_{c}})+1/2cos(\frac{r}{r_{c}})[\pi-2Si(\frac{r}{r_{c}})]\\}$
(43)

where $Ci(z)\equiv\gamma+Len(z)+\int_{0}^{z}(cos(t)-1)dt/t$,
$Si(z)\equiv\int_{0}^{z}(sin(t)dt/t$ and $\gamma\simeq 0.77$ is the Euler-
Mascheroni constant, and the distance $r_{c}$ is defined as follows,

$r_{c}\simeq\frac{M_{pl}^{2}}{2M_{\ast}^{3}}.$ (44)

In this model $r_{c}$ is assumed do be of the order of the present Hubble
size, which is equivalent to the choice $M_{\ast}\sim 10-100MeV$. It is useful
to study the short distance and the long distance behavior of this expression.
At short distance, when $r\ll r_{c}$ we get,

$V(r)=-\frac{1}{8\pi^{2}M_{pl}^{2}}\frac{1}{r}\\{\frac{\pi}{2}+[-1+\gamma+ln(\frac{r}{r_{c}})]\frac{r}{r_{c}}+O(r^{2})\\}$
(45)

Therefore, at short distances the potential has the correct four dimensional
Newtonian $1/r$ scaling. This is subsequently modified by the logarithmic
‘repulsion’ term in the above expression. At large distances $r\gg r_{c}$, the
potential takes the form,

$V(r)=-\frac{1}{8\pi^{2}M_{pl}^{2}}\frac{1}{r}\\{\frac{r_{c}}{r}+O(r^{2})\\}$
(46)

Thus, the long distance potential scales as $1/r^{2}$ in accordance with laws
of five dimensional theory.

###  4.4 Braneworlds with Universal Extra Dimensions

Universal Extra Dimensions model was proposed by Appelquist, Cheng and
Dobrescu [29] in 2001. In this model the extra dimensions are accessible to
all the standard model fields, referred to here as universal dimensions which
may be significantly larger. The key element is the conservation of momentum
in the universal dimensions. In the equivalent four-dimensional theory, this
implies KK number conservation. In particular there are no vertices involving
only one non-zero KK mode, and consequently there are no tree-level
contributions to the electroweak observables. Furthermore, non-zero KK modes
may be produced at colliders only in groups of two or more. Thus, none of the
known bounds on extra dimensions from single KK production at colliders or
from electroweak constraints applies for universal extra dimensions.

The full Lagrangian of this model includes both the bulk and the boundary
Lagrangian. The bulk Lagrangian is determined by the SM parameters after an
appropriate rescaling. The very important property of this model is the
conservation of KK parity that implies the absence of tree level KK
contributions to low energy processes taking place at scales very much less
than $1/R$ In the effective four dimensional theory, in addition to the
ordinary particles of the SM, denoted as zero modes, there are infinite towers
of the KK modes. There is one such tower for each SM boson and two for each SM
fernion, while there also exist physical neutral and churched scalars with
$(n\geq 1)$ that do not have any zero mode partners.

##  5 Conclusion

Many of the major developments in fundamental physics of the past century
arose from identifying and overcoming contradictions between existing ideas.
For example, the incompatibility of Maxwell’s equations and Galilean
invariance led Einstein to propose the special theory of relativity.
Similarly, the inconsistency of special relativity with Newtonian gravity led
him to develop the general theory of relativity. More recently, the
reconciliation of special relativity with quantum mechanics led to the
development of quantum field theory. We are now facing another crisis of the
same character. Namely, general relativity appears to be incompatible with
quantum field theory. Any straight forward attempt to ’quantize’ general
relativity leads to a nonrenormalizable theory. This has lead to theories like
superstrings and braneworlds. Eventhough these theories look rather exotic, at
least for the moment. Yet they lead to important insights and also provide a
framework for addressing a number of phenomenological issues. Further more,
new ideas emerge in approaching fundamental problems which have been puzzling
physicists over the centuries. All this makes the subject interesting and
lively. The question is whether Nature follows any of the routes being
explored in this context?

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Present Address: ]Materials Science Center, National Renewable Energy
Laboratory, Golden, CO 80401, USA.

# Local probe investigation of the spin dynamics in the kagome and inter-
layers of orthorhombic barlowite Cu4(OD)6FBr: 79Br and 63Cu NQR study

Takashi Imai<EMAIL_ADDRESS>Department of Physics and Astronomy, McMaster
University, Hamilton, Ontario, L8S 4M1, Canada  Jiaming Wang  Department of
Physics and Astronomy, McMaster University, Hamilton, Ontario, L8S 4M1, Canada
Rebecca W. Smaha  [  Stanford Institute for Materials and Energy Sciences,
SLAC National Accelerator Laboratory, Menlo Park, CA 94025, USA  Department of
Chemistry, Stanford University, Stanford, CA 94305, USA  Wei He  Stanford
Institute for Materials and Energy Sciences, SLAC National Accelerator
Laboratory, Menlo Park, CA 94025, USA  Department of Materials Science and
Engineering, Stanford University,Stanford, CA 94305, USA  Jiajia Wen  Stanford
Institute for Materials and Energy Sciences, SLAC National Accelerator
Laboratory, Menlo Park, CA 94025, USA  Young S. Lee  Stanford Institute for
Materials and Energy Sciences, SLAC National Accelerator Laboratory, Menlo
Park, CA 94025, USA  Department of Applied Physics, Stanford University,
Stanford, CA 94305, USA

