††thanks: They contribute equally to this work. ††thanks: They contribute
equally to this work.

# Quantum Many-Body Scars and Quantum Criticality

Zhiyuan Yao  Institute for Advanced Study, Tsinghua University, Beijing
100084, China  Lei Pan  Institute for Advanced Study, Tsinghua University,
Beijing 100084, China  Shang Liu<EMAIL_ADDRESS>Department of Physics,
Harvard University, Cambridge, MA, 02138, USA  Hui Zhai<EMAIL_ADDRESS>Institute for Advanced Study, Tsinghua University, Beijing 100084, China

###### Abstract

In this letter, we study the PXP Hamiltonian with an external magnetic field
that exhibits both quantum scar states and quantum criticality. It is known
that this model hosts a series of quantum many-body scar states violating
quantum thermalization at zero magnetic field, and it also exhibits an Ising
quantum phase transition driven by finite magnetic field. Although the former
involves the properties of generic excited states and the latter concerns the
low-energy physics, we discover two surprising connections between them,
inspired by the observation that both states possess log-volume law
entanglement entropies. First, we show that the quantum many-body scar states
can be tracked to a set of quantum critical states, whose nature can be
understood as pair-wisely occupied Fermi sea states. Second, we show that the
partial violation of quantum thermalization diminishes in the quantum critical
regime. We envision that these connections can be extended to general
situations and readily verified in existing cold atom experimental platforms.

Thermalization lies at the center of statistical mechanics, and the eigenstate
thermalization hypothesis (ETH) lays the foundation of quantum statistical
mechanics Deutsch:1991ju ; Srednicki:1994dl ; Rigol:2008bf ; DAlessio:2016gr .
The ETH states that a highly excited eigenstate of a generic interacting
quantum many-body system locally behaves as a thermal ensemble. A direct
consequence of the ETH is that the entanglement entropy of a subsystem equals
its thermal entropy and, therefore, displays the volume law behavior. So far,
most quantum many-body systems are known to obey ETH, with few classes of
exceptions. For example, the ETH can be violated by fine-tuning the system
parameters to exactly solvable points or by adding strong disorder to make the
system many-body localized Gornyi:2005fv ; Basko:2006hh ; Abanin:2019dl .

Figure 1: Schematic of two kinds of log-volume law states in the PXP model
with an external magnetic field. One consists of quantum many-body scar states
(green balls) first discovered at $m=0$, and the other one involves the low-
energy critical states at the Ising quantum critical point $m_{\text{c}}$.
Here $E$ denotes the energy and $m$ is the strength of the external field.
This paper focuses on the relation between these two sets of quantum states.

Recently a new mechanism of ETH violation has been discovered first in a
Rydberg atom quantum simulator, where certain initial states are found to
retain initial local information after sufficiently long time evolution
Bernien:2017bp . Later it is pointed out theoretically that such systems can
be described by the so-called PXP model Turner:2018iz ; Turner:2018in . The
Hamiltonian of this one-dimensional model reads

$\hat{H}=\sum\limits_{i}\hat{P}_{i-1}\hat{\sigma}^{x}_{i}\hat{P}_{i+1}.$ (1)

In the Rydberg atom system, $\hat{\sigma}^{x}_{i}$ couples an atom at site-$i$
between the ground state $|g\rangle$ and the Rydberg excited state
$|e\rangle$. Here $\hat{P}_{i}=(1-\hat{\sigma}^{z}_{i})/2$ with
$\hat{\sigma}_{i}^{z}=|e\rangle\langle e|-|g\rangle\langle g|$ is a projection
operator onto the ground state. The PXP model is an effective model for the
Rydberg blockade when the blockade radius is around one lattice spacing. This
means that excitation to the Rydberg state at a given site is allowed only
when atoms in its two neighboring sites are both in their ground states.
Hence, in addition to the Hamiltonian Eq. 1, there exists an extra constraint
that forbids two neighboring sites being both occupied by the Rydberg excited
state, _i.e._ , for any physically allowed state $|\Psi\rangle$ and for every
site-$i$, it requires

$(1-\hat{P}_{i})(1-\hat{P}_{i+1})|\Psi\rangle=0.$ (2)

There are two unique aspects of quantum thermalization in the PXP model.
First, violation of the ETH in this system neither requires fine-tuning
parameters nor adding disorder. Second, ETH is only partially violated in this
system. That is to say, only a subset of total eigenstates violate ETH, and
the rest of eigenstates still obey ETH. These non-thermal states are called
the quantum many-body scar states, which has attracted considerable attentions
recently Lukin ; Lukin2 ; SXu2019 ; Lukin2019PRL ; Motrunich2019 ;
Papic2020PRB ; PXPgauge ; Papic2021PRX ; Hsieh2020 ; LukinScience2021 ;
Papic2021Review . Later, quantum many-body scar states have also been
discovered in several different models Moudgalya:2018gt ; Moudgalya:2018gj ;
Moudgalya:2020ip ; Moudgalya:2020ez ; Mark:2020dg ; Mark:2020fp ;
Schecter:2019db ; Iadecola2020 ; Katsura2020 ; WWHo2020 ; Lee2020 ; SGA1 ;
SGA2 ; SGA3 , such as the Affleck–Kennedy–Lieb–Tasaki (AKLT) model, the spin-1
XY model, and the generalized Fermi–Hubbard model.

Figure 2: Snapshots of the process of tracking the quantum many-body scar
states (highlighted by red circles) from $m=0$ to $m=m_{\text{c}}$. Here we
present results for system size $L=28$ in the system sector $(k,I)=(0,+)$ at
four typical values of $m$. For a given $m$, the top panel shows expectation
values of local observables $\langle\hat{\sigma}^{z}_{1}\rangle$, and the
corresponding bottom panel displays the bipartite von Neumann entanglement
entropies (see text for definition) of all eigenstates.

The entanglement entropies of these non-thermal scar states display the log-
volume law behavior Turner:2018iz ; Turner:2018in ; Moudgalya:2018gj ;
Motrunich2019 ; Vafek:2017bv ; Schecter:2019db ; SGA2 . This is reminiscent of
quantum critical states at a $(1+1)$D Ising critical point, whose entanglement
entropies also display the same behavior Vidal:2003cn ; Calabrese:2004hl . To
investigate possible connections between these two kinds of states, we need a
model that exhibits both scar states and critical states. To this end, we
consider adding an external magnetic field into the PXP model whose
Hamiltonian reads

$\hat{H}=\sum\limits_{i}(\hat{P}_{i-1}\hat{\sigma}^{x}_{i}\hat{P}_{i+1}-m\hat{\sigma}^{z}_{i})\,,$
(3)

and we focus on the $m>0$ side. The external magnetic field can drive a
quantum phase transition at $m=m_{\text{c}}\approx 0.655$ Note0 ;
Sachdev:2002jl ; Fendley:2004bw ; Rico:2014ik . When $m>m_{\text{c}}$, the
external magnetic field tends to polarize all atoms into the $|e\rangle$
states. However, due to the Hilbert space constraint, at most a half of the
atoms can be in the $|e\rangle$ states. Moreover, in the limit of
$m\rightarrow\infty$, the ground states are two-fold degenerate, which are
$|gegege\dots\rangle$ and $|egegeg\dots\rangle$, and each of them breaks the
$Z_{2}$ symmetry. Thus, this quantum phase transition belongs to the Ising
universality class.

Hence, both scar states and critical states exist in the phase diagram of this
model, as shown in Fig. 1. On the one hand, it has been well studied that the
PXP model at $m=0$ hosts a set of approximately equally spaced scar states,
ranging from the lowest to the highest energy of the model. On the other hand,
around $m=m_{\text{c}}$, the presence of the Ising quantum phase transition
ensures a set of low-energy quantum critical states governed by the Ising
conformal field theory (CFT). Both states display the log-volume law
entanglement entropies but due to different mechanisms. In this letter, we
address the question that whether there exists a connection between these two
sets of seemingly different states, and more generally, whether there exists a
connection between quantum thermalization and quantum criticality.

Tracking Scars to Criticality. Considering a set of eigenstates
$\\{|n\rangle\\}$ of the model Eq. (3) with a given $m$, we then change $m$ to
$m+\delta m$, and the set of eigenstates become $\\{|n^{\prime}\rangle\\}$.
Now for a given eigenstate $|n_{0}\rangle$ in $\\{|n\rangle\\}$, we can find
out a unique state $|n_{0}^{\prime}\rangle$ in $\\{|n^{\prime}\rangle\\}$,
which maximizes $|\langle n^{\prime}_{0}|n_{0}\rangle|$. Then
$|n^{\prime}_{0}\rangle$ is identified as the corresponding state of
$|n_{0}\rangle$. In this way, we can keep tracking an eigenstate as $m$
varies. Here we focus on the situation that $|n_{0}\rangle$ is a scar state
and we track the scar states from $m=0$ to the critical point
$m=m_{\text{c}}$.

