# Nonlinear coupling of phononic resonators induced by surface acoustic waves

Sarah Benchabane, Aymen Jallouli, Laetitia Raguin, Olivier Gaiffe, Jules
Chatellier, Valérie Soumann, Jean-Marc Cote, Roland Salut, and Abdelkrim
Khelif<EMAIL_ADDRESS>FEMTO-ST, Université de Bourgogne
Franche-Comté, CNRS, UFC
15B Avenue des Montboucons, F-25030 Besançon Cedex, France

###### Abstract

The rising need for hybrid physical platforms has triggered a renewed interest
for the development of agile radio-frequency phononic circuits with complex
functionalities. The combination of travelling waves with resonant mechanical
elements appears as an appealing means of harnessing elastic vibration. In
this work, we demonstrate that this combination can be further enriched by the
occurrence of elastic non-linearities induced travelling surface acoustic
waves (SAW) interacting with a pair of otherwise linear micron-scale
mechanical resonators. Reducing the resonator gap distance and increasing the
SAW amplitude results in a frequency softening of the resonator pair response
that lies outside the usual picture of geometrical Duffing non-linearities.
The dynamics of the SAW excitation scheme allows further control of the
resonator motion, notably leading to circular polarization states. These
results paves the way towards versatile high-frequency phononic-MEMS/NEMS
circuits fitting both classical and quantum technologies.

The on-chip manipulation of elastic or mechanical vibrations, whether hosted
and localized within resonators or traveling in the material substrate, has
been leading to significant progress in diverse areas of applied and
fundamental science. Surface acoustic wave (SAW) devices are a striking
example of such rapid developments. They are ubiquitously used in modern
telecommunications systems Morgan and Paige (2007) and are extremely relevant
as sensing components in a number of applications Go et al. (2017); Devkota et
al. (2017). Micro- and nano-mechanical systems, for their part, have imposed
themselves as prevalent means of implementing extremely sensitive sensing
devices Li et al. (2007); Jensen et al. (2008); Chaste et al. (2012); Hanay et
al. (2015). They are characterized by a rich dynamics, involving linear and
nonlinear phenomena Eichler et al. (2011); Villanueva et al. (2013); Truitt et
al. (2013); Mahboob et al. (2015) that opens exciting prospects for mechanical
signal processing. This dynamics can be further enriched in coupled systems
where mechanical mode coupling can be induced in e.g. clamped beams Karabalin
et al. (2008); Okamoto et al. (2013); Faust et al. (2013), pillars Raguin et
al. (2019); Doster et al. (2019) or through intermodal coupling Westra et al.
(2010); Mahboob et al. (2013); Mathew et al. (2016); Matheny et al. (2019).
Over the past few years, both types of mechanical systems have been attracting
renewed interest Delsing et al. (2019) and were exploited to achieve dynamic
and coherent interaction with a variety of physical systems, including holes
and electrons McNeil et al. (2011); Hermelin et al. (2011); Hsiao et al.
(2020), quantum dots Couto et al. (2009); Metcalfe et al. (2010); Kettler et
al. (2020); Vogele et al. (2020) spins MacQuarrie et al. (2013); Golter et al.
(2016); Whiteley et al. (2019), superconducting circuits O’Connell et al.
(2010); Gustafsson et al. (2014); Satzinger et al. (2018); Bienfait et al.
(2019) or and photonic devices Fuhrmann et al. (2011); Balram et al. (2016);
Patel et al. (2017); Riedinger et al. (2018); Kalaee et al. (2019). Enhancing
control over such mechanical systems at the micro- or nano-scale and in the
radio-frequency regime therefore holds promises for the implementation of
complex phononics-based hybrid platforms permitting high-speed operation.
Recent works have proposed to combine NEMS-based architectures with travelling
elastic waves to demonstrate dynamic phononic systems Hatanaka et al. (2014),
that could be further enriched by harnessing mechanical nonlinearities Cha and
Daraio (2018); Kurosu et al. (2018, 2020). The reported approaches exploit
geometric, electrostatic or wave-mixing induced non-linear behaviors, taking
advantage of the high-quality factor resonance of mechanical elements. The
current limit of this approach may however lie in the attainable resonant
frequencies that are heavily conditioned by NEMS classically operating at
maximum frequencies of a few MHz.

In this work, we demonstrate the possibility to implement a nonlinear phononic
platform based on the interaction of travelling SAWs and coupled mechanical
resonators capable to operate in the radio-frequency regime. We analyze a pair
of low aspect ratio cylindrical pillars vibrating on a flexural mode and
demonstrate the obtaining of unexpected non-linear coupled mechanical states
from these otherwise linearly-behaving mechanical resonators. The observed
experimental results, supported by analytical modelling and numerical
simulations, show that these non-linearities are the result of an interplay
between the resonator mode symmetry and the elastic field distribution on the
substrate surface and that they exhibit a strong dependence on the resonator
gap distance and the SAW excitation scheme. The elastic energy distribution
affects in turn the substrate surface displacement, hence disturbing the
resonators dynamics and leading to the occurrence of circular state of motion.
The proposed devices, that operate in the 70-MHz range, are readily scalable
to higher frequencies. They illustrate the relevance of SAW-based phononic
devices for the implementation of high-frequency electromechanical circuits
with complex dynamics.

##  I Results

Figure 1: Experimental frequency responses of SAW-driven phononic
microresonators. (a) Response of a single pillar with a diameter of 4.4 $\mu$m
and a height of about 4 $\mu$m excited by a SAW for different input powers
applied to the IDT (10 dBm, 15 dBm and 20 dBm). The light dotted lines
correspond to data obtained by laser scanning interferometry and acquired
every 10 kHz. The solid lines are obtained by applying a Savitsky-Golay
smoothing filter to the raw experimental data. Inset: Angular plot of the
distribution of the orientation of the flexural mode vibration for frequencies
between 65 and 75 MHz. (b) Frequency response for two pillars within a pair
with a 6-$\mu$m gap distance excited by a longitudinal SAW wave vector under
different drive powers. The solid line corresponds to the resonator located
closest to the SAW source, the dashed line to the resonator located farthest
from the source.