###### Abstract

We report 79Br and 63Cu nuclear quadrupole resonance (NQR) in the paramagnetic
state above $T_{\text{N}}=15$ K of the antiferromagnetic orthorhombic phase of
barlowite Cu4(OD)6FBr consisting of a layered kagome structure. The divergent
behavior of the longitudinal ${}^{79}(1/T_{1})$ and transverse
${}^{79}(1/T_{2})$ relaxation rates observed at 79Br sites evidences that
critical slowing down of Cu spin fluctuations sets in below $\sim 20$ K. This
means that one or more Cu sites, most likely at the interlayer Cu(3,4,5) sites
between the kagome planes, undergo the antiferromagnetic phase transition in a
fairly conventional way. On the other hand, the 63Cu NQR signal intensity is
gradually wiped out below $\sim 30$ K, pointing toward gradual spin freezing
of the kagome layers instead. These contrasting findings suggest significant
roles played by magnetic frustration effects within the kagome layers.

††preprint: APS/123-QED

##  I Introduction

The two-dimensional kagome lattice Heisenberg antiferromagnet is formed by a
network of corner sharing triangles, and believed to be one of the most
promising avenues to realize quantum spin liquids [1, 2, 3]. Theoretically,
the kagome lattice has many competing ground states with nearly identical
energy, and hence its physical properties are sensitive to structural
disorders [4, 5], spin vacancies within the kagome plane [6], and possible
coupling with extra defect spins outside the kagome planes [7]. The candidate
kagome quantum spin liquid material Zn-barlowite (Zn0.95Cu0.05)Cu3(OD)6FBr
(abbreviated as Zn0.95 hereafter) is formed by corner sharing triangles of
Cu2+ ions, and is one of the least structurally disordered spin-1/2 kagome
materials known to date [8, 9, 10, 11, 12, 13, 14]. The nearest-neighbor Cu-Cu
super-exchange interaction across the O2- ion is estimated to be as large as
$J\sim 160$ K [9], yet Zn0.95 exhibits no magnetic phase transition down to
the sub-Kelvin range [10, 11].

On the other hand, $\sim$5 % of the interlayer non-magnetic Zn2+ sites of
Zn0.95 are replaced by Cu2+ defects with spin-1/2 [9]. The interlayer disorder
is believed to play crucial roles in the physical properties of the kagome
planes, such as the drastic increase in the uniform bulk spin susceptibility
in the low temperature range below $\sim 50$ K [7]. In fact, spin singlets
emerge only gradually below $\sim 30$ K in Zn0.95 with a distribution of
inhomogeneous gaps [14]. Moreover, our recent two-dimensional NMR measurements
[15] based on inverse Laplace transform $T_{1}$ analysis technique [16, 17]
demonstrated that $\sim 60$% of 19F sites in Zn0.95 are located within spin
polarized domains induced in the vicinity of the interlayer Cu2+ defect spins.
The precise mechanism of the perturbation caused by the interlayer Cu2+ defect
spins on the kagome planes is yet to be clarified.

In barlowite Cu4(OD)6FBr (abbreviated as Cu4 hereafter) [18, 19, 20, 21, 22,
23, 24, 11, 9, 10, 25], all the interlayer cation sites are occupied by Cu2+
ions at three slightly different locations: Cu(3), Cu(4), and Cu(5) sites, as
shown in Fig. 1(a) [24, 11, 9, 10, 25]. The three-dimensional network of these
Cu2+ sites shown in Fig. 1(b) consists of kagome layers bridged by interlayer
Cu(3,4,5), and undergo a magnetic long range order at $T_{\text{N}}=15$ K with
canted ferromagnetic moments [18, 19, 20, 21, 22, 11, 9]. Thus Cu4 may be
considered as the parent antiferromagnet of Zn0.95, and serves as a model
system to investigate the coupling between the interlayer Cu2+ defects and the
kagome layers in Zn0.95.

Despite the concerted efforts to clarify the nature of magnetism in Cu4, the
complicated structure makes it difficult to reach a consensus in part because
the crystallographic and physical properties of Cu4 are known to depend
slightly on the synthesis route [23, 9]. This study focuses solely on the
orthorhombic variant of Cu4, barlowite 1 phase identified in [9]. In barlowite
1, the equilateral triangles of the kagome planes are slightly elongated along
the a-axis below the orthorhombic structural transition at 265 K, resulting in
two distinct Cu(1) and Cu(2) sites within the kgome planes. The population
ratio between the kagome Cu(1), kagome Cu(2), and the interlayer Cu(3,4,5)
sites is 1 vs. 2 vs. 1. We note that a different synthesis route produces
another variant, barlowite 2 phase of Cu4, which remains hexagonal at low
temperature and exhibits a pinwheel valence bond crystal ground state [9].

