# Determination of the dynamic Young’s modulus of quantum materials in
piezoactuator-driven uniaxial pressure cells using a low-frequency a.c. method

Caitlin I. O’Neil1,2<EMAIL_ADDRESS>Zhenhai Hu1,2 Naoki Kikugawa3
Dmitry A. Sokolov1 Andrew P. Mackenzie1,2 Hilary M. L. Noad1 Elena Gati1
<EMAIL_ADDRESS>Max Planck Institute for Chemical Physics of Solids,
Dresden, Germany  Scottish Universities Physics Alliance, School of Physics
and Astronomy, University of St Andrews, St Andrews, UK  National Institute
for Materials Science, Tsukuba, Japan

###### Abstract

We report on a new technique for measuring the dynamic Young’s modulus, $E$,
of quantum materials at low temperatures as a function of static tuning
strain, $\epsilon$, in piezoactuator-driven pressure cells. In addition to a
static tuning of stress and strain, we apply a small-amplitude, finite-
frequency a.c. (1 Hz$\,\lesssim\,\omega\,\lesssim\,$1000 Hz) uniaxial stress,
$\sigma_{ac}$, to the sample and measure the resulting a.c. strain,
$\epsilon_{ac}$, using a capacitive sensor to obtain the associated modulus
$E$. We demonstrate the performance of the new technique through proof-of-
principle experiments on the unconventional superconductor Sr2RuO4, which is
known for its rich temperature-strain phase diagram. In particular, we show
that the magnitude of $E$, measured using this a.c. technique at low
frequencies, exhibits a pronounced nonlinear elasticity, which is in very good
agreement with previous Young’s modulus measurements on Sr2RuO4 under [1 0 0]
strain using a d.c. method (Noad et al., Science 382, 447-450 (2023)). By
combining the new a.c. Young’s modulus measurements with a.c. elastocaloric
measurements in a single measurement, we demonstrate that these a.c.
techniques are powerful in detecting small anomalies in the elastic properties
of quantum materials. Finally, using the case of Sr2RuO4 as an example, we
demonstrate how the imaginary component of the modulus can provide additional
information about the nature of ordered phases.

##  I Introduction

Recent years have witnessed a surge in the study of elastic properties of
quantum materials, driven by the discovery of novel collective electronic
phases that exhibit a strong coupling to the underlying crystal lattice. A
prominent example is the observation of nematicity in a number of
unconventional superconductors Fernandes  _et al._ (2014); Hinkov  _et al._
(2008). Here, measurements of the elastic constants have revealed a huge
lattice softening Böhmer  _et al._ (2022), which has served as a unique
experimental fingerprint of nematic fluctuations.

Similarly, the strong coupling of electronic and lattice degrees of freedom
makes such electronic systems particularly susceptible to tuning by physical
pressure. Driven by novel developments of pressure-cell technology for tuning
quantum materials by hydrostatic and uniaxial pressure Gati  _et al._ (2020);
Hicks  _et al._ (2014), important discoveries have been made in the field of
quantum materials. Since the application of pressure does not introduce
additional disorder into the system, pressure tuning has been instrumental in
exploring the fundamental properties of quantum materials, like
superconducting $T_{c}$, in clean systems Steppke  _et al._ (2017).

As a result, the combination of the two – the ability to measure elastic
properties while tuning the physical pressure – has proven to be a powerful
addition to the toolbox for studying quantum materials. Recently, measurements
of the lattice elasticity as a function of pressure have led to important
insights into the fundamental question of the role of the lattice in
electronic matter Zacharias  _et al._ (2015a, b); Sarkar  _et al._ (2023) and
to the identification of possible functionalities Sypek  _et al._ (2017). For
example, in correlated quantum materials subjected to pressure tuning,
nonlinear elastic behavior has been observed. This experimental observation
was taken as a strong indication that the lattice profoundly affects the
properties of the electronic system and vice versa. Notable experimental
examples in this respect include the observation of the breakdown of Hooke’s
law Gati  _et al._ (2016, 2018) around the finite-temperature critical
endpoint of the Mott metal-insulator transition in an organic conductor. A
huge lattice softening has also been observedNoad  _et al._ (2023) at the
pressure-induced electronic Lifshitz transition Sunko  _et al._ (2019) in
Sr2RuO4.

The conclusions above were obtained from measurements of stress-strain
relationships under continuously-tuned pressure Hicks  _et al._ (2014); Barber
_et al._ (2019); Gati  _et al._ (2020); Agarmani  _et al._ (2022); Manna  _et
al._ (2012), e.g., in piezoactuator-driven pressure cells Hicks  _et al._
(2014), performed in the demanding cryogenic environment required for the
study of quantum materials. These measurements were made possible by several
recent advances in measuring both the applied stress, $\sigma$, and the
resulting strain, $\epsilon$, with high precision. In general, stresses and
strains are tensor quantities ($\sigma_{ij}$ and $\epsilon_{kl}$) and are
related by the compliance matrix $S_{ijkl}$ (or the inverse elastic constant
matrix $C_{ijkl}$), i.e., $\sigma_{ij}\,=\,\sum_{k,l}S_{ijkl}\epsilon_{kl}$.
When stress, $\sigma$, is applied along a specific direction and the
deformation, $\epsilon$, is measured along the same direction,
$i\,=\,j\,=\,k\,=\,l$ and the corresponding modulus $S_{iiii}$ (or $S_{ii}$ in
Voigt notation) Lüthi (2005) is the Young’s modulus, which we denote as $E$
throughout this manuscript. It is obtained experimentally from the measured
stress-strain relationships via

$E=\frac{d\sigma_{ii}}{d\epsilon_{ii}}.$ (1)

$E$ remains unchanged with $\epsilon$ in linear elastic systems, i.e., in
systems that obey Hooke’s law of elasticity. In contrast, the hallmark of non-
linear elastic systems is that $E$ varies with strain Noad  _et al._ (2023);
Gati  _et al._ (2016).

Figure 1: Schematic representation of a dynamic elastic modulus measurement.
An a.c. stress (dotted line) is applied to a sample, and the resulting a.c.
strain (dashed line) is measured. The phase difference, $\delta$, between the
applied stress and the measured strain characterizes the degree of viscous and
elastic behavior. For $\delta\,=\,0^{\circ}$ ($\delta\,=\,90^{\circ}$), a
system exhibits purely elastic (viscous) behavior. If
$0\,^{\circ}<\,\delta\,<\,90\,^{\circ}$, a material is viscoelastic, i.e., it
shows an elastic response similar to that of solids and liquids
simultaneously. Note that viscoelastic deformation is still fully recoverable,
whereas plastic deformation is characterized by unrecoverable deformation.

In this paper, we introduce a new method to measure $E(\epsilon)$ in
piezoactuator-driven uniaxial pressure cells. In our new approach, we make use
of low-frequency a.c. stresses and strains Ikeda  _et al._ (2019); Li  _et
al._ (2022) to determine the a.c. Young’s modulus. In fact, at ambient
pressure, so-called Dynamical Mechanical Analyzer (DMA) spectroscopy
measurements, in which the real and imaginary elastic moduli, $E^{\prime}$ and
$E^{\prime\prime}$, are determined by the application of low-frequency, low-
amplitude forces and measurements of the resulting strains (see Fig. 1), are
well-established (see e.g. Ref. (21)). In general, as shown schematically in
Fig. 1, a sinusoidal a.c. stress,
$\sigma_{ac}(t)\,=\,\sigma_{ac,0}\sin(\omega_{p}t)$, induces an a.c. strain
response, $\epsilon_{ac}(t)\,=\,\epsilon_{ac,0}\sin(\omega_{p}t-\delta)$, with
a phase lag, $\delta$. $\delta$ can take different values (see Fig. 1): for
purely elastic behavior, an instantaneous strain response is expected
($\delta\,=\,0^{\circ}$ and $E^{\prime}\,\neq\,0,E^{\prime\prime}=0$), whereas
liquids are characterized by a purely viscous behavior with
$\delta\,=\,90^{\circ}$ and $E^{\prime}\,=\,0,E^{\prime\prime}\neq 0$. When
$\delta$ takes values between $0^{\circ}$ and $90^{\circ}$, the stress-strain
response is classified as viscoelastic and both $E^{\prime}$ and
$E^{\prime\prime}$ are finite. These DMA methods are used intensively in the
study of viscoelastic properties Patra  _et al._ (2020); Venkategowda  _et
al._ (2022) of soft materials, tissues, biomaterials, or polymers, e.g., to
extract characteristic energies of glassy freezing processes of structural
entities. In the context of rigid solids, the study of the dynamic moduli,
$E^{\prime}$ and $E^{\prime\prime}$, has mainly been employed to study
ferroelastic phase transitions, where strain acts as the primary order
parameter. From the frequency, amplitude and temperature dependence of
$E^{\prime}$ and $E^{\prime\prime}$, the contribution of microstructural
changes to the macroscopic elastic Young’s modulus has been deduced (e.g., due
to domains). In this respect, the observation of superelasticity in
ferroelastic materials was a particularly relevant discovery Kityk  _et al._
(2000); Schranz and Kityk (2008); Schranz  _et al._ (2012). Here, the stress-
induced movement of domain walls Schranz and Kityk (2008) triggers significant
length changes and thus results in a large ‘superelastic’ softening.

The new setup described in this paper combines the concepts of DMA
measurements with the ability to precisely apply static tuning uniaxial
pressures in piezoactuator-driven uniaxial pressure cells Hicks  _et al._
(2014); Barber  _et al._ (2019). These cells are nowadays widely used in the
field of quantum materials, because they are compatible with a low-temperature
and high-magnetic field environment and allow for in situ d.c. and a.c. stress
tuning Hicks  _et al._ (2014); Ikeda  _et al._ (2019). In this context, DMA-
type measurements are very promising to investigate, e.g., the character of
phase transitions, the role of stress-induced domain dynamics Schranz (1997)
or other collective effects Müller (2011); Wang  _et al._ (2022). In Section
II, we first describe our experimental setup used for the dynamic Young’s
modulus measurements and the data analysis procedure. Then, we show in Sec.
III our proof-of-principle experiments on Sr2RuO4, where we demonstrate that
the magnitude of the Young’s modulus obtained in our setup is in very good
agreement with previous literature results from a d.c. technique Noad  _et
al._ (2023). We conclude the paper by discussing the experimental results of a
finite imaginary part of the Young’s modulus under high uniaxial compression
in the magnetic phase of Sr2RuO4 as an illustration of the DMA-type analysis,
which is now possible with our setup.

##  II Experimental Setup and Data analysis

###  II.1 The uniaxial pressure cell and determination of the static Young’s
modulus

Figure 2: Piezoactuator-controlled uniaxial pressure cell Barber  _et al._
(2019) used to measure the stress-strain relation in quantum materials. In
this cell, a set of piezoelectric actuators (green) generate a displacement
across a gap between moving block A (highlighted in red) and moving block B
(highlighted in blue). The sample carrier, on which the sample is mounted,
connects A and B so that the displacement is transmitted to the sample. The
displacement of the gap is measured by a capacitive sensor mounted under a
shield. In addition, the cell houses a capacitive force sensor, which is used
for d.c. stress-strain measurements. The cell’s spring constant is determined
by that of the moving blocks ($k_{A}$ and $k_{B}$) as well that of the
actuators ($k_{\textrm{P,tot}}$). Figure 3: (a) Simplified schematic diagram
of the uniaxial pressure cell and its working principle. The cell with spring
constant $k_{\textrm{cell}}$ applies varying forces to the ‘load’ spring
$k_{\textrm{meas}}$, whose magnitude is to be determined in the experiment
(see text for details). The displacement of the load spring, $\Delta d$, can
be measured by a capacitive displacement sensor and its magnitude is identical
to the displacement created by the cell; (b) Working curve of applied force,
$F$, vs. generated displacement, $\Delta d_{dc}$, of the piezoactuator-driven
uniaxial pressure cell at a constant supply voltage on the piezoelectric
actuators, $V_{\textrm{dc}}$. The working curve is determined by the
parameters $k_{\textrm{cell}}$, the maximum displacement at this $V_{dc}$
without a load, $\Delta d_{\textrm{dc,0}}$, and the maximum generated force,
$F_{0}$, in case of an infinite load. It is shown by the brown line following
the form $F\,=\,F_{0}-k_{\textrm{cell}}\Delta d_{dc}$. The magnitude of the
load spring constant, $k_{\textrm{meas}}$, determines the force/displacement
provided by the cell at this particular voltage, as seen from the working
curve. This is illustrated by showing two linear stress-strain relations,
corresponding to two different values of $k_{\textrm{meas}}$. The intersection
of these stress-strain relationships with the working curve determine the
applied force and created displacement. The larger $k_{\textrm{meas}}$ is, the
smaller (larger) the created displacement (applied force) will be; (c) Working
curve of the piezoactuator-driven uniaxial pressure cell when an a.c. voltage,
$V_{\textrm{ac}}$, is applied in addition to the d.c. voltage,
$V_{\textrm{dc}}$. The working range is now delineated by the two working
curves at $V_{\textrm{dc}}\,\pm\,V_{\textrm{ac}}$. The alternating
$V_{\textrm{ac}}$ causes an alternating displacement, $\Delta
d_{\textrm{ac,0}}$, created at zero force. The alternating displacement,
$\Delta d_{\textrm{ac}}$, induced by $V_{\textrm{ac}}$ at finite load spring
constant, $k_{\textrm{meas}}$, is related to the magnitude of
$k_{\textrm{meas}}$, see text for details.