###  I.1 Linear regime: single resonators and large gap distances.

The phononic resonators under study consist of micron-scale cylindrical
pillars with an aspect ratio of the order of one, deposited atop a single-
crystal piezoelectric substrate. The resonators are excited by a Rayleigh
surface acoustic wave generated by an interdigital transducer harmonically-
driven at frequencies about 70 MHz. The resulting SAW amplitude depends
linearly on the applied RF power. The mode of interest is a first-order
flexural mode, that is theoretically composed of two degenerate, orthogonally
polarized eigenstates due to the circular cross-section of the resonators. The
frequency response of isolated SAW-coupled resonators were previously shown to
exhibit Fano lineshapes, resulting from the interaction of the impinging SAW
with the localized mechanical resonator Raguin et al. (2019). Figure 1a
reports the out-of-plane vibration amplitude measured out of a single
resonator subjected to a SAW driven by radio-frequency (RF) input powers
ranging from 10 to 20 dBm. A single resonance peak appears at a frequency of
70.7 MHz. The degeneracy lifting expected from fabrication imperfections in
resonators with circular symmetries cannot be observed here. This can be
accounted for by the low quality factor (Q-factor) of the resonance, that is
about equal to 30. When increasing the drive power, the vibration amplitude of
the phononic resonator shows a predictable linear dependency, as expected from
such low quality-factor mechanical structures under the present weak
excitation, with a constant resonant frequency and $Q$-factor.

Similar experiments were conducted on a pillar pair with a 6-$\mu$m gap
distance excited by a SAW with a wave vector directed along the inter-
resonator axis. Figure 1b shows that increasing the SAW drive power again
leaves the frequency position and quality factor of the pillar pair resonances
unaffected. The coupling mechanism is therefore here independent on the SAW
amplitude, as expected for linearly strain-coupled mechanical resonators.

Figure 2:  Experimental frequency and phase responses of a pillar pair excited
by a transverse SAW. The gap distance is 1.5 $\mu$m. (a) Response of resonator
$T1$. (b) Response of resonator $T2$. (c) Experimental phase maps at 12 dBm
for the lower frequency mode (i) at 68.6 MHz (1) and for the higher-frequency
mode (ii) at 71.8 MHz (2); at 20 dBm for the lower frequency mode (i) at 68.3
MHz (3) and for the higher-frequency mode (ii) at 71.5 MHz (4).

###  I.2 Frequency softening in smaller gap distance coupled-resonator
systems.

The previous observations do not hold, however, for shorter gap distances,
where SAW mediated dipole-like interaction is dominant Raguin et al. (2019). A
similar investigation of the resonator response as a function of the drive
power was performed for 1.5-$\mu$m-spaced pillar pairs excited by SAWs
propagating either along the inter-resonator axis (longitudinal excitation) or
in the direction normal to the inter-resonator axis (transverse excitation).
The results obtained in the case of a transverse excitation are reported in
Figure 2a and 2b for each resonator, here denoted $T1$ and $T2$. The resonator
pair frequency response reveals the existence of two vibrational modes. The
first mode (i) is found at about 68 MHz and has a quality factor of about 30
that is comparable with the one observed for a single pillar. A higher
frequency mode (ii) lying slightly below 72 MHz can also be found, with a
quality factor now reaching about 60. The resonance line shape exhibits the
lightly asymmetric profile expected in coupled resonator systems.

Phase maps obtained by laser scanning interferometry and reported in Figure 2c
reveal that mode (i) corresponds to a symmetric mode, while mode (ii)
corresponds to an antisymmetric mode with respect to the propagation axis.
Increasing the drive power in the 10 to 20 dBm range does not significantly
affect the overall modal behaviour of the coupled resonator system. It however
results in the occurrence of a resonance frequency shift for the higher
frequency mode (ii), leading to a downshift of about 400 kHz, while the lower
frequency symmetric mode (i) remains at the same frequency position.

The observed behavior therefore hints at a nonlinear coupling of the first-
order flexural modes of the two phononic resonators, despite each individual
resonator exhibiting an otherwise linear response. The involved non-linear
coupling mechanism is therefore different from those usually exploited in
NEMS, that rely on the coupling of high-quality factor mechanical resonators
with geometrical non linearities Nayfeh and Mook (1995). The higher quality
factor obtained by resonator-to-resonator coupling could help triggering non
linear interactions by lowering the amplitude thresholdrequired to reach the
critical point. However, these interactions here result in a frequency
downshift with increasing drive amplitude, while geometrical nonlinearities
are expected to result in a frequency upshift in such clamped-free resonators.
Such coupling-induced softening nonlinearities were previously reported in
phononic crystals made of local resonators coupled by nonlinear graphene
membranes Midtvedt et al. (2014). In our case, coupling occurs through the
substrate surface and is mediated by the SAW elastic energy distribution, that
is itself conditioned by the resonator-to-resonator coupling conditions. The
observed non-linear behavior can be described as resulting from a softening of
the coupling spring constant induced by the highly-confined SAW. This effect
may allegedly be understood by considering the coupled pillars as forming a
cavity for SAW with a strong energy confinement in the 1.5-$\mu$m-wide gap.
The system becomes resonant, hence promoting energy transfer between the two
resonators as the SAW amplitude increases. Finite element method (FEM)
simulations based on a linear elastic model confirm that the phononic
resonators induce a strong localization of the elastic energy distribution at
the resonator vicinity. The Von Mises stress distributions reported in Figure
3 further highlight that stress localization within the resonator gap is only
observed for the anti-symmetric mode (ii). This observation is in good in
agreement with the experimental device response, as only mode (ii) experiences
a nonlinear frequency downshift.

Figure 3: Finite element method simulations of the Von Mises stress
distribution. The two 1.5 $\mu$m-spaced resonators are excited by a linear
elastic line source. (a) Symmetric mode (i), here found at 70.73 MHz. (b)
Anti-symmetric mode (ii) at 73.34 MHz.  Figure 4: Coupled-oscillator model of
a pillar pair excited by a transverse SAW. (a) The solid lines correspond to
the experimental data for a pillar pair with a 1.5 $\mu$m gap distance excited
using a RF input power of 20 dBm. The dashed lines correspond to the
theoretical frequency response obtained out of the proposed coupled-oscillator
model for an applied force amplitude $F_{0}=10\leavevmode\nobreak\ \mu$N. (b)
Sum of the measured amplitudes of the two resonators for the anti-symmetric
mode. The dots indicate the maximum displacement for each input RF powers, the
dashed line is a fitting to the theoretical backbone curve given in equation
3.