We numerically diagonalize the PXP model of system size $L=28$ under the
periodic boundary condition. Besides the translational symmetry, the system
also has a bond inversion symmetry, and the good quantum numbers associated
with them are momentum $k$ and parity $I$. Quantum many-body scar states exist
only in $(k,I)=(0,+)$ and $(k,I)=(\pi,-)$ sectors where $+$ and $-$ denotes
inversion even and odd respectively. Our results for the $(k,I)=(0,+)$ sector
are shown in Fig. 2, and the results for the $(k,I)=(\pi,-)$ sector are
similar, For this state tracking, we have chosen $\delta m=0.1$ starting from
$m=0$, and for most steps, the maximal overlap is around $0.9$.

As one can see from Fig. 2, there are two prominent features in this tracking
process. First, as regards the energy, around half of the scar states move
towards lower energies when $m$ approaches $m_{\text{c}}$. Second, as regards
the entropy, we employ the von Neumann entanglement entropy
$S_{v}=-\text{Tr}_{A}\rho_{A}\log\rho_{A}$, where $\rho_{A}$ is the reduced
density matrix of the subsystem $A$ after tracing out the rest of the system.
In Fig. 2(e-h), we plot $S_{v}$ when $A$ is taken as half of the entire
system, which is also called the bipartite entanglement entropy, for all
eigenstates in the $(k,I)=(0,+)$ sector. It is clear that in the process of
tracking to the critical point, the tracked scar states always retain
entanglement entropies much lower than the surrounding thermal states.

Figure 3: (a) Comparison between the low-energy spectrum of the PXP model with
an external magnetic field at the criticality (boxes) and the prediction of
the Ising CFT (dashed lines). Filled boxes represents the lowest three tracked
scar states. (b) $S_{v}$ as a function of the size $L_{A}$ of the subsystem
$A$, scaled by the total number of sites $L$. The discrete data points
correspond to the tracked scar states in the dashed box of Fig. 2(h). The
solid lines correspond to the six low-energy and low-entropy states marked by
red circles in (c), and are obtained from the free fermion theory of the
transverse field Ising model. (c) The bipartite entanglement entropy for the
transverse field Ising model. The color bar indicates the number of fermion
quasiparticles of each state. (d) Illustration of the lowest three states
marked by red circles in (c) (also labeled as $|a\rangle$, $|b\rangle$, and
$|c\rangle$ in (a)), which are pair-wisely occupied Fermi sea states of free
fermions.

Low-Entropy States at Criticality. In the dashed box in Fig. 2(h), we
highlight a set of low-energy states at the critical point $m=m_{\text{c}}$,
in which the states marked by red circles are the tracked scar states. We
shall show the nature of these scar states can be understood from the
perspective of quantum criticality utilizing the Ising conformal field theory
(CFT) Francesco:2012fo ; Ginsparg:1988vv . Below we will compare both the
energies and the entanglement entropies of these tracked states at
$m=m_{\text{c}}$ with the prediction of the Ising CFT. To make a proper
comparison, we should caution that the symmetry breaking phase in the PXP
model is an anti-ferromagnetic phase. Therefore, translation of one lattice
spacing in the PXP model is related to the Ising $Z_{2}$ symmetry in the Ising
CFT, and only translation of two lattice spacings in the PXP model corresponds
to the translational symmetry in the Ising CFT. Similarly, the bond-center
inversion in the PXP model is equivalent to the inversion in the Ising CFT
followed by the action of the Ising $Z_{2}$ generator.

Figure 4: (a-c): Time evolution of the local observable
$\langle\hat{\sigma}^{z}_{i}\rangle$ starting from the
$\left|\mathbb{Z}_{2}\right\rangle$ initial state for different $m$. The
dashed lines indicate the corresponding thermal averaged values. (d-f): The
bipartite entanglement entropies of all eigenstates, with color bar
representing the overlaps of the eigenstates with the
$\left|\mathbb{Z}_{2}\right\rangle$ initial state. Here $m=0$ for (a) and (d),
$m=m_{\text{c}}\approx 0.655$ for (b) and (e), and $m=1.0$ for (c) and (f).

Firstly, in Fig. 3(a) we compare the energies of these tracked scar states
with the spectrum of the Ising CFT Note1 , with the latter properly shifted
and rescaled. We find perfect agreement at least for a few lowest energy
states, including three tracked scar states denoted by $|a\rangle$,
$|b\rangle$, and $|c\rangle$ shown in Fig. 3(a). In terms of the standard
notation of the Ising CFT, they are identified as $|1\rangle$,
$|\epsilon\rangle$, and $L_{-2}\tilde{L}_{-2}|1\rangle$, respectively Note2 .
However, this comparison can hardly be extended to higher energy because the
Ising CFT describes continuum models, and the finite size effect of the
lattice model calculation will be more severe as energy increases.

Secondly, we consider the transverse field Ising model sitting at the Ising
criticality. The transverse field Ising model and the PXP model share the same
low-energy physics at criticality due to the universality of critical
behaviors. The transverse field Ising model has the advantage that it can be
written explicitly in terms of free fermions so that many physical quantities,
such as the entanglement entropy, of the eigenstates can be analytically or
semi-analytically computed Peschel:2003gz . Here we properly choose the
overall energy scale of the transverse field Ising model such that its
spectrum matches that of the PXP model at the criticality. Then, we compute
the bipartite entanglement entropy for the transverse field Ising model at the
criticality with $L=10^{3}$ sites, as shown in Fig. 3(c), where an overall
vertical shift has been applied for comparison with the PXP model. As one can
see, the entropy profile is similar to that of the PXP model shown in the
dashed box of Fig. 2(h). In both cases, there exist a series of low-entropy
states (marked by red circles). In Fig. 3(b), we compare the entanglement
entropies of these labeled states in the PXP model with the corresponding ones
in the transverse field Ising model, and we obtain reasonably good agreements
between them, both of which exhibit the log-volume law behavior. Therefore,
these two sets of states share the same nature. Since the latter can be
expressed in terms of the Slater determinants of free fermions, we can now
reveal the nature of these tracked scar states. As shown in Fig. 3(d), they
are just pair-wisely occupied Fermi sea states of free fermions.

Thermalization of the $\mathbb{Z}_{2}$ State. We have shown that the scar
states are stable until reaching the critical point. One physical
manifestation of these scar states is the violation of thermalization of the
$\mathbb{Z}_{2}$ state. Here the $\mathbb{Z}_{2}$ state denotes the
antiferromagnetic state $|gegege\dots\rangle$. This violation is attributed to
the large overlap of the $\mathbb{Z}_{2}$ state with scar eigen-states
Turner:2018iz ; Turner:2018in . Here we study how this violation of
thermalization changes as the parameter $m$ varies across the critical point.

Figure 5: The difference $\delta\sigma^{z}$ between the long-time averaged
local observable $\bar{\sigma}^{z}$ and the thermal equilibrium value
$\sigma^{z}_{\text{th}}$ (see text for detailed definition) as a function of
the magnetic field $m$, for several different system sizes $L$.

Starting from the $\mathbb{Z}_{2}$ initial state, we compute the time
evolution of the local observable
$\langle\Psi(t)|\hat{\sigma}^{z}_{i}|\Psi(t)\rangle$ at site $i$ (below we
take $i=1$ for concreteness), where
$|\Psi(t)\rangle=e^{-i\hat{H}t}|\mathbb{Z}_{2}\rangle$. As shown in Fig.
4(a-c), after a sufficiently long time, this local observable eventually
fluctuates around a stationary value. The stationary value $\bar{\sigma}^{z}$
can be obtained by calculating the long-time average as

$\bar{\sigma}^{z}=\frac{1}{T}\int_{t_{0}}^{t_{0}+T}\langle\Psi(t)|\hat{\sigma}^{z}_{i}|\Psi(t)\rangle
dt,$ (4)

where in practice we take both $t_{0}$ and $T$ as sufficiently long time
scales. We compare $\bar{\sigma}^{z}$ with the thermal equilibrium value
$\sigma^{z}_{\text{th}}$ defined as

$\sigma^{z}_{\text{th}}=\text{Tr}\,\big{(}\rho_{\text{th}}\hat{\sigma}^{z}_{1}\big{)}\,.$
(5)

Here the thermal density matrix $\rho_{\text{th}}$ is given by

$\rho_{\text{th}}=\frac{1}{\mathcal{Z}}e^{-\beta\hat{H}-\lambda\hat{\Pi}},\ \
\mathcal{Z}=\text{Tr}\,e^{-\beta\hat{H}-\lambda\hat{\Pi}},$ (6)

where $\beta$ is the inverse temperature and $\hat{\Pi}$ is the total momentum
operator with $\lambda$ being the associated Lagrange multiplier. The values
of $\beta$ and $\lambda$ are then determined by matching the energy and the
momentum conservation conditions as
$\langle\mathbb{Z}_{2}|\hat{H}|\mathbb{Z}_{2}\rangle=\text{Tr}\,(\hat{H}\rho_{\text{th}})$
and
$\langle\mathbb{Z}_{2}|\hat{\Pi}|\mathbb{Z}_{2}\rangle=\text{Tr}\,(\hat{\Pi}\rho_{\text{th}})$.
The computed values of $\sigma^{z}_{\text{th}}$ are shown by the dashed lines
in Fig. 4(a-c). Then, we compute the quantity
$\delta\sigma^{z}=\bar{\sigma}^{z}-\sigma^{z}_{\text{th}}$, and use this
quantity to quantify the degree of thermalization violation.