We use a uniaxial pressure cell, which is shown schematically in Fig. 2 and is
similar to the one described in Ref. (15), for an in-situ control of the
pressure applied to the sample. Here, we briefly review the working principle
of the pressure cell.

The cell uses piezoelectric actuators to apply the force to the sample. In the
piezocartridge, three piezoelectric actuators are combined to either apply
compressive or tensile forces to the sample, depending on the sign of the
applied voltage. In the present design, the two outer actuators apply
compression to the sample upon application of a positive voltage, whereas the
inner one applies tension (see Fig. 1 of Ref. (15)). Accordingly, we refer to
the two actuators (one actuator) as compression (tension) stacks (stack)
throughout the manuscript.

The piezoelectric actuators drive the motion of the moving block A (see Fig.
2), which is guided by flexures with a low longitudinal spring constant. This
changes the size of the gap between block A and B. When a sample is mounted
between these two blocks, it is either compressed or stretched. In the present
case, we mount our sample on a sample carrier that connects between moving
block A and B (details on the sample carrier design and implications for the
data analysis will be provided in Sec. II.4). The purpose of the sample
carrier is to facilitate sample exchange (see e.g. Ref. (30)). A capacitive
sensor is mounted below the gap between the moving block A and B to measure
the relative displacement of moving blocks, which is related to the sample
strain.

The cell contains a second capacitive sensor that acts as a force sensor. The
moving block B is connected to the frame of the cell by flexures having a
larger longitudinal spring constant compared to the flexures of the moving
block A (see Appendix V for details). The capacitor measures the displacement
of these flexures, which can then be converted into the applied force using
the known spring constant of these flexures.

The d.c. capacitances of the displacement and force sensors can be measured
separately using high-precision capacitance bridges, such as the model AH2550A
from the company Andeen-Hagerling. As described in detail in Ref. (13), these
measurements, together with a precise knowledge of the sample dimensions, can
be used to calculate the applied stress, $\sigma$, and the resulting strain,
$\epsilon$, along the direction of the applied force. Taking the derivative of
$\sigma$ with respect to $\epsilon$ in the post-processing analysis then
yields the Young’s modulus, $E(\epsilon)$, see Eq. 1. Since these measurements
are based on static measurements of $\sigma$ and $\epsilon$, we will refer to
them throughout the text as static Young’s modulus measurements or, in short,
the d.c. method.

###  II.2 Working curve of the piezoactuator-driven stress cell

The working principle of our new a.c. technique is based on the fact that
piezoelectric actuators themselves can be used to measure the spring constant
of the spring on which they exert force, called the ‘load spring’. This
capability is rooted in their characteristic working curve Barber  _et al._
(2019), where key working parameters of piezoelectric actuators, such as the
supplied force and the created displacement depend on the spring constant of
the load spring.

In Fig. 3 (a), we show the simplified version of our experimental setup to
illustrate the concept. The cell with spring constant $k_{\textrm{cell}}$,
which contains the piezoelectric actuators, pushes against a load spring with
spring constant $k_{\textrm{meas}}$. $k_{\textrm{meas}}$ is the key quantity
that is to be determined in the experiment, which can be converted to the
Young’s modulus, $E$, of the sample. The working curve of the cell is
determined by $k_{\textrm{cell}}$ and is shown by the brown line in Fig. 3
(b). At no load, i.e., $k_{\textrm{meas}}\,=\,0$, the cell provides a large
displacement, $\Delta d_{\textrm{dc,0}}$, at essentially no force for a given,
fixed piezoactuator voltage, $V_{\textrm{dc}}$. In case of an infinitely stiff
sample with $k_{\textrm{meas}}\,\rightarrow\,\infty$, the cell provides a
maximum force of $F_{0}=\,k_{\textrm{cell}}\Delta d_{\textrm{dc,0}}$ for the
same $V_{\textrm{dc}}$, but no displacement. Accordingly, the displacement,
$\Delta d_{\textrm{dc}}$, which is generated by the cell and delivered to the
load spring, is a function of $k_{\textrm{meas}}$. For illustration, we also
include in Fig. 3 (b) two stress-strain curves corresponding to two different
load springs, $k_{\textrm{meas,1}}$ and
$k_{\textrm{meas,2}}\,>\,k_{\textrm{meas,1}}$. The applied force
(displacement) is greater (smaller) for $k_{\textrm{meas,2}}$ compared to
$k_{\textrm{meas,1}}.$

This simplified picture describes the working characteristic of a cell at a
fixed piezoactuator voltage for different values of $k_{\textrm{meas}}$. In
our experiment, we want to measure the changes of $k_{\textrm{meas}}$ of a
single sample with changing stress and strain, which is achieved by varying
the static voltage on the piezoelectric actuator.

Specifically, we are interested in obtaining the dynamic moduli from an a.c.
experiment. To this end, we apply a small a.c. voltage to the piezoelectric
actuators, which creates oscillating stresses and strains. Whereas the a.c.
component serves to probe the Young’s modulus, we use the d.c. voltages on the
piezoelectric actuators to tune a given material. The extended working curve
illustrating the situation in our experiment is shown in Fig. 3 (c). The
working range of the cell in the presence of an a.c. voltage,
$V_{\textrm{ac}}$, is now delineated by two parallel lines, corresponding to
the working curves at $V_{\textrm{dc}}\,\pm\,V_{\textrm{ac}}$ (see brown
area). In analogy to the previous discussion, the a.c. displacement is given
by $\Delta d_{\textrm{ac,0}}$, when $k_{\textrm{meas}}\,=\,0$, and will be
zero, when $k_{\textrm{meas}}\,\rightarrow\,\infty$. It follows that $\Delta
d_{\textrm{ac}}$ is directly related to $k_{\textrm{meas}}$ for constant
amplitude of $V_{\textrm{ac}}$. Specifically, the larger $k_{\textrm{meas}}$,
the smaller $\Delta d_{\textrm{ac}}$ will be. Since the a.c. voltage is used
to measure $k_{\textrm{meas}}$, the d.c. voltage can be used independently to
tune the elastic properties of the material under investigation.

###  II.3 Electronic setup for determining the dynamic Young’s modulus using
the new a.c. method

Figure 4: Electronic setup for determining the a.c. displacement, $\Delta
d_{\textrm{ac}}$, which is related to the sample’s Young’s modulus. The
piezocartridge, which drives the uniaxial pressure cell, consists of two
compression stacks and one tension stack. A d.c. voltage is applied to all
stacks to control the static uniaxial pressure on the sample. The d.c. voltage
on the compression stacks is superimposed by a small a.c. voltage with
frequency $\omega_{p}$. The a.c. modulation of the voltages on the compression
stack results in an a.c. change of the capacitance of the displacement sensor,
$C_{\textrm{displ}}(t)\,=\,C_{dc}+C_{ac}(t)$, with same frequency
$\omega_{p}$. To measure the contributions of $C_{\textrm{dc}}$ and
$C_{\textrm{ac}}$ to $C_{\textrm{displ}}$ independently, a home-built
capacitance bridge, consisting of an a.c. voltage supply with frequency
$\omega_{c}$, a resistance $R_{C}$ and two Lock-in amplifiers (Stanford
Research SR860 and SR830), is used. The voltage across the resistor (blue
line) corresponds to the modulated signal of signals with frequencies
$\omega_{p}$ and $\omega_{c}$ (see the orange enveloping curve). As a result,
as shown in the Fourier transformation of the voltage signal, the signal is
composed of signals at $\omega_{c}$ (blue line) as well as at
$|\omega_{c}\,\pm\,\omega_{p}|$ (red line). The SR830 reads out the component
of the voltage signal at $\omega_{c}$, which is related to the d.c.
capacitance value (blue line), whereas the SR860 in Dual Reference mode picks
up the signal at the beat frequency $\omega_{c}-\omega_{p}$, which is
proportional to the a.c. capacitance, see text for details.

Thus, to directly measure $k_{\textrm{meas}}$, our technique relies on the
accurate determination of $\Delta d_{\textrm{ac}}$, which is a mechanical
modulation induced by a finite piezoactuator voltage $V_{\textrm{ac}}$,
through a capacitive measurement. To this end, we have designed a home-built
capacitance bridge, which allows $\Delta d_{\textrm{ac}}$ to be measured
simultaneously with $\Delta d_{\textrm{dc}}$ from a single displacement
capacitor. The ability to measure both is important to determine the probing
strain (related to $\Delta d_{\textrm{ac}}$) separately from the tuning strain
(related to $\Delta d_{\textrm{dc}}$).

The detailed working principle of the bridge is shown in Fig. 4. A small a.c.
voltage is superimposed on the d.c. voltage of one of the stacks. In our case,
the voltage on the compression stacks is superimposed by an a.c. modulation
with frequency $\omega_{p}$, i.e.,
$V(t)\,=\,V_{\textrm{dc}}+V_{\textrm{ac}}\sin(\omega_{p}t)$. This, in turn,
induces a mechanical displacement of the gap, which is measured through a
capacitive displacement sensor with capacitance $C_{\textrm{displ}}$. The
distance of the capacitor plates then follows the form $d\,=\,d_{dc}+\Delta
d_{ac}\sin(\omega_{p}t)$ with $d_{dc}\,=\,d_{dc,0}+\Delta d_{dc}$ and
$d_{dc,0}$ the initial distance of the capacitor plate. Accordingly, the time
dependence of $C_{\textrm{displ}}$ follows as
$C_{\textrm{displ}}\,=\,C_{dc}+C_{ac}\sin(\omega_{p}t)$. In a first
approximation, if $\Delta d_{\textrm{ac}}\,\ll\,d_{\textrm{dc}}$, these
capacitances are related to the displacements by

$\frac{\Delta d_{ac}}{d_{dc}}=\frac{C_{ac}}{C_{dc}}.$ (2)

To extract $C_{\textrm{ac}}$ and $C_{\textrm{dc}}$ from $C_{\textrm{displ}}$,
we use a circuit consisting of a voltage source operating at a second
frequency, $\omega_{c}$, and two lock-in amplifiers from Stanford Research
Systems: specifically, a SR830 model and a SR860 model (see Fig. 4). In
general, applying an a.c. voltage with frequency $\omega_{c}$ to a capacitor
with time-independent value $C_{\textrm{dc}}$ generates an a.c. current with
$\omega_{c}$. In our present case, where $C_{\textrm{displ}}$ is time-
dependent, the returned current, with characteristic frequency $\omega_{c}$,
will be further modulated by $\omega_{p}$. This modulated current is passed
through a resistor with $R_{c}$, where it produces a voltage, $V_{R}$, which
reads as

$\begin{split}V_{R}&=C_{dc}V_{0}R_{c}\omega_{c}\cos{(\omega_{c}t)}+C_{ac}V_{0}R_{c}\\\
&\times[\omega_{c}\cos{(\omega_{c}t)}\sin{(\omega_{p}t)}+\omega_{p}\cos{(\omega_{p}t)}\sin{(\omega_{c}t)}].\end{split}$
(3)

The main task of the two lock-in setup is to perform an electrical
demodulation of the signal Hristov  _et al._ (2018). The first term is
directly proportional to $C_{\textrm{dc}}$ and thus to $d_{\textrm{dc}}$. It
corresponds to the high-frequency signal represented by the blue curve in Fig.
4 (for $\omega_{c}\gg\omega_{p}$). We read out this component of the voltage
in Eq. 3 by locking the SR830 to $\omega_{c}$, such that
$V_{\textrm{SR830}}\,=\,C_{dc}V_{0}R_{c}\omega_{c}$. The second and third
terms in Eq. 3 are products of waves with the characteristic side-band
frequencies $|\omega_{c}\pm\omega_{p}|$ (see also red line in the Fourier
transform in Fig. 4). To read out these voltages, we use the SR860 in Dual
Reference mode, which measures the voltage at the frequency
$|\omega_{c}-\omega_{p}|$. It then follows that

$V_{\textrm{SR860}}=\frac{C_{ac}V_{0}R_{c}|\omega_{c}-\omega_{p}|}{2},$ (4)

which can be directly converted to $C_{\textrm{ac}}$. The signal-to-noise
ratio of $V_{\textrm{SR860}}$ will be larger, the greater the difference
between $\omega_{p}$ and $\omega_{c}$.

Using the known calibration of the displacement sensor, i.e., the functional
form of $C_{\textrm{displ}}$ vs. $d$, $\Delta d_{\textrm{ac}}$ and $\Delta
d_{\textrm{dc}}$ can now be calculated from the measured voltages
$V_{\textrm{SR830}}$ and $V_{\textrm{SR860}}$. In Fig. 5 (a), we show an
example curve of $\Delta d_{\textrm{ac}}$ vs. tuning displacement, $\Delta
d_{\textrm{dc}}$, taken on Sr2RuO4. We will discuss the implications of the
data in detail below in Sec. III. For now, the plot clearly shows that $\Delta
d_{\textrm{ac}}$ is not constant when $\Delta d_{\textrm{dc}}$ is changed, and
that the changes in $\Delta d_{\textrm{ac}}$ can be clearly resolved in our
setup with a resolution of $\approx\,0.2\,$nm. As explained above, the
changing $\Delta d_{\textrm{ac}}$ reflects the changing $k_{\textrm{meas}}$
(or in other words, the changing Young’s modulus, $E$, of the sample).

