# Pocket resonances in low-energy antineutrons reactions with nuclei

Teck-Ghee Lee1<EMAIL_ADDRESS><EMAIL_ADDRESS>1Department of
Physics, Auburn University, Auburn, Alabama 36849, USA.  Orhan Bayrak2
<EMAIL_ADDRESS>2Department of Physics, Akdeniz University, Antalya,
Turkey  Cheuk-Yin Wong3<EMAIL_ADDRESS>3Physics Division, Oak Ridge National
Laboratory111 This manuscript has been authored in part by UT-Battelle, LLC,
under contract DE-AC05-00OR22725 with the US Department of Energy (DOE). The
US government retains and the publisher, by accepting the article for
publication, acknowledges that the US government retains a nonexclusive, paid-
up, irrevocable, worldwide license to publish or reproduce the published form
of this manuscript, or allow others to do so, for US government purposes. DOE
will provide public access to these results of federally sponsored research in
accordance with the DOE Public Access Plan (http://energy.gov/downloads/doe-
public-access-plan), Oak Ridge, Tennessee 37831, USA, Oak Ridge, Tennessee
37831, USA.

###### Abstract

Upon investigating whether the antineutron-nucleus annihilation cross-sections
at very low momenta $p$ satisfy Bethe-Landau’s power law of $\sigma_{\rm
ann}(p)\propto 1/p^{\alpha}$, we uncover unexpected regular oscillatory
structures in the low antineutron energy region from 0.001 to 10 MeV, with
small amplitudes and narrow periodicity in the logarithm of the antineutron
energies, for large-$A$ nuclei such as Pb and Ag. Subsequent semiclassical
analyses of the $S$ matrices reveal that these oscillations are pocket
resonances that arise from quasi-bound states inside the effective potential
pocket with a barrier. They are a continuation of the bound states in the
continuum. Experimental observations of these pocket resonances will provide
vital information on the properties of the optical model potentials and the
nature of the antinucleon-nucleus reactions.

##  I Introduction

One of the greatest mysteries in modern physics is the matter-antimatter
asymmetry in the Universe Sak67 . To unravel this mystery, there has been a
great deal of experimental and theoretical investigations on matter-antimatter
interactions. So far, most of the obtained information centers around
antiproton-nucleus ($\bar{p}$A) reactions and structures Ric20 ; asacusa18 ;
Mau99 ; FAIR09 ; Bia11 ; Bal89 ; Kle05 ; Fri00 ; Fri01 ; Fri14 ; Lee14 ; Lee16
; Lee08 . The corresponding information on antineutron-nucleus ($\bar{n}$A)
reaction Ric20 ; Bres03 , on the other hand, remains comparatively sparse and
limited with the most recent work from the OBELIX collaboration Bar97 ; Ast02
at CERN. Nevertheless, the $\bar{n}A$ annihilation is essential in the process
of quantifying signals from $n\rightarrow\bar{n}$ oscillations in matter Vor20
; Sno19 ; Lad19 ; Phi16 , and the significant connection between the
$\bar{n}A$ interaction-potential and the $n\rightarrow\bar{n}$ oscillations
rates have also been examined theoretically Dov83 ; Kon96 ; Fri08 .

Figure 1: Optical-model cross sections as a function of energy. (a)
${\bar{n}}A$ annihilation, (b) ${\bar{n}}A$ elastic scattering. The red
dotted-curve is the semiclassical result for the ${\bar{n}}$Pb reactions.

It has been generally expected that in the $s$-wave limit the $\bar{n}A$ and
$\bar{p}A$ annihilation cross-sections are to obey Bethe-Landau’s power-law,
$\sigma^{\bar{n}A,\bar{p}A}_{\rm ann}(p)\propto 1/p^{\alpha}$ as a function of
the antineutron momentum $p$, with $\alpha=1$ for $\bar{n}A$ and $\alpha=2$
for $\bar{p}A$ Lan58 . But recent experimental data (see Fig. 5 of
Ref.asacusa18 ) revealed that the $\bar{n}$C and $\bar{p}$C annihilation
cross-sections at low energies appear to follow a similar trend with only
minor differences. Much curious about the puzzle, we recently introduced a new
momentum-dependent optical potential to investigate the behavior of the
$\bar{n}$A, and the $\bar{p}$A annihilation cross-sections on C, Al, Fe, Cu,
Ag, Sn, and Pb nuclei in the momentum range 50 to 500 MeV/$c$ Lee18 , via the
ECIS code Ray81 . The calculated results agree with the OBELIX’s annihilation
cross-section data Bar97 ; Ast02 , but the optical model calculations
indicated that $\alpha\approx 1/2$ for $\bar{n}A$ and $\alpha\approx 1.5$ for
$\bar{p}A$ between 40 and 100 MeV/$c$, leading us to conclude that our low-
energy annihilation reaction in question was yet to reach the $s$-wave limit.

While investigating the energy dependence of the cross-sections at even lower
energies, we uncover, to our surprise, unexpected regular oscillatory
structures with small amplitudes and narrow periods (in the logarithm of the
energy) in the annihilation and elastic cross-sections, in the region from
0.001 to 10 MeV, as shown in Fig. 1. Such oscillations are absent for small
nuclei and gain in strength as the nuclear radius increases. Its amplitude is
larger for the elastic scattering than for the annihilation process. Such
behavior, undoubtedly, contradicts Bethe-Landau’s power-law. It is reminiscent
of a potential resonance behavior. However, the predicted oscillations appear
somewhat different from that of neutron-induced total cross sections for Pb,
Cd, and Ho nuclei, which arises from the Ramsauer’s resonances Ram21 ; Pet62 ;
Mar70  that are broad potential resonances without a barrier whose peak
position moves to higher energies as the nuclear mass increases. They are also
unlike those present in the high-energy 12C+12C fusion, which is due to
successive addition of contributions from even values of partial waves to the
identical-particle fusion cross-section, with increasing energies Esb12 ;
CYW12 ; Row15 .

To understand the nature of these oscillations, we carry out a semiclassical
$S$ matrix analysis for $\bar{n}$Pb reaction in which the oscillations are
quite prominent. As shown in Fig. 1 the structure of the cross-section
oscillations in the semiclassical calculations for $\bar{n}$Pb agrees
approximately with the structure obtained in the optical model calculations,
except for the lowest energy region for which the semiclassical approximation
is not applicable. We wish to show that the cross-section oscillations are
physical and constitute narrow pocket resonances that can be best described as
arising from quasi-bound states inside an effective potential pocket with a
barrier, as depicted in Fig. 5 of Ford and Wheeler FW59 . Such pocket
resonances are expected to be common features of potential scattering in
nuclear, atomic, molecular, and heavy-ion collisions because they are the
continuation of the bound states in the continuum. They move to lower energies
as the mass number of the nucleus increases as is shown in Fig. 1. To
understand the origin of such resonances, a thorough analysis of the reaction
process in detail is therefore needed in what follows.

