# Topological quantum phase transition in individual Fe Atoms on MoS2/Au(111)

G. G. Blesio  Jožef Stefan Institute, Jamova 39, SI-1000 Ljubljana, Slovenia
A. A. Aligia  Instituto de Nanociencia y Nanotecnología CNEA-CONICET, Centro
Atómico Bariloche and Instituto Balseiro, 8400 Bariloche, Argentina

###### Abstract

In a recent experiment [Trishin et al., Phys. Rev. Lett. 127, 236801 (2021)] a
rich physics was observed for Fe atoms on MoS2/Au(111), characterized by three
different behaviors depending on the spectral density of the substrate
$\rho_{c}$: one dominated by a single ion anisotropy, one by the Kondo effect,
and an intermediate one. Based on symmetry and previous works, we show that
the appropriate model to describe the system is the anisotropic two-channel
spin-1 Kondo model (A2CS1KM), which has an underlying topological quantum
phase transition (TQPT) between an ordinary Fermi liquid and a topological one
with a non-trivial value of the Luttinger integral. Solving the model with the
numerical renormalization group (NRG), we show that the different behaviors
can be explained in a unified fashion as a function of $\rho_{c}$, and
correspond to both Fermi liquids and an intermediate regime close to the TQPT
in the topological phase. Further experiments should confirm this transition.

The understanding of the basic physics of systems composed of magnetic atoms
or molecules on metallic surfaces is a subject of great interest in the past
two decades. This is motivated by possible promising technological
applications in the field of spintronics wolf , for which coherent
manipulation of the spin is essential yang1 , and also as a test for
paradigmatic theories in condensed matter, such as the Kondo effect hewson97 ;
kondo64 . In its simplest form, the Kondo effect arises when the free
electrons of a metallic host screen completely the magnetic moment of an
impurity hewson97 .

Several relevant experiments using a scanning-tunneling microscope (STM) have
studied the low-energy electronic structure of systems in which magnetic units
were deposited on noble-metal surfaces. The magnetic units were either
transition-metal atoms with incomplete $d$ shells coau ; mano ; naga ; knorr ;
wahl04 ; limot ; neel ; vita ; choiPRL ; choi , or molecules containing
transition-metal atoms (or other localized electrons) wahl ; gao07 ; roch ;
parks ; mina ; zhang ; kugel14 ; karan ; esat ; iancu ; hira ; ormaza17 ;
orma3 ; verlhac ; yang ; guo ; meng ; gao . In most of these works, several
variants of the Kondo effect are apparent. The theoretical treatments become
involved because of the need to treat accurately the Coulomb repulsion in the
incomplete $d$ shells and in many cases, several localized electrons and
conduction bands are relevant at low energies, leading to non-trivial
interference effects that modify the line-shape in the differential
conductance $G(V)=dI/dV$, where $I$ is the current and $V$ the voltage applied
to the STM tip uj ; meri ; mirages ; lin ; trimer ; serge ; frank ; morr ;
interf ; lim .

One of the simplest models when more that one $d$ orbital is involved, is the
Anderson model with just two localized orbitals coupled by Hund rules to form
a spin 1, hybridized with conduction states of the same symmetry, including
anisotropy. In the integer valence limit, the model reduces to the A2CS1KM
with hard axis anisotropy $D$. Rather surprisingly, it was found a few years
ago that these models have a topological quantum phase transition (TQPT)
between an ordinary Fermi liquid (FL) when $D$ is below a critical value
$D_{c}$ to a topological Fermi liquid (TFL) for $D>D_{c}$ blesio18 ; blesio19
. At the TQPT, the Luttinger integral jumps from 0 (trivial) to $\pi/2$
(topological) with increasing $D$ and therefore, for $D>D_{c}$, the system
cannot be adiabatically connected to a non-interacting system, breaking
Landau’s hypothesis. For this reason the TFL has previously been called “non-
Landau” FL.

For equivalent channels $D_{c}\sim 2.5T_{K}^{0}$, where $T_{K}^{0}$ is the
Kondo temperature for $D=0$, which increases with increasing
$J^{\prime}=\rho_{c}J$, where $\rho_{c}$ is the density of conduction states
and $J$ the exchange coupling. The evolution of the spectral density for
localized states $\rho(\omega)$ with $D$ for constant $T_{K}^{0}$ is shown in
Fig. 11 of Ref. blesio19,  and in Fig. 1 below for constant $D$ and changing
$J^{\prime}$. For small $D/T_{K}^{0}$, the spectral density has the usual
Kondo peak. Increasing $D/T_{K}^{0}$, near the TQPT, $\rho(\omega)$ displays a
very narrow peak mounted on a broader peak (the latter characteristic of the
two-channel spin-1/2 Kondo model blesio18 ). At the TQPT, the narrow peak
changes suddenly to a dip as a consequence of the jump of the Luttinger
integral. For large $D/T_{K}^{0}$, the spectral density has two steps at
$\omega=\pm D$, characteristic of inelastic transitions (the broadening in
Fig. 1 for large $\omega$ is an artifact of the NRG). Similar behavior has
been found in other models leo ; curtin18 ; nishi .

Figure 1: (Color online) Spectral density of localized electrons of the
A2CS1KM as a function of frequency for several values of
$r=J^{\prime}/J^{\prime}_{c}$, where $J^{\prime}_{c}$ is the value of
$J^{\prime}$ at the TQPT. $\rho_{0}$ is the value of $\rho(0)$ predicted by
the ordinary Friedel sum rule with vanishing Luttinger integral nl . Thicker
dashed lines correspond to the values of $r$ chosen for the different curves
in Fig. 2.

This theory has been shown to explain in a consistent way relevant experiments
that have been interpreted previously using alternative (and some times
implausible) explanations. For the system of an isolated iron phthalocyanine
molecule deposited on the Au(111) surface [FePc/Au(111)] a narrow dip in
$G(V)$ near $V=0$ was observed mina , mounted on a broader peak, which
broadens as the Kondo interaction is weakened hira  and is transformed into a
narrow peak with an applied magnetic field yang . These three experiments were
explained using the A2CS1KM with two inequivalent channels nl . A detailed
justification of this model for FePc/Au(111) is given in Ref. sb, .
MnPc/Au(111) also shows the transformation of the narrow dip to a narrow peak
as a function of an applied magnetic field guo . The authors proposed an
interpretation in terms of a quantum phase transition involving localized
singlet states guo . However, this requires that the singlet be below the
triplet by a few meV, while in fact the triplet is energetically favored by a
Hund’s coupling of the order of 1 eV mina .

In the system of nickelocene on Cu(100) [Nc/Cu(100)], as the STM tip is
approached to the molecule, increasing hybridization between localized and
conduction electrons (effectively increasing $T_{K}^{0}$), a transition from a
dip in $G(V)$ to a peak is observed ormaza17 ; mohr  which was discontinuous
in 2/3 of the cases. The authors have tentatively ascribed the change of
behavior to a crossover in the spin of the molecule from 1 in the tunneling
regime to 1/2 in the contact regime. However, the neglected dynamical
correlations should lead to a singlet ground state in both cases, and the
change in the molecular charge is actually insufficient to account for the
large change in the magnetization. Very recently, the different behaviors were
explained in terms of the above mentioned TQPT nc . Similar experiments were
carried out for iron porphyrin molecules on Au(111) meng ; gao . In Ref. meng,
the dip narrows in the contact regime instead of turning to a peak, what
points out to a weaker hybridization of the localized states with the
substrate in comparison to Nc/Cu(100) meng . However the transformation from a
dip to a peak is observed using a Br decorated Au(111) surface gao .