The deviation $\delta\sigma^{z}$ as a function of $m$ is plotted in Fig. 5.
Remarkably, when $m$ increases from zero, we find that $\delta\sigma^{z}$
first decreases before the critical point when $m<m_{\text{c}}$, then vanishes
in the critical regime $m\approx m_{\text{c}}$, and finally increases again
above the critical point when $m>m_{\text{c}}$. This trend can also be seen in
three typical situations in Fig. 4(a-c) for $m<m_{\text{c}}$, $m=0.655\approx
m_{\text{c}}$, and $m>m_{\text{c}}$, respectively.

To understand this feature, we use color plot to illustrate the weight of the
$\mathbb{Z}_{2}$ state on different eigenstates in Fig. 4(d-f), in which the
bipartite entanglement entropies are used to distinguish the scar states from
the thermal states. In Fig. 4(d) and (e), we can see that the main weights of
the $\mathbb{Z}_{2}$ state shift from the scar states to the thermal states
when $m$ increases towards $m_{\text{c}}$. As a result, the deviation from
thermalization becomes smaller. When $m>m_{\text{c}}$, we can see from Fig.
4(f) that the weight of the $\mathbb{Z}_{2}$ state becomes more and more
concentrated on the ground state, because the $\mathbb{Z}_{2}$ state is one of
the symmetry breaking ground states in the large $m$ limit. Consequently, the
deviation from thermalization increases again because the ground state is
usually a non-thermal state. That is to say, the $\mathbb{Z}_{2}$ state does
not thermalize at the $m<m_{\text{c}}$ side because of its large overlaps with
the scar states, and also does not thermalize at the $m>m_{\text{c}}$ side
because of its large overlaps with the symmetry breaking ground state. Hence,
the regime where the $\mathbb{Z}_{2}$ state thermalizes coincides with the
quantum critical regime.

Summary. In summary, we study the PXP model with an external Zeeman field, in
which quantum many-body scar states have been discovered at $m=0$ and an Ising
quantum critical point exists at $m=m_{\text{c}}\approx 0.655$. In this work,
we find two connections between the quantum many-body scar states and the
quantum criticality. First, when a set of scar states are tracked to the
critical point, they become low-energy states and, meanwhile, retain low
entanglement entropies violating the volume law. Moreover, the Ising critical
theory offers a clear physical meaning of these tracked states at the critical
point as pair-wisely occupied Fermi sea states. Second, under sufficiently
long time evolution, the $\mathbb{Z}_{2}$ state thermalizes only in the
critical regime. We highlight that the connections between quantum
thermalization and quantum criticality are highly nontrivial. This is because
the quantum thermalization concerns the properties of generic excited states,
and, on the contrary, the quantum criticality concerns the low-energy
behavior. They are not expected to be correlated at first glance. For future
studies, on the theory side, we envision that this connection discovered in
the PXP model can be extended to other more general situations. On the
experimental side, our prediction can be directly verified in recent Rydberg
atom experiments described by the PXP Hamiltonian. It can also be verified in
the recent cold atom simulation of a lattice gauge model, whose effective
Hamiltonian is mathematically equivalent to the PXP Hamiltonian Surace:2020dy
; Cheng:2021bh .

Acknowledgment. We thank Pengfei Zhang and Yanting Cheng for insightful
discussions. This work is supported by Beijing Outstanding Young Scientist
Program and NSFC Grant No. 11734010.

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myfnsymbols** ††‡‡§§‖∥¶¶

# Investigation of octupole collectivity near the $A=72$ shape-transitional
point

M. Spieker<EMAIL_ADDRESS>Department of Physics, Florida State University,
Tallahassee, Florida 32306, USA  L.A. Riley  Department of Physics and
Astronomy, Ursinus College, Collegeville, PA 19426, USA  P.D. Cottle
Department of Physics, Florida State University, Tallahassee, Florida 32306,
USA  K.W. Kemper  Department of Physics, Florida State University,
Tallahassee, Florida 32306, USA  D. Bazin  Facility for Rare Isotope Beams,
Michigan State University, East Lansing, Michigan 48824, USA  Department of
Physics and Astronomy, Michigan State University, East Lansing, Michigan
48824, USA  S. Biswas  Facility for Rare Isotope Beams, Michigan State
University, East Lansing, Michigan 48824, USA  P.J. Farris  Facility for Rare
Isotope Beams, Michigan State University, East Lansing, Michigan 48824, USA
Department of Physics and Astronomy, Michigan State University, East Lansing,
Michigan 48824, USA  A. Gade  Facility for Rare Isotope Beams, Michigan State
University, East Lansing, Michigan 48824, USA  Department of Physics and
Astronomy, Michigan State University, East Lansing, Michigan 48824, USA  T.
Ginter  Facility for Rare Isotope Beams, Michigan State University, East
Lansing, Michigan 48824, USA  S. Giraud  Facility for Rare Isotope Beams,
Michigan State University, East Lansing, Michigan 48824, USA  J. Li  Facility
for Rare Isotope Beams, Michigan State University, East Lansing, Michigan
48824, USA  S. Noji  Facility for Rare Isotope Beams, Michigan State
University, East Lansing, Michigan 48824, USA  J. Pereira  Facility for Rare
Isotope Beams, Michigan State University, East Lansing, Michigan 48824, USA
M. Smith  Facility for Rare Isotope Beams, Michigan State University, East
Lansing, Michigan 48824, USA  D. Weisshaar  Facility for Rare Isotope Beams,
Michigan State University, East Lansing, Michigan 48824, USA  R.G.T. Zegers
Facility for Rare Isotope Beams, Michigan State University, East Lansing,
Michigan 48824, USA  Department of Physics and Astronomy, Michigan State
University, East Lansing, Michigan 48824, USA

(August 28, 2024)

###### Abstract

Enhanced octupole collectivity is expected in the neutron-deficient Ge, Se and
Kr isotopes with neutron number $N\approx 40$ and has indeed been observed for
70,72Ge. Shape coexistence and configuration mixing are, however, a notorious
challenge for theoretical models trying to reliably predict octupole
collectivity in this mass region, which is known to feature rapid shape
changes with changing nucleon number and spin of the system. To further
investigate the microscopic configurations causing the prolate-oblate-triaxial
shape transition at $A\approx 72$ and their influence on octupole
collectivity, the rare isotopes 72Se and 74,76Kr were studied via inelastic
proton scattering in inverse kinematics. While significantly enhanced octupole
strength of $\sim 32$ Weisskopf units (W.u.) was observed for 72Se, only
strengths of $\sim 15$ W.u. were observed for 74,76Kr. In combination with
existing data, the new data clearly question a simple origin of enhanced
octupole strengths around $N=40$. The present work establishes two regions of
distinct octupole strengths with a sudden strength increase around the $A=72$
shape transitional point.

###### pacs:

##  I Introduction

Much of the study of the structure of atomic nuclei centers on the interplay
between individual nucleons and the emergent collective behavior caused by the
strong interaction between them. Quadrupole-deformed shapes are one of the
emergent phenomena. Among these, axially-symmetric prolate (cigar-like) shapes
are observed more frequently than oblate (disk-like) shapes Hamamoto and
Mottelson (2009); Horoi and Zelevinsky (2010). In addition, axially-asymmetric
(triaxial) shapes are important in some regions of the nuclear chart,
including the Ge-Kr mass region ($Z=32-36$) Lecomte  _et al._ (1978, 1980a);
Kotliński  _et al._ (1990); Guo  _et al._ (2007); Girod  _et al._ (2009);
Rodríguez (2014); Nikšić  _et al._ (2014); Ayangeakaa  _et al._ (2016);
Henderson  _et al._ (2019); Ayangeakaa  _et al._ (2019). In some of the nuclei
in this region, both axially symmetric and asymmetric shapes appear to coexist
at comparably low excitation energies and lead to complex quantum-state mixing
Heyde and Wood (2011); Görgen and Korten (2016); Garrett  _et al._ (2022). The
delicate interplay between the different configurations influences several
experimental observables connected to the quadrupole degree of freedom and,
furthermore, causes rapid shape changes observed with both isospin and spin
Gade  _et al._ (2005); Clément  _et al._ (2007); Ljungvall  _et al._ (2008);
Iwasaki  _et al._ (2014); Henderson  _et al._ (2018); Wimmer  _et al._ (2020,
2021).