As a possible description of this SAW-coupled resonator system, we propose to
use a tentative simplified model based on two single-mode linear oscillators
with respective masses $m_{i}$ ($i=1,2$) coupled by a non linear coupling
strength $k_{nl}$. The corresponding equation of motion can be written as:

$\displaystyle
m_{1}\ddot{x_{1}}+\gamma\dot{x_{1}}+kx_{1}+\kappa_{12}(x_{1}-x_{2})+\kappa_{nl}(x_{1}-x_{2})^{3}$
$\displaystyle=F_{0}\cos(\Omega t),$ (1a) $\displaystyle
m_{2}\ddot{x_{2}}+\gamma\dot{x_{2}}+kx_{2}+\kappa_{12}(x_{2}-x_{1})+\kappa_{nl}(x_{2}-x_{1})^{3}=0,$
(1b)

where $x_{1}$ and $x_{2}$ are the displacement amplitudes for each resonator,
$\gamma$ is the linear damping, $k$ is the resonator spring constant and
$\kappa_{12}$ is the linear coupling constant. The unperturbed frequencies of
the resonators are defined as $\omega_{0}^{(i)}=\sqrt{k/m_{i}}$. The two
effective masses are defined from the physical mass of the resonators
($m_{i}=\rho V_{i}$, $\rho$ being the mass density and $V_{i}$ the volume of
the resonators). A difference in mass of about 5%, corresponding to an
estimated difference in height of 200 nm between the two resonators, was
introduced in the model. The frequency softening effect is introduced by
assuming a negative $\kappa_{nl}$ constant. The term on the right-hand side
corresponds to the SAW excitation, modelled as an external periodic force of
strength $F_{0}$ and frequency $\Omega$. This external force is applied to
only one of the resonators to avoid artificially forcing a relative phase
condition between the two oscillators. Figure 4a displays the experimental
response of the two resonators for a drive power of 20 dBm, along with the
corresponding simulated vibration amplitude. The spring constant $k=116$
[kN/m] and the linear coupling constant $\kappa_{12}=5.3$ [kN/m] used in the
model are estimated from the two resonance frequencies measured for mode (i)
and mode (ii) of the pillar pair (see Supplementary Note II). In order to
estimate $\kappa_{nl}$, we rewrite the system of equations (1a-1b) assuming
equal resonator masses ($m_{1}=m_{2}=m$; Supplementary Note III):

$\ddot{u}+\mu\dot{u}+\omega_{0}^{2}u+\alpha u^{3}\\\ =F\cos(\Omega t),$ (2)

where $\omega_{0}=\sqrt{\frac{k+2\kappa_{12}}{M}}$ , $\mu=\gamma/M$ and
$\alpha=2\kappa_{nl}/M$.
$u=x_{1}-x_{2}$ is the amplitude difference between the two resonators. The
natural frequency response curve of the nonlinear system can then be described
by expressing its backbone curve that follows a typical parabola-shaped curve
as a function of the maximum amplitude of $u$:

$\Omega=\omega_{0}+\frac{3}{8}\frac{\alpha}{\omega_{0}}a^{2}$ (3)

where $a$ represents the amplitude of $u$. Fitting the experimental resonance
frequency at different drive power to this backbone curve equation, as
presented in Figure 4b, yields a nonlinear coupling constant $\kappa_{nl}$ of
the order of -0.016 [kN/m3]. The nonlinear coupling term
$\pm\kappa_{nl}(x_{1}-x_{2})^{3}$ described in equation (2) as
$\pm\kappa_{nl}(x_{1}-x_{2})^{3}$ affecting only the out-of-phase mode, the
proposed model captures the mode symmetry dependence of the observed nonlinear
effect, with a symmetrical, in-phase mode frequency position remaining
independent of the drive SAW amplitude.

Figure 5:  Experimental frequency and phase responses of a pillar pair excited
by a longitudinal SAW. The gap distance is 1.5 $\mu$m. (a) Response of
resonator L1. (b) Response of resonator L2. (c) Experimental phase maps at 12
dBm for the lower frequency mode (iii) at 71.9 MHz (1) and for the higher-
frequency mode (iv) at 73.4 MHz (2); at 20 dBm for the lower frequency mode
(iii) at 71.24 MHz (3) and for the higher-frequency mode (iv) at 72.9 MHz (4).

The occurrence of non-linearities was confirmed by investigating a similar
pillar pair subjected to a longitudinal excitation. The measured frequency
responses, reported in Figure 5, show that the resonator pair, composed by two
pillars $L1$ and $L2$, again hosts two sets of modes appearing at the
respective frequencies of 71.5 MHz (mode ((iii) and 73 MHz (mode (iv)) at 10
dBm. Increasing the applied drive power now leads to a frequency downshift for
both resonances. The higher frequency resonance shift is comparable to the one
observed in the transverse case and is estimated to be slightly lower than 400
kHz. The resonance frequency of mode (iv) is also downshifted, with a shift
now of the order of 600 kHz, as displayed in Figure 6a. But one of the most
interesting feature of this frequency response lies in the observed change in
resonance profile, that presents an increasing asymmetry as the amplitude
increases. This effect leads to a frequency instability area, which is further
confirmed by performing reverse frequency sweeps. Figure 6b compares upward
and downward frequency sweeps for pillar $L1$, situated closer to the
excitation source at 20 dBm. A typical hysteresis cycle is observed, showing
an amplitude jump. This behavior confirms the presence of non-linearities in
the proposed system, a rather unexpected result given the geometrical
characteristics and the operating frequencies of the involved mechanical
resonators.

Figure 6: Hysteresis cycle. (a) Frequency response of resonator $L1$ for
varying input powers. (b) Comparison of up and down frequency sweeps for the
first resonance of pillar $L1$ (closest to the excitation source) at an RF
input power of 20 dBm. The frequency shift linked to the hysteresis cycle
appears at about 70.9 MHz.