In the earlier studies of Cu4 barlowite, it has been agreed that the
interlayer Cu(3,4,5) sites possess a robust ordered moment of $\sim
0.5\mu_{\text{B}}$ or even greater, where $\mu_{\text{B}}$ represents a Bohr
magneton [21, 22]; it is a typical magnitude for quasi two dimensional
Heisenberg antiferromagnet with quantum reduction [26]. On the other hand, the
average moments within the kagome planes at Cu(1) or Cu(2) sites may be small
[21, 22]. However, no firm consensus has been reached yet on the unique spin
structure established below $T_{\text{N}}$, and for the barlowite 2 variant,
recent work suggests instead that the kagome Cu(1) and Cu(2) sites may host
the q = 0 structure with sizable moments [9], which is generally favored by
the frustrated kagome lattice to accommodate a high degree of degeneracy.
Equally perplexing is that the magnetically ordered state of Cu4 appears to
emerge below $T_{\text{N}}$ through a conventional three-dimensional long
range order in a $\mu$SR study [11], while an early 79Br NMR work emphasized
somewhat unconventional nature of the ordered state that seems slowly freezing
across $T_{\text{N}}$ [20]. This apparent discrepancy may be caused by
different samples, and underscores the importance of working on a phase pure
barlowite sample.

In this article, we report a detailed 79Br and 63Cu nuclear quadrupole
resonance (NQR) investigation of Cu4 in zero external magnetic field using a
phase-pure, well-characterized powder sample of orthorhombic barlowite 1 [9],
and compare the results with the 19F NMR data we recently reported for the
same sample specimen [15]. From the 79Br NQR measurements, we demonstrate that
at least one of the crystallographically inequivalent Cu sites, most likely
the interlayer Cu(3,4,5) sites, exhibit the typical signature of three-
dimensional antiferromagnetic long range order, i.e. the critical slowing down
of spin fluctuations below $\sim 20$ K that precedes the long range order at
$T_{\text{N}}$. In contrast, the gradual loss of 63Cu NQR signal intensity
suggests that gradual spin freezing sets in below $\sim 30$ K within the
kagome layers. These contrasting findings suggest an interesting possibility
that the frustrated kagome planes in barlowite 1 are forced into the ordered
state when the interlayer Cu(3,4,5) sites undergo a robust long range order.
This is reminiscent of the successive magnetic transitions observed for
barlowite 2, where ordering of the interlayer Cu moments upon cooling drives
the ordering of the kagome Cu moments [9].

Figure 1: (a) The crystal structure of the orthorhombic “barlowite 1” phase of
Cu4(OD)6FBr [9] viewed along the b-axis, visualized using Vesta [27]: Cu
(blue), O (red), D (gray), F (green), and Br (brown). Black line shows the
unit cell. Distorted kagome planes are formed by Cu(1,2), O, and D sites,
while Cu(3,4,5), F and Br sites are located between two adjacent kagome
planes. (b) A tilted skeleton view depicting only the Cu sites, showing four
interlayers formed by Cu(3,4,5) sites that sandwich three slightly distorted
kagome layers. (c) Local geometry of the Br site surrounded by two Cu(1) and
four Cu(2) sites in adjacent kagome planes as well as three Cu(3,4,5) sites
within the same interlayer. For clarity, only Br and Cu sites are shown.
Figure 2: (a) 79,81Br and 63,65Cu NQR lineshapes observed for Cu4(OD)6FBr at
representative temperatures with the separation time $\tau=10~{}\mu$s between
the $\pi$/2 and $\pi$ pulses. The origin of the vertical axis is shifted at
different temperatures for clarity. (b) Deconvolution of the 63,65Cu NQR
lineshape observed at 30 K ($\bullet$, same data as in panel (a)) into 63Cu
only ($\blacktriangle$) and 65Cu only ($\filleddiamond$) NQR lineshapes. See
the main text for the details of the procedures for deconvolution. (c) 79,81Br
and 63,65Cu NQR lineshapes observed for (Zn0.95Cu0.05)Cu3(OD)6FBr at 4.2 K
with somewhat longer $\tau=15~{}\mu$s [14]. The * mark shows the small signals
observed at the same frequency as 63Cu(B) sites in Cu4. All solid and dashed
lines are a guide for eyes.  Figure 3: (Main panel) The transverse spin echo
decay curve $M(2\tau)$ observed for the 79Br sites of Cu4. The solid curves
are the best fit with an empirical form
$M(2\tau)=M(0)~{}exp(-[2\tau/^{79}T_{2}]^{\alpha})$, where $\alpha$ was
consistently found to be $\sim 1.3$ from the best fit. We corrected the
intensity for the Boltzmann factor by multiplying temperature $T$, and set the
overall intensity by normalizing the 60 K intensity extrapolated to $2\tau=0$
as 1. No other normalization was made at different temperatures, but the spin
echo intensity in the limit of $2\tau=0$ is conserved at different
temperatures above $T_{\text{N}}$. (Inset) The transverse relaxation rate
${}^{79}(1/T_{2})$ observed at 79Br sites of Cu4. Throughout this article, the
region above $\sim 60$ K with light blue shading marks the temperature range
where the slow lattice dynamics enhance the NMR relaxation rates of the
quadrupolar nuclei [20, 28].