A variant of the interaction between magnetic units and metallic surfaces has
been realized introducing thin decoupling layers between both, weakening the
hybridization between localized and conduction states hein ; hir ; loth ; paul
; tri . In particular, Trishin et al. considered a system consisting of an Fe
atom on top of a monolayer of MoS2 deposited in turn on a Au(111) surface tri
. Particularly interesting in this system is the fact that MoS2 on Au(111)
forms a Moiré structure, which implies strong local variations of the density
of conduction electrons. Therefore, depending on the specific position at
which the Fe adatom is located, dramatic variations of the effective exchange
between electron localized at the Fe adatom and conduction electrons in the
rest of the system takes place. The differential conductance $G(V)$ were
recorded on nearly 40 different Fe atoms and six of them are presented in Ref.
tri, . We denote the corresponding spectra by the letters (a) to (f) used in
Fig. 2 of Ref. tri, . Five of them [(b) to (f)] are reproduced here in Fig. 2.
The spectra (a) to (d) contain a dip of variable width in $G(V)$ near $V=0$.
Spectrum (e) contains a narrow dip mounted on a broader peak, and spectrum (f)
consists of a single Kondo peak.

Taking into account that the rigorous study of the A2CS1KM is both recent and
difficult, it is understandable that the theoretical part of Ref. tri,  was
based on more traditional one-channel approaches, to explain the different
regimes [perturbation theory for (b) to (d) ternes , Lorentz peak plus Frota
dip (without justification) for (e) and Frota peak for (f)]. However, not only
more than one channel is expected (see below) but also a recent experimental
study of the same group for Mn atoms, concludes that more than one channel is
coupled to the atom tri2 . Since Mn is next to Fe in the periodic table, one
expects the same to be valid for Fe atoms.

In this work we show that all spectra can be semiquantitatively explained
using the A2CS1KM with equivalent channels, fixed $D$ and varying
$J^{\prime}=\rho_{c}J$, assumed the same for both channels. The minor
discrepancies are likely due to the dependence on the energy of the conduction
density of states $\rho_{c}$ which we neglect. Since alternative perturbative
or diagrammatic approximations fail to describe the TQPT blesio19 , we use the
NRG.

The model for the system can be justified on general physical grounds.
Density-functional-theory (DFT) calculations of Fe atoms on free- standing
MoS2 indicate that the spin state of the atoms is either $S=1$ wang  or $S=2$
chen . However, for $S=2$, one would expect a second jump in $G(V)$ at larger
$|V|$ in the regime of low $J$, which is not observed experimentally [Fig.
3(h) of Ref. tri, ]. In addition, experiment as well as the DFT calculations,
indicate that the Fe atom are located in positions with symmetry corresponding
to the point group $C_{3v}$. Therefore, the Fe $3d$ orbitals are split in one
$A_{1}$ singlet and two $E$ doublets tkro . Choosing the coordinates in such a
way that $z$ is perpendicular to the surface, the $3d$ orbital with symmetry
$3z^{2}-r^{2}$ transforms as $A_{1}$, one $E$ doublet has the form tkro

$\displaystyle|x\rangle$ $\displaystyle=$
$\displaystyle\alpha|xz\rangle+\beta|\left(x^{2}-y^{2}\right)/2\rangle,$
$\displaystyle|y\rangle$ $\displaystyle=$
$\displaystyle\alpha|yz\rangle-\beta|xy\rangle,$ (1)

and the other $E$ doublet has the same form interchanging $\alpha$ with
$-\beta$ (so that all states are orthogonal).

It is likely that the spin 1 is formed occupying the two states of an $E$
doublet. Then it is expected that the spin-orbit coupling (SOC) originates a
hard axis anisotropy $D(S_{z})^{2}$ with $D>0$ nc  and that each of the
degenerate orbitals of the doublet hybridizes with conduction states of the
same symmetry blesio18 ; blesio19 ; nc . In the integer valence limit, this
reasoning leads naturally to the A2CS1KM with the same exchange interaction
for both channels. In the other case in which one of the orbitals of the spin
1 corresponds to the $A_{1}$ symmetry, the SOC favors a particular linear
combination of the $E$ states for the other orbital sb  and the low-energy
model is also the A2CS1KM but with different exchange interactions for both
symmetries. This case corresponds to iron phthalocyanine on Au(111) nl ; sb .
For the system studied here, we find that very different exchange constants
cannot explain the experiments, and then we assume that they are equal.
Therefore, the model takes the form

$\displaystyle H_{K}$ $\displaystyle=$
$\displaystyle\sum_{k\tau\sigma}\varepsilon_{k\tau}c_{k\tau\sigma}^{\dagger}c_{k\tau\sigma}+\sum_{k\tau\sigma\sigma^{\prime}}\frac{J}{2}c_{k\tau\sigma}^{\dagger}(\vec{\sigma})_{\sigma\sigma^{\prime}}c_{k\tau^{\prime}\sigma^{\prime}}\cdot\vec{S}$
(2) $\displaystyle+DS_{z}^{2},$

where $c_{k\tau\sigma}^{\dagger}$ creates a conduction electron with point-
group symmetry $\tau$, spin $\sigma$ and remaining quantum numbers $k$. The
first term describes the substrate bands, the second the Kondo exchange with
the localized spin $\vec{S}$ with exchange couplings $J$, and the last term is
the single-ion uniaxial magnetic anisotropy.

The numerical calculations were performed with the Ljubljana code of the NRG
zitko09 ; nrglj . We assume flat conduction bands extending from $-W$ to $W$
for both symmetries. The density of these states is therefore
$\rho_{c}=1/(2W)$. We take $W=1$ eV, $D=2.7$ meV and assume that $J$ varies
depending on the Fe position. In the real system $\rho_{c}$ depends on the
position of the Fe adatom, but the relevant parameter is the adimensional
product $J^{\prime}=\rho_{c}J$. The TQPT is at $J^{\prime}_{c}\sim 0.135$. For
the sake of clarity, the different curves are shown for different values of
the ratio $r=J^{\prime}/J^{\prime}_{c}$, which is the main variable in what
follows.

We assume that the structure at low voltage $V$ in the differential
conductance $G(V)=dI/dV$, is determined by the localized and conduction
electrons of symmetry $\tau$ included in the model, and that the STM tip
senses mainly the localized $3d$ states with some admixture of conduction
states. Then, the contribution of the model to $G(V)$ at zero temperature is
(except for a factor $f$ included below) nl

$\displaystyle
G_{m}(V)=-\left[(1-q^{2})\text{Im}G_{\tau\sigma}^{d}(\omega)+2q\text{Re}G_{\tau\sigma}^{d}(\omega)\right],$
(3)

where the $G_{\tau\sigma}^{d}(\omega)$ is the Green function of localized
electrons for symmetry $\tau$ and spin $\sigma$ (which depends only on
$J^{\prime}$) and $q$ is a measure of the contribution of the conduction
states. In the experimental spectrum, there is also a linear background due to
the contribution of other states, and $G_{m}(V)$ is affected by a factor $f$
which depends on the distance of the STM tip to the system. Thus we take

$G(V)=fG_{m}(V)+A+B\omega,$ (4)

This expression contains only five parameters, but the shape of each curve
depends mainly on $r$ and to a lesser extent on $q$. The fact that both $r$
and $q$ increase in going from (b) to (f) is consistent with the reported
greater density of conduction states.