As an aside, we would like to remark that the precise determination of $\Delta
d_{\textrm{ac}}$ and hence $\epsilon_{\textrm{ac}}$, which we demonstrate
here, is also crucial for quantitative measurements of the elastocaloric
effect Ikeda  _et al._ (2019); Li  _et al._ (2022), where the temperature
change, $\Delta T$, induced by a finite $\epsilon_{\textrm{ac}}$, is measured.

###  II.4 Data Analysis

We now discuss how to convert the measured $\Delta d_{\textrm{ac}}(\Delta
d_{\textrm{dc}})$ into absolute values of the Young’s modulus $E(\epsilon)$.
This requires two main steps: (i) converting $\Delta d_{\textrm{ac}}$ to the
measured spring constant, $k_{\textrm{meas}}$, and (ii) extracting the
sample’s spring constant from $k_{\textrm{meas}}$. This second step is
necessary because $k_{\textrm{meas}}$ contains contributions from, e.g., the
mounting glue and the sample carrier, in addition to the contribution of the
sample. This part of the analysis is also required when analyzing the data
from the d.c. method and is described in detail in the Supplementary
Information of Ref. (13).

Figure 5:  (a) Experimental data of the a.c. displacement, $\Delta
d_{\textrm{ac}}$, as a function of the d.c. tuning displacement, $\Delta
d_{\textrm{dc}}$, induced by a the application of a d.c. and an a.c. voltage
on the piezoelectric actuators. $d_{ac}$ and $d_{dc}$ were obtained by reading
out the displacement capacitive sensor in the piezoactuator-driven pressure
cell with our home-built capacitance bridge. This example dataset was recorded
on Sr2RuO4 at a temperature of $T\,=\,$ 5 K, a frequency $f_{p}$ of 167 Hz and
an a.c. amplitude $V_{ac}$ of 5 V; (b) Preparation and mounting of samples for
stress-strain measurements in uniaxial pressure cells. The sample (in the
present case: Sr2RuO4) is cut into a narrow neck with wide anchor tabs using a
Plasma Focused Ion Beam. The anchors are epoxied to the sample carrier. The
sample shape ensures a rapid crossover from the low-stress region in the
anchor tabs to the high-stress region in the neck and minimizes the stress in
the mounting epoxy; (c) Due to the shape of the sample, the sample’s spring
constant can be modeled to a good approximation as a set of two discrete
springs in series, i.e., the anchor (spring constant $k_{\textrm{anch}}$) and
the neck spring (spring constant $k_{\textrm{neck}}$). In addition, the small
carrier spring constant, $k_{\textrm{car}}$, contributes to the measured
spring constant, $k_{\textrm{meas}}$, and is in parallel to
$k_{\textrm{anch}}$ and $k_{\textrm{neck}}$.

For the first step, we refer back to the working curve of the cell, which has
been introduced in Sec. II.3 and is shown in Fig. 3 (c). For example, we can
compare the a.c. displacement in case of an empty cell, $\Delta
d_{\textrm{ac,0}}$, with the displacement, $\Delta d_{\textrm{ac}}$, when the
cell works against a load spring with $k_{\textrm{meas}}$. Clearly, $\Delta
d_{\textrm{ac}}\,<\,\Delta d_{\textrm{ac,0}}$ and it can be deduced that the
ratio determines $k_{\textrm{meas}}$ via

$k_{\textrm{meas}}=k_{\textrm{cell}}\left(\frac{\Delta
d_{\textrm{ac,0}}}{\Delta d_{\textrm{ac}}}-1\right).$ (5)

Thus, $k_{\textrm{meas}}$ can be calculated for all values of $\Delta
d_{\textrm{ac}}$, when $\Delta d_{\textrm{ac,0}}$ is measured, e.g. when the
sample is broken after an experiment. Alternatively, any reference point with
finite $k_{\textrm{meas}}$ can be chosen, as long as the value of
$k_{\textrm{meas}}$ at a particular strain is well known.

The key parameter for converting $\Delta d_{\textrm{ac}}$ to
$k_{\textrm{meas}}$ is the value of the cell spring constant,
$k_{\textrm{cell}}$, which cannot be determined during the experiment itself.
In general, it can be determined from calibration experiments and from
simulations, as shown in Ref. (15) for a cell of similar design. Following the
same procedure, we determined the cell spring constant of the present cell to
be $k_{\textrm{cell}}\,=\,(3.4\,\pm\,0.5)$ N/$\mu$m (see Sec. V for details).
Since titanium, the material of the cell, becomes stiffer upon cooling and
calibration experiments were performed at room temperature, we applied a
correction factor of 15% to $k_{\textrm{cell}}$, when analyzing low-
temperature data Barber  _et al._ (2019). The exact value of
$k_{\textrm{cell}}$ will likely vary from cell to cell, even if the design is
technically the same. Once $k_{\textrm{cell}}$ is determined for a specific
cell, e.g., by calibration measurements, it is not expected to change from
experiment to experiment and is thus not a free parameter in the analysis.
Equation 5 also shows that the ability to resolve changes in
$k_{\textrm{meas}}$ also depends on $k_{\textrm{cell}}$. As expected from the
working curve, the setup works best when $k_{\textrm{cell}}$ and
$k_{\textrm{meas}}$ are of similar magnitude.

The second step in obtaining $E(\epsilon)$ from the a.c. data is to extract
the spring constant of the sample from $k_{\textrm{meas}}$. Following the
protocol established in Ref. (13), the samples are cut into a narrow neck with
wide anchor tabs using a Xe Plasma Focused Ion Beam (see Fig. 5 (b)). The
necking creates a rapid crossover between regions of low stress in the anchors
and high stress in the neck. The sample can therefore be approximated by two
springs in series, which we label as $k_{\textrm{neck}}$ and
$k_{\textrm{anch}}$. We also include the spring constant of the mounting glue
in $k_{\textrm{anch}}$. In the two-spring approximation, $k_{\textrm{anch}}$
can be calculated from the total measured signal $k_{meas}$ around a reference
strain and an independently measured value of $E$ at that reference strain
Noad  _et al._ (2023). In the case of Sr2RuO4 over the range of displacements
considered here, Ref. (13) shows that $k_{\textrm{anch}}$ can indeed be taken
as stress-independent to a good approximation.

In addition, the sample carrier has flexures with spring constants
$k_{\textrm{car}}$, which are in series with $k_{\textrm{neck}}$ and
$k_{\textrm{anch}}$, as shown schematically in Fig. 5 (c). It follows that

$k_{\textrm{neck}}(\epsilon)=\left[\frac{1}{k_{\textrm{meas}}(\epsilon)-k_{\textrm{car}}}-\frac{1}{k_{\textrm{anch}}}\right]^{-1}.$
(6)

The spring constant of the carrier, $k_{car}\,\approx\,0.02$ N/$\mu$m, is per
design much smaller than typical values of $k_{\textrm{meas}}$ and can also be
calibrated experimentally.

Finally, the Young’s modulus is obtained from exact knowledge of the length of
the necked region, $l_{\textrm{neck}}$, and the cross-sectional area of the
sample, $A_{\textrm{cs}}$, via
$E\,=\,k_{\textrm{neck}}l_{\textrm{neck}}/A_{\textrm{cs}}$. The strain in the
neck, $\epsilon_{xx}$, follows from the measured $d_{dc}$ values via
$(k_{\text{meas}}/k_{\text{neck}})\,\times\,({\Delta}d_{dc}/l_{\text{neck}})$.
The sample of Sr2RuO4 used for our proof-of-principle measurements shown in
Fig. 5 (b) was cut to dimensions of $A_{\textrm{cs}}\,=\,$102 $\mu$m $\times$
120 $\mu$m and $l_{\textrm{neck}}\,=\,$717 $\mu$m, with the long edge of the
neck oriented along the x = [1 0 0] direction of the crystal, so that
$k_{\textrm{neck}}\,(\epsilon_{xx}\,=\,0)\,=\,$2.72 N/$\mu$m. The anchor
spring constant in our experiments was $k_{\textrm{anch}}\,=\,$2.36 N/$\mu$m.

###  II.5 Working Parameters of the a.c. Young’s modulus setup

In the following, we specify the working range of the a.c. Young’s modulus
setup, including accessible frequency, amplitude and temperature range.

The frequency range of operation is determined by the choice of piezoelectric
actuators and the mechanical resonances of the cell. The multi-layer ceramic
actuators PIC  typically used in the pressure cells operate well in the low-
frequency range 1 Hz $\lesssim\,f_{p}\,=\,\omega_{p}/(2\pi)\,\lesssim\,$1 kHz.
Future technical developments might be able to extend the frequency range
significantly, e.g., by choosing different type of actuators.

The second relevant frequency is that of the capacitance bridge, $f_{c}$. All
measurements in this manuscript were taken with
$f_{c}\,=\,\omega_{c}/(2\pi)\,=\,2.297\,$kHz, which is of the same magnitude
as the frequency of 1 kHz used in the Andeen-Hagerling AH2550A capacitance
bridges, employed in the earlier d.c. work.

In terms of the amplitude, the maximum voltage that can be applied to the
actuators is temperature dependent. At lowest temperatures, the voltage range
can typically be extendedBarber  _et al._ (2019) to -300 V to 400 V. With such
large d.c. voltages, tuning strains of $\,\pm\,$1-2% may be achieved for a
sample of the previously-mentioned dimensions. Since the a.c. voltage produces
the probing strain, its magnitude must be chosen to be much smaller than the
d.c. voltages. In our proof-of-principle studies on Sr2RuO4, we used a.c.
voltages, $V_{ac}$, up to 5 V at lowest temperatures, corresponding to typical
values of stress and strain amplitude of $\sigma_{ac}\,\approx\,10^{-2}\,$GPa
and $\epsilon_{ac}\,\approx\,1\,\times\,10^{-4}\,$. $V_{ac}$ and $f_{ac}$ were
chosen to optimize signal-to-noise ratio, while ensuring that phase transition
features were not significantly smeared out by a large $V_{ac}$.

The piezoacutator-driven uniaxial pressure cells are designed to operate down
to very low temperatures, even down to dilution-fridge temperatures.
Typically, the lowest-accessible temperature is limited by the cooling power
of the fridge and by the heating created by the piezoelectric actuators. In
the a.c. setup, heating of the actuators becomes a serious problem at higher
frequencies and/or higher a.c. voltage amplitudes. Thus, in practical terms,
the lowest accessible temperature may be a trade-off with the
frequency/amplitude range of interest.

The a.c. Young’s modulus measurements can also be performed in a finite
magnetic field, which then allows the simultaneous study of the effect of
different tuning parameters on the elastic properties of quantum materials.

##  III Proof-of-Principle Results

To demonstrate the functionality of our new a.c. Young’s modulus setup, we
performed proof-of-principle experiments on the ruthenate Sr2RuO4, whose
Young’s modulus under finite strain has recently been documented with high
precision by d.c. stress-strain measurements Noad  _et al._ (2023). Before
discussing our proof-of-principle a.c. data taken on Sr2RuO4, we first present
the main aspects of the phase diagram of Sr2RuO4 under uniaxial pressure that
are relevant for the present work.

The unconventional superconductor Sr2RuO4 has been extensively studied in
uniaxial pressure experiments Steppke  _et al._ (2017); Sunko  _et al._
(2019); Noad  _et al._ (2023); Jerzembeck  _et al._ (2022); Grinenko  _et al._
(2021); Li  _et al._ (2022) in the last decade. These studies have uncovered a
rich phase diagram under uniaxial stress, $\sigma_{xx}$, applied along the [1
0 0] axis of the tetragonal lattice. Upon increasing compression, the Fermi
surface of Sr2RuO4, consisting of so-called $\alpha$, $\beta$ and $\gamma$
sheets (see sketches on top of Fig. 6), becomes distorted. The $\gamma$ sheet
shows the strongest changes. When a compressive [1 0 0] strain of
$\epsilon_{xx}\,\approx\,-0.45\%$ is applied, the $\gamma$ sheet undergoes a
Lifshitz transition, at which the Fermi surface drastically changes its
topology Sunko  _et al._ (2019) from a closed to an open configuration.
Although the Lifshitz transition involves only a fraction of the conduction
electrons, it has recently been shown by Noad et al. Noad  _et al._ (2023)
that these conduction electrons drive a very large lattice softening. The
significant renormalization of the Young’s modulus upon crossing the Lifshitz
transition, that is reported with high accuracy, provides an excellent testbed
to benchmark our new a.c. method. In addition, a recent set of experiments
Grinenko  _et al._ (2021); Li  _et al._ (2022), including thermodynamic
measurements of the elastocaloric effect, suggests that Sr2RuO4 undergoes a
transition into a magnetically-ordered state for higher compression beyond the
Lifshitz strain and for temperatures $T\,\lesssim\,$8 K. This additional phase
transition is expected to lead to anomalous behavior in $E(\epsilon_{xx})$.

In the following, we first discuss the results of the magnitude of the dynamic
Young’s modulus of Sr2RuO4. For low-enough frequencies, it may be expected
that the magnitude of the a.c. Young’s modulus agrees with the one inferred
from d.c. measurements. Afterwards, we discuss our measurements of the phase
shift in the dynamic signal as we tune the material into its magnetically
ordered regime.