##  II Experimental

We synthesized a deuterated powder sample of the barlowite 1 phase of Cu4
based on the recipe described elsewhere [9]. We refer readers to [9] for
detailed bulk characterization of the barlowite 1 phase based on the
measurements of powder X-ray diffraction, magnetization, and specific heat. We
conducted 63,65Cu and 79,81Br (all with nuclear spin 3/2) NQR measurements
with standard pulsed NMR spectrometers using the same Cu4 sample investigated
previously with 19F NMR [15]. For the measurements of the nuclear spin-lattice
relaxation rate $1/T_{1}$, we used a separation time $\tau=6~{}\mu$s between
the $\pi$/2 and $\pi$ radio frequency pulses so that we capture as much
signals with short transverse relaxation time $T_{2}$ as possible. For the NQR
lineshape measurements, we used a somewhat longer pulse separation time
$\tau=10~{}\mu$s to reduce the ring down in the lower frequency range below
$\sim$30 MHz.

We measured the $1/T_{1}$ recovery curve $M_{\text{z}}(t)$ by applying an
inversion pulse prior to the spin echo sequence, and fitted the observed
results with a stretched exponential form,
$M_{\text{z}}(t)=M_{\text{0}}-A~{}\text{exp}(-(3t/T_{1})^{\beta})$, where the
factor 3 originates from the matrix element for nuclear spin $I=3/2$; besides
$1/T_{1}$, the saturated nuclear magnetization $M_{\text{0}}$, the inverted
nuclear magnetization $A$, and the stretched exponent $\beta$ are the free
parameters of the fit. We found that the distribution in the magnitude of
$1/T_{1}$ is fairly small and $\beta\simeq 1$ in most of the temperature range
of our concern between $T_{\text{N}}=15$ K and 60 K. The only exception was
one of the two Cu peaks below 30 K, where the glassy nature of spin dynamics
strongly manifests itself and the signal intensity is gradually wiped out.

##  III Results and Discussion

###  III.1 79Br NQR lineshapes

In Fig. 2(a), we summarize the 79,81Br and 63,65Cu NQR lineshapes observed at
representative temperatures in zero applied magnetic field. The overall
intensity of each lineshape is corrected for the Boltzmann factor by
multiplying temperature $T$, and also for the frequency-dependent sensitivity
by dividing the observed intensity by the frequency squared, $f^{2}$. The
natural abundance of 79Br and 81Br nuclei is 51% and 49%, respectively, and
nearly equal NQR intensities observed at 30 K for 79Br and 81Br confirms that
our procedures work well for the estimation of the relative intensities of
different peaks.

Since the 79,81Br NQR peak frequencies are proportional to their nuclear
quadrupole moment ${}^{79,81}Q$, the ratio between the peak frequency of 81Br
and 79Br is expected to be ${}^{81}f/^{79}f=~{}^{81}Q/^{79}Q=0.837$. Likewise,
the ratio between the peak frequencies of 65Cu and 63Cu would be
${}^{65}f/^{63}f=~{}^{65}Q/^{63}Q=0.927$. Therefore, we can distinguish the
79,81Br and and 63,65Cu peaks as marked in Fig. 2(a). The observed 79,81Br
peak frequencies are similar to an earlier report [20], but our 79,81Br peaks
are well separated from each other; it assures us that our sample of barlowite
1 phase has less structural disorder. The intensity of 65Cu peak is smaller
than that of 63Cu peak, because the natural abundance of 65Cu (31%) is much
smaller than that of 63Cu (69%).

Below $T_{\text{N}}$, the static hyperfine magnetic fields arising from
ordered Cu moments perturb the energy levels of the nuclear spins and shift
the resonance frequencies. At 4.2 K, we confirmed that the Zeeman-perturbed
NQR lineshape extends above 60 MHz. One can sometimes gain local information
about the magnitude and orientation of the ordered moments belonging to
multiple sub-lattices from comprehensive measurements and analysis of the
lineshapes below $T_{\text{N}}$ [29]. However, the analysis of the Zeeman-
perturbed NQR lineshapes is a highly complicated problem if the hyperfine
fields are comparable to NQR frequencies and perturbation analysis is not
applicable [30], which appears to be the case here. Accordingly, the ordered
state below $T_{\text{N}}$ is beyond the scope of the present work.