###  I.3 Circularly-polarized resonances.

The experimental field maps of Figure 5c further reveal that the input SAW
amplitude also affects the nature of the resonator polarization states and
breaks the flexural mode symmetry previously observed in transversely-coupled
resonators. In the transverse excitation case reported in Figure 2, only pure
flexural modes are observed and the out-of-plane component of the displacement
field exhibits a well-defined nodal line, as seen in the phase maps in Figure
2c. The corresponding phase states are further depicted in Figure 7a and b
that represent the radial phase of the two resonators extracted from these
field maps for an input power of 20 dBm. The two resonators are shown to
oscillate either in-phase (Fig. 7a) or out-of-phase (Fig. 7b) depending on the
considered vibration mode. But a longitudinal excitation leads to the
occurrence of circular polarization states. In the case of mode (iii), only
one out of two resonators is circularly polarized, while the second one keeps
a well-defined flexural behavior (Fig. 7c). The phase state is table over the
power range, as shown in the phase maps labeled (1) and (2) in Figure 5c taken
at a drive power of 12 dBm. The higher-frequency mode (iv), that appears at a
frequency of about 72.9 MHz at 20 dBm, is however characterized by circularly-
polarized states with opposite handedness (Figure 7d). This behavior points at
a cross-coupling between the two orthogonal polarization states theoretically
composing the first-order flexural mode of such resonators with cylindrical
symmetry.

Figure 7: Typical phase states for the two proposed excitation schemes. The
measurements are taken at a drive power of 20 dBm. (a) Symmetric mode at 68.3
MHz and (b) anti-symmetric mode at 71.5 MHz for a transverse excitation. (c)
Illustration of rotating polarization states obtained for one out of two
resonators in the case of a longitudinal excitation ($f=71.24$ MHz) and (d)
for both resonators, with opposite handedness for an excitation frequency of
71.9 MHz.

The occurrence of these circular states is concomitant with an increase in
drive power; they may not, however, be indisputably be attributed to nonlinear
effects. If non-linearities were already shown to induce elliptical
polarization states in singly-clamped mechanical resonators Nayfeh and Mook
(1995), e.g. in carbon nanotubes Perisanu et al. (2010); Vincent et al.
(2019); Conley et al. (2008), the elliptical transition is obtained at very
high drive powers and can only been obtained out of high quality-factor
mechanical resonators. For low Q-factors, this elliptical transition is
considered trivial and hence independent of any non-linear behavior. But
linear systems can also yield circular polarization states, provided that the
resonators are subjected to force field with high vorticities Gloppe et al.
(2014). In the present case, the nature of the Rayleigh SAW involves a
linearly polarized excitation in the substrate plane. The vibration of the
phononic resonator may however results in a mechanical back-action on the
surface acoustic wave field at the resonator vicinity, resulting in coupling
of the shear polarization components of the surface displacements. This, in
turn, induces cross-coupling of the two orthogonal polarizations of the
resonator, hence giving rise to the observed circular polarization states.
This back-action process is, again, triggered by the higher force field
induced by the cavity formed in the resonator-to-resonator gap, therefore
leading to a power-dependant mechanism.

In conclusion, we demonstrated the occurrence of surface-acoustic wave induced
non-linearities in pairs of otherwise linear coupled phononic microresonators.
These non-linearities are characterized by a frequency softening of the pillar
pair response for short gap distances, while isolated resonators retain a
linear behavior. The characteristic features of the non-linear response are
also conditioned by the incident SAW wave vector direction. Coupled resonators
submitted to a transverse SAW excitation are well described by assuming a
negative, cubic Duffing-like term applied to the resonator-to-resonator
coupling constant, leading to a linear coupling rate of about 5 kN/m and a
nonlinear coupling constant of the order of 10-2 kN/m3. In the case of a
longitudinal excitation, in addition to the observed nonlinear dynamics, the
increased surface displacements at the resonator vicinity results in a
feedback mechanism between the mechanical motion of SAW-coupled phononic
resonators and the substrate surface that acts as both a clamping element and
a source of mechanical excitation. This interaction leads to a cross-coupling
of the resonator eigenmodes, and hence on the occurrence of rotating
polarization states. These results illustrate the rich dynamics involved in
these phononic systems, where linear and non-linear effects can be combined to
achieve coherent control of both the frequency response and modal behavior of
high-frequency phononic microresonators.

## Methods

The phononic resonators under study consists of ion-beam-induced deposited
platinum (IBID-Pt) cylinders. These pillars are deposited on a lithium niobate
substrate (LiNbO3 in the Y-crystallographic orientation. The substrate surface
hosts a set of interdigitated transducers (IDTs) allowing for Rayleigh SAW
excitation. The IDTs are made of a array of aluminum electrodes deposited
following a chirped pattern, with a mechanical period varying from 34 $\mu$m
to 68 $\mu$m, leading to the excitation of a Rayleigh wave over a frequency
spectrum centered at about 73 MHz with a fractional bandwidth of 80%. The IDTs
are driven by a signal synthesizer allowing to control the radio-frequency
(RF) signal power from -140 to +20 dBm, leading to the generation of a
propagating Rayleigh wave with an out-of-plane displacement field reaching a
maximum of 0.6 nm at 20 dBm at the substrate surface. These propagating
Rayleigh waves are in turn used to excite the phononic resonators. The
amplitude and phase of the out-of-plane component of the elastic displacement
field is then imaged by optical interferometry using a Mach-Zehnder heterodyne
laser scanning interferometer. The resonator geometry is therefore dictated by
the trade-off imposed by the need to perform optical characterizations under a
Rayleigh SAW excitation, imposing diameters in the micron-range and aspect
ratio of the order of one, at least for the plain cylindrical geometry
considered in the present work. In the present case, the resonator diameter
was set to 4.4 $\mu$m, the height is comprised between 4.0 and 4.2 $\mu$m
depending on the considered sample. As SAW-induced resonator coupling is
expected to lead to altered coupling schemes as a function of both the
separation distance and the chosen excitation conditions Raguin et al. (2019),
the behavior of single resonators and of resonator pairs with gap distances
$w$ equal to 1.5 $\mu$m 6 $\mu$m were investigated. Both transverse and
longitudinal excitation scheme were considered, where transverse denotes an
elastic wave front orthogonal to the inter-resonator axis while longitudinal
refers to an elastic wave front parallel with the inter-resonator axis.

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## Acknowledgements

This project has received funding from the European Research Council (ERC)
under the European Union’s Horizon 2020 research and innovation programme
(grant agreement No 865724). This work has also received support from the
Agence Nationale de la Recherche under grant ANR-14-CE26-0003-01-PHOREST and
the French RENATECH network with its FEMTO-ST technological facility. A. J.
was supported by the EIPHI Graduate School (contract “ANR-17-EURE-0002”).