###  III.1 Results: Magnitude of the dynamic Young’s modulus of Sr2RuO4,
determined from the a.c. technique

Figure 6: (a) Young’s Modulus, $E$, of Sr2RuO4 as a function of strain,
$\epsilon_{xx}$, at a temperature of $T\,=\,4$ K, extracted from the new a.c.
method (open symbols) and compared to results from the d.c. method (closed
symbols) on the same sample. The data in the a.c. method was taken at a
frequency of $f_{p}\,=\,$167 Hz and at a piezoactuator a.c. voltage of
$V_{ac}\,=\,5\,$V; (b) Elastocaloric temperature oscillation amplitude,
$\Delta T$, measured in the same experiment as the Young’s modulus data shown
in (a). In both panels, the dashed line indicates the strain at which Sr2RuO4
undergoes a Lifshitz transition of the Fermi surface, which is schematically
shown in the insets. The dotted line indicates the strain at which Sr2RuO4
undergoes a transition to magnetic order.

In Fig. 6 (a) we compare the results of Young’s modulus measurements conducted
at a temperature of 4 K with the d.c. method Noad  _et al._ (2023) (closed
symbols) vs. the new a.c. method (open symbols), taken on the exact same
sample of Sr2RuO4 in the same experimental run. Clearly, the two data sets as
a function of tuning strain, $\epsilon_{xx}$, are in very good agreement. The
quantitative agreement is achieved by using
$k_{\textrm{cell}}\,=\,(3.7\,\times\,1.15)\,$N/$\mu$m in the analysis (see
Sec. II.4). This value is within the error bars of the simulation results for
$k_{\textrm{cell}}$ (see Sec. V).

Importantly, the data show a set of anomalies associated with the rich phase
diagram of Sr2RuO4 under [1 0 0] stress. The pronounced softening of $E$ at
the electronic Lifshitz transition at around $-0.45\%$ strain is clearly
resolved. Upon further increasing compression, a second, albeit smaller,
anomaly can be discerned in the $E(\epsilon_{xx})$ data around
$\epsilon_{xx}\,\approx\,-0.64\%$. Although the anomaly is visible in both
d.c. and a.c. data sets, it is slightly more evident in the a.c. data. That
this small drop in $E(\epsilon_{xx})$ does indeed correspond to a
thermodynamic phase transition, becomes clear when comparing to the results of
elastocaloric measurements, which were performed simultaneously to the
$E(\epsilon_{xx})$ measurements. The elastocaloric temperature amplitude,
$\Delta T$, manifests an anomaly at the same strain as where the small drop in
$E$ occurs. In the previous work by Li et al., this feature was associated
with the transition into the magnetically ordered phase Grinenko  _et al._
(2021) of Sr2RuO4.

The a.c. Young’s modulus data in Fig. 6 were taken at $f_{p}\,=\,167\,$Hz,
$f_{c}\,=\,2.297\,$kHz, and $V_{ac}\,=\,5\,$V. In Fig. 7, we now show our data
at different $f_{p}$ (a) and different $V_{ac}$ (b) to demonstrate
experimentally that our setup is operational over wider ranges of $f_{p}$ and
$V_{ac}$. Both data sets demonstrate that the magnitude of Young’s modulus is
essentially independent of the exact frequency or amplitude within the ranges
of 17 Hz $\leq\,f_{p}\,\leq\,927$ Hz and 2 V$\,\leq\,V_{ac}\,\leq\,$5 V.
Specifically, in Fig. 7 (a), the feature of the Lifshitz transition, which is
the prominent feature at $T\,=\,5\,$K, is clearly visible in all data sets at
different $f_{p}$ and the absolute value of $E$ agrees between the data sets
within the signal-to-noise ratio. However, the latter is smaller at higher
frequency, as expected, since the absolute signal depends on
$\omega_{c}-\omega_{p}$, as seen in Eq. 4.

We now demonstrate that the chosen $V_{ac}$ are small enough to not smear out
phase transitions. To this end, we show in Fig. 7 (b) the amplitude dependence
of the Young’s modulus at a temperature of 2.5 K, where the feature associated
with the magnetic phase at $\epsilon_{xx}\,\sim\,-0.62\,\%$ is more
pronounced, and thus serves as a good benchmark for this analysis. For the
range of $V_{ac}$ between 2 V and 5 V, corresponding to a range of
$\sigma_{ac}$ ($\epsilon_{ac}$) between 0.011 GPa and 0.027 GPa
($6\,\times\,10^{-5}$ and $1.5\,\times\,10^{-4}$), no significant smearing of
Young’s modulus anomalies can be detected. Overall, all the data agree
quantitatively very well, and only the signal-to-noise ratio changes with
decreasing $V_{ac}$, as expected.

Figure 7: Dependence of the Young’s modulus, $E$, of Sr2RuO4 on frequency (a)
and stress amplitude (b), as obtained by the new a.c. method. The data are
plotted as a function of strain, $\epsilon_{xx}$. In (a), different actuator
frequencies, $f_{p}$, are applied in the range between 17 Hz and 927 Hz at a
temperature of 5 K and an a.c. voltage of 5 V. This voltage induces a stress
amplitude of $\sigma_{ac}\,=\,$0.027 GPa. In (b), the data for different a.c.
voltage amplitudes, $V_{ac}$, between 2 V and 5 V are shown at a temperature
of 2 K and an actuator frequency of 167 Hz. The corresponding stress
amplitudes, $\sigma_{ac}$, vary between 0.011 GPa and 0.027 GPa. All data
shown were taken with a capacitance-bridge frequency of
$f_{c}\,=\,2.297\,$kHz.

###  III.2 Results: Phase of the dynamic Young’s modulus of Sr2RuO4

Figure 8: Magnitude (a) and phase (b) of the dynamic Young’s modulus of
Sr2RuO4 under in-plane strain, $\epsilon_{xx}$. Data of the dynamic modulus
are shown at temperatures of 2.5 K and 8 K. The strain region, in which
Sr2RuO4 is magnetically ordered at 2.5 K, is indicated by the grey area. The
transition into the magnetically-ordered phase at 2.5 K and
$\epsilon_{xx}\,\approx\,-0.62\%$ is also clearly identified in the
elastocaloric temperature amplitude, $\Delta T$, which is also included in (a)
on the right axis. At 8 K, Sr2RuO4 remains non-magnetic over the strain range
shown here Li  _et al._ (2022). The dashed line indicates the position of the
Lifshitz transition at 2.5 K. All data shown were taken at a frequency of
$f_{p}\,=\,$167 Hz and an amplitude of $V_{ac}=$ 5 V. Error bars for the phase
value are exemplarily indicated in (b) at strain of $\sim\,-0.7\%$.

We now turn to the additional phase information that is provided by performing
our measurements in the a.c. mode. As introduced in Sec. I, dynamic
measurements reveal information on the real and imaginary part of the Young’s
modulus ($E^{\prime}$ and $E^{\prime\prime}$). In particular the latter is of
interest to investigate dissipative processes. In the following, we describe
our results in this respect by presenting the Young’s modulus in terms of its
magnitude and phase. Phase and magnitude are related to the real and imaginary
part of the modulus via
$E^{\prime}\,=\,\frac{|\sigma_{ac,0}|}{|\epsilon_{ac,0}|}\cos{\delta}$ and
$E^{\prime\prime}\,=\,\frac{|\sigma_{ac,0}|}{|\epsilon_{ac,0}|}\sin{\delta}$,
i.e., a finite $\delta$ corresponds to a finite $E^{\prime\prime}$.

Figure 8 summarizes our measurement results of $\delta$ in our experiments on
Sr2RuO4 at a probing frequency of $f_{p}\,=\,167$ Hz and $V_{ac}\,=\,5\,$V. To
this end, we now compare the results at two different temperatures,
$T\,=\,2.5\,$K and $T\,=\,$8 K. Whereas Sr2RuO4 was reported to enter a
magnetically-ordered phase at high compression at $T\,=\,$2.5 K, it remains
non-magnetic at 8 K Grinenko  _et al._ (2021); Li  _et al._ (2022). This is
fully consistent with our thermodynamic data, shown in Fig. 8 (a). At low
compression, the magnitude of $E$ as a function of $\epsilon_{xx}$ shows the
softening at the Lifshitz transition, and the softening is more pronounced at
low temperatures, consistent with the earlier reports Noad  _et al._ (2023).
For higher compression, the data taken at 2.5 K reveal an additional feature
at $\epsilon_{xx}\,\approx\,-0.61\%$, associated with the transition into the
magnetic state. In contrast, the data set taken at 8 K shows no features of
additional phase transitions besides the Lifshitz transition. The grey area in
Fig. 8 marks the region of magnetic order at 2.5 K, as determined by the
simultaneous measurements of $E$ and the elastocaloric effect (see grey line,
right axis).

In Fig. 8 (b), we show the behavior of the phase, $\delta$, as a function of
$\epsilon_{xx}$ for the same temperatures and at the same frequency. At 8 K,
no change of $\delta$ with strain is observed within the signal-to-noise level
over the entire strain range. In contrast, at 2.5 K, an increase of $\delta$
is observed at the strain, where Sr2RuO4 undergoes the transition into the
magnetically-ordered state. Although the changes of $\delta$ are very small,
they exceed the signal to noise by more than a factor of two.

The result of a finite phase between applied stress and resulting strain in
the magnetic phase of Sr2RuO4 is a key new insight accessible by extending
stress-strain measurements to finite frequencies. It implies that there is an
energy dissipation during the (un-)loading stress cycle. Since this phenomenon
occurs at low frequencies, compared to typical spin relaxation times, it is
likely to be related to interactions between the magnetic order and the
crystal lattice, such as domain walls or other collective effects. In
elemental chromium (Cr), for example, it was found that the pressure-dependent
spin-density wave vector shows a certain degree of irreversibility between
increasing and decreasing pressure Ruesink  _et al._ (1980). This observation
was interpreted in terms of crystal-lattice domain-wall distortions Fenton
(1980), which lock the wave vector for small distortions. A similar mechanism
may be at work here in Sr2RuO4. Further knowledge of the ordering vector and
its strain dependence Kim  _et al._ (2023) in Sr2RuO4 will be crucial for
understanding the low-frequency response of its elastic constants.

##  IV Conclusion and Outlook

In this work, we described a new experimental technique to determine the
dynamic Young’s modulus as a function of pressure, frequency and temperature
in piezoactuator-driven uniaxial pressure cells. Our setup exploits the
ability of piezoelectric actuators to generate finite-frequency stresses and
strains in the Hz-kHz range through the application of an a.c. voltage. Using
the ruthenate Sr2RuO4 as a test-bed material for proof-of-principle
measurements, we have shown that the Young’s modulus data from our low-
frequency a.c. setup are in very good agreement with data from static Young’s
modulus measurements. Our a.c. setup is well suited for detecting small
anomalies in the strain dependence of the modulus. Furthermore, our a.c. data
contains information on the phase between applied stress and resulting strain,
which we find to be finite in the magnetic phase of Sr2RuO4 under high
compression.

Our setup opens up the possibility to study the finite-frequency elastic
response function in quantum materials which are subjected to time-varying
external stress fields and large, static tuning stresses. This approach is
akin to a.c. susceptibility studies Topping and Blundell (2018); Grigera  _et
al._ (2003), where a time-dependent magnetic field acts as the driving
external force to probe dynamics. Our method provides a new perspective on the
viscoelastic behavior and lattice dynamics of solids. Following the
fluctuation-dissipation theoremKubo (1966), the viscoelastic response is
related to the low-frequency lattice dynamics of systems whenever they are
amenable to stress tuning. It can be expected that the dynamics are governed
by a range of intriguing phenomena, such as the movement of domain walls under
strain, slow order-parameter dynamics and collective effects in general.

###### Acknowledgements.

We thank You-Sheng Li and Clifford W. Hicks for highly useful insights from
technical discussions. Financial support by the Max Planck Society is
gratefully acknowledged. In addition, we gratefully acknowledge funding
through the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation)
through TRR 288—422213477 (Project A10) and the SFB 1143 (project-id
247310070; project C09). Research in Dresden benefits from the environment
provided by the DFG Cluster of Excellence ct.qmat (EXC 2147, project ID
390858490). C.I.O and Z.H. acknowledges the support of a St Leonards
scholarship from the University of St. Andrews. N.K. work is supported by JSPS
KAKENHI (No. JP18K04715, No. JP21H01033, and No. JP22K19093).

##  V Appendix: Determination of the spring constant of the cell

We follow the report of Barber et al. Barber  _et al._ (2019) to determine the
spring constant of our cell, which is similar in design, but not identical, to
theirs. For the finite-element simulations, we used the software COMSOL COM .
Since the cell is made out of titanium, we use the room-temperature Young’s
modulus of 103 GPa and a Poisson’s ratio of 0.33 for the simulations.

The spring constant of the cell contains various contributions (see Fig. 2):
(i) the spring constant of the piezoelectric actuators, $k_{\textrm{p,tot}}$,
(ii) the spring constant of the moving block B (the force block), $k_{B}$, and
(iii) the spring constant of the moving block A, $k_{A}$.

According to the data sheet of the actuators PIC , the spring constant of a
single stack of type P-885.51 used here is $k_{p}\,=\,50\,$N$/\mu$m at room
temperature. Combining the two compression actuators in parallel to each other
and in series with the tension actuator gives
$k_{\textrm{p,tot}}\,=\,33\,$N/$\mu$m.

To simulate $k_{B}$, we apply a force of 10 N to moving block B at the
position of the carrier mounting holes and measure its displacement. The
resulting spring constant in the simulations was $k_{B}\,=\,(7.6\,\pm\,0.3)$
N/$\mu$m. We also determined this spring constant experimentally, by hanging
various weights from block B and measuring the resulting displacement. This
procedure yielded an experimental value of $(7.7\,\pm\,0.1)$ N/$\mu$m, which
is, within the error bars, consistent with the simulated value.