##  II Semiclassical analyses of the oscillations of the annihilation and
elastic cross sections

Semiclassical descriptions of the potential scattering problem have been
discussed in great details by Ford and Wheeler FW59  using the Wentzel-
Kramers-Brillouin (WKB) approximation, and later consolidated by other
pioneering works Mil68 ; BM72 ; BT77 ; Lee78 ; Ohk87 . However, we shall
follow the Brink-Takigawa formulation BT77  for complex-potential scattering
that was also applied by Lee, Marty, and Ohkubo BT77 ; Lee78 ; Ohk87 .

Figure 2:  Real part of the $\bar{n}$Pb interaction potential Re($V_{\rm
eff}(r,E)$) at a sample energy $E$, where only the potential curves for even
orbital angular momentum $L$ are displayed. The $r_{1,2,3}$ denote three
turning points for Re($V_{\rm eff}(r,E)$).

Accordingly, we consider the interference between a barrier wave reflected at
the potential barrier and modified by the tunneling effect, and an internal
wave that penetrates the barrier, into the potential pocket, and reemerges
through the barrier out to $r\to\infty$.

For a given $E$ and a partial wave $L$, the total phase shift
$\delta=\delta(E,L)$ is related to the WKB phase function $\delta_{\rm WKB}$,
the action phase angle $\delta_{ij}$ between the turning points $r_{i}$ and
$r_{j}$, tunneling phase $\phi$, tunneling phase angle $\xi$ and tunneling
coefficient $w$ by BT77 ; Lee78 ; Ohk87 ,

$\displaystyle\delta=\delta_{\rm
WKB}-\phi+\tan^{-1}\\{w(\xi)\tan(\delta_{32}-\phi)\\},$ (1)
$\displaystyle\delta_{\rm
WKB}=\frac{\pi}{2}(L+\frac{1}{2})-kr_{1}+\int_{r_{1}}^{\infty}(K-k)dr,~{}~{}~{}~{}~{}~{}$
(2) $\displaystyle\delta_{ij}=\int_{r_{i}}^{r_{j}}Kdr,$ (3)
$\displaystyle\phi=\frac{1}{2}\left(\xi\ln(\xi/e)-\arg\Gamma(\frac{1}{2}+i\xi)\right),$
(4) $\displaystyle\xi=-\frac{i}{\pi}\delta_{21},$ (5) $\displaystyle
w(\xi)=\frac{\sqrt{(1+e^{-2\pi\xi})}-1}{\sqrt{(1+e^{-2\pi\xi})}+1},$ (6)

where $K$ = $\sqrt{2\mu[E-V_{\rm eff}(r,E)]/\hbar^{2}}$, $k=\sqrt{2\mu
E/\hbar^{2}}$, $E$ is the center-of-mass energy, and $\mu$ is the reduced mass
of the collision pair. The turning points $r_{1,2,3}$ are the roots of
$E-V_{\rm eff}(r,E)=0$, as displayed in Fig. 2. In our case, $V_{\rm
eff}(r,E)$ is the Langer’s modified interaction potential,

$\displaystyle V_{\rm
eff}(r,E)=\frac{-(V(E)+iW)}{1+\exp\\{(r-R)/a_{o}\\}}+\frac{\hbar^{2}(L+1/2)^{2}}{2\mu
r^{2}}.$ (7)

We adapt the same momentum-dependent optical potential used in our earlier
work Lee18  in the present semiclassical theory. The values for the parameters
$r_{o}=R/A^{1/3}$, $a_{o}$ and volume terms $V(E)$ and $W$ are listed in Table
1, 2, and 3 in Lee18 . To keep the analyses simple, we consider only the
nuclear volume terms in the present semiclassical theory.

Figure 2 illustrates the real part of the potential Re($V_{\rm eff}(r,E))$ as
a function of internuclear distance $r$ and different $L$ at a sample energy
$E$ for the $\bar{n}$Pb interaction. With the nuclear potential that has a
negative imaginary potential $W$, the turning points $r_{1,3}$ are below the
real axis of the complex-plane and $r_{2}$ lies above, and the $V_{\rm
eff}(r,E)$ for a given $E$ must be an analytic function of $r$. The integrands
in the action phase angle $\delta_{ij}$ are multiple-valued quantities
involving branch cuts. The integration paths in the complex plane need to be
chosen so that the action phase angles $\delta_{ij}$ and $\xi$ are positive,
as discussed in BT77 .

As mentioned earlier in the Introduction, the semiclassical cross-sections for
$\bar{n}$Pb reaction give an approximate representation of the optical model
results, except for the lowest energy region. Our task is to carry out a
detailed semiclassical partial wave analysis to pinpoint the origin of the
cross-section oscillations.

So we concentrate on the interference between the barrier waves and the
internal waves and split the total $S$ matrix element into the barrier wave
contribution $S_{B}$, and the internal wave contribution $S_{I}$:

$\displaystyle S$ $\displaystyle=$ $\displaystyle e^{2i\delta}=S_{B}+S_{I},$
(8) $\displaystyle S_{B}$ $\displaystyle=$
$\displaystyle\frac{e^{2i\delta_{\rm WKB}}}{N},$ (9) $\displaystyle S_{I}$
$\displaystyle=$ $\displaystyle\left[\frac{e^{2i\delta_{\rm
WKB}}}{N}\right]e^{-2\pi\xi}\left[\frac{e^{2i\delta_{\rm 32}}}{N}\right]$ (10)
$\displaystyle\times\left[1+\frac{e^{2i\delta_{\rm
32}}}{N}+\left(\frac{e^{2i\delta_{\rm
32}}}{N}\right)^{2}+\left(\frac{e^{2i\delta_{\rm
32}}}{N}\right)^{3}+\cdots\right]$ $\displaystyle=$
$\displaystyle\left[\frac{e^{2i\delta_{\rm
WKB}}}{N}\right]e^{-2\pi\xi}\left[\frac{e^{2i\delta_{\rm
32}}}{N}\right]\left[\frac{1}{1+e^{2i\delta_{\rm 32}}/N}\right],$

where $N=\sqrt{2\pi}\exp(-2\pi\xi+i\xi\ln(\xi/e))/\Gamma(\frac{1}{2}+i\xi)$ is
the barrier-penetration factor. The factor $1/(1+e^{2i\delta_{32}}/N)$ in the
$S_{I}$ term is from Padé approximation, where $e^{2i\delta_{32}}/N$ is the
amplitude of penetration-weighted internal-wave. Thus, the factor
$e^{2i\delta_{32}}/(N+e^{2i\delta_{32}})$ essentially describes the sum of
amplitudes of multiple reflections of internal-wave inside the potential
pocket BT77 ; Lee78 . Knowing these $S$-matrices, $\sigma^{\rm tot}_{\rm
ann}=\sum_{L}\sigma_{\rm ann}(L)=({\pi}/{k^{2}})\sum_{L}(2L+1)(1-|S|^{2})$ and
$\sigma^{\rm tot}_{\rm el}=\sum_{L}\sigma_{\rm
el}(L)=({\pi}/{k^{2}})\sum_{L}(2L+1)(|1-S|^{2})$ define the cross sections for
annihilation and elastic scattering, can be readily evaluated.