CALT-TH 2023-027

# A gap between holographic and quantum mechanical extreme rays of the
subadditivity cone

Temple He  Walter Burke Institute for Theoretical Physics
California Institute of Technology, Pasadena, CA 91125 USA
<EMAIL_ADDRESS>Veronika E. Hubeny  Center for Quantum Mathematics and
Physics (QMAP)
Department of Physics & Astronomy, University of California, Davis, CA 95616
USA<EMAIL_ADDRESS>Massimiliano Rota  Center for Quantum
Mathematics and Physics (QMAP)
Department of Physics & Astronomy, University of California, Davis, CA 95616
USA  Institute for Theoretical Physics, University of Amsterdam, Science Park
904, Postbus 94485, 1090 GL Amsterdam, The Netherlands<EMAIL_ADDRESS>
###### Abstract

We show via explicit construction that for six or more parties, there exist
extreme rays of the subadditivity cone that can be realized by quantum states,
but not by holographic states. This is a counterexample to a conjecture first
formulated in Hernández-Cuenca  _et al._ (2022), and implies the existence of
deep holographic constraints that restrict the allowed patterns of
independence among various subsystems beyond the universal quantum mechanical
restrictions.

††preprint: APS/123-QED

##  I  Introduction

A central question in the context of the gauge/gravity duality Maldacena
(1998); Gubser  _et al._ (1998); Witten (1998) is to understand how the bulk
classical geometry is encoded in the entanglement structure of the boundary
state, and one might hope to extract useful information about such encoding by
investigating properties of the von Neumann entropy which are specific to this
setting. The discovery of monogamy of mutual information (MMI) Hayden  _et
al._ (2013); Wall (2014) showed that for geometric states, i.e., states of
holographic CFTs which are dual to classical geometries, the HRRT prescription
Ryu and Takayanagi (2006); Hubeny  _et al._ (2007) implies that the entropies
of spatial subsystems in the boundary CFT satisfy constraints that in general
do not hold for arbitrary quantum systems. Since then, new holographic entropy
inequalities have been found, and the holographic entropy cone (HEC) Bao  _et
al._ (2015) has been studied extensively Marolf  _et al._ (2017); Rota and
Weinberg (2018); Cui  _et al._ (2019); Hubeny  _et al._ (2018); Bao and Mezei
(2018); Hubeny  _et al._ (2019); Hernández Cuenca (2019); Czech and Dong
(2019); He  _et al._ (2019, 2020); Avis and Hernández-Cuenca (2023); Czech and
Wang (2023).

As the number of parties ${\sf N}$ increases, the search for new inequalities
quickly becomes computationally unfeasible due to the fact that the
combinatorics governing the number of inequalities typically grows doubly
exponentially as a function of ${\sf N}$. Furthermore, fixing ${\sf N}$ is
immaterial in QFT, since one can always imagine further partitioning the ${\sf
N}$ regions into smaller subregions. For these reasons, Hernández-Cuenca  _et
al._ (2022) took a different approach to the characterization of the HEC.
Rather than looking for the explicit expression of the inequalities at some
given ${\sf N}$, Hernández-Cuenca  _et al._ (2022) attempted to provide a more
implicit description of the HEC for an arbitrary number of parties by relating
it to the quantum entropy cone (QEC) Pippenger (2003), and to distill the
essential information that would allow for its reconstruction (at least in
principle). Drawing from the ideas of Hubeny  _et al._ (2018, 2019),
Hernández-Cuenca  _et al._ (2022) suggested that this essential information is
the solution to the holographic marginal independence problem (HMIP)
Hernández-Cuenca  _et al._ (2019).

The HMIP is the restriction of the more general quantum marginal independence
problem (QMIP), introduced in Hernández-Cuenca  _et al._ (2019), to geometric
states. The QMIP asks the following question: Given an ${\sf N}$-party system
and a complete specification of the presence of correlation (or conversely the
lack thereof) among the various subsystems, is there a density matrix that
satisfies these constraints? This problem can be conveniently formalized using
the polyhedral cone in entropy space carved out by all instances of
subadditivity (SA) at given ${\sf N}$, called the subadditivity cone (SAC).
The SAC is an outer bound to the HEC, and a pattern of marginal independence
(PMI) is defined as the linear subspace spanned by one of its faces 111This is
the definition of a PMI given in Hernández-Cuenca  _et al._ (2022), whereas
the original definition from Hernández-Cuenca  _et al._ (2019) was weaker..
The HMIP then asks which faces of the SAC, and therefore which PMIs, can be
reached by the HEC.

The analysis of Hernández-Cuenca  _et al._ (2022) suggested that the HEC can
be reconstructed if the solution to the extremal version of the HMIP is known,
i.e., if it is known which extreme rays of the SAC (or equivalently
1-dimensional PMIs) can be realized by geometric states. Specifically,
Hernández-Cuenca  _et al._ (2022) provided strong evidence that the extreme
rays of the HEC can simply be obtained from 1-dimensional PMIs involving more
subsystems by particular projections that correspond to coarse-grainings
222Notice that this implies that to reconstruct the extreme rays of the ${\sf
N}$-party HEC, one needs to know the extreme rays of the SAC for some ${\sf
N}^{\prime}\geq{\sf N}$. We will return to this point in the next section..
Holographic entropy inequalities can then be derived, at least in principle,
using standard algorithms to convert the description of a polyhedral cone in
terms of extreme rays into one given by facets 333Since no efficient algorithm
is known, such explicit derivation of the inequalities remains computationally
unfeasible for large ${\sf N}$..

If this reconstruction is indeed possible, the characterization of the HEC
would reduce to the characterization of the set of extreme rays of the SAC
that can be realized by geometric states. A natural guess suggested in
Hernández-Cuenca  _et al._ (2022), which is consistent with the SAC for all
${\sf N}\leq 5$, is that these are all the extreme rays that can be realized
in quantum mechanics. The aim of this letter is to show that this is not the
case, implying that even if the reconstruction argued in Hernández-Cuenca  _et
al._ (2022) is possible, extremal quantum marginal independence alone is not
enough to fully characterize the HEC, since there exist deeper constraints
which restrict the set of extremal PMIs that can be realized in holography.