The moving block A is designed to have a small spring constant in the
direction of the applied force and a large spring constant in the orthogonal
directions, to avoid torque on the sample. These rotational spring constants
dominate $k_{A}$. For the simulations of $k_{A}$, we consider the application
of force of 10 N to an infinitely stiff sample. Since the sample is infinitely
stiff, there is zero displacement between the carrier mounting points on block
A and B ($\Delta d\,=\,0$) upon the application of 10 N. Due to the finite
rotational spring constant of block A, the actuators must apply a slightly
higher force than 10 N. Our simulations showed that a force of 10.45 N is
needed. We then evaluated the difference in actual displacement between the
carrier mounting holes on block A and the actuator attachment area. Note that,
even though there is zero relative displacement across the gap, block A is
still displaced because block B moves under applied force (see Ref. (15) for
further details on this simulation). The present simulation gives an estimate
of $k_{A}\,=\,(7.7\pm 2)$ N/$\mu$m.

Taking $k_{\textrm{p,tot}}$, $k_{A}$ and $k_{B}$ together in series yields
$k_{\textrm{cell}}=(3.4\,\pm\,0.5)$ N/$\mu$m at room temperature.

## References

  * Fernandes  _et al._ (2014) R. M. Fernandes, A. V. Chubukov, and J. Schmalian, Nat. Phys. 10, 97 (2014).
  * Hinkov  _et al._ (2008) V. Hinkov, D. Haug, B. Fauqué, P. Bourges, Y. Sidis, A. Ivanov, C. Bernhard, C. T. Lin, and B. Keimer, Science 319, 597 (2008).
  * Böhmer  _et al._ (2022) A. E. Böhmer, J.-H. Chu, S. Lederer, and M. Yi, Nat. Phys. 18, 1412 (2022).
  * Gati  _et al._ (2020) E. Gati, L. Xiang, S. L. Bud’ko, and P. C. Canfield, Annal. Phys. (Berl.) 532, 2000248 (2020).
  * Hicks  _et al._ (2014) C. W. Hicks, M. E. Barber, S. D. Edkins, D. O. Brodsky, and A. P. Mackenzie, Rev. Sci. Instrum. 85, 065003 (2014).
  * Steppke  _et al._ (2017) A. Steppke, L. Zhao, M. E. Barber, T. Scaffidi, F. Jerzembeck, H. Rosner, A. S. Gibbs, Y. Maeno, S. H. Simon, A. P. Mackenzie, and C. W. Hicks, Science 355, eaaf9398 (2017).
  * Zacharias  _et al._ (2015a) M. Zacharias, I. Paul, and M. Garst, Phys. Rev. Lett. 115, 025703 (2015a).
  * Zacharias  _et al._ (2015b) M. Zacharias, A. Rosch, and M. Garst, Eur. Phys. J. Spec. Top. 224, 1021 (2015b).
  * Sarkar  _et al._ (2023) S. Sarkar, L. Franke, N. Grivas, and M. Garst, Phys. Rev. B 108, 235126 (2023).
  * Sypek  _et al._ (2017) J. T. Sypek, H. Yu, K. J. Dusoe, G. Drachuck, H. Patel, A. M. Giroux, A. I. Goldman, A. Kreyssig, P. C. Canfield, S. L. Bud’ko, C. R. Weinberger, and S.-W. Lee, Nat. Comm. 8, 1083 (2017).
  * Gati  _et al._ (2016) E. Gati, M. Garst, R. S. Manna, U. Tutsch, B. Wolf, L. Bartosch, H. Schubert, T. Sasaki, J. A. Schlueter, and M. Lang, Sci. Adv. 2, e1601646 (2016).
  * Gati  _et al._ (2018) E. Gati, U. Tutsch, A. Naji, M. Garst, S. Köhler, H. Schubert, T. Sasaki, and M. Lang, Crystals 8, 38 (2018).
  * Noad  _et al._ (2023) H. M. L. Noad, K. Ishida, Y.-S. Li, E. Gati, V. Stangier, N. Kikugawa, D. A. Sokolov, M. Nicklas, B. Kim, I. I. Mazin, M. Garst, J. Schmalian, A. P. Mackenzie, and C. W. Hicks, Science 382, 447 (2023).
  * Sunko  _et al._ (2019) V. Sunko, E. A. Morales, I. Markovic, M. E. Barber, D. Milosavljevic, F. Mazzola, D. A. Sokolov, N. Kikugawa, C. Cacho, P. Dudin, H. Rosner, C. W. Hicks, P. D. C. King, and A. P. Mackenzie, npj Quantum Mater. 4, 46 (2019).
  * Barber  _et al._ (2019) M. E. Barber, A. Steppke, A. P. Mackenzie, and C. W. Hicks, Rev. Sci. Instrum. 90, 023904 (2019).
  * Agarmani  _et al._ (2022) Y. Agarmani, S. Hartmann, J. Zimmermann, E. Gati, C. Delleske, U. Tutsch, B. Wolf, and M. Lang, Rev. Sci. Instruments. 93, 113902 (2022).
  * Manna  _et al._ (2012) R. S. Manna, B. Wolf, M. de Souza, and M. Lang, Rev. Sci. Instruments. 83, 085111 (2012).
  * Lüthi (2005) B. Lüthi,  _Physical Acoustics in the Solid State_, 1st ed., Springer Series in Solid-State Sciences, Vol. 148 (Springer, Berlin Heidelberg, 2005).
  * Ikeda  _et al._ (2019) M. S. Ikeda, J. A. W. Straquadine, A. T. Hristov, T. Worasaran, J. C. Palmstrom, M. Sorensen, P. Walmsley, and I. R. Fisher, Rev. Sci. Instruments. 90, 083902 (2019).
  * Li  _et al._ (2022) Y.-S. Li, M. Garst, J. Schmalian, S. Ghosh, N. Kikugawa, D. A. Sokolov, C. W. Hicks, F. Jerzembeck, M. S. Ikeda, Z. Hu, B. J. Ramshaw, A. W. Rost, M. Nicklas, and A. P. Mackenzie, Nature 607, 276 (2022).
  * Salje and Schranz (2011) E. K. H. Salje and W. Schranz, Z. Kristallog. 226, 1 (2011).
  * Patra  _et al._ (2020) S. Patra, P. M. Ajayan, and T. N. Narayanan, Oxford Open Mater. Sci. 1, itaa001 (2020).
  * Venkategowda  _et al._ (2022) T. Venkategowda, L. Manjunatha, and P. Anilkumar, Mater. Today: Proc. 54, 395 (2022).
  * Kityk  _et al._ (2000) A. V. Kityk, W. Schranz, P. Sondergeld, D. Havlik, E. K. H. Salje, and J. F. Scott, Phys. Rev. B 61, 946 (2000).
  * Schranz and Kityk (2008) W. Schranz and A. V. Kityk, Ferroelectrics 375, 178 (2008).
  * Schranz  _et al._ (2012) W. Schranz, H. Kabelka, A. Sarras, and M. Burock, Appl. Phys. Lett. 101, 141913 (2012).
  * Schranz (1997) W. Schranz, Phase Transitions 64, 103 (1997).
  * Müller (2011) J. Müller, ChemPhysChem 12, 1222 (2011).
  * Wang  _et al._ (2022) S. Wang, J. Chen, L. Wu, and Y. Zhao, Phys. Rev. Lett. 128, 095702 (2022).
  * Jerzembeck  _et al._ (2022) F. Jerzembeck, H. S. Roising, A. Steppke, H. Rosner, D. A. Sokolov, N. Kikugawa, T. Scaffidi, S. H. Simon, A. P. Mackenzie, and C. W. Hicks, Nat. Commun. 13, 4596 (2022).
  * Hristov  _et al._ (2018) A. T. Hristov, J. C. Palmstrom, J. A. W. Straquadine, T. A. Merz, H. Y. Hwang, and I. R. Fisher, Rev. Sci. Instruments. 89, 103901 (2018).
  * (32) Physik Instrumente P-885.51 PICMA ®Stack Multilayer Piezo Actuator.
  * Grinenko  _et al._ (2021) V. Grinenko, S. Ghosh, R. Sarkar, J.-C. Orain, A. Nikitin, M. Elender, D. Das, Z. Guguchia, F. Brueckner, M. E. Barber, J. Park, N. Kikugawa, D. A. Sokolov, J. S. Bobowski, T. Miyoshi, Y. Maeno, A. P. Mackenzie, H. Luetkens, C. W. Hicks, and H.-H. Klauss, Nat. Phys. 17, 748 (2021).
  * Ruesink  _et al._ (1980) D. W. Ruesink, J. M. Perz, and I. M. Templeton, Phys. Rev. Lett. 45, 734 (1980).
  * Fenton (1980) E. W. Fenton, Phys. Rev. Lett. 45, 736 (1980).
  * Kim  _et al._ (2023) B. Kim, S. Khmelevskyi, C. Franchini, and I. I. Mazin, Phys. Rev. Lett. 130, 026702 (2023).
  * Topping and Blundell (2018) C. V. Topping and S. J. Blundell, J. Phys. Condens. Matter 31, 013001 (2018).
  * Grigera  _et al._ (2003) S. A. Grigera, R. A. Borzi, A. P. Mackenzie, S. R. Julian, R. S. Perry, and Y. Maeno, Phys. Rev. B 67, 214427 (2003).
  * Kubo (1966) R. Kubo, Rep. Prog. Phys. 29, 255 (1966).
  * (40) See http://www.comsol.com for COMSOL Multiphysics® v. 6.2a, COMSOL AB, Stockholm, Sweden.


Averages of b-hadron, c-hadron, and τ-lepton properties
as of summer 2016
Heavy Flavor Averaging Group (HFLAV):

1]Y. Amhis[1]LAL, Université Paris-Sud, CNRS/IN2P3, Orsay, France
2]Sw. Banerjee[2]University of Louisville, Louisville, Kentucky, USA
3]E. Ben-Haim[3]LPNHE, Université Pierre et Marie Curie, Université Paris Diderot, CNRS/IN2P3, Paris, France
4]F. Bernlochner[4]University of Bonn, Bonn, Germany
5]A. Bozek[5]H. Niewodniczanski Institute of Nuclear Physics, Kraków, Poland
6]C. Bozzi[6]Universita e INFN, Ferrara, Ferrara, Italy
5,7]M. Chrząszcz[7]Physik-Institut, Universität Zürich, Zürich, Switzerland
4]J. Dingfelder
4]S. Duell
8]M. Gersabeck[8]School of Physics and Astronomy, University of Manchester, Manchester, UK
9]T. Gershon[9]Department of Physics, University of Warwick, Coventry, UK
10]D. Gerstel[10]Aix Marseille Univ, CNRS/IN2P3, CPPM, Marseille, France
11]P. Goldenzweig[11]Institut für Experimentelle Kernphysik, Karlsruher Institut für Technologie, Karlsruhe, Germany
12]R. Harr[12]Wayne State University, Detroit, Michigan, USA
13]K. Hayasaka[13]Niigata University, Niigata, Japan
14]H. Hayashii[14]Nara Women's University, Nara, Japan
15]M. Kenzie[15]Cavendish Laboratory, University of Cambridge, Cambridge, UK
16]T. Kuhr[16]Ludwig-Maximilians-University, Munich, Germany
10]O. Leroy
17,18]A. Lusiani[17]Scuola Normale Superiore, Pisa, Italy[18]Sezione INFN di Pisa, Pisa, Italy
19]X.R. Lyu[19]University of Chinese Academy of Sciences, Beijing, China
13]K. Miyabayashi
20]P. Naik[20]H.H. Wills Physics Laboratory, University of Bristol, Bristol, UK
21]T. Nanut[21]J. Stefan Institute, Ljubljana, Slovenia
22]A. Oyanguren Campos[22]Instituto de Fisica Corpuscular, Centro Mixto Universidad de Valencia - CSIC, Valencia, Spain
23]M. Patel[23]Imperial College London, London, UK
24]D. Pedrini [24]INFN Sezione di Milano-Bicocca, Milano, Italy
25]M. Petrič[25]European Organization for Nuclear Research (CERN), Switzerland
18]M. Rama
26]M. Roney[26]University of Victoria, Victoria, British Columbia, Canada
27]M. Rotondo[27]Laboratori Nazionali dell'INFN di Frascati, Frascati, Italy
28]O. Schneider[28]Institute of Physics, Ecole Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland
29]C. Schwanda[29]Institute of High Energy Physics, Vienna, Austria
30]A. J. Schwartz[30]University of Cincinnati, Cincinnati, Ohio, USA
10]J. Serrano
31,32]B. Shwartz[31]Budker Institute of Nuclear Physics (SB RAS), Novosibirsk, Russia[32]Novosibirsk State University, Novosibirsk, Russia
33]R. Tesarek[33]Fermi National Accelerator Laboratory, Batavia, Illinois, USA
18]D. Tonelli
34,35]K. Trabelsi[34]High Energy Accelerator Research Organization (KEK), Tsukuba, Japan[35]SOKENDAI (The Graduate University for Advanced Studies), Hayama, Japan
36]P. Urquijo[36]School of Physics, University of Melbourne, Melbourne, Victoria, Australia
37]R. Van Kooten[37]Indiana University, Bloomington, Indiana, USA
38]J. Yelton[38]University of Florida, Gainesville, Florida, USA
21,39]A. Zupanc[39]Faculty of Mathematics and Physics, University of Ljubljana, Ljubljana, Slovenia

This article reports world averages of measurements of $b$-hadron, $c$-hadron,
and $\tau$-lepton properties obtained by the Heavy Flavor Averaging Group using results available through summer 2016.
For the averaging, common input parameters used in the various analyses are adjusted (rescaled) to common values, and known correlations are taken into account.
The averages include branching fractions, lifetimes, neutral meson mixing
parameters,  violation parameters, parameters of semileptonic decays, and
Cabbibo-Kobayashi-Maskawa matrix elements.