Partial wave analysis of the annihilation oscillations in terms of the
interference of the barrier and the internal waves gives Ohk87

$\displaystyle\sigma_{\rm ann}^{\rm tot}=\Sigma^{\rm BI}_{\rm ann}+\Sigma^{\rm
int}_{\rm ann}=\sum\limits_{L=0}^{\infty}(\sigma^{\rm BI}_{\rm
ann}(L)+\sigma^{\rm int}_{\rm ann}(L)),$ (11) $\displaystyle\sigma^{\rm
BI}_{\rm ann}(L)=\frac{\pi}{k^{2}}(2L+1)(1-|S_{B}|^{2}-|S_{I}|^{2}),$ (12)
$\displaystyle\sigma^{\rm int}_{\rm ann}(L)=\frac{\pi}{k^{2}}(2L+1)(-2{\rm
Re}\\{S_{B}S^{\ast}_{I}\\}),$ (13)

where $\sigma_{\rm ann}^{\rm int}(L)$ arises from the interference between the
internal and barrier waves, and $\sigma_{\rm ann}^{\rm BI}(L)$ from the
remaining non-interference term.

Figure 3:  Semiclassical results for $\bar{n}$Pb annihilation as a function of
energy. (a) is the total cross section of Eq.(11), (b) is the partial-wave
cross section of Eq.(11), (c) is the non-interference cross section of Eq.(12)
and (d) is the interference cross section of Eq.(13)

.

We show in Fig. 3(a) the results of the separation of the total annihilation
cross section $\sigma_{\rm ann}^{\rm tot}$ of Eq. (11) into the non-
interference part, $\Sigma^{\rm BI}_{\rm ann}$, and the interference part,
$\Sigma^{\rm int}_{\rm ann}$ as a function of $E$. The non-interference
$\Sigma^{\rm BI}_{\rm ann}$ produces the familiar $1/p$-like dependence with
decreasing energy without any oscillations. In contrast, the interference part
$\Sigma^{\rm int}_{\rm ann}$ does oscillate, giving rise to the total (sum of
all partial-wave) annihilation cross section $\sigma^{\rm tot}_{\rm ann}$ that
oscillates on the smooth $1/p$ background as shown in the plot.

Figure 3(b) shows the decomposition of the total annihilation cross section,
$\sigma^{\rm tot}_{\rm ann}$, into different partial waves contributions,
$\sigma_{\rm ann}(L)$. Except for the $L$ = 0 partial-wave that resembles the
$1/p$-like behavior, the other partial waves display a double peak. Fig. 3(c)
shows the $\sigma_{\rm ann}^{\rm BI}(L)$ for different $L$ as given by
Eq.(12), indicating a monotonically decreasing cross section for $L=0$ and a
single broad peak for higher partial waves.

Figure 4: For different partial waves $L$, the energy dependence of (a) the
real part of the barrier penetration factor in Eq. (14), (b) the real part of
the pocket factor in Eq. (14), (c) the quantity $[-{\rm
Re}\\{S_{B}S^{\ast}_{I}\\}]$, and (d) the quantity Re($\delta_{32}$).

Figure 3(d) displays $\sigma_{\rm ann}^{\rm int}(L)$ for different $L$ as
given by Eq.(13). Inspection of Eq.(9), (10) and (13) reveals that the
oscillations of $\sigma_{\rm ann}^{\rm int}(L)$ are governed by $[-2{\rm
Re}(S_{B}S^{\ast}_{I})]$ in which $[-(S_{B}S^{\ast}_{I})]$ can be factorized
into

$-S_{B}S^{\ast}_{I}\\!=\underbrace{\left[\frac{e^{2i\delta_{\rm
WKB}}}{N}\right]\left[\frac{e^{2i(\delta_{\rm WKB}+\delta_{\rm
21})}}{N}\right]^{\ast}}_{\rm
barrier~{}penetration~{}factor}\underbrace{\left[\frac{-e^{2i\delta_{\rm
32}}/N}{1+e^{2i\delta_{\rm 32}}/N}\right]^{*}}_{\rm pocket~{}factor}\\!,$ (14)

which depends on $\delta_{\rm WKB}$, $\delta_{21}$, and $\delta_{32}$. The
real part of the above barrier penetration factor and pocket factor in Eq.(14)
for various $L$ are displayed in Fig. 4 as a function of $E$. In the
$r_{2}<r<r_{1}$ region below the barrier, $\delta_{21}$ are predominantly
imaginary, resulting in the pentration probability
$\exp(2i\delta_{21})\approx\exp(-2\pi\xi)$ with $\xi>0$. The ($\delta_{\rm
WKB}-\delta^{*}_{\rm WKB})$ difference of the $r>$ $r_{1}$ outer-most region
are also predominantly imaginary, rendering $\exp[2i(\delta_{\rm
WKB}-\delta^{*}_{\rm WKB})]\approx\exp(-2\eta)$ with $\eta>0$. We can
therefore recast Eq.(13) into

$\sigma^{\rm int}_{\rm
ann}(L)\\!\sim\\!\frac{\pi}{k^{2}}(2L\\!+\\!1)|S_{B}||S_{I}|e^{-2(\pi\xi+\eta)}[-2\cos(2\delta_{32})].\\!$
(15)