Figure 2: (Color online) Differential conductance corresponding to the
experimental curves (b) to (f) in Ref. tri,  (blue full lines) and the results
for our model (red dashed lines). Note that (a) is not included and the top
figure corresponds to (b).

In Fig. 2 we represent the experimental curves (b) to (f) obtained from Fig. 2
of Ref. tri,  and the corresponding result from our theory. We have not
included the curve (a), which is similar to a rectangular dip formed by two
step-like functions, because these steps are overbroadened in our calculations
due to technical reasons that limit the resolution of the NRG at large
energies bulla . We have not attempted to fit the data but rather we have
chosen parameters (in particular $r$) that provide a semiquantitative
agreement. The assumption of a constant density of conduction states and the
effect of other orbitals not contained in our model should affect a realistic
comparison with experiment. Our aim is to show that the same model can explain
the different behaviors observed.

Experimentally, the figures can be divided into those belonging to a regime
dominated by the anisotropy $D$ in which Kondo interactions can be treated
pertubatively [curves (a) to (c)], curve (f) where Kondo exchange $J$
dominates that can be fitted by a simple Frota line shape and an intermediate
regime [curves (d) and (e)]. For curve (f) a Frota fitting is better than our
result, mainly because the former has the freedom of locating the position of
the peak, which our theory does not have. This can be improved allowing
intermediate valence. It is known that a smaller occupancy of the localized
states shifts the Kondo peak to higher energies. A higher $J^{\prime}$ implies
a higher hybridization between localized and conduction states and this
implies a higher degree of intermediate valence. However, we want to keep the
model simple and our goal is to explain all regimes within the same footing
rather than to improve minor details for each curve. The dips obtained by our
theory for the curves (d) and (e) are deeper than the ones experimentally
observed. This might also be an effect of intermediate valence which increases
the minimum possible value of the localized spectral density nc .

The intermediate regime, in particular curve (e) with a narrow dip mounted on
a broader dip, results naturally from the A2CS1KM (see Fig. 1 for $r$ slightly
smaller than 1), but is very hard to explain with alternative theories. Before
the study of the A2CS1KM, Minamitani et al. found a similar behavior in $G(V)$
in FePc/Au(111), which was interpreted as a two-stage Kondo effect: the broad
peak would be originated by the Kondo effect for one channel ($3z^{2}-r^{2}$
symmetry), and the narrow dip as the second-stage Kondo effect for states of a
different symmetry (denoted by $\pi$) mina ; joaq . A peak indicates dominance
of localized states in the tunneling to the STM tip [low $q$ in Eq. (4)] for
$3z^{2}-r^{2}$ symmetry and dominance of the tunneling to conduction states
[large $q$ in Eq. (4)] for $\pi$ orbitals. However with this explanation one
would expect a narrowing of both features as the molecule is raised from the
surface, contrary to more recent experiments hira , which together with the
dependence on the magnetic field yang  could be explained using the A2CS1KM nl
. For Fe atoms on MoS2/Au(111), the presence of a decoupling layer weakens the
tunneling of conduction electrons to the STM tip, and therefore one expects
low $q$ (as used in this work) and that $G(V)$ reflects mostly the density of
Fe states.

To support further the relevance of the A2CS1KM to the system, it would be
interesting to perform experiments for $\rho_{c}$ slightly larger than the
corresponding value for curve (e) and still smaller than the corresponding one
for curve (f). In this interval there might be a spectrum with a narrow peak
mounted on a broader peak corresponding to $r$ slightly larger than $1$. In
any case, for the curve (e), an applied magnetic field parallel to the
anisotropy direction should turn the dip into a peak in a continuous way nl .
This is another smoking gun for the A2CS1KM or its extension to intermediate
valence, since alternative meaningful explanations do not exist.

In summary, we provide strong arguments that indicate that the appropriate
model to describe individual Fe atoms on MoS2/Au(111) is the anisotropic two-
channel spin-1 Kondo model, possibly with some degree of intermediate valence
in the regime in which the Kondo exchange dominates. The different observed
curves of differential conductance can be explained by the theory modifying
the spectral density of the substrate. The model has a topological quantum
phase transition (TQPT) for anisotropy $D_{c}\sim 2.5T_{K}^{0}$, where
$T_{K}^{0}$ is the Kondo temperature for $D=0$. We expect that this work
stimulates further experiments for conditions near the transition. A fine
tuning of the Fe positions or changes in the substrate, might render possible
to observe a sudden change from a narrow dip to a narrow peak mounted on a
broader peak signaling the TQPT. This sudden change has not been observed yet
in similar materials. In any case, applying a magnetic field in the
topological phase near the TQPT [curve (e)] should change the narrow dip into
a narrow peak confirming the validity of the model.

We thank Felix von Oppen and Katharina Franke for useful discussions. AAA is
supported by PICT 2017-2726 and PICT 2018-01546 of the ANPCyT, Argentina. GGB
is supported by Slovenian Research Agency (ARRS) under Grant no. P1-0044 and
J1-2458. Part of computation were performed on the supercomputer Vega at the
Institute of Information Science (IZUM) in Maribor, Slovenia.

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# Unidirectional Microwave Transduction with Chirality Selected Short-
Wavelength Magnon Excitations

Yi Li<EMAIL_ADDRESS>Materials Science Division, Argonne National Laboratory,
Argonne, IL 60439, USA  Tzu-Hsiang Lo  Department of Materials Science and
Engineering, UIUC, Urbana, IL 61820, USA  Jinho Lim  Department of Materials
Science and Engineering, UIUC, Urbana, IL 61820, USA  John E. Pearson
Materials Science Division, Argonne National Laboratory, Argonne, IL 60439,
USA  Ralu Divan  Center for Nanoscale Materials, Argonne National Laboratory,
Argonne, IL 60439, USA  Wei Zhang  Department of Physics and Astronomy,
University of North Carolina, Chapel Hill, NC 27599, USA  Ulrich Welp
Materials Science Division, Argonne National Laboratory, Argonne, IL 60439,
USA  Wai-Kwong Kwok  Materials Science Division, Argonne National Laboratory,
Argonne, IL 60439, USA  Axel Hoffmann<EMAIL_ADDRESS>Department of
Materials Science and Engineering, UIUC, Urbana, IL 61820, USA  Valentine
Novosad<EMAIL_ADDRESS>Materials Science Division, Argonne National
Laboratory, Argonne, IL 60439, USA

###### Abstract

Nonreciprocal magnon propagation has recently become a highly potential
approach of developing chip-embedded microwave isolators for advanced
information processing. However, it is challenging to achieve large
nonreciprocity in miniaturized magnetic thin-film devices because of the
difficulty of distinguishing propagating surface spin waves along the opposite
directions when the film thickness is small. In this work, we experimentally
realize unidirectional microwave transduction with sub-micron-wavelength
propagating magnons in a yttrium iron garnet (YIG) thin film delay line. We
achieve a non-decaying isolation of 30 dB with a broad field-tunable band-pass
frequency range up to 14 GHz. The large isolation is due to the selection of
chiral magnetostatic surface spin waves with the Oersted field generated from
the coplanar waveguide antenna. Increasing the geometry ratio between the
antenna width and YIG thickness drastically reduces the nonreciprocity and
introduces additional magnon transmission bands. Our results pave the way for
on-chip microwave isolation and tunable delay line with short-wavelength
magnonic excitations.