§ SUMMARY

This article provides updated world averages of measurements of $b$-hadron, $c$-hadron, and $\tau$-lepton properties using results available through Summer 2016.
A small selection of highlights of the results described in
Sections <ref>–<ref> is given in Table <ref>.

Selected world averages.
Where two uncertainties are given the first is statistical and the second is systematic, except where indicated otherwise.

2cSelected world averages – continued from previous page.

-̱hadron lifetimes

$\bar{\tau}(\Bs) = 1/\Gs$

$\tau(B^0_{s\rm L})$

$\tau(B^0_{s\rm H})$

2|l| and  mixing / violation parameters

2|l|Parameters related to Unitarity Triangle angles

$\stwob \equiv \sin\! 2\phi_1$     $\phantom{-}0.691 \pm 0.017$

$\beta \equiv \phi_1$              $\phantom{\degrees}\left( 21.9 \pm 0.7 \right)\degrees$

$-\etacp S_{\phi \KS}$             $0.74\,^{+0.11}_{-0.13}$

$-\etacp S_{\etapr \Kz}$           $\phantom{-}0.63 \pm 0.06$

$-\etacp S_{\KS \KS \KS}$          $\phantom{-}0.72 \pm 0.19$

$\phi_s(\phi\phi)$                 $-0.17 \pm 0.15 \pm 0.03 \, {\rm rad}$

$-\etacp S_{\jpsi \piz}$           $\phantom{-}0.93 \pm 0.15$

$-\etacp S_{\Dp\Dm}$               $\phantom{-}0.84 \pm 0.12$

$-\etacp S_{\jpsi \rhoz}$          $\phantom{-}0.66 \, ^{+0.13}_{-0.12}\,^{+0.09}_{-0.03}$

$S_{K^* \gamma}$                   $-0.16 \pm 0.22$

$\left( S_{\pi^+\pi^-}, C_{\pi^+\pi^-} \right)$     $\left( -0.68 \pm 0.04, -0.27 \pm 0.04 \right)$

$\left( S_{\rho^+\rho^-}, C_{\rho^+\rho^-} \right)$                 $\left( -0.14 \pm 0.13, \phantom{-}0.00 \pm 0.09\right)$

$a(D^{*\pm}\pi^{\mp})$             $-0.039 \pm 0.010$

$A^{}_{\CP}(B\ra D^{}_{\CP+}K)$           $\phantom{-}0.111 \pm 0.018$

$A_{\rm ADS}(B\ra D^{}_{K\pi}K)$         $-0.415 \pm 0.055$

$\gamma \equiv \phi_3$                   $\phantom{\degrees}(74.0\,^{+5.8}_{-6.4})\degrees$

Semileptonic decay parameters

${\cal B}(\Bzb\to D^{*+}\ell^-\nub_\ell)$     $(4.88\pm 0.10)\%$

${\cal B}(\B^-\to D^{*0}\ell^-\nub_\ell)$     $(5.59\pm 0.19)\%$

$\eta_{\rm EW}{\cal F}(1)\vcb$     $(35.61\pm 0.43)\times 10^{-3}$

$\vcb$ from $\bar B\to D^*\ell^-\bar\nu_\ell$     $(39.05\pm 0.47_{\rm exp}\pm 0.58_{\rm th})\times 10^{-3}$

${\cal B}(\Bzb\to D^+\ell^-\nub_\ell)$     $(2.20\pm 0.10)\%$

${\cal B}(\B^-\to D^0\ell^-\nub_\ell)$     $(2.33\pm 0.10)\%$

$\eta_{\rm EW}{\cal G}(1)\vcb$     $(41.57 \pm 1.00)\times 10^{-3}$

$\vcb$ from $\bar B\to D\ell^-\bar\nu_\ell$     $(39.18 \pm 0.94_{\rm exp}\pm 0.36_{\rm th})\times 10^{-3}$

${\cal B}(\bar B\to X_c\ell^-\bar\nu_\ell)$     $(10.65\pm 0.16)\%$

${\cal B}(\bar B\to X\ell^-\bar\nu_\ell)$     $(10.86\pm 0.16)\%$

$\vcb$ from $\bar B\to X\ell^-\bar\nu_\ell$     $(42.19\pm 0.78)\times 10^{-3}$

${\cal B}(\Bb\to\pi\ell^-\nub_\ell)$     $(1.50\pm 0.06)\times 10^{-4}$

$\vub$ from $\Bb\to\pi\ell^-\nub_\ell$     $(3.67\pm 0.15)\times

$\vub$ from $\Bb\to X_u\ell^-\nub_\ell$     $(4.52\pm 0.15_{\rm exp}\pm 0.13_{\rm th})\times 10^{-3}$

$\vub/\vcb$ from $\Lambda_b^0\to p\mu^-\nub_\mu/\Lambda_b^0\to \Lambda_c^+\mu^-\nub_\mu$     $0.080\pm 0.004_{\rm exp}\pm 0.004_{\rm th}$

${\cal R}(D)={\cal B}(B\to D\tau\nu_\tau)/{\cal B}(B\to
D\ell\nu_\ell)$     $0.403\pm 0.047$

${\cal R}(D^*)={\cal B}(B\to D^*\tau\nu_\tau)/{\cal B}(B\to
D^*\ell\nu_\ell)$     $0.310\pm 0.017$

-̱hadron to charmed hadron decays

${\cal B}(\Bzb \to D^+ \pi^-)$     $(2.65\pm 0.15) \times 10^{-3}$

${\cal B}(B^- \to D^0 \pi^-)$     $(4.75 \pm 0.19) \times 10^{-3}$

${\cal B}(\Bsb \to D_s^+ \pi^-)$     $(3.03 \pm 0.25) \times 10^{-3}$

${\cal B}(\Lambda_b^0 \to \Lambda_c^+ \pi^-)$     $(4.30^{+0.36}_{-0.35}) \times 10^{-3}$

${\cal B}(\Bzb \to J/\psi \bar{K}^0)$     $(0.863 \pm 0.035) \times 10^{-3}$

${\cal B}(B^- \to J/\psi K^-)$     $(1.028 \pm 0.040) \times 10^{-3}$

${\cal B}(\Bsb \to J/\psi \phi)$     $(1.00 \pm 0.09) \times 10^{-3}$

Rare decays

${\cal B}(\Bs \to \mu^+\mu^-)$     $\left( 2.8\,^{+0.07}_{-0.06} \right) \times 10^{-9}$

${\cal B}(\Bz \to \mu^+\mu^-)$     $\left( 0.39\,^{+0.16}_{-0.14} \right) \times 10^{-9}$

${\cal B}(B \to X_s \gamma)$  ($E_{\gamma}>1.6~\gev$)     $(3.32 \pm 0.16) \times 10^{-4}$

${\cal B}(\Bp \to \tau^+ \nu)$     $(1.06 \pm 0.19) \times 10^{-4}$

$R_K = \mathcal{B}(\Bp \to K^+\mu^+\mu^-)/\mathcal{B}(\Bp \to K^+e^+e^-)$
2* $\aerr{0.745}{0.090}{0.074}{0.036}$

in $1.0<m^2_{\ell^+\ell^-}<6.0~{\mathrm{Ge\kern -0.1em V}^2/c^4}$

$A_{\CP}(\Bd\to K^+\pi^-)$, $A_{\CP}(B^+\to K^+\pi^0)$      $-0.082 \pm 0.006$, $0.040 \pm 0.021$

$A_{\CP}(\Bs\to K^-\pi^+)$     $0.26 \pm 0.04$

Longitudinal polarisation of $\Bd \to \phi  \Kstarz$     $0.497 \pm 0.017$

Longitudinal polarisation of $\Bs \to \phi  \phi$     $0.361 \pm 0.022$

Observables in $\Bz \to K^{*0}\mu^+\mu^-$ decays     2*See Sec. <ref>

in bins of $q^2 = m^2(\mu^+\mu^-)$

$D^0$ mixing and violation parameters

$x$      $(0.32\,\pm 0.14)\%$

$y$      $(0.69\,^{+0.06}_{-0.07})\%$

$\delta^{}_{K\pi}$      $(15.2\,^{+7.6}_{-10.0})^\circ$

$A^{}_D$      $(-0.88\,\pm 0.99)\%$

$|q/p|$     $0.89\,^{+0.08}_{-0.07}$

$\phi$      $(-12.9\,^{+9.9}_{-8.7})^\circ$

$x^{}_{12}$ (no direct violation)      $(0.41\,^{+0.14}_{-0.15})\%$

$y^{}_{12}$ (no direct violation)      $(0.61\,\pm 0.07)\%$

$\phi^{}_{12}$ (no direct violation)      $(-0.17\,\pm 1.8)^\circ$

$a^{\rm ind}_{\CP}$     $(0.030 \pm 0.026)\%$

$\Delta a^{\rm dir}_{\CP}$     $(-0.134 \pm 0.070)\%$

Leptonic $D$ decays

$f^{}_D$         $(203.7\,\pm 4.9)$ MeV

$f^{}_{D_s}$      $(257.1\,\pm 4.6)$ MeV

$|V^{}_{cd}|$     $0.2164\,\pm 0.0050_{\rm exp} \pm 0.0015_{\rm LQCD}$

$|V^{}_{cs}|$     $1.006\,\pm 0.018_{\rm exp} \pm 0.005_{\rm LQCD}$

Benchmark charm branching fractions

${\cal B}(\Lambda^+_c\ra pK^-\pi^+)$     $(6.46\,\pm 0.24)\%$

${\cal B}(D^0\ra K^-\pi^+)$
$(3.962\,\pm 0.017\,\pm 0.038\,\pm 0.027_{\rm FSR})\%$

${\cal B}(D^0\ra K^+\pi^-)/{\cal B}(D^0\ra K^-\pi^+)$

${\cal B}(D^+_s\ra K^+K^-\pi^+)$
$(5.44\,\pm 0.09\,\pm 0.11)\%$

$\tau$ parameters, lepton universality, and $|V_{us}|$

$g^{}_{\tau}/g^{}_{\mu}$        gtaubygmu_tau

$g^{}_{\tau}/g^{}_{e}$          gtaubyge_tau

$g^{}_\mu/g^{}_e$               gmubyge_tau

${\cal B}_e^{\text{uni}}$       Be_univ%

$R_{\text{had}}$                R_tau

$|V_{us}|$ from sum of strange branching fractions                                       Vus

$|V_{us}|$ from ${\cal{B}}(\tau^- \to K^-\nu^{}_\tau)/ {\cal{B}}(\tau^- \to \pi^-\nu^{}_\tau)$     Vus_tauKpi

$|V_{us}|$ average                                                                             Vus_tau

Since the previous version of this document <cit.>,
the -̱hadron lifetime and mixing averages have mostly made gradual
progress in precision.
Notable exceptions with significant improvement are
the averages for the mass difference in the $\Bd$–$\Bzb$ system () and the
violation parameter in $\Bs$–$\Bsb$ system ($|q_{\particle{s}}/p_{\particle{s}}|$).
In total eleven new results
(of which ten from the LHC Run 1 data and one from the Tevatron data)
have been incorporated in these averages.
On the other hand, all results that remained unpublished and
for which there is no publication plan,
have been removed from the averages.

The lifetime hierarchy for the most abundant weakly decaying -̱hadron species
is well established, with impressive precisions of 5 fs or less
for the most common , and mesons,
and compatible with the expectations from the Heavy Quark Expansion.
However, statistics are still lacking for ḇaryons heavier
than (, , $\Omega_b$, and all other yet-to-be-discovered
ḇaryons), but this will surely come from the LHC with sufficient time.
A sizable value of the decay width difference in the $\Bs$–$\Bsb$ system
is measured with a relative precision of 7% and is well predicted by the
Standard Model (SM). In contrast,
the experimental results for the decay width difference in the
$\Bd$–$\Bzb$ system are not yet precise enough to distinguish
the small (expected) value from zero.
The mass differences in both systems are known very accurately, to the (few)
per mil level. On the other hand, violation in the mixing of either system
has not been observed yet, with asymmetries known within a couple per mil but
still consistent both with zero and their SM predictions.
A similar conclusion holds for the violation induced
by mixing in the $b\to c\bar{c}s$ transition, although in this case
the experimental precision on the corresponding weak phase is an order
of magnitude larger, but now becoming just smaller than the SM central value.
Many measurements are still dominated by statistical uncertainties and will improve once new results from the LHC Run 2 become available.

The measurement of $\sin 2\beta \equiv \sin 2\phi_1$ from $b \to
c\bar{c}s$ transitions such as $\Bz \to \jpsi\KS$ has reached $<2.5\,\%$
precision: $\sin 2\beta \equiv \sin 2\phi_1 = 0.691 \pm 0.017$.
Measurements of the same parameter using different quark-level processes
provide a consistency test of the Standard Model and allow insight into
possible new physics.