Consequently we find that the maxima of the annihilation cross sections
$\sigma_{\rm ann}^{\rm int}(L)$ in Fig. 3(d) as “annihilation resonances” are
located at the maxima of the real part of the pocket factor of Eq. (14) in
Fig. 4(b) and the maxima of $[-{\rm Re}(S_{B}S_{I}^{*})]$ in Fig. 4(c), as
marked by the vertical dashed lines in Fig. 4(c) and 4(d) for some of the
resonances. Even though the barrier penetration factor of Fig. 4(a) increases
approximately stepwise with increasing energy and modifies the oscillation
amplitude of the pocket factor of Fig. 4(b), the final positions of the
oscillation maxima as shown in Fig. 4(c) remain approximately unchanged. The
maxima of the pocket factor, according to Eq.(15), occur whenever
$\cos(2\delta_{32})=-1$ (or $2\delta_{32}=(2n+1)\pi$). The subsequent maxima
of $[-{\rm Re}\\{S_{B}S^{\ast}_{I}\\}]$ are indeed located at
$\delta_{32}\approx$ 8.5$\pi$, 7.5$\pi$ and 4.5$\pi$, etc., respectively, as
indicated by horizontal lines in Fig. 4(d). The quantity $\delta_{32}$ varies
by one $\pi$ unit between resonances of the same $L$ with
$\partial\delta_{32}/\partial n\approx-\pi$. We further find that wherever the
horizontal lines intersect the $\delta_{32}$ curves of Fig. 4(d), each
intersection point approximately, and respectively, matches each maximum of
$[-{\rm Re}\\{S_{B}S^{\ast}_{I}\\}]$ in Fig. 4(c) and each maximum of
$\sigma_{\rm ann}^{\rm int}(L)$ in Fig. 3(d). This means that the condition
for an annihilation resonance at a cross-section maximum at the energy $E$ is

$\displaystyle\delta_{32}(E,L)\approx(n_{L}-n_{L}^{\prime}+\frac{1}{2})\pi,$
(16)

where $n_{L}$ is the number of bound and quasi-bound radial states lower than
$E$ for the angular momentum $L$ and $n_{L}^{\prime}$ (= 0, 1, 2, $\cdots$) is
the additional resonance number above the first resonance of the same $L$. The
above sign of $n_{L}^{\prime}$ in Eq. (16) depends on $\partial V/\partial E$
as a function of $E$.

Since $\delta_{32}(E,L)$ is the action phase angle integrating over the
confining potential pocket of $V_{\rm eff}(r,E))$, condition of Eq.(16) is
just the quantization condition for a quasi-bound state inside the pocket.
Thus, the “pocket resonance” is, in fact, a continuation of the bound states
in the continuum trapped in a pocket with a barrier as depicted in Fig. 5 of
Ford and Wheeler FW59 . The condition of Eq. (16) for the resonances suggests
that pocket resonances occur quite generally in potential scattering problems.
Their observations however depend on the degree of absorption as we shall
discuss later.

Figure 5:  Semiclassical results for $\bar{n}$Pb elastic scattering as a
function of energy. (a) is the total cross section of Eq.(17), (b) is the
partial-wave cross section of Eq.(17) for different $L$, (c) is the non-
interference cross section of Eq.(18), (d) is the interference cross section
of Eq.(19) and (e) is the Re$(\delta_{\rm WKB}+\delta_{32})$ for different
$L$.

We now examine the elastic oscillations with

$\displaystyle\sigma_{\rm el}^{\rm tot}=\Sigma_{\rm el}^{\rm BI}+\Sigma_{\rm
el}^{\rm int}=\sum\limits_{L=0}^{\infty}(\sigma^{\rm BI}_{\rm
el}(L)+\sigma^{\rm int}_{\rm el}(L)),$ (17) $\displaystyle\sigma^{\rm BI}_{\rm
el}(L)=\frac{\pi}{k^{2}}{(2L+1)(1+|S_{I}|^{2}+|S_{B}|^{2})},$ (18)
$\displaystyle\sigma^{\rm int}_{\rm el}(L)=\frac{\pi}{k^{2}}{(2L+1)(2{\rm
Re}\\{S_{I}^{*}S_{B}-S_{I}^{*}-S_{B}\\}}).$ (19)

Figure 5(a) indicates that there is a significant cancellation of $\Sigma_{\rm
el}^{\rm BI}$ by $\Sigma_{\rm el}^{\rm int}$, where $\Sigma_{\rm el}^{\rm
int}$ has an oscillating behavior, resulting in an oscillating total elastic
cross section $\sigma_{\rm el}^{\rm tot}$ as a function of energy as shown in
Fig. 5(a).

Figure 5(b) shows the decomposition of the total elastic $\sigma^{\rm
tot}_{\rm el}$ into different partial waves contributions $\sigma_{\rm
el}(L)$. Only the $L$ = 0 partial wave has three, narrow, sinusoidal
oscillations with decreasing amplitudes. The $L>$ 0 curves, however, have only
a single maximum with a similar shape. The first, broad maximum seen in
$\sigma^{\rm tot}_{\rm el}$ near 0.05 MeV closely corresponds to the first
maximum of $L$ = 0 curve at the same energy. The subsequent, second maximum of
$\sigma^{\rm tot}_{\rm el}$ that is near 0.2 MeV corresponds to the
contributions from the $L$ = 1 curve and the second maximum of $L$ = 0 curve.
Beyond the second maximum, each consecutive maximum now appears to be
associated with a specific partial wave of $L$ = 2, 3, 4, and so forth. Fig.
5(c) shows that $\sigma^{\rm BI}_{\rm el}(L)$ for different $L$ from Eq.(18)
decrease monotonically as energy increases.

Figure 5(d) is more interesting as it exhibits the oscillating $\sigma^{\rm
int}_{\rm el}(L)$ as a function of $E$ for different $L$. The oscillations are
similar to those in the annihilation case. Although Eq.(19) is more
complicated than Eq.(13) for the annihilation, we have found from our
numerical calculations that the $[-2{\rm Re}{(S_{I})}]$ term dominates in the
evaluation of $\sigma_{\rm el}^{\rm int}(L)$ over the energy range.
Furthermore, the maximum of $\sigma^{\rm int}_{\rm el}(L)$ as an “elastic
resonance” for various partial waves is located at the same energy as the
maximum of $[-{2\rm Re}{(S_{I})}]$. The phase of $S_{I}$, according to
Eq.(10), is given predominantly by $(\delta_{\rm WKB}+\delta_{32})$. Numerical
calculations of $(\delta_{\rm WKB}+\delta_{32})$ indicate that the difference
of the phases between elastic resonances is $\pi$, i.e., $\partial(\delta_{\rm
WKB}+\delta_{32})/\partial n\approx-\pi$.