The intensities of the 79,81Br NQR peaks are conserved below 60 K down to 30
K. This is consistent with the general expectations for paramagnetic
insulators, because the number of nuclear spins detected in our spin echo
measurements is conserved. The intensity of the 79Br peak at 16 K appears
somewhat smaller, simply because the transverse relaxation rate
${}^{79}(1/T_{2})$ of 79Br sites is enhanced near $T_{\text{N}}$, as
summarized in the inset of Fig. 3. Notice that the extrapolation of the
transverse nuclear magnetization $M(2\tau)$ to $2\tau=0$ in the main panel of
Fig. 3 is actually conserved at 16 K. We also observed enhancement of
${}^{79}(1/T_{2})$ above 60 K in the region marked with blue shading in the
inset of Fig. 3. It is due to gradual freezing of the lattice distortion
commonly observed for Cu4 [20], Zn0.95 [28] and herbertsmithite ZnCu3(OH)6Cl2
[31, 32, 33].

In the case of Zn0.95, the spin echo decay curve $M(2\tau)$ at 79Br sites
develops a highly damped oscillation due to indirect nuclear spin-spin
coupling effect below 50 K [28]. It indicated Br-Br dimer formation for up to
$\sim 50$% of Br sites, as spin singlets gradually emerge in the kagome planes
with inhomogeneous gaps [14]. Such oscillatory behaviors in $M(2\tau)$ were
previously observed for other spin singlet materials as well [34, 35, 36], but
we find no such oscillations in the present case.

###  III.2 63Cu NQR lineshapes

Let us now turn our attention to two sets of 63,65Cu NQR peaks in Fig. 2(a).
In Fig. 2(b), we use the following standard procedures to deconvolute the
63,65Cu NQR lineshapes observed at 30 K into 63Cu only and 65Cu only NQR
lineshapes: First, from the 63Cu NQR signals observed at ${}^{63}f\simeq 37$
MHz near the upper end of the frequency range, we estimate the corresponding
65Cu NQR frequency ${}^{65}f=(^{65}Q/^{63}Q)\times^{63}f$ ($\simeq 34.3$ MHz).
We also estimate the intensity of the 65Cu NQR signal at ${}^{65}f$ from the
isotope ratio, $0.31/0.69=0.45$. Then one can plot the deconvoluted 65Cu NQR
lineshape around $\sim 34.3$ MHz, as shown with gray diamonds. We can repeat
this procedure down to ${}^{63}f\simeq 35$ MHz, below which we need to
subtract the estimated 65Cu NQR intensity from the overall intensity to
estimate the 63Cu contribution shown with blue triangles, then use the latter
to estimate the corresponding 65Cu NQR signal at lower frequency.

By repeating these procedures down to $\sim 29$ MHz, we identified two
distinct 63Cu peaks in Fig. 2(b): 63Cu(A) peak at $\sim$32 MHz and 63Cu(B)
peak at $\sim$35.5 MHz. The bare integrated intensity ratio between the narrow
32 MHz peak and broad 35.5 MHz peak is $1\pm 0.15$. Once we take into account
the slightly faster transverse relaxation rate ${}^{63}(1/T_{2})$ of the 32
MHz peak as shown in Fig.4(c), we estimate the intensity ratio to be $1.3\pm
0.2$. Recall that there are five non-equivalent Cu sites in Cu4, although the
differences are small between Cu(1) and Cu(2) sites as well as between Cu(3),
Cu(4), and Cu(5) sites. Additional 63,65Cu signals may have evaded our
detection due to their very large $1/T_{1}$ and $1/T_{2}$ ($\propto 1/J$ [37])
enhanced by small super-exchange interactions $J$ with neighboring Cu sites.

The 63Cu(A) and 63Cu(B) peak frequencies are comparable to the main peak
frequency $\sim$33.7 MHz observed for the kagome Cu sites of Zn0.95 (see Fig.
2(c) for comparison). Recalling that Cu(2) sites are twice more abundant than
Cu(1) sites, a plausible scenario is that 63Cu(A) and 63Cu(B) peaks arise from
the kagome Cu(1) and Cu(2) sites, respectively, but the NQR signals arising
from some of these sites are not observable depending on the interlayer
Cu(3,4,5) configuration nearby. Note that the interlayer Cu(3,4,5) sites are
located at slightly different locations, and their coupling with the kagome
planes may be locally different.

On the other hand, as shown in Fig. 5(c), the magnitude of ${}^{63}(1/T_{1})$
observed at Cu(B) sites is by a factor of $\sim 3$ greater than at Cu(A)
sites. Moreover, the Cu(B) peak frequency is nearly identical with that of the
small signals observed for Zn0.95, in which only $\sim$5% of the interlayer Zn
sites occupied by Cu (see * in Fig.2(c)). Therefore, an alternate scenario is
that the Cu(A) peak arises from some of the Cu(1) and/or Cu(2) sites, whereas
the Cu(B) peak arises from some of the Cu(3,4,5) sites. In any case, the
uncertainty in the site assignments of Cu NQR signals hardly affects our key
conclusions below.