Recent advances in information technologies such as quantum information Blais
_et al._ (2021), microelectronics Razavi (2013) and 5G/6G networks Al-Falahy
and Alani (2017) call for disruptive innovations in microwave signal
processing. In particular, circuit-integrated microwave isolators are highly
desirable for many applications as they filter unwanted microwave backflow. In
quantum information, filtering environmental noise from the output ports is
essential for protecting quantum states and entanglement Suter and Álvarez
(2016). Being able to embed isolators on chip will significantly reduce the
device volume compared to currently used bulk ferrite-based isolators, and
enable non-Hermitian engineering of dynamic systems at a circuit level El-
Ganainy  _et al._ (2017); Hurst and Flebus .

Magnonics offers opportunities for implementing unidirectional microwave
transduction with miniaturized geometry Harris  _et al._ (2009); Serga  _et
al._ (2010); Barman and et al. (2021). Because of their special dispersion
relations Herring and Kittel (1951), magnons support short-wavelength
excitations down to nanometer scale at microwave bandwidth Liu  _et al._
(2018); Shiota  _et al._ (2020); Heinz  _et al._ (2020); Baumgaertl  _et al._
(2020); Wang  _et al._ (2021); Che  _et al._ (2022) along with superior
frequency tunability with an external magnetic field. In addition, magnons
exhibit nonreciprocity based on many unique properties of magnetic
excitations, including intrinsic chirality selection in propagating
magnetostatic surface spin waves (MSSW) Khalili Amiri  _et al._ (2007);
Schneider  _et al._ (2008); Sekiguchi  _et al._ (2010); Jamali  _et al._
(2013); Wong  _et al._ (2014); Deorani  _et al._ (2014); Nakayama  _et al._
(2015); Shibata  _et al._ (2018), wavevector-dependent spin wave dispersion
shifting Moon  _et al._ (2013); T. Nembach  _et al._ (2015); Lan  _et al._
(2015); Lee  _et al._ (2016); Brächer  _et al._ (2017); Gallardo  _et al._
(2019); Verba  _et al._ (2019); Chen  _et al._ (2019); Wang  _et al._ (2020);
Ishibashi  _et al._ (2020); Han  _et al._ (2021), and non-Hermitian circuit
engineering Wang  _et al._ (2019); Zhang  _et al._ (2020). This allows for
compact integration of miniaturized, broad-band and highly tunable isolators
in microwave circuits. Furthermore, the realization of miniaturized magnonic
isolator is also important for spin wave computing, where magnons are used to
carry, transport and process information Chumak  _et al._ (2015); Mahmoud  _et
al._ (2020). Unidirectional flow of magnon information enables the isolation
of input and output and ensures deterministic logic output Sato  _et al._
(2013); Talmelli  _et al._ (2020).

Figure 1: (a) Optical microscope image of the YIG delay line with
$t_{\text{YIG}}=100$ nm, $w=200$ nm and $d=10$ $\mu$m. (b) Nonreciprocal
microwave transmission spectra of the device shown in (a) at
$\mu_{0}H_{B}=0.22$ T, measured by a vector network analyzer. (c-d) Repeated
measurements of (b) at different magnetic fields, showing a broad band
nonreciprocity with a constant isolation of 30 dB.

One major challenge of chip-integrated magnonic microwave isolators is the
degradation of nonreciprocity in magnetic thin-film devices. As the film
thickness is decreased down to nanometer levels, the $+k$ and $-k$ MSSWs
modes, which are localized at the top and bottom surfaces of the film, will
permeate to the entire thickness and be both excited by antennas at the top
surface, leading to suppressed isolation that are typically less than 10 dB.
Other approaches of realizing large magnon nonreciprocity usually suffer from
the narrow bandwidth Wang  _et al._ (2019); Xu  _et al._ (2020); Shah  _et
al._ (2020). Alternatively, spin wave nonreciprocity can be achieved by
chirality selection from well-designed microwave antennas Deorani  _et al._
(2014); Au  _et al._ (2012); Kruglyak (2021), where $+k$ and $-k$ MSSWs
exhibit clockwise and counter-clockwise mode profiles depending on the
orientation of the magnetization vector. This technique is not restricted by
the film thickness and thus can be applied for highly efficient spin wave
isolation with proper geometric design.

Figure 2: (a-d) Nonreciprocal microwave transmission spectra with different
geometries: (a) $w=200$ nm, $t_{\text{YIG}}=100$ nm; (b) $w=500$ nm,
$t_{\text{YIG}}=100$ nm; (c) $w=200$ nm, $t_{\text{YIG}}=200$ nm; (d) $w=500$
nm, $t_{\text{YIG}}=200$ nm. The biasing fields are all $\mu_{0}H_{B}=0.22$ T
for comparison. (e) OOMMF simulations of the top CPW Oersted field chirality
and the $\pm k$ MSSW mode chirality. (f) Oersted field distribution for top:
$w/t_{\text{YIG}}=0.5$ and bottom: $w/t_{\text{YIG}}=5$. (g) Summary of
experimental magnon isolations as a function of aspect ratio
$w/t_{\text{YIG}}$, along with MuMax simulation results.

In this work, we obtain unidirectional magnon excitations with over 30 dB
isolation in yttrium iron garnet (YIG) thin film delay lines with thickness
down to 100 nm. We find the leading mechanism of nonreciprocity to be the
Oersted field chirality generated from the coplanar waveguide (CPW) antenna,
which shows a sinc-function-like spatial profile and selectively couples to
the MSSW mode propagating unidirectionally. The isolation quickly decreases
from 35 dB to 10 dB as the ratio of the antenna width to the YIG film
thickness increases from 1 to 5, which is due to the development of sharp
kinks for the Oersted field at the electrode edges. In addition, we also
utilize the time-domain functionality of the measurement system to obtain the
delay time, group velocity and the nature of the additional microwave
transduction bands as the antenna and film geometries deviate from the optimal
condition. The demonstration and physical understanding of high-isolation,
high-tunability and thin-film-based compact microwave isolator are critical
for extending microwave engineering with magnonic devices, and bring new
potential for on-chip noise reduction in quantum information applications.

The structure of the delay line is shown in Figure 1(a). A pair of
nanofabricated ground-signal-ground (GSG) CPW antennas are patterned on top of
a YIG thin-film stripe with a thickness of $t_{\text{YIG}}=100$ or 200 nm. The
three electrodes of the GSG antennas and the gaps between them have an equal
width of $w=200$ or $500$ nm and a length of 30 $\mu$m, leading to the
excitation of spin waves with a wavelength around $\lambda=4w$ Sushruth  _et
al._ (2020). The separation of the two CPWs is $d=5$, 10 or 20 $\mu$m. The
external magnetic field is applied parallel to the two CPWs in order to define
the MSSW propagation modes between the two antennas.

Figure 1(b) shows the microwave transmission spectra for $t_{\text{YIG}}=100$
nm, $w=200$ nm and $d=10$ $\mu$m at $\mu_{0}H_{B}=0.22$ T, which are measured
with a vector network analyzer (VNA). For magnons that propagate from left to
right ($S_{21}$) as shown in Fig. 1(a), a single microwave transmission band
is observed around 9 GHz and stands well above the microwave background. By
reversing the measurement direction from right to left ($S_{12}$), the
transmission band disappears. The $S_{21}$ band corresponds to the MSSW modes
with the wavevector $k$ orthogonal to the magnetization $M$. By flipping the
biasing field, the magnon transmission direction is also reversed sup . With
different external biasing fields, the transmission band can be tuned
continuously along the frequency axis, as shown in Fig. 1(c). A consistent
isolation of 30 dB and a 3-dB bandwidth of 0.15 GHz are measured with
negligible change in a broad frequency band from 8 to 15 GHz.