Together with Fig. 5(d), the plot of 5(e) shows the condition for the elastic
resonances at the energy $E$ arising from the pocket is

$\displaystyle(\delta_{\rm
WKB}+\delta_{32})(E,L)\approx(n_{L}-n_{L}^{\prime}+\frac{1}{2})\pi,$ (20)

at which $\cos[2(\delta_{\rm WKB}+\delta_{32})(E,L)]=-1$, substantiating the
quantization condition for the elastic pocket resonance similar to that for
the annihilation pocket resonance in Eq.(16), but with the addition of
$\delta_{\rm WKB}$ to $\delta_{32}$. And because of this additional
$\delta_{\rm WKB}$ phase the location of the elastic pocket resonance energy
$E$ is slightly shifted from the annihilation pocket resonance energy for the
same $n$ and $L$.

Another interesting feature for both the annihilation and elastic resonances
is that the resonance energies shift toward lower energies with increasing
nuclear radius or mass, as is clear from Fig. 1(b) in comparing $\bar{n}$Ag
and $\bar{n}$Pb elastic cross-sections. This arises because the pocket
resonances are the continuation of the bound states in the continuum. Note
that the opposite behavior occurs where the Ramsauer resonance shifts toward
higher energies with increasing nuclear radius in the neutron total cross-
section data for Cu, Cd, and Pb Bow61  and Ho Mar70  nuclei from 2 MeV to 125
MeV. The cause of the broad resonance and its shifts to higher energies have
been explained by Peterson Pet62  and Marshak $et.~{}al.$ Mar70  using the
concept of Ramsauer’s interference Ram21 .

As the incident energy further decreases below 0.03 MeV, we see the optical
model predicted annihilation cross-sections for C, Ag, and Pb nuclei merge
into one single curve. Parametrizing these curves with $\sigma^{\bar{n}A}_{\rm
ann}(p)\propto 1/p^{\alpha_{\bar{N}}}$ in this energy region (i.e., 1.0 $\leq
p\leq$ 2.0 MeV/$c$), we found that
$\alpha_{\bar{N}}=\partial\ln(\sigma^{\bar{n}A}_{\rm ann})/\partial\ln(p)$ for
C, Ag and Pb nuclei to be $\sim$1.0, $\sim$0.9, and $\sim$0.8, respectively,
which are surely close to Bethe-Landau’s $s$-wave prediction of
$\alpha_{\bar{n}}$ = 1. On the elastic scattering at energies below 0.03 MeV,
we notice that the scattering becomes isotropic and energy-independent.
Consequently, the “$s$-wave” cross-section is purely geometrical effect and a
constant, which can be described by a black-nucleus model with $\sigma_{\rm
el}^{\rm tot}=\pi R^{2}$ Bla52 .

It is important to point out the sensitivity of the pocket resonances on the
optical model parameters $W$, $V$, and $R$ of Eq. (7). For the pocket
resonances to exist, the lifetime of the incident particle inside the
potential pocket, $\hbar/2W$, must be longer than the passage time for the
particle to traverse from the entrance to the exit in the pocket, of order
$2R/v$, where $v$ is the particle velocity in the pocket. This corresponds to
the requirement of $W$ to be approximately less than
$W_{o}\sim\hbar\sqrt{2(E+|V|)/m_{\bar{n}}}/4R$. For the $\bar{n}$Pb reaction,
$W_{o}\sim$ 2.3 MeV (estimated) for potential parameters extrapolated to the
pocket resonance energy region.

The available $\bar{n}$Pb annihilation cross-section data are consistent with
the value of $W=2.8$ MeV Lee18 , which is comparable to the above estimate of
onset value $W_{o}$. Hence, the pocket resonances are slightly diminished in
oscillatory magnitudes but remain to be present for the $\bar{n}$Pb reaction
in extrapolation, as shown in Fig. 1. Other choices of $W$ and radius
parameter would not lead to a good description of the experimental
annihilation data. The oscillatory amplitudes of pocket resonances will be
much sharper for very small values of $W$ but decrease as $W$ increases, and
the oscillatory pocket resonances will be absent if $W$ is substantially
large.

It has been known from the works of Brink and Takigawa BT77 ; Lee78  that
there is an anomalous large-angle scattering (ALAS) elastic scattering that is
sensitive to the imaginary part $W$ of the optical potential. The elastic
scattering at large angles at a pocket resonance requires the passage of the
incident particle from the entrance to the exit of the pocket and is likewise
sensitive to the degree of absorption represented by $W$. We find that as a
pocket resonance is associated with a specific angular momentum $L$ which
shows up as a dip in a particular angle in the elastic angular distribution at
large angles, the shape of the dip, and the swing of the angular distribution
at larger scattering angles is sensitively affected by the strength of $W$.
There is also the ALAS phenomenon associated with the pocket resonance as
energy is varied across the resonances energy region, similar to the other
ALAS observed in Ref.BT77 ; Lee78  .

##  III Conclusions

Optical-model calculations show oscillatory structures in $\bar{n}A$
annihilation and elastic cross-sections in the low $\bar{n}$ energy range
0.001 to 10 MeV. The cross-sections oscillate in the logarithm of the
antineutron energies with small amplitudes and narrow periods for large
nuclei. This surprising behavior contradicts the generally expected Bethe-
Landau’s power-law in the $s$-wave limit.

Semiclassical $S$-matrix analysis provides new insight into the nature and the
origin of the oscillations. Two important contributions to the structures in
the reaction cross sections are found: (1) For annihilation and elastic
scattering, the oscillations are attributed to the interference between the
internal and barrier waves, while their smooth, $1/p$, backgrounds are mainly
due to the non-interference term. (2) Delving deeper into the interference
term for both reactions, we identified the maxima of the cross-sections as
resonances occurred within the potential pocket. The condition for an
annihilation resonance is the quantization rule:
$\delta_{32}(E,L)\approx(n_{L}-n_{L}^{\prime}+1/2)\pi$, and for an elastic
resonance: $(\delta_{\rm
WKB}+\delta_{32})(E,L)\approx(n_{L}-n_{L}^{\prime}+1/2)\pi$ where $n_{L}$ is
the the number of bound and quasi-bound radial states in the pocket below $E$
for the partial wave $L$, and $n_{L}^{\prime}$ is the additional resonance
number. So the existence of resonances is connected to the potential pocket
and barrier that depends sensitively on optical potential parameters.
Experimental observations of these pocket resonances, if they occur, will
provide valuable information on the properties of the optical model potentials
and the nature of the reaction process. While here we have focused on
$\bar{n}A$ reactions, similar potential pocket resonances are also expected
for reactions involving the use of an optical potential in nuclear, atomic,
molecular, and heavy-ion collisions such as in 12C+12C reactions Tang20 ,
which is an important reactions in nuclear astrophysics.

## Acknowledgment

CYW’s research is supported in part by the Division of Nuclear Physics, U.S.
Department of Energy under Contract DE-AC05-00OR22725.