The structure of the paper is as follows. In §II we review the definition of
the HEC from Bao  _et al._ (2015), the notion of holographic marginal
independence from Hernández-Cuenca  _et al._ (2019), and the reconstruction of
the HEC from the solution to the extremal HMIP argued in Hernández-Cuenca  _et
al._ (2022). In §III we review the machinery from He  _et al._ (2022), which
allows us to efficiently derive extreme rays of the SAC that satisfy strong
subadditivity (SSA) and therefore have a chance of being realizable by quantum
states. We will use these techniques to find such an extreme ray that violates
MMI, and therefore cannot be realized by any geometric state. Nevertheless, we
will demonstrate in §IV that this extreme ray is in fact realized by a quantum
state. Finally, in §V we comment on the implications of this result for the
characterization of the HEC for an arbitrary number of parties.

##  II The holographic entropy cone from marginal independence

###  II.1 Definition of the holographic entropy cone

The HEC was introduced in Bao  _et al._ (2015), following the analogous
program for arbitrary quantum states Pippenger (2003), as a convenient
framework to analyze the set of inequalities implied by the Ryu–Takayanagi
(RT) formula Ryu and Takayanagi (2006). For our purposes, it will be
sufficient to view the HEC as the convex cone of entropy vectors which are
realized by graph models of holographic entanglement, which we now briefly
review. For the original definition of the HEC, and for a detailed explanation
of how a graph model is related to the RT surfaces that compute the entropies
of a collection of boundary spatial subsystems, we refer the reader to Bao
_et al._ (2015) (see also Marolf  _et al._ (2017); Hubeny  _et al._ (2018,
2019) for additional subtleties).

An ${\sf N}$-party graph model is a simple weighted graph $G=(V,E)$ with
positive weights, a specification of a subset $\partial V\subseteq V$ of
vertices called boundary vertices, and a surjective (but not necessarily
injective) map $\xi:\partial V\rightarrow[{\sf N}+1]=\\{1,2,\ldots,{\sf
N}+1\\}$, where $1,\ldots,{\sf N}$ label the parties and ${\sf N}+1$ labels
the purifier. Given a graph model, one associates to it an entropy vector
$\vec{\sf S}\in\mathbb{R}^{\sf{D}}$, with $\sf{D}=2^{\sf N}-1$, as follows.
For a non-empty subset $\mathscr{I}\subseteq[{\sf N}]$, a cut “homologous to
$\mathscr{I}$” ($\mathscr{I}$-cut) is a subset $V_{\mathscr{I}}\subset V$ such
that $\partial V\cap V_{\mathscr{I}}=\xi^{-1}(\mathscr{I})$, where $\xi^{-1}$
denotes the pre-image of $\xi$. The cost of any such cut is the sum of the
weights of the edges that connect a vertex in $V_{\mathscr{I}}$ to one in
$V_{\mathscr{I}}^{c}$, the complement of $V_{\mathscr{I}}$ in $V$. The entropy
${\sf S}_{\mathscr{I}}$ is then defined as the cost of the $\mathscr{I}$-cut
with minimal cost.

It is straightforward to see that the set of entropy vectors obtained from all
such graph models at any fixed ${\sf N}$ is a convex cone. This follows from
the fact that given any two graph models $G_{1},G_{2}$ with entropy vectors
$\vec{\sf S}_{1},\vec{\sf S}_{2}$, the conical combination $\vec{\sf
S}=\alpha\vec{\sf S}_{1}+\beta\vec{\sf S}_{2}$, with $\alpha,\beta>0$, is
realized by the graph $G=\alpha G_{1}\oplus\beta G_{2}$, where $\alpha G_{1}$
is a graph model obtained from $G_{1}$ by rescaling the weights with the
coefficient $\alpha$ (and similarly for $\beta G_{2}$), and $\oplus$ denotes
the disjoint union. As shown in Bao  _et al._ (2015), this cone is identical
to the HEC, and it is polyhedral for any ${\sf N}$, i.e., it is specified by a
finite number of inequalities, or equivalently, by a finite number of extreme
rays. It was further shown in Hayden  _et al._ (2016) that, for any ${\sf N}$,
the cone of graph models, or equivalently the HEC, is contained in the QEC
444This is not a priori obvious because the original definition of the HEC in
Bao  _et al._ (2015) is a purely geometric one, and it does not assume that
the bulk geometry corresponds to a CFT state Marolf  _et al._ (2017)..

###  II.2 Formalization of the HMIP and its extreme version

An obvious outer bound to the QEC, and therefore also the HEC, is the SAC,
i.e., the polyhedral cone carved out by all instances of SA:

${\sf S}_{\underline{\mathscr{I}}}+{\sf S}_{\underline{\mathscr{K}}}-{\sf
S}_{\underline{\mathscr{I}}\underline{\mathscr{K}}}\geq
0\quad\forall\,\underline{\mathscr{I}},\underline{\mathscr{K}}\subset[{\sf
N}+1],\quad\underline{\mathscr{I}}\cap\underline{\mathscr{K}}=\varnothing,$
(1)

where underlined indices indicate subsystems that can include the purifier
${\sf N}+1$ (as opposed to $\mathscr{I}\subseteq[{\sf N}]$). Thus, the Araki-
Lieb inequality conveniently takes the form of an SA involving the purifier.

Consider now a face $\mathscr{F}$ of the SAC and a vector $\vec{\sf
S}\in\text{int}(\mathscr{F})$. Notice that the collection of SA instances
which are saturated by $\vec{\sf S}$ is independent from the specific choice
of this vector. Furthermore, the saturation of SA is equivalent to the
vanishing of the mutual information (MI) ${\sf
I}(\underline{\mathscr{I}}:\underline{\mathscr{K}})$, which in quantum
mechanics is attained if and only if the density matrix factorizes, i.e., if
the subsystems $\underline{\mathscr{I}}$ and $\underline{\mathscr{K}}$ are
independent. We can then interpret the linear subspace spanned by
$\mathscr{F}$ as corresponding to a specification of which subsystems are
independent and which manifest some correlation, while remaining agnostic
about the specific values of the entropies. This subspace is in fact a PMI as
defined in Hernández-Cuenca  _et al._ (2022).