In addition to quadrupole excitations, octupole excitations are observed
throughout the nuclear chart Butler and Nazarewicz (1996); Robledo and Bertsch
(2011); Butler (2016); Cao  _et al._ (2020). Due to the presence of the
$2p_{3/2}$ and $1g_{9/2}$ orbitals for both protons and neutrons around the
Fermi surface, enhanced electric octupole $B(E3)$ transition strengths are
expected for the neutron-deficient Ge, Se and Kr isotopes. Previous
experimental studies established that the low energy octupole state (LEOS)
fragments into two or more $J^{\pi}=3^{-}_{i}$ states with the
$B(E3;3^{-}_{i}\rightarrow 0^{+}_{1})$ strengths summing up to approximately
15 Weisskopf units (W.u.) Kibedi and Spear (2002); Lecomte  _et al._ (1980b);
Matsuki  _et al._ (1982); Rosier  _et al._ (1986); Ogino (1986); Schürmann
_et al._ (1987). However, 70,72Ge are notable exceptions as a very sudden
$B(E3;3^{-}_{1}\rightarrow 0^{+}_{1})$ strength increase to around 30 W.u. is
observed Lecomte  _et al._ (1980b); Rosier  _et al._ (1986); Schürmann  _et
al._ (1987). Chuu et al. were able to describe this $B(E3)$ strength increase
in the Ge isotopes with the Interacting-Boson-Model plus Interacting-Boson-
Fermion-Model approach (IBM+IBFM); however, without considering shape
coexistence Chuu  _et al._ (1993). In their study, they attributed the sudden
increase in octupole collectivity to a maximum contribution of the collective
$f$-boson configuration to the total wave function. The contribution of the
$f_{5/2}-g_{9/2}$ fermion-pair configuration turned out to be negligibly small
at $N=40$. Interestingly though, the sudden strength increase is not observed
for the $N=40$ isotone 74Se Ogino (1986). When comparing to the other isotonic
chains, there also appears to be nothing particularly special about proton
number $Z=32$ in terms of octupole collectivity Kibedi and Spear (2002). This
questions previous conclusions about a simple origin of enhanced octupole
collectivity at $N=40$ drawn in, e.g., Refs. Chuu  _et al._ (1993); Wang  _et
al._ (2022); Cottle (1990). Instead, the idea that octupole collectivity is
more sensitive to quadrupole distortions in the Ge-Kr region than in other
mass regions might be correct Nazarewicz (1990). Up to now, the sharp
difference in octupole collectivity between 70,72Ge and the rest of the nuclei
in this region has remained a puzzle. Shape coexistence and strong
configuration mixing generally complicate the theoretical description of
octupole strengths (see, e.g., the remarks in Robledo and Bertsch (2012)). To
more systematically approach this challenge, first exploratory calculations
within the framework of the configuration-mixing $sdf$ IBM mapping approach,
which is based on microscopic self-consistent mean-field calculations
employing universal energy density functionals and takes shape coexistence
explicitly into account, have recently been performed Nomura (2022). In
addition to enhanced strength for 72Ge, $B(E3;3^{-}_{1}\rightarrow 0^{+}_{1})$
strengths of around 30 W.u. have been predicted in the rare isotopes 74,76Kr.

In this work, we report on measurements of the previously unknown
$B(E3;3^{-}_{i}\rightarrow 0^{+}_{1})$ strengths in the rare isotopes 72Se
($Z=34$) and 74,76Kr ($Z=36$), which are the $N=38$ and $N=40$ isotones of
70,72Ge. To measure the octupole strengths in these nuclei, inelastic proton
scattering experiments in inverse kinematics were performed. Inelastic proton
scattering has proven to be a very powerful tool to study the fragmentation of
the LEOS among a few to several excited 3- states for different structures of
the ground state Ogino (1986); Cottle  _et al._ (1988, 1990); Kennedy  _et
al._ (1992). In combination with data available for stable nuclides, the new
data clearly show that a simple picture of enhanced octupole correlations
around the octupole magic number $N=40$ cannot be claimed.

##  II Experiment

The experiments were performed at the Coupled Cyclotron Facility of the
National Superconducting Cyclotron Laboratory (NSCL) at Michigan State
University Gade and Sherrill (2016) at secondary beam energies corresponding
to proton energies of around 100 MeV in the center-of-mass frame. At these
energies, both proton and neutron contributions to the wave function are
probed almost equally Cottle (2001). The secondary 76Kr (79 $\%$ purity), 74Kr
(51 $\%$ purity) and 72Se (6 $\%$ purity) beams were produced from a 150 MeV/u
78Kr primary beam in projectile fragmentation on a 308-mg/cm2 thick 9Be
target. The A1900 fragment separator Morrissey  _et al._ (2003), using a
240-mg/cm2 Al degrader, was tuned to select the fragments of interest in
flight using two separate magnetic settings. For the magnetic setting centered
on 74Kr, the secondary 72Se beam was part of the cocktail beam. All three
secondary beams could be unambiguously distinguished from the other components
in the cocktail beam via the time-of-flight difference measured between two
plastic scintillators located at the exit of the A1900 and the object position
of the S800 analysis beam line. Downstream, the NSCL/Ursinus Liquid Hydrogen
(LH2) Target was located at the target position of the S800 spectrograph. The
projectilelike reaction residues entering the S800 focal plane were identified
event-by-event from their energy loss and time of flight Bazin  _et al._
(2003).

Figure 1: (color online) Doppler-corrected, in-beam $\gamma$-ray spectra for
76Kr (top), 74Kr (middle) and 72Se (bottom). Data are shown in black. geant4
simulations performed with ucgretina Riley  _et al._ (2021) are presented in
blue. A prompt background consisting of two exponential functions was included
in the simulation. As an example for the $\gamma\gamma$ coincidences placing
the $3^{-}_{1}$ state in 74Kr, the inset in the middle panel shows the
coincidence spectrum when gated on the 1953-keV, $3^{-}_{1}\rightarrow
2^{+}_{1}$ $\gamma$-ray transition. A clear coincidence with the 456-keV,
$2^{+}_{1}\rightarrow 0^{+}_{1}$ $\gamma$-ray transition is observed.

The GRETINA $\gamma$-ray tracking array Paschalis  _et al._ (2013); Weisshaar
_et al._ (2017) was used to detect $\gamma$ rays emitted by the reaction
residues in flight ($v/c\approx 0.4$). Eight GRETINA modules, containing four,
36-fold segmented HPGe detectors each, were mounted in the north half of the
mounting shell to accommodate the LH2 target. Event-by-event Doppler
reconstruction of the residues’ $\gamma$-ray energies was performed based on
the angle of the $\gamma$-ray emission determined from the main interaction
point in the Ge crystal and including trajectory reconstruction of the
residues through the S800 spectrograph Weisshaar  _et al._ (2017). Fig. 1
shows the experimental Doppler-corrected, in-beam $\gamma$-ray spectra for
72Se and 74,76Kr together with the corresponding spectra simulated with
ucgretina Riley  _et al._ (2021). For the simulation, the known experimental
kinematics, target thickness, setup geometry and $\gamma$-ray detection
efficiency were used as inputs. Pressure differences across the Kapton
entrance and exit windows of the LH2 cell cause them to bulge outwards. To
determine the target thickness, this effect was taken into account and its
contribution quantified by simulating the kinetic-energy distribution of the
outgoing beam with the procedure described in Riley  _et al._ (2019). An areal
density of 69(3) mg/cm2 was determined. Assuming the population of different
excited states in the reaction, the experimental $\gamma$-ray yields were
determined by fitting a superposition of the simulated $\gamma$-decay spectra
of individual excited states to the experimental spectrum. For each excited
state, $\gamma$-decay branching was explicitly taken into account if known
from previous experiments ENSDF ; Giannatiempo  _et al._ (2005); McCutchan
_et al._ (2011); Dunlop  _et al._ (2013); Mukherjee  _et al._ (2022). The
$\gamma$-decay intensities were varied within the reported uncertainties. As
$\gamma$-ray cascades are included in the simulation, the obtained yields are
corrected for observed feeders. The yields are used to calculate the inelastic
proton scattering cross sections to excited states in 72Se and 74,76Kr by
normalizing them to the number of incoming beam particles and the number of
target nuclei. For the $2^{+}_{1}$ states of 72Se and 74,76Kr, the
$(p,p^{\prime})$ cross sections are 17(4) mb, 28(5) mb and 43(2) mb,
respectively. For the $3^{-}_{1}$ states of 74,76Kr and the $3^{-}_{2}$ state
of 72Se, they are 4.6(8) mb, 5.9(3) mb and 13(3) mb. Stated uncertainties
include statistical uncertainties, the stability of the secondary beam
composition, uncertainties coming from the choice of software gates and the
target thickness.