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July 14, 2022

# Numerical Study of S=1/2 Heisenberg Antiferromagnet
on the Floret Pentagonal Lattice

Rito Furuchi1 Hiroki Nakano1 and T$\hat{\rm{o}}$ru Sakai1,2 1 Graduate School
of Science1 Graduate School of Science University of Hyogo University of Hyogo
Kamigori Kamigori Hyogo 678-1297 Hyogo 678-1297 Japan
2 National Institutes for Quantum and Radiological Science and Technology
Japan
2 National Institutes for Quantum and Radiological Science and Technology
SPring-8 Sayo SPring-8 Sayo Hyogo 679-5148 Hyogo 679-5148 Japan
Japan
<EMAIL_ADDRESS>
###### Abstract

The $S=1/2$ Heisenberg antiferromagnet on the floret-pentagonal lattice with
two kinds of interaction strength is studied by the numerical-diagonalization
method. It is known that, near the five-ninth of the saturation magnetization,
this system shows a magnetization jump that is not accompanied by
magnetization plateaux. We focus our attention on the behavior of this system
around the five-ninth of the saturation magnetization; the changes of the
magnetization jump and plateau at and around this magnetization are clarified
from the diagonalization data for finite-size systems up to 45 sites.

$S=1/2$ Heisenberg antiferromagnet,The floret-pentagonal lattice,
Magnetization process, Numerical diagonalization

##  1 Introduction

Frustration in magnetic materials has attracted much attention of many
condensed-matter physicists. The frustration is typically caused by a local
structure of antiferromagnetic interactions forming a polygon with an odd
number of sides – an odd-gon – . The simplest case is the triangular
structure; magnetism on various lattices including such triangular structure,
for example, the triangular lattice and the kagome lattice, have been
extensively and intensively studied. The next possibility for the odd-gon is a
pentagon. However, a relatively much smaller number of investigations have
been carried out for the magnetism of a system on a pentagonal lattice. As
such a two-dimensional pentagonal lattice, the Cairo-pentagonal-lattice
Heisenberg antiferromagnet was studied and the system shows the characteristic
magnetization process with magnetization plateaux and jumps[1, 2, 3]. Note
here that candidate materials for the Cairo-pentagonal-lattice antiferromagnet
were studied recently[4, 5, 6, 7, 8]. As systems including local pentagonal
structure, studies concerning spherical kagome cluster[9], dodecahedral
cluster[10, 11], and icosidodecahedron cluster [12, 13] are also known.

Recently, the floret-pentagonal-lattice (FPL) antiferromagnet was
investigated[14] as the second two-dimensional case among pentagonal lattices.
The FPL Heisenberg antferromagnet shows magnetization plateaux in its
magnetization process at one-ninth, one-third, and seven-ninth of the
saturation magnetization. The magnetization plateaux are related to the number
of spin sites in each unit cell of this lattice, namely, nine. Spin sites of
the FPL are divided into two groups; a coordination number of sites on one
group is six and that on the other group is three. Let us consider
antiferromagnetic interaction bonding a site of coordination number to be six
($J_{1}$) and antiferromagnetic interaction not bonding such sites ($J_{2}$).
Reference References reported that the magnetization plateaux at one-third and
seven-ninth of the saturation magnetization get smaller and close when
$J_{2}/J_{1}$ is increased.

Let us we focus our attention on the behavior of the FPL antiferromagnet at
and near the five-ninth of the saturation magnetization. In the uniform case,
the FPL antiferromagnet does not show a magnetization plateau at this
magnetization; on the other hand, the system shows a peculiar magnetization
jump near this magnetization. Under these circumstances, the purpose of this
study is to clarify the behavior near this magnetization during the variation
of $J_{2}/J_{1}$ by the Lanczos-diagonalization method for finite-size
clusters of this system. We successfully capture the appearance the
magnetization jump near this magnetization in a specific range of
$J_{2}/J_{1}$. We also find that the magnetization plateau at five-ninth of
the saturation appears for $J_{2}/J_{1}$ that is larger than a specific value
of this ratio.

This paper is organized as follows. In the next section, the model Hamiltonian
will be introduced. The method of calculations will also be explained. The
third section is devoted to the presentation and discussion of our results
concerning the magnetization jump near the five-ninth of the saturation. In
the fourth section, we discuss the appearance of the magnetization plateau at
this magnetization. In the final section, a summary of our results and some
remarks will be given.

##  2 Model and calculation

We investigate the $S=1/2$ Heisenberg antiferromagnet on the FPL. The
Hamiltonian of the investigated model is described by ${\cal H}={\cal
H}_{0}+{\cal H}_{\mathrm{zeeman}}$ where

$\displaystyle{\cal H}_{0}$ $\displaystyle=$
$\displaystyle\sum_{\rm{solid\hskip
2.0ptbonds}}J_{1}\mbox{\boldmath$S$}_{i}\cdot\mbox{\boldmath$S$}_{j}+\sum_{\rm{dotted\hskip
2.0ptbonds}}J_{2}\mbox{\boldmath$S$}_{i}\cdot\mbox{\boldmath$S$}_{j},\ {\cal
H}_{\rm zeeman}=-h\sum_{j}S_{j}^{z}.$ (1)

Here, $\mbox{\boldmath$S$}_{i}$ denotes the $S=1/2$ spin operator at site $i$
illustrated by the closed circle at a vertex in Fig. 1(a). In the first term
of Eq. (1), the sum is taken for the bonds depicted by the solid line in Fig.
1(a), and in the second term, the sum is taken for the bonds depicted by the
dotted line. Energies are measured in units of $J_{1}$ in Eq. (1). Since we
examine the case of antiferromagnetic interaction, we put $J_{1}=1$, $J_{2}>0$
hereafter. We define $\eta=J_{2}/J_{1}$ as the ratio of these interactions. In
the case $\eta=1$, all interactions are equivalent, when the behavior of this
system was studied in Ref. References. In that case, some nontrivial phenomena
due to frustration were reported. In the present study, we investigate what
happens when $\eta$ is varied concerning the phenomena that were observed in
Ref. References.

Figure 1: Panel (a) shows the floret-pentagonal lattice and finite-size
clusters treated in this study. Solid and broken lines denote interaction with
amplitude $J_{1}$ and $J_{2}$, respectively. Violet, blue, black, red, and
green solid lines correspond to the finite-size clusters for $N=9$, $18$,
$27$, $36$, and $45$, respectively. Panel (b) shows groups of vertices:
$\alpha$, $\beta$, and $\gamma$.