To further explore the role of geometric structure in nonreciprocity, we
compare the transmission spectra for a few different antenna geometries, with
the results shown in Figs. 2(a-d). Experimentally we see that by increasing
$w$ from 200 nm to 500 nm, there is a major increase of the isolated band
($S_{12}$) from below the noise background to being clearly visible, leading
to a suppressed nonreciprocity of 15 dB in Fig. 2(b) and 10 dB in Fig. 2(d).
In addition, a new side band appears at a higher frequency which is marked as
mode 2. This corresponds to the second harmonic of the main MSSW mode.

In order to understand such a large magnon nonreciprocity, we calculate the
Oersted field distribution of the GSG antenna as well as the $+k$ and $-k$
MSSW magnon profiles with the experimental geometry. As shown in Fig. 2(e),
the Oersted field and the $+k$ mode share the same counter-clockwise spatial
chirality, whereas the $-k$ mode exhibit a clockwise chirality Lee  _et al._
(2016). This means the GSG antenna selectively excites the $+k$ mode and is
decoupled from the $-k$ mode, leading to chirality induced nonreciprocal
magnon excitations. Furthermore, by increasing the aspect ratio between the
antenna width and YIG thickness, $w/t_{\text{YIG}}$, from 0.5 to 5, the
Oersted field profile changes from a sinc-like smooth profile to a profile
with multiple sharp kinks at the edges of the antenna electrodes [Fig. 2(f)],
leading to a finite overlap with the $-k$ mode and a reduction of
nonreciprocity. This effect also increases the efficiency of the second
harmonic excitation; see the Supplemental Information for more details sup .

We also conduct numerical simulations of the magnon isolations using Mumax in
order to compare with experiments. Interestingly, we find a large discrepancy
between experiments and simulations, as plotted in Fig. 2(g). The simulations
show a slow reduction in isolation from 24 dB to 18 dB as the aspect ratio
$w/t_{\text{YIG}}$ increases from 0.5 to 5. However, the experimental results
show a much faster reduction in isolation from 36 dB to 10 dB for
$w/t_{\text{YIG}}$ increasing from 1 to 5. We attribute the disagreement to
the different simulation conditions from the experiments as we used a single
pixel through the entire thickness to speed up the calculation. This may lead
to omission of subtle magnetic profile distributions when the CPW antenna
width is comparable to the film thickness. In most previous studies on CPW
geometry Deorani  _et al._ (2014); Nakayama  _et al._ (2015); Shibata  _et
al._ (2018), the antenna widths are usually much larger than the film
thickness and the chirality selection of the Oersted field has been
suppressed. We also note that the skin effect can be ruled out because we do
not observe strong frequency dependence of the isolation. The skin depth of
gold at 10 GHz is about 785 nm ski , which is also much larger than the
antenna widths of this work so evenly distributed microwave current in the
antenna should be a valid assumption.

Figure 3: (a) Comparision of microwave transmission spectra with $w=200$ nm,
$t_{\text{YIG}}=100$ nm, for three different antenna separations $d$. (b) IFFT
of (a) with a transformation window from 3 to 15 GHz. The biasing fields for
(a) and (b) are set as $\mu_{0}H_{B}=0.27$ T. (c) Maximal transmission
amplitude $S_{21,\text{peak}}$ as a function of the peak position
$\omega_{\text{peak}}$. (d) 3 dB frequency linewidth $\Delta\omega_{3dB}$ as a
function of $\omega_{\text{peak}}$. (c) and (d) are extracted from the
measurements of (a) at different fields. (e) Group velocity obtained from the
magnon time delay, and plotted as a function of $\omega_{\text{peak}}$. The
stars are the group velocity obtained from the phase changing rate in the
frequency spectra measured at $\mu_{0}H_{B}=0.22$ T, same as in Fig. 1(b). (f)
3 dB linewidth of the IFFT peak as a function of $\omega_{\text{peak}}$. (e)
and (f) are extracted from the measurements of (b) at different fields.

To examine the performance of the nonreciprocal magnonic delay line in the
time domain, we also perform time-domain analysis using inverse Fast Fourier
Transformation (IFFT) function of the VNA. IFFT simulates the input of a sinc
function pulse holding an equal weight of all the frequency components set by
the frequency window. As a demonstration, we focus on the delay lines with
$w=200$ nm and $t_{\text{YIG}}=100$ nm. Figs. 3(a) and (b) show the frequency
and time domain measurements of the major transmission band ($S_{21}$) at
different antenna separations ($d=5$, 10 and 20 $\mu$m). The frequency window
for the IFFT analysis of Fig. 3(b) is set to 3 to 15 GHz in order to capture
the transmission of the entire frequency band. In the frequency domain, nearly
identical transmission bands are measured at different $d$, with only a
decreasing transmission amplitude due to the magnon decay with additional
propagating distance. In the time domain, the magnon transmission peaks are
followed by an initial microwave pulse at 20 ns which is due to the direct
microwave radiation between the two antennas serving as the microwave
background in 3(a). The time delay of the magnon signal from the microwave
radiation pulse scales linearly as a function of $d$, yielding a time delay of
96 ns at $d=20$ $\mu$m or a magnon group velocity of $v_{g}=208$ m/s at
$\omega_{\text{peak}}/2\pi=10$ GHz. The values of the group velocities are
also confirmed from the $S_{21}$ phase changing rate in the frequency domain,
as marked by the stars in Fig. 3(e); see the Supplemental Materials for
details sup . No magnon signals are measured for the opposite microwave
transmission direction ($S_{12}$).

Figure 4: Decomposition of magnonic mode traces with different IFFT frequency
windows. (a-d) are converted from IFFT of the frequency domain spectra of Fig.
2(a-d), respectively, with $\mu_{0}H_{B}=0.22$ T. (a) Mode 1 is obtained with
a frequency window of 8.5-9.5 GHz. (b) Modes 1 and 2 are obtained from
frequency ranges as 1: 8.45 to 8.85 GHz, and 2: 8.85 to 9.2 GHz. (c) Modes 1
and 3 are obtained from frequency ranges as 1: 9.0 to 10 GHz, and 3: 8.5 to
9.0 GHz. (d) Modes 1, 2, 3 are obtained from frequency ranges as 1: 8.88 to
9.1 GHz, 2: 9.1 to 9.6 GHz, and 3: 8.5 to 8.88 GHz. The amplitudes are
measured in voltage ($V_{21}$).

The field dependence of the extracted magnon transmission parameters are
plotted as a function of peak transmission frequency ($\omega_{\text{peak}}$)
in Figs. 3(c-f). For the frequency domain, all the three devices show nearly
frequency-independent peak transmission amplitudes ($S_{21,\text{peak}}$) and
linewidth ($\Delta\omega_{3dB}$) from 8 to 14 GHz. The $d$-dependence of
$S_{21,\text{peak}}$ yields a magnon decay rate of 0.5 dB/$\mu$m or an
exponential decay length of 8.7 $\mu$m. The value of $\Delta\omega_{3dB}/2\pi$
is around 0.12 GHz. Note that it is a function of the GSG antenna geometry and
is independent of frequency. For the time domain, the extracted group
velocities from time delays are consistent for the three devices and show a
slow decrease with frequency, which is the characteristic of MSSW modes.
Comparing with analytical calculations, the group velocity values fall between
the dipolar and exchange regimes sup , which is in accordance with the range
of sub-micron magnon wavelengths.