Figure 4: The spin echo decay curve $M(2\tau)$ observed for Cu4 at (a) 63Cu(A)
sites at $\sim 32$ MHz and (b) 63Cu(B) sites at $\sim 35.5$ MHz. The spin echo
intensity is corrected for the Boltzmann factor by multiplying temperature
$T$, and normalized for the intensity observed at 50 K. The straight lines are
the best exponential fit with
$M(2\tau)=M(2\tau=0)\times\text{exp}(-2\tau/^{63}T_{2})$. Notice that the
extrapolation of the fit to $2\tau=0$ is not conserved below 30 K. (c) The
transverse relaxation rate ${}^{63}(1/T_{2})$ observed at 63Cu sites, as
determined from the exponential fit in (a) and (b). (d) The signal intensity
$M(2\tau=0)$, corrected for both the Boltzmann factor and the transverse
relaxation effects.

###  III.3 Cu spin dynamics probed by 79Br NQR

In Fig. 5(a), we summarize the temperature dependence of ${}^{79}(1/T_{1})$
measured at the 79Br NQR peak of Cu4 in comparison to the results observed for
Zn0.95 [14]. The interlayer 79Br sites of barlowite are surrounded by two
Cu(1) and four Cu(2) sites in two adjacent kagome planes and three Cu(3,4,5)
sites within the same interlayer, as shown in Fig. 1(c). In view of the
spatially extended nature of the atomic orbitals of 79Br, the hyperfine
couplings with all these Cu sites probably contribute significantly.
Therefore, ${}^{79}(1/T_{1})$ is likely to probe low frequency Cu spin
dynamics at $\sim 28.6$ MHz at all the Cu sites within the kagome layers and
interlayers.

The gradual increase of ${}^{79}(1/T_{1})$ observed for Cu4 below $\sim 60$ K
is typical when short range spin correlation grows. This is consistent with
the fact that the 79Br spin echo decay curve of Cu4 does not exhibit the
damped oscillation often seen when spin dimers form [28]. Below $\sim 20$ K,
${}^{79}(1/T_{1})$ in Fig. 5(a) as well as ${}^{79}(1/T_{2})$ in the inset of
Fig. 2 quickly grow toward $T_{\text{N}}$. These divergent behaviors of
${}^{79}(1/T_{1})$ and ${}^{79}(1/T_{2})$ slightly above $T_{\text{N}}$ are
the prototypical signature of critical slowing down of spin fluctuations
toward the three-dimensional antiferromagnetic long range order in quasi-two
dimensional antiferromagnets [38, 39]. In other words, as far as the
${}^{79}(1/T_{1})$ and ${}^{79}(1/T_{2})$ results are concerned, Cu4 appears
to be a prototypical three dimensional antiferromagnet. Also note that the
stretched exponent at 79Br sites hardly deviates from $\beta=1$ down to
$T_{\text{N}}$ as shown in Fig. 5(d), implying that ${}^{79}(1/T_{1})$ is not
significantly distributed. Therefore, the ${}^{79}(1/T_{1})$ results show
nearly homogeneous slowing down of at least one of the Cu sites, but not
necessarily all Cu sites. For example, as long as the hyperfine coupling
between the 79Br nuclear spins and the interlayer Cu(3,4,5) sites is not
negligibly small, ${}^{79}(1/T_{1})$ and ${}^{79}(1/T_{2})$ would show the
signature of critical slowing down toward $T_{\text{N}}$ when the interlayer
Cu(3,4,5) sites undergo a conventional antiferromagnetic long range order,
even if Cu(1) and Cu(2) sites do not enter a long range ordered state and
remain paramagnetic.

Figure 5: (a) ${}^{79}(1/T_{1})$ observed at 79Br sites of Cu4 ($\bullet$) and
Zn0.95 ($\times$, from [14]). (b) ${}^{19}(1/T_{1})$ observed for 19F sites of
Cu4 ($\triangle$) and Zn0.95 (+) in 2.4 T, both from [15]. Note that the
upturn of ${}^{19}(1/T_{1})$ caused by critical slowing down is absent below
20 K. Since 19F nuclear spin is 1/2 and hence immune from the fluctuations of
the electric field gradient, ${}^{19}(1/T_{1})$ does not show an upturn above
60 K, either [14]. (c) ${}^{63}(1/T_{1})$ observed at 63Cu(A) ($\diamond$) and
63Cu(B) ($\square$) peaks of Cu4. The data points below 30 K should be
considered the lower bound of the highly distributed ${}^{63}(1/T_{1})$
throughout the sample volume, because the 63Cu sites with larger relaxation
rates do not contribute to the spin echo signals detected with a finite delay
time $\tau=6~{}\mu$s. For comparison, we also show ${}^{63}(1/T_{1})$ at the
paramagnetic Cu site (P) of Zn0.95 (x) and spin singlet Cu site (S) of Zn0.95
(+) as determined by inverse Laplace transform (ILT) $T_{1}$ analysis of the
recovery curve [14]. (d) The stretched exponent $\beta$ observed for Cu4 at
79Br ($\bullet$), 19F ($\triangle$), 63Cu(A) ($\diamond$) and 63Cu(B)
($\square$) sites, shown using the same symbols as in panel (a) - (c).