All results among hadronic $b \to s$ penguin dominated decays of mesons are currently consistent with the Standard Model expectations.
Measurements of violation parameters in $\Bs \to \phi\phi$ allow a similar comparison to the value of $\phiccbars$; again, results are consistent with the SM expectation (which in this case is very close to zero).
Among measurements related to the Unitarity Triangle angle $\alpha \equiv \phi_2$, results from the $\rho\rho$ system allow constraints at the level of $\approx
These remain the strongest constraints, although results from all of , and LHCb lead to good precision on the violation parameters in $\Bz \to \pip\pim$ decays.
Knowledge of the third angle $\gamma \equiv \phi_3$ also continues to improve, with the current world average being $(74.0\,^{+5.8}_{-6.4})\degrees$.
The precision is expected to improve further as more data becomes available at LHCb and Belle II.

In semileptonic $B$ meson decays, the anomalies reported in the last
version of the document have remained:
The discrepancy between  measured with inclusive and exclusive decays is of the order of $3\sigma$ ($3.2\sigma$ for  from $\Bb\to D^*\ell^-\bar\nu_\ell$, $2.4\sigma$ for  from $\Bb\to D\ell^-\bar\nu_\ell$).
The difference between  measured with
inclusive decays $\Bb\to X_u\ell^-\nub_\ell$ and  from $\Bb\to\pi\ell^-\nub_\ell$ has risen to $3.6\sigma$.
An important new contribution to the determination of the values of  and  comes from exclusive $b$-baryon decays.
The largest anomaly however is observed in $B\to D^{(*)}\tau\nu_\tau$ decays: The combined discrepancy of the measured values of ${\cal R}(D^*)$ and ${\cal R}(D)$ to their standard model expectations is found to be $3.9\sigma$.

The most important new measurements of rare $b$-hadron decays are coming from the LHC.
Precision measurements of $\Bs$ decays are particularly noteworthy, including several measurements of the longitudinal polarisation fraction from LHCb.
ATLAS, CMS and LHCb have significantly improved the sensitivity to the $B^0_{(s)}\to\mumu$ decays.
Recently, CMS and LHCb published a combined analysis that allowed the first observation of the $\Bs\to\mumu$ decay to be obtained, and provided three standard deviations evidence of the $\Bz\to\mumu$ decay.
The results are compatible with the SM predictions, and yield constrains on the parameter space of new physics models.
CMS and LHCb have also performed angular analyses of the $\Bz\to\Kstarz\mumu$ decay, complementing, extending and improving on the precision of results from and Belle.
One of the observables measured by LHCb, $P_5^{\prime}$, differs from the SM prediction by $3.7\sigma$ in one of the $m^2_{\mumu}$ intervals; results from Belle on this observable are consistent but less precise.
Improved measurements from LHCb and other experiments are keenly anticipated.
A measurement of the ratio of branching fractions of $\Bp \to K^+\mu^+\mu^-$ and $\Bp \to K^+e^+e^-$ decays ($R_K$) has been made by LHCb.
In the low $m^2_{\ell^+\ell^-}$ region, it differs from the standard model prediction by $2.6\sigma$.
Among the violating observables in rare decays, the “$K\pi$ puzzle” persists, and important new results have appeared in three-body decays.
LHCb has produced many other results on a wide variety of decays, including $b$-baryon and -meson decays.
Belle and  continue to produce new results though their output rates are dwindling.
It will still be some years before we see new results from the upgraded SuperKEKB $B$ factory and the Belle II experiment.

About 800 $b$ to charm results from , Belle, CDF, D0, LHCb, CMS, and ATLAS reported in more than 200 papers
are compiled in a list of over 600 averages.

The huge samples of $b$ hadrons that are available in contemporary experiments allows measurements of decays to states with open or hidden charm content with unprecedented precision.
In addition to improvements in precision for branching fractions of $\Bzb$ and $B^-$ mesons, many new decay modes have been discovered.
In addition, there is a rapidly increasing set of measurements available for $\Bsb$ and $B_c^-$ mesons as well as for $b$ baryon decays.

In the charm sector, $\Dz$–$\Dzb$ mixing is now well-established and
the emphasis has shifted to searching for  violation. Measurements
of 49 observables from the E791, FOCUS, Belle, , CLEO, BESIII, CDF,
and LHCb experiments are input into a global fit for 10 underlying parameters,
and the no-mixing hypothesis is excluded at a confidence level $>11.5\sigma$.
The mixing parameters $x$ and $y$ individually differ from zero by
$1.9\sigma$ and $9.4\sigma$, respectively. The world average value for
the observable $y_{\CP}$ is positive, indicating that the -even
state is shorter-lived as in the $\Kz$–$\Kzb$ system.
The  violation parameters $|q/p|$ and $\phi$ are consistent with the
no- violation hypothesis within $1\sigma$. Thus there is no evidence
for  violation arising from mixing ($|q/p|\neq 1$) or
from a phase difference between the mixing amplitude and
a direct decay amplitude ($\phi\neq 0$). In addition,
the most recent data indicates no direct  violation in
$D^0\ra K^+K^-/\pi^+\pi^-$ decays; performing a global
fit to all relevant measurements gives
$\Delta a^{\rm dir}_{\CP}= (-0.134\pm 0.070)\%$.
The world's most precise measurements of $|V^{}_{cd}|$ and $|V^{}_{cs}|$
are obtained from leptonic $D^+\ra \mu^+\nu$ and $D^+_s\ra\mu^+\nu/\tau^+\nu$
decays, respectively.
These measurements have theoretical uncertainties arising from decay constants.

However, calculations of decay constants within lattice QCD have improved such
that the theory error is $<1/3$ the experimental errors of the measurements.

Since 2016, HFLAV provides the branching fraction fit averages for the
PDG Review of Particle Physics. For the PDG, a unitarity constrained
variant of the fit is performed, using only inputs that are published
and included in the PDG. Two preliminary results used in the HFLAV 2014
report have been removed both in the HFLAV and in the PDG variants of the
fit. A few minor imperfections of the 2014 fit have been corrected. There
are no non-negligible changes to the lepton universality tests and to
the determinations from the branching fractions. There is
still a large discrepancy between from , from kaons and
from and CKM matrix unitarity. On this topic, recent
studies <cit.>
claim to get a more reliable theory uncertainty on
with a revised calculation method that uses also the $\tau$
spectral functions.
Just one more lepton-flavour-violating branching
fraction upper limit has been published, which does not change the
computed combined related limit. The list of limits and their combinations has
been revised to remove old preliminary results.

§ ACKNOWLEDGMENTS

We are grateful for the strong support of the
ATLAS, , , BESIII, CLEO(c), CDF, CMS,
and LHCb collaborations, without whom this
compilation of results and world averages would not have
been possible. The success of these experiments in turn would
not have been possible without the excellent operations of the
BEPC, CESR, KEKB, LHC, PEP-II, and Tevatron accelerators.

We also recognise the interplay between theoretical and experimental
communities that has provided a stimulus for many of the measurements
in this document.

Our averages and this compilation have benefitted greatly from
contributions to the Heavy Flavor Averaging Group from numerous
individuals. We thank David Kirkby, Yoshihide Sakai, Simon Eidelman,
Soeren Prell, and Gianluca Cavoto for their past leadership of HFLAV.
We are grateful to Paolo Gambino and Bob Kowalewski for assistance with
averages that appear in Section <ref>;
to Ruslan Chistov, Lawrence Gibbons, Bostjan Golob, Milind Purohit,
and Patrick Roudeau for significant contributions to
Section <ref>;
and to Michel Davier for providing valuable input to Section <ref>.
We especially thank the following for their careful review of the
text in preparing this paper for publication:

Vanya Belyaev,
Guennadi Borissov,
Jolanta Brodzicka,
Paoti Chang,
Paolo Gambino,
Martin Jung,
Andreas Kronfeld,
George Lafferty,
Alexander Lenz,
Zoltan Ligeti,
David London,
Vera Lüth,
Alexey Petrov,
Antonio Pich, and
Fernando Martinez Vidal.
Finally, we thank the SLAC National Accelerator Laboratory for
providing crucial computing resources and support to HFLAV.


# On the number of star-shaped classes in optimal colorings of Kneser graphs

Hamid Reza Daneshpajouh  Hamid Reza Daneshpajouh, University of Nottingham
Ningbo China, 199 Taikang E Rd, Yinzhou, Ningbo, Zhejiang, China, 315104
<EMAIL_ADDRESS>
###### Abstract.

A family of sets is called star-shaped if all the members of the family have a
point in common. The main aim of this paper is to provide a negative answer to
the following question raised by James Aisenberg et al [Short proofs of the
kneser–Lovász coloring principle, Information and Computation, 261:296–310,
2018.], for the case $k=2$.

Question. Do there exist $(n-2k+2)$-colorings of the $(n,k)$-Kneser graphs
with more than $k-1$ many non-starshaped color classes?

###### Key words and phrases:

chromatic number; Kneser graph; line graph

## Introduction

In this paper all graphs are finite, undirected and simple. A (proper)
coloring of a graph is an assignment of colors to its vertices such that no
two adjacent vertices have the same color. The chromatic number $\chi(H)$ of a
graph $H$ is the smallest number of colors in a coloring of $H$. For a given
coloring of a graph $H$, a subgraph of $H$ is called colorful if all its
vertices receives different colors. The $(n,k)$-Kneser graph $KG(n,k)$ is a
graph whose vertices correspond to all $k$-subsets of the set
$[n]=\\{1,2\ldots,n\\}$, and where two vertices are adjacent if and only if
the two corresponding sets are disjoint. Finally, a family $\mathcal{F}$ of
sets is called star-shaped if $\bigcap\mathcal{F}\neq\emptyset$.

Almost seventy years ago, a purely combinatorial conjecture was raised by
Kneser which in the language of graph theory is equivalent to say
$\chi(KG(n,k))=n-2k+2$ for $n\geqslant 2k-1$. About two decades later, Lovász
[12] proved Kneser’s conjecture by taking an extraordinary approach, using
tools from algebraic topology. Since then, many attentions have been drawn to
study various problems related to this conjecture, including a large number of
new proofs such as [4, 9, 11, 13], and many generalizations such as [2, 3, 10,
14, 5, 8, 6]. In particular, James Aisenberg et al [1] gave a new proof of
this conjecture uses a simple counting argument based on the Hilton–Milner
theorem, for all but finitely many cases111Here, we assume that $k$ is a fixed
number..

Indeed, their main idea was based on that fact that if there was an
$(n-2k+1)$-coloring of $KG(n,k)$, then it would contain a star-shaped color
class [1, Lemma 8] provided that $n$ is enough large222For instance
$n\geqslant k^{4}$ works.. Then removing this vertex would lead to a
$((n-1)-2k+1)$-coloring of $KG(n-1,k)$. So, to verify the conjecture one just
needs to check the validity of the conjecture for finitely many base cases
which can be checked, for instance, using the topological arguments provided
by Lovász for these cases333If $k$ is small enough ($k=2$, or $3$), then the
base cases can be checked by hand or a computer. However, for $k\geqslant 4$
we do not know whether there exist a “purely combinatorial way” to establish
the conjecture..

Actually, they showed if there was such a coloring then it would contain many
star-shaped color classes [1, Lemma 9], again provided $n$ is enough large.
Motivated by this result, they showed that there is an optimal coloring ( a
coloring with $(n-2k+2)$ colors) of $KG(n,k)$ with $(k-1)$ star-shaped color
classes provided that $n\geqslant 3k-3$ and then they raised the following
question in this regard.

######  Question 1 ([1]).

Do there exist $(n-2k+2)$-colorings of the $(n,k)$-Kneser graphs with more
than $k-1$ many non star-shaped color classes?

The main objective of this paper is to give a negative answer to the above
question for the case $k=2$. As a corollary, this result leads to a purely
combinatorial proof for the $\mathcal{K}_{l,m}$-Theorem [15, Theorem 2] for
the case $KG(n,2)$. More precisely, for every non-negative integers $k,l$ with
$k+l+m=n-2$, and any optimal proper coloring of $KG(n,2)$, there is a colorful
complete bipartite $\mathcal{K}_{l,m}$ subgraph in $KG(n,2)$. A similar result
for existence of colorful complete tripartite sungraph in $KG(n,2)$ will be
provided as as well.

##  1\. Main Results

First note that we can view $KG(n,2)$ as the complement of the line graph of
the complete graph $\mathcal{K}_{n}$ and therefore proper colorings of
$KG(n,2)$ are exactly partitions of the edge set of the complete graphs
$\mathcal{K}_{n}$ into stars and triangles. More generally, as mentioned here
[7], if $G$ is the complement of the line graph of a graph $H$, then proper
colorings of $G$ are exactly partitions of the edge set of $H$ into stars and
triangles. A star is a tree with a vertex, the center of the star, connected
to all other vertices. A single-edge tree is in particular a star. In this
particular case, we will always assume that exactly one of the two vertices
has been identified as the center, so as to be in a position of always speak
of the center of a star without ambiguity. A triangle is a circuit of length
$3$. We call such a partition of $H$ into stars and triangles an
$\operatorname{ST}$-partition.

######  Lemma 1.

Consider an optimal $\operatorname{ST}$-partition $\mathcal{P}$ of a graph
$H$. Then, at most one vertex of each triangle in $\mathcal{P}$ can be the
center of a star.