IFFT analysis also allow us to access the magnon pulse width ($\Delta
t_{\text{3dB}}$), which are in the range of 4-6 ns for the three devices as
shown in Fig. 3(f). Because the simulated sinc pulse has a much smaller width
($<0.1$ ns for a frequency window between 3 and 15 GHz), the measured values
of $\Delta t_{\text{3dB}}$ pose a fundamental limit of the pulse width of the
delay line structure. The main source of broadening is the finite spatial
width of the antenna for the magnon pulse to pass by. If we take the width of
the entire antenna as $4w=800$ nm and the group velocity $v_{g}$ as 200 m/s,
the total traveling time of the magnon pulse across the antenna is
$4w/v_{g}=4$ ns, which is close to the lower bound of $\Delta t_{\text{3dB}}$
in Fig. 3(f). The increase of $\Delta t_{\text{3dB}}$ with $d$ is due to the
distribution of $v_{g}$ within the finite bandwidth, $\Delta\omega_{3dB}$,
leading to the expansion of the magnon wave package as it propagates. As an
estimate, the additional magnon pulse broadening $\Delta t_{k}$ can be
expressed as $\Delta t_{k}=(\Delta\omega_{\text{3dB}}/\omega)t$; see the
Supplemental Materials for detailed discussion sup . From Fig. 3(d),
$\Delta\omega_{\text{3dB}}/\omega\approx 1$%. This yields $\Delta t_{k}\approx
1$ ns by changing $d$ from 5 nm to 20 nm, which agrees with the change of
$\Delta t_{\text{3dB}}$ in Fig. 3(f).

Finally, we use IFFT analysis to identify the nature of additional bands for
different antenna geometries in Figs. 2(a-d), which are marked as modes 1, 2
and 3. Fig. 4 shows the time-domain profiles of different modes for each
geometry. The frequency windows are limited to the edges of each magnon band
in order to filter out the contributions from other bands. Mode 1 is known to
be the main transmission band. For mode 2 which appears in Figs. 4(b) and (d),
the peak position has a longer delay compared with mode 1. This supports that
mode 2 is the second harmonic because for MSSW modes the $\omega$-$k$
dispersion curve softens at higher $k$ and the group velocity becomes smaller,
resulting in longer travel time of propagating magnons. Mode 3 which appears
in Fig. 4(c) and (d) exhibits a long time span of more than 50 ns. From the
$S_{21}$ phase changing rate as discussed in the Supplemental Materials sup ,
we determine mode 3 to be the obliquely launched backward-volume magnetostatic
spin waves (BVMSWs) with canted wavevector due to spin wave diffraction
Stigloher  _et al._ (2016); Vlaminck  _et al._ . From the phase analysis for
Fig. 2(d), the high-frequency edge of mode 3 exhibits a near-zero group
velocity due to the flat $\omega$-$k$ dispersion at a certain canted angle,
causing the magnon pulse to take long time to be transmitted Chen  _et al._
(1994); Demokritov  _et al._ (2001); Chumak  _et al._ (2009). We conclude from
the time domain analysis that purer magnon modes arise from narrower GSG
antenna widths, which is desirable for minimizing loss and decoherence during
the pulse microwave processing with the nonreciprocal magnon delay line.

In summary, we present a systematic study of YIG-thin-film magnon delay lines
with broad-band isolation above 30 dB. We identify the source of
nonreciprocity as the selective coupling of the chiral Oersted field from the
GSG antenna to the propagating MSSW modes in only one direction. Using time-
domain IFFT analysis, we determine important parameters of the delay line,
including time delay, group velocity, bandwidth and time domain broadening. We
also identify the nature of the additional magnon transmission bands as the
second harmonic and the canted BVMSWs band, which can be suppressed by
limiting the aspect ratio of the antenna and the YIG thickness. Our results
show a new promise of nonreciprocal magnonic delay lines for processing pulse
microwave signals with excellent noise isolation, which may be implemented as
chip-embedded microwave isolators for spin wave computing and quantum
information processing. We note that our demonstration of nonreciprocal spin
wave propagation can be also combined with the recent work of spin-torque spin
wave amplification Breitbach  _et al._ for implementing unidirectional
magnonic microwave nano-amplifiers.

Methods. The devices were fabricated using YIG thin films grown on Gd3Ga5O12
(GGG) substrate by liquid phase epitaxy (LPE), which are commercially
available Mat . In the first step, chromium hard masks were lithographically
patterned on YIG films and used for etching the YIG delay lines. Ar+ ion
milling was used for efficiently etching the YIG layer. The Cr masks can be
removed by Cr wet etch without damaging the YIG devices. The edges of the two
ends are 45 degree away from the long sides in order to minimize magnon
reflection. The width of the YIG stripe is 56 $\mu$m and the lengths of the
CPW antennas are 30 $\mu$m. In the second step, nano-CPWs were fabricated on
top of the YIG devices using e-beam lithography, with Ti(5 nm)/Au(50 nm)
electrodes grown by e-beam evaporation to minimize side walls. In the third
step, the extended CPWs were fabricated using photolithography. The electrode
thicknesses are Ti(5 nm)/Au(150 nm) for $t_{\text{YIG}}=100$ nm and Ti(5
nm)/Au(250 nm) for $t_{\text{YIG}}=200$ nm in order to minimize the impedance
mismatch across the side walls of the YIG devices.

Acknowledgement. The work was supported by the U.S. DOE, Office of Science,
Basic Energy Sciences, Materials Sciences and Engineering Division, with parts
of the manuscript preparation, first-principle calculations, device design,
and sample fabrication and characterization supported under contract No. DE-
SC0022060. U. W, W.-K. K. and V. N. acknowledge support by the U.S. DOE,
Office of Science, Basic Energy Sciences, Materials Sciences and Engineering
Division. We acknowledge Jiang-Chao Qian, Zhihao Jiang, Andre Schleife,
Wolfgang Pfaff and Jian-Min Zuo from UIUC for their helpful discussions.

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# The quantum mass gap of extremal black holes

Shahar Hod  The Ruppin Academic Center, Emeq Hefer 40250, Israel  The Hadassah
Institute, Jerusalem 91010, Israel

###### Abstract

Using sophisticated string theory calculations, Maldacena and Susskind have
intriguingly shown that near-extremal black holes are characterized by a
finite mass gap above the corresponding zero-temperature (extremal) black-hole
configuration. In the present compact paper we explicitly prove that the
minimum energy gap ${\cal E}_{\text{gap}}=\hbar^{2}/M^{3}$, which
characterizes the mass spectra of near-extremal charged Reissner-Nordström
black holes, can be inferred from a simple semi-classical analysis.

##  I Introduction

Extremal black holes are of fundamental importance in candidate theories of
quantum gravity since they mark the boundary between black-hole configurations
and horizonless naked singularities Cohc ; Hisc ; Clarc ; BekRosc ; Bradc ;
Hub ; Hod1c ; Hod2c ; Hod3c . These unique solutions of the Einstein field
equations, which are characterized by the minimally allowed mass (radius) for
a given amount of black-hole charge or angular momentum Noteunits , have
attracted much attention from physicists and mathematicians during the last
five decades.