Figure 1(a) also shows the shape of finite size clusters treated in the
present study. The number of spin sites is denoted as $N$. Note that the
finite-size clusters for $N=9$, $27$, and $36$ are rhombic, whereas those for
$N=18$ and $45$ are not rhombic. Although these nonrhombic clusters therefore
show symmetries that are different from the FPL, calculations of $N=18$ and
$45$ contribute to deepen our understanding of the FPL antiferromagnet. In all
the finite-size clusters, the periodic boundary condition is employed. Note
also that the magnetization process for a 45-site cluster was first reported
in the case of the kagome-lattice antiferromagnet[15] and that the present
study for the FPL antiferromagnet is the second report of a 45-site
magnetization process to the best of our knowledge.

The FPL has originally been known in the tiling problem[16]. Figure 1(b) shows
the grouping of vertices of this lattice. A unit cell of the FPL contains nine
vertices, which are divided into two groups at first. One is a group of
vertices of the type characterized by the coordination number $z=6$ and the
other group consists of vertices with $z=3$. The former vertex is called
$\alpha$ sites, and the latter is further divided into two groups: those
linked by a bond with $\alpha$ are called $\beta$ sites, and those not linked
are called $\gamma$ sites.

In this study, the ground-state energy of ${\cal H}_{0}$ is calculated in the
subspace characterized by $M$ defined by $\sum_{j}S_{j}^{z}$. The calculation
to obtain the energy by diagonalizing ${\cal H}_{0}$ is based on the Lanczos
algorithm and/or the Householder algorithm. The energy is denoted by $E(N,M)$.
The saturation value of $M$ is defined as $M_{\rm s}(=SN)$, until which $M$
increases discretely with $\delta M=1$. The magnetization process is
determined so that the magnetization increases from $M$ to $M+1$ at magnetic
field $h=E(N,M+1)-E(N,M)$ when the magnetization monotonically increases with
$h$. When the monotonic increase disappears, there appears a jump; the Maxwell
construction should be carried out to capture the behavior of the jump. We
evaluate local magnetization $m_{\rm loc}$ defined as
$(1/N_{\xi})\sum_{j\in\xi}\langle S_{j}^{z}\rangle$, where $\xi$ takes
$\alpha$, $\beta$, and $\gamma$; $\langle{\cal O}\rangle$ represents the
expectation value of an operator ${\cal O}$ with respect to the lowest-energy
state within the subspace with a fixed $M$ of interest. For simplicity, here,
we define $m=M/M_{\rm{s}}$ as the normalized magnetization. Part of Lanczos
diagonalizations has been performed using the MPI-parallelized code, which was
originally developed in the research of the Haldane gaps[17]. The usefulness
of our program was demonstrated in several large-scale parallelized
calculations[18, 19, 20, 21].

##  3 Magnetization jump

Figure 2: Panels (a) and (b) show the magnetization process and local
magnetization for $\eta=1.56$ and $1.44$, respectively. The green, red, and
black lines and dots represent results for $N=45$, $36$, and $27$ systems,
respectively. The main panel shows a magnified view around the magnetization
jump; the solid lines depict the magnetization process after the Maxwell
construction is applied. The dashed lines depict the results before the
Maxwell construction is applied, which explicitly show unrealized states. The
lower-right inset depicts the magnetization process in the entire range up to
the saturation. The upper-left inset shows the averaged local magnetization in
each system for the rhombic clusters for $N=36$ and 27. The averaging is
carried out among each of groups, $\alpha$, $\beta$, and $\gamma$ sites that
are represented by triangles, squares, and circles, respectively. Open symbols
indicate skipped states because these states do not become a ground state even
under any magnetic fields although closed symbols indicate realized states.

Recall here that in the system with $\eta=1$, a jump for the $N=36$ cluster
appears as a skipped case of $m=11/18$[14]. This jump is significantly
different from jumps that have been reported in many other frustrate systems.
Such magnetization jumps are associated with a specific magnetization plateau.
However, the jump of the $N=36$ FPL-antiferromagnetic cluster at $m=11/18$
appears away from both the plateaux at $m=1/3$ and 7/9. In this section, we
clarify what happens in the magnetization process of this model by observing
the change of this jump when $\eta$ is varied.

Now, let us consider the cases for larger $\eta$; results of magnetization
processes for $N=27$, $36$, and $45$ are depicted in Fig. 2 for $\eta=1.56$
and 1.44 in panel (a) and (b), respectively. Figure 2 presents the
magnetization processes in the entire range in the lower-right inset and a
zoom-in view near the jump in the main panel. Let us examine the change of the
magnetization process when $\eta$ is decreased. Recall before observing Fig. 2
that the $m=7/9$ plateau begins to open around $\eta\sim 1.6$[14]. For
$\eta=1.56$, there appears a magnetization jump at the lower-edge of the
$m=7/9$ plateau in Fig. 2(a). For $\eta=1.44$, next, the magnetization jump
departs from the $m=7/9$ plateau.

Next, let us observe the relationship between the change of the magnetization
jump and averaged local magnetizations $m_{\rm loc}$ in the upper-left insets
of Fig. 2. In particular, a marked behavior appears in the results of $m_{\rm
loc}$ for $\alpha$ sites. For $\eta=1.56$, the clear discontinuous behavior of
$m_{\rm loc}$ in $\alpha$-site results is observed between data at $m=7/9$ and
those for $0.6<m<0.7$. For $\eta=1.44$, on the other hand, states for $m\sim
0.6$ actually are realized as the ground states under a specific magnetic
field. The results of $m_{\rm loc}$ of $\alpha$ sites in these states and
$m_{\rm loc}$ at $m=7/9$ show a continuous behavior. At the same time, there
still exists a discontinuous behavior across the jump around $m\sim 0.6$. On
the other hand, there are no significant changes in the spin states around
$m\sim 0.5$.

From observing the behavior of this system in Fig. 2, the jump originally
begins to appear in an association with the opening of the plateau at $m=7/9$
when $\eta$ is decreased from a large-$\eta$ side. In even smaller $\eta$,
there appear new states between the jump and the $m=7/9$ plateau under the
situation that the jump still survives. Therefore, the new states play an
essential role in the formation of the magnetization jump away from any
magnetization plateaux.

Figure 3: The $\eta$-dependence of the magnetic fields before and after the
Maxwell construction ($h_{\rm{min}}$, $h_{\rm{max}}$, and $h_{\rm{MC}}$) for
the appearing target jump, where the upper-left inset of panel (a) illustrates
$h_{\rm{min}}$, $h_{\rm{max}}$, and $h_{\rm{MC}}$. Main panel (a) shows the
$\eta$-dependence of $h_{\rm{min}}/J_{1}$ by open triangles and closed squares
and $h_{\rm{max}}/J_{1}$ by open inverted triangles and closed diamonds; the
lower-right inset shows that of $h_{\rm{MC}}/J_{1}$ by closed circles and
crosses. Green and red symbols represent results for $N=45$ and $36$ systems,
respectively. Panel (b) shows the $\eta$-dependence of
$h_{\rm{min}}/h_{\rm{MC}}$ and $h_{\rm{max}}/h_{\rm{MC}}$.