##  III Results and Discussion

To calculate reduced transition probabilities $B(E\lambda)$ from deformation
parameters $\beta_{\lambda}$ ($\lambda=2,3,4,...$), reaction calculations were
performed with the coupled-channels program chuck3 Kunz and Comfort  using the
global optical-model parameters of Koning and Delaroche (2003). Only single-
step excitation was considered. As described in Refs. Kennedy  _et al._
(1992); Kibedi and Spear (2002); Pritychenko  _et al._ (2014), the deformation
parameters $\beta_{\lambda}$ can be calculated by scaling the theoretical
cross sections to the experimentally determined ones. For the $2^{+}_{1}$
states, deformation parameters of $\beta_{2}=0.40\pm 0.02\ \mathrm{(stat.)}\pm
0.03\ \mathrm{(sys.)}$ for 76Kr, $\beta_{2}=0.35\pm 0.06\ \mathrm{(stat.)}\pm
0.02\ \mathrm{(sys.)}$ for 74Kr, and $\beta_{2}=0.26\pm 0.06\
\mathrm{(stat.)}\pm 0.02\ \mathrm{(sys.)}$ for 72Se were determined.
Systematic uncertainties stem from the theoretically expected variation of the
cross section over the LH2 target thickness. Within uncertainties, the
$\beta_{2}$ values for the Kr isotopes are in excellent agreement with the
adopted values of 0.3920(66) and 0.363(9) Pritychenko  _et al._ (2014),
respectively. For 72Se, the value agrees with the adopted value of 0.215(5)
within uncertainties. For 74,76Kr, the inferred $B(E2;2^{+}_{1}\rightarrow
0^{+}_{1})$ values are shown in Fig. 2 (a) including data for the Kr isotopes
between $N=34$ and $N=50$. As can be seen, the $(p,p^{\prime})$ data confirm
the trend of a decreasing $B(E2)$ strength when passing $N=40$ ($A=76$) and
agree with the strengths determined with other probes Pritychenko  _et al._
(2014); Wimmer  _et al._ (2020); Gillespie  _et al._ (2021); Wimmer  _et al._
(2021); Sakamoto  _et al._ (1979), validating both the reaction calculations
and feeding correction.

Figure 2: Experimental systematics of (a) $\mathrm{B(E2;2^{+}_{1}\rightarrow
0^{+}_{1})}$, (b) $\mathrm{B(E3;3^{-}_{i}\rightarrow 0^{+}_{1})}$ strengths
and (c) excitation energies $\mathrm{E_{x}}$ for the $3^{-}_{1}$ states and
$3^{-}_{2,(p,p^{\prime})}$ states in Kr isotopes. In panels (b) and (c), data
obtained in this work are shown in black and previously reported data in grey
Matsuki  _et al._ (1982). The notation $3^{-}_{2,(p,p^{\prime})}$ is used to
indicate that they are the second-strongest fragment observed in
$(p,p^{\prime})$. The $3^{-}_{2,(p,p^{\prime})}$ states in 74,76Kr, observed
in this work, are tentatively assigned as discussed in the text. Additional
$3^{-}$ states ENSDF  important for the discussion are presented with a cross
in panel (c). As seen in panel (c), the $3^{-}_{2,(p,p^{\prime})}$ states do
not necessarily correspond to the second $3^{-}$ state. Other $B(E2)$ data are
from Pritychenko  _et al._ (2014); Wimmer  _et al._ (2020); Gillespie  _et
al._ (2021); Wimmer  _et al._ (2021); Sakamoto  _et al._ (1979).

The 2257-keV, $3^{-}_{1}$ state of 76Kr was previously observed in several
experiments including the $(p,t)$ experiment of Matsuki  _et al._ (1981). No
excited $3^{-}$ states were known in 74Kr prior to this work. The first two
excited $3^{-}$ states of 72Se at 2406 keV and 2434 keV, respectively, were
observed in a number of experiments ENSDF ; Abriola and Sonzogni (2010).
However, for none of the three nuclei, $B(E3)$ strengths were previously
measured. In this work, the well-known 1834-keV, $3^{-}_{1}\rightarrow
2^{+}_{1}$ $\gamma$-ray transition in 76Kr is prominently observed (see Fig.
1). The $J^{\pi}=3^{-}_{1}$ state of 74Kr is newly assigned based on the
striking similarity of its 1953-keV, $3^{-}_{1}\rightarrow 2^{+}_{1}$
$\gamma$-ray transition to the corresponding one in 76Kr (also see Fig. 1).
$\gamma\gamma$ coincidences confirm the placement and establish the 74Kr,
$J^{\pi}=3^{-}_{1}$ state at 2409(3) keV (see inset in Fig. 1). Reduced
transition strengths $B(E3;3^{-}_{1}\rightarrow 0^{+}_{1})$ of $15.0\pm
0.9\mathrm{(stat.)}\pm 1.8\mathrm{(sys.)}$ W.u. for 76Kr and $13\pm
2\mathrm{(stat.)}\pm 2\mathrm{(sys.)}$ W.u. for 74Kr were determined,
respectively. As can be seen in Fig. 2 (b), they match the rather constant
$B(E3;3^{-}_{1}\rightarrow 0^{+}_{1})$ values of around 15 W.u. observed in
the stable Kr isotopes Matsuki  _et al._ (1982). The excitation-energy
systematics in the Kr isotopes are presented in Fig. 2 (c). Interestingly,
rather than the $3^{-}_{1}$ state, the $3^{-}_{2}$ state is the most strongly
populated $3^{-}$ state in 72Se (see 1572-keV, $3^{-}_{2}\rightarrow
2^{+}_{1}$ $\gamma$-ray transition in the bottom panel of Fig. 1). A
significantly larger $B(E3;3^{-}_{2}\rightarrow 0^{+}_{1})$ strength of $32\pm
7\mathrm{(stat.)}\pm 4\mathrm{(sys.)}$ W.u. is determined. For the $3^{-}_{1}$
state, only an upper limit of $B(E3;3^{-}_{1}\rightarrow 0^{+}_{1})\leq 4.5$
W.u. can be reported. In agreement with newer experiments on 72Se McCutchan
_et al._ (2011); Mukherjee  _et al._ (2022), the $3^{-}_{2}\rightarrow
0^{+}_{1}$ ground-state branch is not observed. Note that in 74,76Kr, the
$3^{-}_{1}\rightarrow 0^{+}_{1}$ $\gamma$-decay branch was also not observed
and, thus, not considered for the calculation of the experimental
$(p,p^{\prime})$ cross sections.

Given that the $3^{-}_{2}$ state is more strongly populated than the
$3^{-}_{1}$ state in 72Se, it is worth noting that Ref. Matsuki  _et al._
(1982) established a pronounced variation of the $B(E3;3^{-}_{2}\rightarrow
0^{+}_{1})$ strength with neutron number in the stable Kr isotopes. This was
linked to the emergence of quadrupole deformation possibly fragmenting the
strength. The data are shown in Fig. 2 (b). When inspecting Figs. 2 (a) and
(b), it is apparent that the $B(E3;3^{-}_{2}\rightarrow 0^{+}_{1})$ follows
the $B(E2;2^{+}_{1}\rightarrow 0^{+}_{1})$ strength increase. There is a
caveat though. In the stable isotope 78Kr, the $3^{-}_{2}$ is not the one
reported in Matsuki  _et al._ (1982). There exists a lower-lying $3^{-}$ state
at 2678 keV ENSDF , which was most likely not observed in $(p,p^{\prime})$
because of its small $B(E3)$ strength [see Fig. 2 (c)]. The fragmentation of
the $B(E3)$ strength appears, thus, nontrivial as quadrupole deformation
begins to manifest. The observations made for 72Se support this point. For the
Kr isotopes, we will consequently use the notation $3^{-}_{2,(p,p^{\prime})}$
for the second $3^{-}$ state observed in $(p,p^{\prime})$ and carrying non-
negligible $B(E3)$ strength. In unstable 76Kr, the situation is comparable to
78Kr. Matsuki et al. reported a possible $3^{-}$ state at 2601(15) keV Matsuki
_et al._ (1981), which they observed in the 78Kr$(p,t)$76Kr reaction. In a
follow-up publication, they presented a firmly identified $3^{-}$ state at
2872(15) keV as the $3^{-}_{2}$ candidate Matsuki  _et al._ (1982). We do not
observe any resolved $\gamma$ rays in our spectra, which could be attributed
to the population of the 2872-keV state in $(p,p^{\prime})$. Its $B(E3)$
strength must either be small or its $\gamma$-decay behavior be very complex,
i.e., the yield be shared between several $\gamma$-decay branches. Both
scenarios may prevent its detection. A $J^{\pi}=3^{-}$ assignment is possible
for a state at 2581 keV though. Previously, Giannatiempo et al. had argued for
a $J^{\pi}=2^{+}$ assignment based on the state’s $\gamma$-decay behavior and
the deduced $\log(ft)$ value Giannatiempo  _et al._ (2005). The $\log(ft)=7.1$
value is, however, exactly the same as for the known $3^{-}_{1}$ state in
76Kr. The corresponding feeding intensity is also low suggesting a forbidden
decay from the $J^{\pi}=1^{-}$ parent ground state of 76Rb. Furthermore, the
$\gamma$ decays to the $2^{+}_{2}$ ($I_{\gamma}=78$ $\%$) and $4^{+}_{1}$
($I_{\gamma}=22$ $\%$) states allow for a $J^{\pi}=3^{-}$ assignment. This
state might, thus, correspond to the 2601-keV state reported by Matsuki et
al., where the $(p,t)$ angular distribution favors an $l=3$ transfer. We
observe the population of the 2581-keV state in $(p,p^{\prime})$. The
$(3^{-}_{2})\rightarrow 2^{+}_{2}$ and $(3^{-}_{2})\rightarrow 4^{+}_{1}$
transitions are highlighted in Fig. 1. As mentioned earlier, the
$\gamma$-decay intensities reported in Ref. Giannatiempo  _et al._ (2005) were
used for the ucgretina simulation. Even though the spin-parity assignment is
tentative, the deduced $B(E3;3^{-}_{2,(p,p^{\prime})}\rightarrow
0^{+}_{1})=6.2\pm 0.7\mathrm{(stat.)}\pm 0.7\mathrm{(sys.)}$ W.u. fits well
into the systematics. For 74Kr, we observe a new $\gamma$-ray transition of
2062(5) keV (see Fig. 1), which is in coincidence with the 456-keV,
$2^{+}_{1}\rightarrow 0^{+}_{1}$ transition. Thus, there is evidence for a
previously unobserved level at 2518(5) keV. If this state is indeed a $3^{-}$
state, then this would establish two excited $3^{-}$ states within $\sim 100$
keV. We, thus, want to emphasize that the first two excited $3^{-}$ states of
the $N=38$ isotone 72Se are within 28 keV ENSDF . The 2518-keV $\gamma$-ray
yield corresponds to a $B(E3;3^{-}_{2,(p,p^{\prime})}\rightarrow
0^{+}_{1})=4.3\pm 1.1\mathrm{(stat.)}\pm 0.5\mathrm{(sys.)}$ W.u., which is
again in excellent agreement with the general trend seen in Fig. 2 (b).
Interestingly, for both 74,76Kr and even though tentatively assigned, the
$B(E3;3^{-}_{2,(p,p^{\prime})}\rightarrow 0^{+}_{1})$ strength agrees with the
upper limit of 4.5 W.u. determined for the $B(E3;3^{-}_{1}\rightarrow
0^{+}_{1})$ strength in 72Se.