In particular, extremal black holes play a key role in various types of
gedanken experiments that have been designed to challenge the Penrose cosmic
censorship conjecture Pen ; HawPen ; Cohc ; Hisc ; Clarc ; BekRosc ; Bradc ;
Hub ; Hod1c ; Hod2c ; Hod3c . Likewise, extremal black holes play a
fundamental role in various attempts to prove the validity of the intriguing
weak gravity conjecture Arkwk ; Chewk ; Kamwk ; Browk ; Kapwk ; Hebwk ; Koowk
; Klawk ; Hodwk . In addition, the thermodynamic and statistical properties of
extremal black holes have been extensively studied by many physicists in order
to gain some insights on the origin of the Bekenstein-Hawking entropy Strob ;
Calb ; Horb ; Hawnull ; Teitnull ; Ghonull ; Hodnull .

One of the most intriguing predictions of string theory is that near-extremal
black holes are characterized by a finite (non-zero) energy gap of the first
excited state above the extremal (zero-temperature) configuration Gap1 . In
particular, it has been proved Gap2  that charged Reissner-Nordström (RN)
black holes are characterized by the finite mass gap Noteq0 ; Notemq

${\cal E}_{\text{gap}}\equiv M-M_{\text{extremal}}={{\hbar^{2}}\over{M^{3}}}\
\ \ \ ;\ \ \ \ M_{\text{extremal}}=Q\ $ (1)

of the first excited state above the extremal $M_{\text{extremal}}=Q$ black-
hole configuration.

It is important to emphasize that the authors of the physically interesting
work Gap2  have derived the suggested energy gap (1) using a physical argument
which is based on holography without using explicit string theory
calculations. In the present paper we shall argue that an energy gap of the
form (1) arises naturally in a gedanken experiment that involves neither
string theory calculations nor holography.

The main goal of the present compact paper is to gain some physical insights
on the origins of the physically intriguing mass gap (1). In particular, below
we shall demonstrate that the interesting mass gap (1) can be inferred from a
simple semi-classical gedanken experiment [we would like to stress the fact
that we do not rule out the possible existence of an energy gap for extremal
black holes which may be smaller than the suggested gap (1)]. To this end, we
shall analyze the physical and mathematical properties of composed extremal-
RN-black-hole-massive-shell configurations.

##  II Description of the system

We consider a physical system which is composed of a charged RN black hole of
mass $M$ and electric charge $Q$ and a concentric spherically symmetric
massive neutral shell of radius $R$ and energy (energy as measured by
asymptotic observers) $E(R)$. According to the Birkhoff theorem, the spacetime
inside the spherically symmetric shell is described by the curved RN line
element Whb ; Notesch

$ds^{2}=-\Big{(}1-{{2M}\over{r}}+{{Q^{2}}\over{r^{2}}}\Big{)}dt^{2}+\Big{(}1-{{2M}\over{r}}+{{Q^{2}}\over{r^{2}}}\Big{)}^{-1}dr^{2}+r^{2}(d\theta^{2}+\sin^{2}\theta
d\phi^{2})\ .$ (2)

In addition, according to the Birkhoff theorem, the curved spacetime outside
the spherically symmetric shell of radius $R$ is described by the curved RN
line element

$ds^{2}=-\Big{\\{}1-{{2[M+E(R)]}\over{r}}+{{Q^{2}}\over{r^{2}}}\Big{\\}}dt^{2}+\Big{\\{}1-{{2[M+E(R)]}\over{r}}+{{Q^{2}}\over{r^{2}}}\Big{\\}}^{-1}dr^{2}+r^{2}(d\theta^{2}+\sin^{2}\theta
d\phi^{2})\ .$ (3)

The (outer and inner) horizon radii of the central charged black hole are
characterized by the simple relation

$r_{\pm}=M\pm(M^{2}-Q^{2})^{1/2}\ .$ (4)

The Bekenstein-Hawking temperature of the black hole is given by the compact
expression Tem1 ; Tem2

$T_{\text{BH}}={{(r_{+}-r_{-})\hbar}\over{2r^{2}_{+}}}\ .$ (5)

In the present paper we shall focus our attention on extremal black holes.
These zero-temperature ($T_{\text{BH}}=0$) charged black-hole configurations,
which mark the boundary between black-hole spacetimes and horizonless naked
singularities, are characterized by the simple relation

$Q=M\ .$ (6)

As explicitly shown in Hub , the energy $E(R)$ of the massive shell as
measured by far away asymptotic observers can be deduced by solving the
equation of motion

$\sqrt{g_{\text{in}}(r)+\dot{R}^{2}}-\sqrt{g_{\text{out}}(r)+\dot{R}^{2}}=-{{m}\over{r}}\
$ (7)

of the shell in the curved black-hole spacetime, where [see Eqs. (2) and (3)
with $Q=M$]

$g_{\text{in}}(r)=\Big{(}1-{{M}\over{r}}\Big{)}^{2}\ ,$ (8)
$g_{\text{out}}(r)=1-{{2[M+E(R)]}\over{r}}+{{M^{2}}\over{r^{2}}}\ ,$ (9)

and $\dot{R}\equiv dR/d\tau$ Notetau .

##  III The minimal mass gap ${\cal E}_{\text{gap}}\equiv M-Q$ of near-
extremal black holes

In the present section we shall use analytical techniques in order to provide
supporting evidence for the existence of a finite mass gap,

${\cal E}_{\text{gap}}\equiv M-Q\ ,$ (10)

for a near-extremal RN black hole above the extremal limit (6). To this end,
we shall analyze a gedanken experiment in which a shell of proper mass $m$,
which is concentric with an extremal charged RN black hole, is lowered towards
the central black hole. At some point, the shell would merge with the original
extremal black hole to form a larger black hole [see Eq. (14) below], thus
increasing the mass (energy) of the central black hole. Below we shall try to
minimize the energy increase (10).

In order to determine the minimal mass gap of near-extremal black holes above
the extremal limit (6), one should try to deliver to the central extremal
black hole the smallest possible (non-zero) amount of energy. We shall
therefore lower the massive shell slowly towards the central black hole with
an infinitesimally small radial velocity

$\dot{R}\to 0\ .$ (11)

As we shall explicitly show below, most (but not all) of the mass-energy of
the shell as measured by asymptotic observers would be red-shifted by the
gravitational field of the central black hole during this adiabatic lowering
process. It is important to emphasize that the physical apparatus which is
required in order to perform the adiabatic lowering process of the shell
towards the black hole would probably need to withstand extremely high
tensions Bekchal . Thus, it should be realized that the present gedanken
experiment is a highly idealized one and technical problems may prevent the
actual performance of the experiment.

Taking cognizance of Eqs. (7), (8), (9), and (11), one finds that the total
energy (as measured by far away asymptotic observers) of the massive shell in
the curved black-hole spacetime is given by the simple functional expression

$E(R)=m\cdot\Big{(}1-{{M}\over{R}}\Big{)}-{{m^{2}}\over{2R}}\ ,$ (12)

where $R$ is the radius of the shell. The first term on the r.h.s of (12)
represents the red-shifted mass-energy of the shell in the curved black-hole
spacetime. The second term on the r.h.s of (12) represents the gravitational
self-energy (an energy term proportional to $m^{2}$) of the massive shell.