It is also interesting to note that our ${}^{79}(1/T_{1})$ results are
noticeably different from the high field NMR results reported in an earlier
work [20]. The magnitude of our ${}^{79}(1/T_{1})$ is larger by as much as a
factor of $\sim 3$, and the enhancement toward $T_{\text{N}}$ seems more
pronounced than in the previous work. As noted above, the physical and
structural properties of Cu4 are known to be sensitive to synthesis conditions
[23, 40, 9], which might explain the apparent discrepancies. We do note that
79,81Br NQR peaks are barely distinguishable in the earlier report [20], which
indicates a higher level of structural disorder in the sample used for the
previous report.

###  III.4 Comparison with 19F sites

In Fig. 5(b), we reproduce ${}^{19}(1/T_{1})$ measured in 2.4 T at the 19F
sites of Cu4 and Zn0.95 [15]. ${}^{19}(1/T_{1})$ of Cu4 gradually grows from
250 K [15], but exhibits only a broad hump centered at $T_{\text{N}}$ with no
hint of critical slowing down below $\sim 20$ K. To understand this apparently
contradicting results, it is useful to recall that the atomic orbitals of 19F
sites are spatially less extended than those of 79Br. Therefore, the hyperfine
couplings of 19F sites may be limited to the six nearest neighbor O sites
located at a distance of $\sim 2.7$ Å in each of the two adjacent kagome
layers, which in turn bond with Cu(1) and Cu(2) kagome sites. (The distance
between the 19F and closest interlayer Cu(3,4,5) sites within the same
interlayer is much greater, 3.6 Å, and no O2- sites exist in between them to
transfer the spin from the Cu(3,4,5) sites to 19F sites.) In this scenario, if
critical slowing down is setting in below 20 K only at the interlayer
Cu(3,4,5) sites, ${}^{19}(1/T_{1})$ may not show a sharp peak at
$T_{\text{N}}$.

An alternate scenario to account for the absence of the sharp peak in
${}^{19}(1/T_{1})$ at $T_{\text{N}}$ would rely on a complete geometrical
cancellation of the transferred hyperfine magnetic fields at 19F sites due to
the wave-vector dependence in the hyperfine form factor [41]. If all the Cu
sites form a commensurate antiferromagnetic structure, in principle, the
fluctuating transferred hyperfine fields at the high symmetry 19F sites can
cancel out each other and $1/T_{1}$ may not exhibit any sharp anomaly at
$T_{\text{N}}$, as previously demonstrated for the case of square-lattice
Heisenberg antiferromagnet [42]. We cannot logically rule out such a scenario
in the present case. However, as shown in the next subsection, 63Cu NQR
results show hints of spin freezing below $\sim 30$ K at one or more Cu sites
rather than critical slowing down.

###  III.5 63Cu signal intensity wipeout, ${}^{63}(1/T_{1})$,
${}^{63}(1/T_{2})$

In Fig. 5(c), we summarize the temperature dependence of ${}^{63}(1/T_{1})$.
${}^{63}(1/T_{1})$ at the Cu(A) peak shows qualitatively similar temperature
dependence as ${}^{79}(1/T_{1})$ above 20 K. Near $T_{\text{N}}$, however,
${}^{63}(1/T_{1})$ at 63Cu(A) peak as well as ${}^{63}(1/T_{2})$ shown in Fig.
4(c) appear to decrease toward $T_{\text{N}}$. This apparent lack of the
signature of antiferromagnetic order, however, does not necessarily mean that
Cu spins at Cu(A) peak do not order below $T_{\text{N}}$. We note that the
63Cu(A) NQR peak progressively disappears below 30 K, as shown in Fig.2(a).
Even if we take into account the transverse relaxation effect by extrapolating
the spin echo decay curves in Fig. 4(a) to $2\tau=0$, the extrapolated
intensity $M(2\tau=0)$ decreases dramatically toward 16 K as summarized in
Fig. 4(d). Analogous signal intensity loss is often seen when spin freezing
sets in inhomogeneously with highly distributed ${}^{63}(1/T_{1})$ and
${}^{63}(1/T_{2})$ [30, 43, 17].

When spin fluctuations at some Cu(A) sites slow down locally in an
inhomogeneous manner, ${}^{63}(1/T_{1})$ and ${}^{63}(1/T_{2})$ at these sites
are enhanced. Then these Cu(A) sites no longer contribute to the spin echo
signals detected at a finite delay time $\tau=6~{}\mu$s used for
${}^{63}(1/T_{1})$ measurements [43]. In other words, ${}^{63}(1/T_{1})$ and
${}^{63}(1/T_{2})$ deduced below 30 K reflect only the Cu(A) sites which are
yet to begin to freeze, and hence the data points of ${}^{63}(1/T_{1})$ and
${}^{63}(1/T_{2})$ plotted below 30 K should be considered only the lower
bound of their distribution in space. Besides the signal intensity loss, the
deviation of the stretched exponent $\beta$ from 1 near $T_{\text{N}}$ in Fig.
5(d) underscores the inhomogeneous nature of spin fluctuations at Cu(A) sites
as well. We also confirmed from the inverse Laplace transform analysis of the
$1/T_{1}$ recovery curve that unlike the case of Zn0.95 [14] no split off peak
of spin singlets emerges in the density distribution function $P(1/T_{1})$
when the distribution in ${}^{63}(1/T_{1})$ grows.