Based on our new data, we establish that the sudden $B(E3)$ strength increase
at $N=40$ is exclusively observed for 72Ge ($Z=32$). For the $N=38$ isotones,
it is observed for 70Ge ($Z=32$) and 72Se $(Z=34)$ but not for 74Kr ($Z=36$).
A simple picture of enhanced octupole correlations around the octupole magic
number $N=40$ can consequently not be claimed. The almost degenerate, low-
lying $3^{-}$ states and the fact that – in contrast to 70,72Ge, 74Se and
74,76Kr – the $3^{-}_{2}$ state is the strongest fragment in 72Se also suggest
that two microscopic configurations could cross beyond $A=74$.

Figure 3: (color online) Experimental (a) $B(E3;3^{-}_{i}\rightarrow
0^{+}_{1})$ strength of the strongest fragment and (b) summed
$\sum_{i}B(E3;3^{-}_{i}\rightarrow 0^{+}_{1})$ strengths in Ge, Se and Kr
isotopes up to 5 MeV (open symbols with uncertainties). Except for 72Se and
74,76Kr, the data are from Kibedi and Spear (2002); Rosier  _et al._ (1986);
Schürmann  _et al._ (1987); Ogino (1986); Matsuki  _et al._ (1982). For all
nuclei but 72Se, the strongest fragment is the $3^{-}_{1}$ state. Theoretical
predictions for the $B(E3;3^{-}_{1}\rightarrow 0^{+}_{1})$ strengths of 72Ge,
74Se, and 74,76Kr were added to panel (a) [solid symbols]. These were obtained
with the recently introduced configuration-mixing $sdf$ IBM mapping approach
Nomura (2022). The color coding and corresponding symbols are the same for the
experimental data and theoretical predictions. Additionally, predictions made
within IBM+IBFM approach of Ref. Chuu  _et al._ (1993) for the Ge isotopes
were added to panel (a) [dashed line]. The blue and grey bands correspond to
the 14(4) W.u. and 18(4) W.u. averages mentioned in the text.

To obtain a clearer picture, the $B(E3;3^{-}_{i}\rightarrow 0^{+}_{1})$ of the
strongest fragment and summed $B(E3;3^{-}_{i}\rightarrow 0^{+}_{1})$ strengths
in the Ge-Kr mass region are compiled in Fig. 3. Except for 72Se and 74,76Kr,
the summed strengths were deduced from the available proton and alpha
inelastic scattering experiments on the stable Ge, Se and Kr isotopes Rosier
_et al._ (1986); Schürmann  _et al._ (1987); Ogino (1986); Matsuki  _et al._
(1982). The data draw an intriguing picture of two distinct regions. The first
region extends from the spherical $N=50$ neutron shell closure all the way
down to $A=74$. In this region, weighted averages of $B(E3)=14(4)$ W.u. and
$\sum B(E3)=18(4)$ W.u. are determined from the data. The quoted uncertainties
correspond to the standard deviation. The new data on 74,76Kr, with their
dominant prolate ground-state configuration Clément  _et al._ (2007), fit
perfectly into this group. Then, the sudden jump of the $B(E3)$ strength is
observed at $A=72$. The location of this “jump” coincides with the transition
from a prolate to an oblate ground-state configuration at $A\approx 72$ Gade
_et al._ (2005); Ljungvall  _et al._ (2008); Iwasaki  _et al._ (2014); Wimmer
_et al._ (2020); Henderson  _et al._ (2018). However, based on experimental
data, triaxial configurations appear to be important at $A=72$, too Ayangeakaa
_et al._ (2016). Most importantly, the “jump” is not observed at a fixed
proton or neutron number as might be naively expected and, thus, seems to be
more intimately connected to specific structure changes.

Predictions for the $B(E3;3^{-}_{1}\rightarrow 0^{+}_{1})$ strengths of 72Ge,
74Se and 74,76Kr, obtained with the pioneering configuration-mixing $sdf$ IBM
mapping approach Nomura (2022), were added to Fig. 3 (a). Before discussing
these predictions, it should be mentioned that the experimentally determined
magnitude of the quadrupole moments $Q_{2^{+}_{1}}$ are well reproduced while
for all nuclei but 74Kr the predicted signs disagree with the data. A similar
observation was made in Ref. Henderson  _et al._ (2018). As the self-
consistent mean-field results already predict pronounced oblate minima, this
gets propagated to the mapped IBM wave functions. However, Ref. Nomura (2022)
also shows that the ground-state wave functions of all considered nuclei are
strongly mixed with spherical 0p-0h, oblate 2p-2h and prolate 4p-4h
configurations contributing. This highlights the complexity of this mass
region. Still, for a more meaningful comparison in terms of octupole
collectivity, a functional should be employed, which can reproduce the signs
of the experimental quadrupole moments as the correct ground-state structure
is critical Robledo and Bertsch (2012). The consistent overprediction of the
$B(E3)$ strengths, with the exception of 72Ge, could consequently be an
artifact of the incorrect and dominantly oblate ground-state structure. Most
$3^{-}_{1}$ $sdf$-IBM wave functions are also predicted to be dominated by the
oblate 2p-2h intruder configuration. More importantly though, the calculations
show that enhanced octupole collectivity is indeed expected for the oblate
configuration. This qualitatively agrees with the significant $B(E3)$ strength
increase seen at $A=72$, where oblate configurations start to strongly mix
into or even dominate the ground state wave function (see, e.g., the work of
Refs. Gade  _et al._ (2005); Clément  _et al._ (2007); Ljungvall  _et al._
(2008); Iwasaki  _et al._ (2014); Henderson  _et al._ (2018); Wimmer  _et al._
(2020, 2021); Hinohara  _et al._ (2010); Sato and Hinohara (2011); Rodríguez
(2014)). For completeness, we added the IBM+IBFM results of Ref. Chuu  _et
al._ (1993) to Fig. 3 (a). As the parameters are, however, explicitly fitted
to the Ge isotopes, no clear microscopic information for the entire Ge-Kr mass
region can be extracted. Considering the projected shell model calculations of
Ref. Wu  _et al._ (2017), which predict a dominant two-quasiparticle (2QP)
character for the lowest negative-parity rotational bands in the Kr isotopes,
it is possible that for nuclei with $A>72$ the contribution of the
$f_{5/2}-g_{9/2}$ fermion-pair configuration to the total wave function
increases and leads to decreased octupole collectivity as in the Ge isotopes.
The inspection of the predicted structures reveals, however, that the 2QP
Nilsson configurations in 72-76Kr originate from the spherical $2p_{3/2}$ and
$1g_{9/2}$ orbitals, i.e., the octupole-collectivity driving orbitals Wu  _et
al._ (2017). While the experimental signature is clear, the theoretical
picture in the Ge-Kr mass region remains a puzzle.