It is worth noting that if the massive shell could be lowered slowly all the
way down to the horizon ($r_{\text{H}}=M$) of the central extremal RN black
hole, then the mass term $m\cdot(1-{{M}/{R}})$ in (12) would be completely
red-shifted in this process [see, in particular, the energy expression (12)
with $R=r_{\text{H}}=M$]. However, as we shall now prove explicitly, the
adiabatic lowering process of the massive shell towards the central black hole
must stop in the regime $R>M$ before the shell reaches the original horizon
$r_{\text{H}}=M$ of the central extremal black hole.

In particular, it is easy to verify that the lowering process of the shell
towards the central black hole produces a new (and larger) horizon, which is
characterized by the relation $r_{\text{NH}}>M$, before the massive shell
reaches the radius $r_{\text{H}}=M$ of the original black-hole horizon. Taking
cognizance of the curved line element (3), one finds that the physical
condition for the formation of a new engulfing horizon in the composed black-
hole-massive-shell system is given by the characteristic functional expression

$1-{{2[M+E(R)]}\over{R}}+{{Q^{2}}\over{R^{2}}}=0\ \ \ \ \text{with}\ \ \ \
Q=M\ .$ (13)

Substituting the $R$-dependent energy (12) of the shell into the
characteristic black-hole relation (13), one deduces that a new and larger
horizon, which engulfs both the original central black hole and the massive
shell, is formed in the composed spherically symmetric black-hole-massive-
shell system when the radius of the shell reaches the critical value

$R\to r_{\text{NH}}\equiv M+m\ .$ (14)

The radius (14) of the new horizon in the composed black-hole-shell system is
larger than the original radius $r_{\text{H}}=M$ of the central extremal black
hole.

Before proceeding, it is worth stressing the fact that the analytically
derived expression (14) provides the minimum possible radius of the new
engulfing horizon that results when a shell of mass $m$ is added to the
central extremal RN black hole. In particular, the total energy of a massive
shell which has a non-vanishing kinetic energy (a non-vanishing radial
velocity) is larger than the one given by the energy expression (12). Hence, a
massive shell which falls towards the central extremal black hole with a non-
vanishing radial velocity would form an engulfing new horizon which is larger
than the one given by the relation (14) Notelb .

Substituting Eq. (14) into Eq. (12) one finds that, for a given mass $M$ of
the original central black hole, the energy gap ${\cal E}_{\text{gap}}$ (the
minimum energy which is delivered to the extremal black hole due to the
assimilation of the massive shell) is given by the remarkably simple
functional expression

${\cal E}_{\text{gap}}=E_{\text{min}}(m;M)={{m^{2}}\over{2(M+m)}}\ .$ (15)

The energy expression (15) implies that the minimum energy
$E_{\text{min}}(m;M)$ which is delivered to the extremal RN black hole in the
present gedanken experiment can be minimized by minimizing the mass $m$ of the
shell. We therefore raise here the following physically important question:
How small can the mass $m$ of the shell be?

In order to address this physically interesting question, we can think of the
shell as representing a spherically symmetric localized wave packet of a
massive particle. It is well known that, according to quantum theory and
special relativity, a wave packet representing a particle of mass $m$ cannot
be localized to better than its Compton length: $R\geq\hbar/2m$ Com1 ; Com2 .
This relation implies that the circumference $C=2\pi R$ of the spherically
symmetric wave packet is bounded from below by the relation

$2\pi R\geq{{h}\over{2m}}\ .$ (16)

One therefore concludes that a spherically symmetric compact wave packet of
circumference radius $R$ representing a particle of mass $m$ is characterized
by the lower bound

$m\geq{{\hbar}\over{2R}}\ .$ (17)

Since in our case the radius of the shell at the point of assimilation is
$R_{\text{min}}=M+m$ [see Eq. (14)], one finds the simple expression Noteap1

$m_{\text{min}}={{\hbar}\over{2M}}\cdot[1+O(\hbar/M^{2})]\ $ (18)

for the minimally allowed mass of the shell which is consistent with quantum
theory and special relativity. Substituting (18) into (15), one finds the
expression

${\cal E}_{\text{gap}}=E_{\text{min}}={{\hbar^{2}}\over{8M^{3}}}\ $ (19)

for the energy gap of the central extremal black hole in the regime
$M^{2}\gg\hbar$ of large black holes. It is important to emphasize that we
cannot rule out the possible existence of a different physical process that
may yield an energy gap which may be smaller than the analytically derived
energy gap (19).

##  IV Summary

Using a sophisticated string theory analysis, Maldacena and Susskind Gap1
have nicely shown that near-extremal black holes are characterized by a finite
mass gap of the first excited state above the extremal (zero-temperature)
black-hole configuration. As explicitly proved in the physically interesting
work Gap2 , the same energy gap can be inferred using a physical argument
which is based on holography without using explicit string theory
calculations.

The main goal of the present compact paper was to gain some physical insights
on the origin of the intriguing mass gap (1). In particular, we have
demonstrated that the suggested finite mass gap (1) can be deduced from a
simple semi-classical gedanken experiment in which a spherically symmetric
massive shell is lowered towards a central extremal Reissner-Nordström black
hole which eventually absorbs the shell. In order to determine the minimum
mass gap of the system above the extremal black-hole limit (6), we have tried
to minimize the energy which is delivered to the black hole by the shell. We
have therefore assumed that the massive shell is lowered adiabatically
(slowly) towards the central black hole.

The main results derived in the present paper and their physical implications
are as follows:
(1) It has been shown that during the lowering process of the shell, most (but
not all) of its mass-energy as measured by far away asymptotic observers is
red-shifted by the gravitational field of the central black hole.
Intriguingly, however, we have explicitly proved that at some point during the
lowering process of the shell, a new and larger horizon, which engulfs both
the original black hole and the massive shell, is formed. In particular, the
engulfing new horizon with $r_{\text{NH}}>M$ [see Eq. (14)] always forms
before the massive shell reaches the horizon $r_{\text{H}}=M$ of the original
extremal black hole.
(2) The formation of the new and larger horizon prevents one from totally red-
shifting the mass-energy of the lowered shell. Thus, for a given proper mass
$m$ of the shell, there is always a finite (non-zero) lower bound
$E_{\text{min}}(m;M)=m^{2}/[2(M+m)]$ [see Eq. (15)] on the energy which is
delivered to the central extremal black hole.
(3) Taking cognizance of the fact that a combination of quantum theory and
special relativity sets the lower bound $R\geq\hbar/2m$ on the proper
circumference radius of a shell of proper mass $m$, we have derived the semi-
classical expression

${\cal E}_{\text{gap}}=M-Q={{\hbar^{2}}\over{8M^{3}}}\ $ (20)

for the minimum energy which is delivered to the central extremal black hole
in the present gedanken experiment.

Interestingly, the results of the present semi-classical gedanken experiment
support the validity of the previously derived Gap1 ; Gap2  quantum energy gap
(1) of extremal black holes in a wider context than string theory and
holography. As emphasized above, our analysis cannot rule out the possible
existence of another physical process that may yield an energy gap for
extremal black holes which may be smaller than the characteristic gap (20).

ACKNOWLEDGMENTS

This research is supported by the Carmel Science Foundation. I thank Yael
Oren, Arbel M. Ongo, Ayelet B. Lata, and Alona B. Tea for stimulating
discussions.

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