The signal intensity loss at Cu(B) peak toward $T_{\text{N}}$ is far less
pronounced than at Cu(A) peak, as shown in Fig. 4(d). The apparent reduction
of the Cu(B) peak intensity toward $T_{\text{N}}$ in Fig. 2(a) is primarily
because of the enhancement of ${}^{63}(1/T_{2})$, as seen in Fig. 4(b) and
(c). This means that most if not all Cu(B) sites continue to contribute to the
measurements of ${}^{63}(1/T_{1})$ and ${}^{63}(1/T_{2})$. Indeed
${}^{63}(1/T_{2})$ at the Cu(B) peak in Fig. 4(c) exhibits a mild divergent
behavior toward $T_{\text{N}}$, although ${}^{63}(1/T_{1})$ increases only
slightly in the same temperature range. Thus, the signature of spin freezing
at the Cu(B) sites is less pronounced.

##  IV Summary and Conclusions

We have reported a comprehensive 79Br and 63Cu NQR investigation of
antiferromagnetic orthorhombic barlowite 1 phase of Cu4 above
$T_{\text{N}}=15$ K, and compared the results with our earlier 19F NMR
results. The results of ${}^{79}(1/T_{1})$ and ${}^{79}(1/T_{2})$ at 79Br
sites should be considered reflective of the spin dynamics at all the
structurally inequivalent Cu sites. Their divergent behaviors below 20 K
toward $T_{\text{N}}$ indicate that at least one of the inequivalent Cu sites
enters the long range ordered state in a fairly conventional manner in the
sense that critical slowing down of low frequency Cu spin fluctuations
precedes the phase transition.

On the other hand, ${}^{19}(1/T_{1})$ at 19F sites, which is more likely to
probe only the kagome Cu(1,2) sites due to their proximity, shows only a broad
hump at $T_{\text{N}}$ without any hint of critical slowing down below 20 K.
Two sets of observable Cu NQR signals show little evidence for critical
slowing down, either. Instead, the Cu NQR signal intensity is gradually wiped
out below 30 K, which is the typical signature of inhomogeneous spin freezing.
Combining all the observations, the most likely scenario is: (a) Cu(A) and
Cu(B) peaks arise from Cu(1) and Cu(2) sites within the kagome planes, (b) it
is the Cu NQR signals at Cu(3,4,5) that are missing in the lineshape in Fig.
2, (c) Cu(3,4,5) sites enter the long range ordered state in a rather
conventional manner, and hence 79Br sites exhibit the signature of critical
slowing down, but (d) kagome Cu(1) and Cu(2) sites enter the ordered state in
a glassy manner, resulting in the 63Cu NQR signal intensity loss below 30 K.

The contrasting behaviors inferred for the interlayer and kagome layers
suggest that magnetic frustration effects within the kagome layers play
significant roles in the spin dynamics of the barlowite 1 phase of Cu4, even
though elongation of the kagome layers along the a-axis partially alleviates
the frustration effects. Although Cu4 is a three-dimensional antiferromagnet,
the spin-1/2 kagome planes are in close proximity to other competing ground
states, such as valence bond crystals and quantum spin liquids. Here, the
robust ordering of the interlayer spins may be the deciding factor in inducing
the ordering of the kagome moments. This scenario is consistent with the
earlier finding that the single crystal sample of barlowite 2 phase with
slightly different crystal structure exhibits a potential signature of the
pinwheel valence bond phase below 10 K due to frustration effects, before the
kagome planes enter the ${\bf q}={\bf 0}$ long range ordered state at
$T_{\text{N}}=6$ K, induced by ordering of the interlayer moments [9]. It
would be interesting to compare the NQR results between barlowite 1 and
barlowite 2 phases, but the latter has been successfully synthesized only in
the form of small single crystals unsuitable for NQR measurements [9].

###### Acknowledgements.

The work at McMaster was supported by NSERC (T.I.). The work at Stanford and
SLAC (sample synthesis and characterization) was supported by the U.S.
Department of Energy (DOE), Office of Science, Basic Energy Sciences,
Materials Sciences and Engineering Division, under contract no. DE-
AC02-76SF00515 (Y.S.L. and J.W.). R.W.S. was supported by a NSF Graduate
Research Fellowship (DGE-1656518). T.I. acknowledges that the manuscript was
finalized at Kavli Institute for Theoretical Physics, University of California
at Santa Barbara, during his participation in the qmagnets23 program.

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