##  IV Summary

In summary, we have performed inelastic proton scattering experiments in
inverse kinematics on the rare isotopes 72Se and 74,76Kr to measure their
$B(E3)$ strengths. While significantly enhanced octupole strength of $\sim 32$
W.u. was established for 72Se ($Z=34,N=38$), much smaller strength of $\sim
15$ W.u. was observed for 74,76Kr ($Z=36,N=38,40$). Based on our new data, we
establish that the sudden $B(E3)$ strength increase at $N=40$ is exclusively
observed for 72Ge ($Z=32$). For the $N=38$ isotones, it is observed for 70Ge
($Z=32$) and 72Se $(Z=34)$ but not for 74Kr ($Z=36$). The almost degenerate,
low-lying $3^{-}$ states and the fact that – in contrast to 70,72Ge, 74Se and
74,76Kr – the $3^{-}_{2}$ state is the strongest fragment in 72Se also suggest
that two microscopic configurations could cross beyond $A=74$. In combination
with previously existing data, the new data clearly question a simple origin
of enhanced octupole strengths around $N=40$. Instead, the present work
establishes two regions of distinct octupole strengths with a sudden strength
increase around the $A=72$ prolate-oblate-triaxial shape transitional point.
Theoretical calculations performed in the framework of the configuration-
mixing $sdf$ IBM mapping approach predict enhanced $B(E3)$ strengths built on
the oblate minimum in this mass region, but fall short on correctly describing
the ground-state structure of the considered nuclei. Future experiments at
next-generation rare isotope beam facilities must test whether, as in 70Ge,
enhanced octupole strengths can also be observed in the $A=70$ isobars 70Se
and 70Kr. To investigate how far the region of enhanced octupole collectivity
extends, strengths should also be determined for the even lighter Ge, Se and
Kr isotopes. To arrive at a sound understanding of the experimental data, more
microscopic calculations, along the lines of Ref. Nomura (2022) and which
incorporate configuration mixing as well as triaxial degrees of freedom, are
called for.

###### Acknowledgements.

This work was supported by the National Science Foundation (NSF) under Grant
No. PHY-2012522 (WoU-MMA: Studies of Nuclear Structure and Nuclear
Astrophysics), Grant No. PHY-1565546 (NSCL), Grant No. PHY-2209429 (Windows on
the Universe: Nuclear Astrophysics at FRIB), and by the Department of Energy,
Office of Science, Office of Nuclear Physics, Grant No. DE-SC0020451 (MSU).
GRETINA was funded by the Department of Energy, Office of Science. The
operation of the array at NSCL was supported by the DOE under Grant No. DE-
SC0019034. M.S. acknowledges support through the FRIB Visiting Scholar Program
for Experimental Science 2020.

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# Stress Engineering of Thermal Fluctuation of Magnetization and Noise Spectra
in Low Barrier Nanomagnets Used as Analog and Binary Stochastic Neurons

Rahnuma Rahman and Supriyo Bandyopadhyay<EMAIL_ADDRESS>www.people.vcu.edu/$∼$sbandy Department of Electrical and Computer
Engineering, Virginia Commonwealth University, Richmond, VA 23284, USA

###### Abstract

A single-domain nanomagnet, shaped like a thin elliptical disk with small
eccentricity, has a double well potential profile with two degenerate energy
minima separated by a small barrier of a few $kT$ ($k$ = Boltzmann constant
and $T$ = absolute temperature). The two minima correspond to the
magnetization pointing along the two mutually anti-parallel directions along
the major axis of the ellipse. At room temperature, the magnetization will
fluctuate randomly between the two minima mimicking telegraph noise. This
makes the nanomagnet act as a “binary” stochastic neuron (BSN) with the
neuronal state encoded in the magnetization orientation. If the nanomagnet is
magnetostrictive, then the barrier can be gradually depressed further by
applying (electrically generated) uniaxial stress along the ellipse’s major
axis, thereby gradually eroding the double well shape. When the barrier almost
vanishes, the magnetization begins to randomly assume any arbitrary
orientation (not just along the major axis), making the nanomagnet act as an
“analog” stochastic neuron (ASN). The fluctuation then begins to increasingly
resemble white noise. The full-width-at-half-maximum (FWHM) of the noise auto-
correlation function decreases with increasing stress, as the fluctuation
gradually transforms from telegraph noise to white noise. The noise spectral
density exhibits a 1/fβ spectrum (at high frequencies) with $\beta$ decreasing
with increasing stress, which is again characteristic of the transition from
telegraph to white noise. Stress can thus not only reconfigure a BSN to an
ASN, which has its own applications, but it can also perform “noise
engineering”, i.e., tune the auto-correlation function and power spectral
density. That can have applications in signal processing.

††preprint: APS/123-QED

##  I Introduction

Binary and analog stochastic neurons are powerful hardware accelerators for
probabilistic computers that are adept at solving either combinatorial
optimization problems in binary space [1, 2, 3, 4, 5] or temporal sequence
learning/prediction in analog space [6, 7]. Both types of neuron can be
implemented with a single-domain low barrier nanomagnet, such as one shaped
into a thin elliptical disk with small eccentricity. The magnetization
orientation encodes the neuron’s state. If the energy barrier within the
nanomagnet is low compared to the thermal energy $kT$, but still high enough
that the potential profile has the character of a double well with two
degenerate energy minima, then the magnetization will fluctuate randomly
between the two minima and the behavior will be that of a binary stochastic
neuron (BSN) whose state is always either +1 or -1, albeit varying randomly in
time. The probability of being in either state can be tuned by injecting a
spin polarized current into the nanomagnet [1] or by applying a voltage to
induce voltage controlled magnetic anisotropy [8]. If, on the other hand, the
energy barrier is depressed enough with some external agent to erode the
double well feature, then the magnetization will be equally likely to point in
any direction, i.e., the neuron’s state can assume (randomly) any value
between +1 and -1. That makes it an analog stochastic neuron (ASN). The
external agent thus reconfigures a BSN to an ASN.

If the nanomagnet is magnetostrictive, then the external agent can be
electrically generated uniaxial stress of the right sign applied along the
major axis of the elliptical disk [9, 10]. The sign of the stress (tensile or
compressive) must be such that the sign of the product of the stress and
magnetostriction is negative [9]. Such a stress will depress the energy
barrier and transform a BSN into an ASN, thereby providing a powerful route to
reconfigurability in probablistic computers. The myriad uses of strain-
engineered reconfigurable stochastic neurons have been examined in [9, 10].

Here, we address a different topic, namely how the fluctuation characteristic
of the magnetization (neuronal state) changes as the internal energy barrier
within the low-barrier nanomagnet (LBM) is gradually depressed with stress to
reconfigure a BSN to an ASN. When no stress is applied and the energy barrier
is high enough to sport a double-well appearance, the fluctuation of the
magnetization mimics telegraph noise [see Fig. 1]. As the energy barrier is
gradually lowered, the fluctuation begins to change from telegraph noise to
white noise as shown in Fig. 1. Throughout this range, the noise spectral
density has a 1/fβ spectrum at high frequencies. The value of $\beta$
decreases with increasing stress (it will become 0 in the limit of pure white
noise and approach 2 in the limit of pure telegraph noise). Thus, a stress-
engineered low barrier nanomagnet is not only a reconfigurable stochastic
neuron, but also a tunable noise source with tunable power spectrum – a
controllable nanomachine – that may have applications in communications such
as noise radar technology [11] and automatic speaker classification [12].

Figure 1: Temporal fluctuations in the magnetization component along the major
axis of a Co nanomagnet shaped like an elliptical disk, with major axis = 100
nm, minor axis = 99 nm and thickness = 5 nm. The fluctuations are shown at
different values of uniaxial tensile stress applied along the major axis. Note
that the noise gradually transitions from telegraph to nearly white with
increasing stress which increasingly depresses the energy barrier within the
nanomagnet. Reproduced from [9] with permission from the Institute of Physics.

##  II Materials and Methods

To study the noise engineering paradigm, we simulated the magnetization
dynamics in a single domain low barrier nanomagnet at room temperature under
different (barrier lowering) stresses using the Landau-Lifshitz-Gilbert-
Langevin (LLGL) equation with a thermal noise term. The nanomagnet is a thin
elliptical disk of cobalt with major axis 100 nm, minor axis 99 nm and
thickness 5 nm. For cobalt, the saturation magnetization $M_{s}$ = 106 A/m,
the magnetostriction coefficient $\lambda_{s}$ = -35 ppm and the Gilbert
damping coefficient $\alpha$ = 0.01.

The coupled LLGL equations governing the temporal evolutions of the scalar
components of the magnetization $m_{x}(t)$, $m_{y}(t)$ and $m_{z}(t)$ – all
normalized to the saturation magnetization $M_{s}$ – were solved with finite
difference method [13, 14]. The effect of uniaxial stress was modeled via a
magnetic field term and the effect of thermal noise via another (random)
magnetic field term. In all cases, the initial condition was that the
magnetization was aligned close to the major axis of the nanomagnet. The time
step used in the simulation was 0.1 ps.