###### Proof.

We proceed by a contradiction. Suppose that $T$ is a triangle in $\mathcal{P}$
with at least two vertices $x$, $y$ such that each of them is also the center
of a star, name it $S_{x}$, and $S_{y}$ respectively. Then we can add the two
edges of $T$ incident to $x$ to $S_{x}$, and the remaining edge of $T$ to
$S_{y}$, and finally remove $T$ from $S$. This leads to a new
$\operatorname{ST}$-partition with fewer elements which contradicts the
optimality of $\mathcal{P}$. ∎

######  Lemma 2.

Consider an optimal $\operatorname{ST}$-partition $\mathcal{P}$ of a graph
$H$. If there is a triangle $T$ in $\mathcal{P}$, then each vertex $x$ of $H$
which does not belong to any triangle of $\mathcal{P}$ and it is connected to
at least two vertices of $T$ is the center of a star.

###### Proof.

We will use a proof by contradiction. Suppose $x$ is not the center of any
star. If $a,b$ are two vertices of $T$ which are connected to $x$, then each
of the edges $\\{a,x\\}$ and $\\{b,x\\}$ must belong to a star or a triangle
in $\mathcal{P}$. These imply that $a$ and $b$ are both must be the center of
stars, as no triangle in $\mathcal{P}$ having $x$ as its vertices, which
contradicts the previous lemma. ∎

######  Theorem 1.

There is no optimal ST-partition of the complete graph $\mathcal{K}_{n}$ with
more than one triangle. Moreover, if $n\geqslant 3$, any optimal ST-partition
of $\mathcal{K}_{n}$ contains exactly one triangle.

###### Proof.

Suppose the contrary and choose the minimum integer $n$ with the property that
there is an optimal ST-partition $\mathcal{P}$ of the complete graph
$\mathcal{K}_{n}$ with more than one triangle. Let $T_{1},\ldots,T_{k}$ be the
list of all the triangles in $\mathcal{P}$ where $k\geqslant 2$. Let
$A=\bigcup_{i=1}^{k}V(T_{i})$ be the set of the vertices of these triangles,
$B\subseteq[n]\setminus A$ be the set of the rest vertices which are centers
of stars, and finally put $C=[n]\setminus(A\cup B)$. By Lemma 2, we have
$C=\emptyset$. Moreover, Lemma 1 implies $B=\emptyset$. Indeed, if there was a
vertex of $\mathcal{K}_{n}$ in $B$, then removing this vertex leads to an
optimal ST-partition with at least two triangles for $\mathcal{K}_{n-1}$,
which would contradict the minimality of $n$, and therefore we must have
$B=\emptyset$. In summery, we have shown that any vertex of $\mathcal{K}_{n}$
belongs to at least one of these triangles $T_{1},\ldots,T_{k}$. Now, we
consider two cases:

Case I. First, we consider the case that there are exactly two triangles in
$\mathcal{P}$, say $T_{1}$, and $T_{2}$. We have two possibilities.

  1. (1)

$T_{1}$, and $T_{2}$ are vertex disjoint

Note that in this case we have $n=6$ and therefore $|\mathcal{P}|=4$ (since
$\mathcal{P}$ is an optimal ST-partition of $\mathcal{K}_{6}$). But on the
other hand, we have

$|P|\geqslant 2+3=5$

as $T_{1},T_{2}\in P$ and at least three vertices of $T_{1}\cup T_{2}$ are
centers. To see the latter claim, consider the edges $e_{1},e_{2},e_{3}$ of
$\mathcal{K}_{6}$ which are shown in below

Each of these edges must belong to a triangle or a start in $\mathcal{P}$. But
none of them can belong to a triangle as there are no other triangle in
$\mathcal{P}$ except $T_{1}$ and $T_{2}$. So, each of them must belong to a
star, and hence one of the endpoints of each of $e_{1},e_{2},$ and $e_{3}$
must be the center of a star. So, there are at least three stars in
$\mathcal{P}$ as these edges are pairwise disjoint. Hence $\mathcal{P}$ must
have at least $5$ elements which contradicts the optimality of $\mathcal{P}$.

  2. (2)

If $T_{1}$, and $T_{2}$ have a vertex in common,

then we have

$|P|\geqslant 2+2=4,$

as $T_{1},T_{2}\in P$, and at least two vertices of $T_{1}\cup T_{2}$ are
centers. Again to see why the latter claim is true, it is enough to consider
the edges $e_{1}$ and $e_{2}$ which are depicted below

and repeat the same argument. So, the size of $\mathcal{P}$ is at least four.
But, this is again impossible as $P$ is an optimal ST-partition of
$\mathcal{K}_{5}$ which implies $|\mathcal{P}|=3$.

Case II. Finally, suppose there are at least 3 triangles in $\mathcal{P}$.
Then, this implies that no vertex of a triangle is a center of a star. Indeed,
if there was a triangle $T$ in $\mathcal{P}$ which one of its vertices was a
center of a star, then all the other triangles in $\mathcal{P}$ except at most
one of them is attached to this triangle at $x$ (otherwise removing this
vertex would lead to an optimal ST-partition for $\mathcal{K}_{n-1}$ with at
least two triangles which would contradict the minimality of $n$) as shown in
below

Now, consider the edges $e_{1}$ and $e_{2}$ of $\mathcal{K}_{n}$ as shown
below.

At most one of these edges can be covered with that only one possible triangle
$\mathcal{P}$ which is not attached to $x$. So, at least one of the end points
of $e_{1}$ or $e_{2}$ must be the center of a star. But then, we would have a
triangle with two of its vertices ($x$ and of the endpoints of $e_{1}$ or $x$
and of the endpoints of $e_{2}$ ) are the center of stars which would
contradict Lemma 1. So, all member of $\mathcal{P}$ are triangles. Thus, we
must have

$3(n-2)=\binom{n}{2},$

as $\mathcal{P}$ is an optimal ST-partition of edges of $\mathcal{K}_{n}$.
This equation just has two solutions $n=3$ or $4$, but on the other hand
having at least 3 triangles in $\mathcal{P}$ implies $n\geqslant 6$ which is
again a contradiction.

So, we have shown that any optimal ST-partition of $\mathcal{K}_{n}$ contains
at most one triangle. Moreover, we show that every optimal ST-partition of
$\mathcal{K}_{n}$ contains exactly one triangle provided that $n\geqslant 3$.
Indeed, the size of any ST-partition $\mathcal{P}$ of $\mathcal{K}_{n}$
without a triangle is at least $n-1$ as at least one of the endpoints of each
edge must be the center of a star. Thus, any optimal ST-partition
$\mathcal{P}$ of $\mathcal{K}_{n}$ where $n\geqslant 3$ has exactly one
triangle (as $|\mathcal{P}|=n-2$ for $n\geqslant 3$).

∎

######  Corollary 1.

Let $n\geqslant 3$ be a natural number. For every non-negative integers $l$
and $m$ with $l+m=n-2$, and any optimal proper coloring of $KG(n,2)$, there is
a colorful complete tripartite $\mathcal{K}_{l,m}$ subgraph in $KG(n,2)$.

###### Proof.

The assertion is obviously true for $n=3$. So, without loss of generality we
can assume $n\geqslant 4$. Let $\mathcal{P}$ be an arbitrary optimal ST-
partition of $\mathcal{K}_{n}$. By Theorem 1, $\mathcal{P}$ contains exactly
one triangle, say $T$ with vertex set $\\{1,2,3\\}$. Moreover, no vertex of
this triangle is a center of a star. Since otherwise, we could delete this
vertex and find a minimal ST-partition for $\mathcal{K}_{n-1}$ without any
triangle. Then, this would imply that $n-1\leqslant 2$, which contradicts our
assumption. Now, $\mathcal{K}_{A,B}$ is the desired subgraph in $KG(n,2)$
where

$\displaystyle A=\\{\\{1,i\\}:4\leqslant i\leqslant l+3\\}$ $\displaystyle
B=\\{\\{2,j\\}:l+4\leqslant j\leqslant n\\}\cup\\{\\{2,3\\}\\}$

as each $i\neq 1,2,3$ is a center of a star and no vertex of $T$ is a center.
∎

Above proof reveals that we can have a similar result about the existence of
colorful complete tripartite subgraph in $KG(n,2)$ as well.

######  Corollary 2.

Let $n\geqslant 6$ be a natural number. For every integers $k,l,m\geqslant 1$
with $k+l+m=n-3$, and any optimal proper coloring of $KG(n,2)$, there is a
colorful complete tripartite $\mathcal{K}_{k,l,m}$ subgraph in $KG(n,2)$.

###### Proof.

Again, by Theorem 1, $\mathcal{P}$ contains exactly one triangle, say $T$ with
vertex set $\\{1,2,3\\}$ and also as mentioned before no vertex of this
triangle can be a center of a star. Now, $\mathcal{K}_{A,B,C}$ is the desired
subgraph in $KG(n,2)$ where

$\displaystyle A=\\{\\{1,i\\}:4\leqslant i\leqslant k+3\\}$ $\displaystyle
B=\\{\\{2,i\\}:k+4\leqslant i\leqslant k+l+3\\}$ $\displaystyle
C=\\{\\{3,i\\}:k+l+4\leqslant i\leqslant n\\}$

as each $i\neq 1,2,3$ is a center of a star and no vertex of $T$ is a center.
∎

######  Remark.

Note that in the previous corollary the assumption $k+l+m=n-3$ cannot be
replaced with $k+l+m=n-2$. To see this, consider the following coloring of
$KG(n,2)$

$c(\\{i<j\\})=\begin{cases}3\quad&\text{if}\,j\leqslant 3\\\
j\quad&\text{if}\,i\leqslant 3\,\&\,j\geqslant 4\\\
i\quad&\text{if}\,i\geqslant 4\\\ \end{cases}$

We can see this coloring as a coloring of edges of the complete graph
$\mathcal{K}_{n}$. With having this interpretation in mind, one can check that
there is no colorful cycle in $\mathcal{K}_{n}$, i.e., a cycle whose all edges
receives different colors. Now, if there was a colorful complete tripartite
subgraph $\mathcal{K}_{A,B,C}$ in $KG(n,2)$ where $|A|=k,|B|=l,|C|=m$ with
$k,l,m\geqslant 1$ and $m+l+k=n-2$ , then , as none of $A,B$, and $C$ form a
cycle in $K_{n}$, we must have

$|\bigcup A|\geqslant k+1,|\bigcup B|\geqslant l+1,|\bigcup C|\geqslant m+1.$

Now, since the sets $\bigcup A,\bigcup B,\bigcup C$ are pairwise disjoint, we
must have $(k+1)+(l+1)+(m+1)\leqslant n$ which implies $m+l+k\leqslant n-3$.
Then, this would contradict our assumption that $m+l+k=n-2$.

### Acknowledgments

The idea of this paper was inspired while the author was at Université Paris
Est as a postdoctoral researcher under the supervision of Frédéric Meunier.
The author would like to express his deepest gratitude to him for his
invaluable advice, continuous support, his generous hospitality, and his
valuable comments on the first draft of this paper. Also a part of this work
was done when the author was at the School of Mathematics of IPM as a guest
researcher. Special thanks go to IPM as in that period of time the research of
the author was supported by a grant from IPM.

## References

  * [1] James Aisenberg, Maria Luisa Bonet, Sam Buss, Adrian Crăciun, and Gabriel Istrate.  Short proofs of the kneser–lovász coloring principle.  Information and Computation, 261:296–310, 2018.
  * [2] Meysam Alishahi and Hossein Hajiabolhassan.  On the chromatic number of general kneser hypergraphs.  Journal of Combinatorial Theory, Series B, 115:186–209, 2015.
  * [3] Noga Alon, Peter Frankl, and László Lovász.  The chromatic number of kneser hypergraphs.  Transactions of the American Mathematical Society, 298(1):359–370, 1986.
  * [4] Hamid Reza Daneshpajouh.  New construction of graphs with high chromatic number and small clique number.  Discrete & Computational Geometry, 59(1):238–245, 2018.
  * [5] Hamid Reza Daneshpajouh.  On the chromatic number of generalized kneser hypergraphs.  European Journal of Combinatorics, 81:150–155, 2019.
  * [6] Hamid Reza Daneshpajouh, Roman Karasev, and Alexey Yu Volovikov.  Hedetniemi’s conjecture from the topological viewpoint.  arXiv preprint arXiv:1806.04963, 2018.
  * [7] Hamid Reza Daneshpajouh, Frédéric Meunier, and Guilhem Mizrahi.  Colorings of complements of line graphs.  Journal of Graph Theory, 98(2):216–233, 2021.
  * [8] Hamid Reza Daneshpajouh and József Osztényi.  On the neighborhood complex of $\vec{s}$-stable kneser graphs.  Discrete Mathematics, 344(4):112302, 2021.
  * [9] Vladimir L Dol’nikov.  A certain combinatorial inequality.  Siberian Mathematics Journal, 29:375–397, 1988.
  * [10] Florian Frick.  Intersection patterns of finite sets and of convex sets.  Proceedings of the American Mathematical Society, 145(7):2827–2842, 2017.
  * [11] Joshua E Greene.  A new short proof of kneser’s conjecture.  The American mathematical monthly, 109(10):918–920, 2002.
  * [12] László Lovász.  Kneser’s conjecture, chromatic number and homotopy.  Journal of Combinatorial Theory, Series A, 25:319–324, 1978.
  * [13] Jiří Matoušek.  A combinatorial proof of kneser’s conjecture.  Combinatorica, 24(1):163–170, 2004.
  * [14] Frédéric Meunier.  The chromatic number of almost-stable Kneser hypergraphs.  Journal of Combinatorial Theory, Series A, 118:1820–1828, 2011\.
  * [15] Gábor Simonyi and Gábor Tardos.  Colorful subgraphs in kneser-like graphs.  European Journal of Combinatorics, 28(8):2188–2200, 2007.