The QMIP and HMIP then ask which PMIs can be realized by quantum states and
graph models, respectively. More specifically, they ask which PMIs correspond
to faces of the SAC such that there exist in the interior at least one entropy
vector realized by a quantum state or, respectively, a graph model. In the
following sections we will be interested in the extremal version of these
problems, which we denote by EQMIP and EHMIP, and only focus on the
1-dimensional PMIs. These correspond to the 1-dimensional faces of the SAC,
i.e., its extreme rays.

###  II.3 Reconstructing the HEC from the solution to the EHMIP

The intuition that the solution to the HMIP could provide sufficient
information for the derivation of the HEC was first suggested in Hubeny  _et
al._ (2018, 2019), based on the observation that the PMI of a choice of
boundary regions in a geometric state is captured by the connectivity of the
RT surfaces, while the specific value of the entropy is immaterial in QFT
because it depends on the choice of cut-offs. This intuition was then further
developed in Hernández-Cuenca  _et al._ (2022), which formulated the following
conjecture and checked that it holds for ${\sf N}\leq 5$ 555Most of the
machinery developed in Hernández-Cuenca  _et al._ (2022) was used to show that
1 would follow from other purely graph theoretic conjectures, but this
machinery is not necessary for the purpose of this letter.:

######  Conjecture 1.

For any extreme ray $\vec{\sf R}_{{\sf N}}$ of the ${\sf N}$-party _HEC_ ,
there exists for some ${\sf N}^{\prime}\geq{\sf N}$ an extreme ray $\vec{\sf
R}^{\prime}_{{\sf N}^{\prime}}$ of the ${\sf N}^{\prime}$-party _SAC_ such
that $\vec{\sf R}^{\prime}_{{\sf N}^{\prime}}$ can be realized by a graph
model and

$\Lambda_{{\sf N}^{\prime}\rightarrow{\sf N}}\vec{\sf R}^{\prime}_{{\sf
N}^{\prime}}={\sf R}_{{\sf N}},$ (2)

where $\Lambda_{{\sf N}^{\prime}\rightarrow{\sf N}}$ is a map associated to a
coarse-graining of the ${\sf N}^{\prime}$ parties into ${\sf N}$ blocks _6
66The precise expression of $\Lambda_{{\sf N}^{\prime}\rightarrow{\sf N}}$ is
not necessary for this discussion, but for clarity we give a simple example of
such a projection. Consider the 3-party entropy vector $\vec{\sf
S}=(1,1,1;2,2,2;1)$ for $A,B,C$ (the entropies are ordered conventionally as
exemplified in (4) for ${\sf N}=6$), and define the new parties
$A^{\prime}=A,\,B^{\prime}=BC$. The 2-party entropy vector for these coarse-
grained parties is then $\vec{\sf S}^{\prime}=(1,2;1)$, and it is obtained
from $\vec{\sf S}$ by simply dropping the components that separate $B$ from
$C$, i.e., $B,C,AB,AC$._.

If proven to be true, 1 would have important implications for the
characterization of the HEC. It would imply that for any ${\sf N}$, there
exists some finite ${\sf N}^{\prime}\geq{\sf N}$ such that the ${\sf N}$-party
HEC can be obtained as the conical hull of all possible coarse-grainings of
the extreme ray of the ${\sf N}^{\prime}$-party SAC realizable by graph models
Hernández-Cuenca  _et al._ (2022). In other words, to reconstruct the ${\sf
N}$-party HEC it would be sufficient to know the solution to the EHMIP for a
certain ${\sf N}^{\prime}\geq{\sf N}$ which depends on ${\sf N}$ (it was shown
in Hernández-Cuenca  _et al._ (2022) that for ${\sf N}=3,4$ it suffices to
have ${\sf N}^{\prime}={\sf N}$, whereas for ${\sf N}=5$ one needs ${\sf
N}^{\prime}=8$).

While a proof of 1 is still lacking, and it is far beyond the scope of this
letter, considering the strong evidence given in Hernández-Cuenca  _et al._
(2022), it is natural to focus on the solution to the EHMIP, since this
distills the essential information underlying the HEC. The immediate question
then is whether there is some physical principle that identifies the SAC
extreme rays realizable by graph models. A possibility suggested in Hernández-
Cuenca  _et al._ (2022), which holds for ${\sf N}\leq 5$, is that there is in
fact nothing special about graph models, and that the EHMIP and EQMIP have the
same solution:

######  Conjecture 2.

For any ${\sf N}$, all extreme rays of the ${\sf N}$-party _SAC_ that can be
realized by quantum states can also be realized by graph models.

In the rest of this work we will construct a counterexample to 2. Its
implication for the characterization of the HEC will then be discussed in §V.

##  III Deriving extreme rays of the SAC which satisfy SSA

As we mentioned, 2 holds for ${\sf N}\leq 5$, so to look for a counterexample
we need to consider at least ${\sf N}=6$. However, in this case the
combinatorics of the faces of the SAC is sufficiently complicated that an
explicit derivation of all extreme rays is not feasible even using state of
the art algorithms Bruns  _et al._ . Instead, we should restrict ourselves to
the relatively few extreme rays which satisfy SSA, since these are the only
ones that can possibly be realized by quantum states.

A first useful result in this direction was recently obtained in He  _et al._
(2022).

######  Theorem 1.

For any ${\sf N}$, all extreme rays of the _SAC_ that can possibly be realized
by quantum states (other than the ones realized by Bell pairs) belong to the
face that spans the subspace given by

${\sf I}(\ell:\ell^{\prime})=0\quad\forall\ell,\ell^{\prime}\in[{\sf N}+1].$
(3)

###### Proof.

See Corollary 1 in He  _et al._ (2022). ∎

While 1 gives a considerable speed-up in the computation of the extreme rays
of the SAC that satisfy SSA 777The computation of all the extreme rays of the
SAC for ${\sf N}=5$ takes several days on a standard laptop, but the extreme
rays of the face identified by 1 only takes a few minutes., it is still not
sufficient to obtain all such extreme rays for ${\sf N}=6$. To obtain these
rays, one can generalize 1 and derive new constraints using extreme rays that
are already known, in a similar fashion to how He  _et al._ (2022) proved 1
using the extreme rays realized by Bell pairs. This program is currently being
explored systematically in He  _et al._ , but for the purpose of providing a
counterexample to 2, it suffices to construct a single extreme ray with the
requisite conditions.