The coupled LLGL equations describing the temporal evolution of the three
components of the magnetization are:

$\displaystyle{{dm_{x}(t)}\over{dt}}$ $\displaystyle=$
$\displaystyle-\gamma\left[H_{z}(t)m_{y}(t)-H_{y}(t)m_{z}(t)\right]$
$\displaystyle-\alpha\gamma[H_{y}(t)m_{x}(t)m_{y}(t)-H_{x}(t)m_{y}^{2}(t)$
$\displaystyle-H_{x}(t)m_{z}^{2}(t)+H_{z}(t)m_{x}(t)m_{z}(t)]$
$\displaystyle{{dm_{y}(t)}\over{dt}}$ $\displaystyle=$
$\displaystyle-\gamma\left[H_{x}(t)m_{z}(t)-H_{z}(t)m_{x}(t)\right]$
$\displaystyle-\alpha\gamma[H_{z}(t)m_{y}(t)m_{z}(t)-H_{y}(t)m_{z}^{2}(t)$
$\displaystyle-H_{y}(t)m_{x}^{2}(t)+H_{x}(t)m_{x}(t)m_{y}(t)]$
$\displaystyle{{dm_{z}(t)}\over{dt}}$ $\displaystyle=$
$\displaystyle-\gamma\left[H_{y}(t)m_{x}(t)-H_{x}(t)m_{y}(t)\right]$
$\displaystyle-\alpha\gamma[H_{x}(t)m_{z}(t)m_{x}(t)-H_{z}(t)m_{x}^{2}(t)$
$\displaystyle-H_{z}(t)m_{y}^{2}(t)+H_{y}(t)m_{y}(t)m_{z}(t)]$

where $\alpha$ is the Gilbert damping factor of the nanomagnet material,
$\gamma$ is the gyromagnetic factor (a constant), $m_{i}(t)$ is the $i$-th
component of the normalized magnetization at time $t$, and $H_{i}(t)$ is the
$i$-th component of the effective magnetic field experienced by the nanomagnet
at time $t$. The major axis of the nanomagnet is along the y-direction and the
minor axis is along the x-direction.

The effective magnetic field components are given by

$\displaystyle H_{x}(t)$ $\displaystyle=$ $\displaystyle-
M_{s}N_{x}m_{x}(t)+h_{x}^{noise}(t)$ $\displaystyle H_{y}(t)$ $\displaystyle=$
$\displaystyle-
M_{s}N_{y}m_{y}(t)+h_{y}^{noise}(t)+{{3}\over{\mu_{0}M_{s}}}\lambda_{s}\sigma
m_{y}(t)$ $\displaystyle H_{z}(t)$ $\displaystyle=$ $\displaystyle-
M_{s}N_{z}m_{z}(t)+h_{z}^{noise}(t),$ (2)

where $N_{x}$, $N_{y}$ and $N_{z}$ are the demagnetization factors along the
minor axis, major axis and out-of-plane direction (they depend on the
dimenstions of the major axis, minor axis and thickness) while
$h_{i}^{noise}(t)=\sqrt{{{2\alpha
kT}\over{\gamma\left(1+\alpha^{2}\right)\mu_{0}M_{s}\Omega\Delta
t}}}G_{(0,1)}^{i}(t)$ with $G_{(0,1)}^{i}(t)$ ($i=x,y,z$) being three
uncorrelated Gaussians of zero mean and unit standard deviation, $\Omega$ is
the nanomagnet volume, $\sigma$ is the uniaxial stress applied along the major
axis, and $\Delta t$ is the attempt period which is the time step of the
simulation (0.1 ps).

As the magnetization fluctuates randomly owing to thermal noise, the component
along the major axis $m_{y}(t)$ fluctuates. We have calculated the auto-
correlation function $C(\tau)$ of $m_{y}(t)$ using the usual prescription

$C(\tau)=\int_{-\infty}^{\infty}m_{y}(t)m_{y}(t+\tau)dt.$ (3)

We then use the Wiener-Khinchin theorem to extract the noise power spectral
density spectrum $S(f)$:

$S(f)=Re\left[\int_{-\infty}^{\infty}C(\tau)e^{-2\pi
f\tau}d\tau\right]=2\int_{0}^{\infty}C(\tau)cos(2\pi f\tau)d\tau.$ (4)

##  III Results and Discussion

The auto-correlation functions $C(\tau)$ are plotted in Fig. 2 for different
stress values, while the corresponding power spectral density spectra are
shown in Fig. 3.

At zero or low stress values, the magnetization fluctuation resembles
telegraph noise [see Fig. 1]. Telegraph noise consists of a random signal
$\zeta(t)$ $\in$ [-1, +1] and the number of arrivals in an interval
[$t_{1},t_{2}$] is Poissonian with distribution $\lambda(t_{1}-t_{2})$. The
auto-correlation function of telegraph noise is ideally
$C(\tau)=e^{-2\lambda\tau}$ [15]. In Fig. 2, we have plotted the auto-
correlation function in both linear and log-linear scale. From the latter
plot, we have extracted an effective Poisson parameter $\lambda_{eff}$ from
the slope. These are listed in Table 1. We emphasize that the fluctuation
resembles telegraph noise only at zero or low stress values and hence
$\lambda_{eff}$ will correspond to the Poisson parameter only at low stress
value and not at higher stress value when the noise begins to deviate from the
character of telegraph noise.

We also list the full-width-at-half-maximum (FWHM) of the auto-correlation
functions in Table 1.

Table 1: The parameter $\lambda_{eff}$ and the FWHM of the auto-correlation function at different stress values  Stress (MPa) |  $\lambda_{eff}$ (MHz) | FWHM ($\mu$s)
---|---|---
0 | 2.67 | 0.200
2 | 3.68 | 0.070
5 | 5.41 | 0.025
6 | 5.41 | 0.009

Two features immediately stand out in the above table. First, $\lambda_{eff}$
increases with increasing stress, indicating an increase in the arrival rate.
This is consistent with Fig. 1, where we clearly see that the flips per second
(or the rate of flips) is increasing with increasing stress. This happens
because stress depresses the barrier, making it easier for the magnetization
state to hop over the barrier. It has a very important consequence for
probabilistic computing with stochastic neurons. For autonomous clockless
computing, the computational speed depends on the flips per second (fps) [16],
and increasing that quantity with stress will increase the computational
speed. In the past, we have shown that the fps can be increased by choosing
ferromagnets with low saturation magnetization [17], which, like stress, has
the effect of lowering the energy barrier within the nanomagnet. Here, we show
that this objective can be also achieved by applying stress. The former
approach is not reconfigurable since the fps could not be changed once the
nanomagnet has been fabricated, but here one can change the fps at will with
electrically generated stress.

The second feature to note in Fig. 2 is that the FWHM decreases with
increasing stress and the auto-correlation function gradually approaches a
$\delta$-function as the stress is increased. The auto-correlation function of
white noise is a $\delta$-function. Hence, as we increase stress and lower the
energy barrier within the nanomagnet, the magnetization fluctuation gradually
transforms from telegraph noise to white noise.

We also point out two features in the spectral density plots shown in Fig. 3.
First, it is evident from the spectra that the noise begins to develop higher
frequency components with increasing stress. This is also a manifestation of
the fact that the flips per second are increasing with stress which lowers the
energy barrier within the nanomagnet, and this is also a telltale sign of the
noise gradually transforming from telegraph to white. The other feature is
less obvious. We have calculated the integrated noise power
$\int_{0}^{\infty}S(f)df$ at various values of stress and tabulated them in
Table 2. Within numerical inaccuracies, this quantity is invariant under
stress. This is expected. Stress does not inject or extract any power from the
system and hence the integrated noise should be independent of stress.

Figure 2: Calculated auto-correlation functions of the fluctuations in the
magnetization component along the major axis of the nanomagnet at a
temperature of 300 K for different values [0, 2, 5 and 6 MPa] of the uniaxial
tensile stress (applied along the major axis of the nanomagnet). The plots are
shown in both linear and log-linear scale. Also shown are the full-width-at-
half-maximum (FWHM) of the auto-correlation functions and the parameter
$\lambda_{eff}$ at different stress values.

Figure 3: Calculated noise power spectral density of the fluctuations in the magnetization component along the major axis of the nanomagnet at a temperature of 300 K for different values [0, 2, 5 and 6 MPa] of the uniaxial tensile stress (applied along the major axis of the nanomagnet). The power is expressed in arbitrary units. Table 2: Integrated noise power $\int_{0}^{\infty}S(f)df$ at different stress values  Stress (MPa) |  $\int_{0}^{\infty}S(f)df$ (arb. units)
---|---
0 | 0.2090
2 | 0.2149
5 | 0.1983
6 | 0.1836

Finally, we note that the power spectral density of ideal telegraph noise has
the form of a Cauchy density function [15] and hence will have the form

$S(f)=\frac{\lambda}{\pi f^{2}+\lambda^{2}},$ (5)

which, in the high frequency limit, becomes approximately

$S(f)\approx\frac{\lambda}{\pi}\left[\frac{1}{f^{2}}\right].$ (6)

Because the noise has telegraph character at zero or low stress values and
increasingly resembles white noise at high stress values, we expect the noise
spectra at high frequencies to exhibit a 1/$f^{\beta}$ dependence where
$\beta$ will be close to 2 at zero stress value and decrease at higher stress
values (for white noise, $\beta$ = 0).

We fitted the noise power spectra at high frequencies with 1/$f^{\beta}$ and
found that the $\beta$ values vary from 2 (no stress) to 1.88 (6 MPa stress).
The $\beta$ value does decrease with stress, but never quite approaches zero,
which would be characteristic of white noise.

##  IV Conclusion

Low barrier nanomagnets shaped like thin elliptical disks with low
eccentricity are popular hardware accelerators for stochastic neurons, where
the randomly fluctuating magnetization orientation encodes the neuronal state.
Here, we have shown that this construct has another application, namely an
engineered noise source. The magnetization fluctuation at room temperature can
be made to transform from telegraph noise to white noise by applying uniaxial
stress of the right sign along the major axis. This changes the auto-
correlation function and the noise power spectral density, which has
applications in communication engineering.

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