To clarify the range of $\eta$ where the magnetization jump discussed in the
previous paragraph, let us present the $\eta$-dependence of magnetic fields at
the jump before and after the Maxwell construction, where the fields are
denoted by $h_{\rm{min}}$, $h_{\rm{max}}$, and $h_{\rm{MC}}$ illustrated in
the upper-left inset of Fig. 3(a); results are depicted in Fig. 3(a). When we
focus our attention on the edges of this range, there appears the jump in the
region of $\eta$ up to $\eta=1.618$ for the $N=45$ and $\eta=1.611$ for the
$N=36$. The size difference of this value of $\eta$ is quite small. On the
other hand, the jump appears in the region down to $\eta\sim 1.1$ for the
$N=45$ and $\eta=0.92$ for the $N=36$; the size dependence is relatively
larger than the other edge.

Next, let us observe the behavior inside the range in Fig. 3. One can find a
change around $\eta\sim 1.5$ concerning the moving of these fields
characterizing the jump $h_{\rm{min}}/J_{1}$, $h_{\rm{max}}/J_{1}$, and
$h_{\rm{MC}}/J_{1}$. Note markedly that there is no significant difference in
$\eta\sim 1.5$ of the change among $h_{\rm{min}}/J_{1}$, $h_{\rm{max}}/J_{1}$,
and $h_{\rm{MC}}/J_{1}$. In particular, $h_{\rm{MC}}/J_{1}$ clearly shows the
difference between the regions $\eta<1.5$ and $\eta>1.5$, namely, the gradient
in $\eta<1.5$ differs from that in $\eta>1.5$. To clarify the differenece, let
us observe $h_{\rm{max}}/h_{\rm{MC}}$ and $h_{\rm{min}}/h_{\rm{MC}}$ show in
Fig. 3(b). These behaviors observed in Fig. 3 suggest that the properties of
the magnetization jump are different between $\eta>1.5$ and $\eta<1.5$. It is
reasonable that the change of $h_{\rm{min}}$ around $\eta\sim 1.5$ comes from
the appearance of the new states between the jump and the $m=7/9$ plateau;
however, there also appears the change of $h_{\rm{max}}$ almost at the same
$\eta$, which suggests that the properties of the magnetization jump change.
It is still unclear at the present why the change of the jump appears only in
the present model; the reason should be studied in future studies.

Figure 4:  Panel (a) depicts the magnetization process for $\eta=2$ where the
green, red, and black lines represent results for $N=45$, $36$, and $27$
systems respectively. In the main panel (b), $\eta$-dependence of the width of
$m=5/9$ in the magnetization process of $N=36$ cluster is depicted. Closed and
open circles correspond to the cases when the $m=5/9$ states to be realized
and unrealized under a specific magnetic field, respectively. Namely, the
cases of the open circles indicate the appearance of the magnetization jump.
Inset of panel (b) depicts size-dependence of the $m=5/9$ widths. The red
(black) lines and symbols represent the results for $\eta=1$ ($\eta=2$).

##  4 Appearance of the $m=5/9$ plateau

Let us review the appearance of plateaux in the case of all the interactions
are equivalent, namely $\eta=1$. Reference References reported that for
$\eta=1$, no indication is detected for the plateau at $m=5/9$ although there
appear plateaux at $m=7/9$, 1/3, and 1/9.

Now, let us consider the behavior at $m=5/9$ when $\eta$ is varied. First, we
present the magnetization process for $\eta=2$; results for $N=45$, 36, and 27
are depicted in Fig. 4(a). One easily finds an existing plateau at $m=5/9$. We
define $h_{\rm L}$ ($h_{\rm R}$) as the lower-field (higher-field) edge of
$m=5/9$. In the following, let us examine the width of $m=5/9$ namely, $h_{\rm
L}-h_{\rm R}$.

In the inset of Fig. 4(b), next, the $N$-dependence of this width for $\eta=2$
together with the case of $\eta=1$. Although the width for $\eta=2$ gradually
decreases as $N$ is increased, an extrapolated value of the width to the
thermodynamic limit seems nonzero, which suggests the $m=5/9$ plateau
certainly opens. For $\eta=1$, in contrast, finite-size widths are much
smaller than those for $\eta=2$; an extrapolated value of these finite-size
results seems to vanish. These different situations from $\eta=2$ and 1
suggest that the $m=5/9$ plateau closes in a value of intermediate $\eta$ when
$\eta$ is decreased from $\eta=2$.

In order to observe the behavior, let us observe results of the width for
$N=36$ between $\eta=1$ and 2 in the main panel of Fig. 4(b). With decreasing
$\eta$, the width gradually deceases down to $\eta\sim 1.5$. At $\eta\sim
1.4$, finally, the case of $m=5/9$ encounters the magnetization jump moving
from higher $m$ and becomes skipped in the magnetization process. Below
$\eta\sim 1.2$, the case of $m=5/9$ recovers its width; however, the width
still shows a small value due to a finite-size effect. It is noticeable that
the magnetization plateau at $m=5/9$ appears for large $\eta$ whereas it
disappears for small $\eta$ and that the situation of $m=5/9$ is clearly
different from those of $m=7/9$ and 1/3 reported in Ref. References.

##  5 Summary

We have studied the $S=1/2$ Heisenberg antiferromagnet on the floret-
pentagonal lattice by using the numerical diagonalization method. The model is
controlled by the ratio of two interactions, each of which is determined by
the different coordination numbers of spin sites. Our numerical-
diagonalization calculations up to the 45-site cluster have clarified the
behavior around the five-ninth of the saturation magnetization. Further
investigations concerning the system on various pentagonal lattices will
contribute much to our understanding frustration effects in magnetic
materials.

## Acknowledgments

This work was partly supported by JSPS KAKENHI Grant Numbers 16K05419,
16H01080(J-Physics), 18H04330(J-Physics), JP20K03866, and JP20H05274.
Nonhybrid thread-parallel calculations in numerical diagonalizations were
based on TITPACK version 2 coded by H. Nishimori. In this research, we used
the computational resources of the supercomputer Fugaku provided by RIKEN
through the HPCI System Research projects (Project IDs: hp200173, hp210068,
hp210127, hp210201, and hp220043). Some of the computations were performed
using facilities of the Institute for Solid State Physics, The University of
Tokyo.

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