Labeling the six parties by $A,B,C,D,E,F$, and ordering the components of an
entropy vector lexicographically, as in

$(A,\ldots,F;AB,AC,\ldots,EF;ABC,\ldots;ABCDEF),$ (4)

the example we consider is

$\displaystyle\begin{split}{\sf
R}_{6}=(&2,1,1,1,2,2;\,3,3,3,4,4,2,2,3,3,2,3,3,3,3,4;\\\
&2,4,5,5,4,5,5,3,5,4,3,4,4,4,4,5,4,4,5,3;\,3,\\\
&4,4,4,4,3,4,4,3,3,3,5,4,4,4;\,3,3,2,2,2,3;\,1).\end{split}$ (5)

The reader can easily verify that (5) is indeed an extreme ray of the 6-party
SAC by first checking that it satisfies all instances of SA, and that the set
of vanishing MI instances has rank ${\sf{D}}-1=62$, where ${\sf{D}}=2^{6}-1$
is the dimension of the ${\sf N}=6$ entropy space. Furthermore, as the reader
can check, (5) violates one instance of MMI, in particular

$-{\sf I}_{3}\,(A:BC:DE)=-2\not\geq 0,$ (6)

where

$\displaystyle{\sf I}_{3}\,(X:Y:Z):=$ $\displaystyle\qquad{\sf S}_{X}+{\sf
S}_{Y}+{\sf S}_{Z}-{\sf S}_{XY}-{\sf S}_{XZ}-{\sf S}_{YZ}+{\sf S}_{XYZ}.$ (7)

This implies that (5) cannot be realized by a graph model.

To complete the proof that 2 is false, we now need to show that even though
(5) cannot be realized by a graph model, it is nevertheless the entropy vector
of a quantum state. This is the goal of the next section.

##  IV  Quantum state realization

To show that (5) is the entropy vector of a quantum state we will use the
hypergraph models introduced in Bao  _et al._ (2020). Hypergraph models are
defined analogously to the graph models presented above, with the only
difference being that in addition to edges one also allows for hyperedges
connecting three or more vertices. Therefore we only need to clarify under
what circumstances the weight of a hyperedge contributes to the cost of an
$\mathscr{I}$-cut. As for standard edges, given an $\mathscr{I}$-cut
$V_{\mathscr{I}}$, and a hyperedge $h$ (thought of as a collection of
vertices), the weight of $h$ contributes to the cost of the cut if and only if
$h$ contains at least one vertex in both $V_{\mathscr{I}}$ and
$V_{\mathscr{I}}^{c}$. As usual, the entropy of $\mathscr{I}$ is then given by
the cost of the $\mathscr{I}$-cut with minimal cost.

We can now try to construct a hypergraph model whose entropy vector is the
extreme ray (5). There is currently no systematic procedure to construct a
hypergraph (or even graph) realization of a given entropy vector, but a
convenient starting point is the observation from the previous section that
(5) violates only a single instance of MMI (cf. (6)). The prototypical example
of a quantum state that violates MMI is the GHZ state, which is realized by a
hypergraph with just a single hyperedge. To realize (5) we then start from a
hypergraph with a single weight 2 hyperedge (2 is the value of the instance of
${\sf I}_{3}$ in (6) obtained from (5)) connecting four vertices labeling the
coarse-grained subsystems $A,BC,DE,FO$. With a few manipulations we then
arrive at the hypergraph shown in Figure 1.

$2$$2$$A$$B$$C$$D$$E$$F$$O$ Figure 1: The hypergraph model that realizes the
entropy vector (5). The (yellow) blob describes the only hyperedge in the
model, which connects the boundary vertex $A$ and the three bulk vertices
(red), and has weight 2. All other edges (blue) have weight 1, except for the
$E$ leaf which has weight 2.

We are now ready to prove the main result of this letter.

######  Theorem 2.

The extreme ray of the 6-party _SAC_ given in (5) is the entropy vector of a
quantum state.

###### Proof.

We leave it as a simple exercise for the reader to explicitly verify that the
entropy vector of the hypergraph model shown in Figure 1 is precisely (5). The
fact that (5) is the entropy vector of a quantum state then follows
immediately from the result of Walter and Witteveen (2021), which showed that
(similar to the case of a standard graph) any entropy vector realizable by a
hypergraph model is the entropy vector of a quantum stabilizer state. ∎

##  V  Discussion

We conclude with a few comments about the implications of the failure of 2 for
the reconstruction and the physical interpretation of the HEC. If 1 is true,
the HEC can be fully reconstructed from the solution to the EHMIP. In that
case, if 2 were also true, the HEC would be the largest possible polyhedral
cone compatible with this reconstruction procedure and quantum mechanics. One
should then question the actual physical meaning of holographic entropy
inequalities, since they would just follow from the bound on ${\sf N}$, which
is artificial in QFT. The failure of 2 proves that this is not the case, even
if the reconstruction procedure of 1 can indeed be achieved. In that case, one
should then try to understand what distinguishes the extreme rays of the SAC
that can be realized by graph models from the larger set of quantum mechanical
ones.

A first step in this direction is the derivation of all extreme rays which are
compatible with SSA for ${\sf N}=6$ He  _et al._ . Since this is the smallest
value of ${\sf N}$ where the solutions to the EQMIP and EHMIP differ, it is a
useful testing ground to develop intuition. Next, one should determine which
of these rays can be realized in quantum mechanics, and as demonstrated here,
the hypergraph construction could be a useful tool in this direction.
Ultimately, one may then attempt to use the construction suggested in Bao  _et
al._ (2020), or the techniques from Hayden  _et al._ (2016); Walter and
Witteveen (2021) to explicitly construct the corresponding quantum states and
find a new characterization from the perspective of quantum information
theory. We leave these questions for future investigation.

Acknowledgments: V.H and M.R. would like to thank the Centro de Ciencias de
Benasque Pedro Pascual for hospitality during the program “Gravity - New
perspectives from strings and higher dimensions”, where this work was carried
out. T.H. is supported by the Heising-Simons Foundation “Observational
Signatures of Quantum Gravity” collaboration grant 2021-2817, the U.S.
Department of Energy grant DE-SC0011632, and the Walter Burke Institute for
Theoretical Physics. V.H. has been supported in part by the U.S. Department of
Energy grant DE-SC0009999 and by funds from the University of California. M.R
is supported by funds from the University of California.

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