# The Role of Vacuum Fluctuations and Symmetry in the Hydrogen Atom in Quantum
Mechanics and SED

G. Jordan Maclay<EMAIL_ADDRESS>Quantum Fields LLC, St.
Charles IL 60174

(June 2019)

###### Abstract

SED has had success modeling black body radiation, the harmonic oscillator,
the Casimir effect, van der Waals forces, diamagnetism, and uniform
acceleration of electrodynamic systems using the stochastic zero-point
fluctuations of the electromagnetic field with classical mechanics. However
the hydrogen atom, with its 1/r potential remains a critical challenge. Cole
and Zou in 2003 and Nieuwenhuizen and Liska in 2015 found that the SED field
prevented the electron orbit from collapsing into the proton but eventually
the atom became ionized. We look at the issues of the H atom and SED from the
perspective of symmetry of the quantum mechanical Hamiltonian which is used to
obtain the quantum mechanical results, and the Abraham-Lorentz equation, which
is a force equation that includes the effects of radiation reaction and is
used to obtain the SED simulations. We contrast the physical computed effects
of the quantized electromagnetic vacuum flucuations with the role of the real
stochastic electromagnetic field.

Key words: stochastic, electrodynamics, SED, hydrogen, symmetry, vacuum
fluctuations, simulation, QED. PACS:11.10, 05.20, 05.30,03.65

##  I Introduction

The hydrogen atom has been the testing ground for theoretical atomic physics
for over a hundred years. The original quantum mechanics was motivated to
model the H atom and explain its spectrum herz . That theory was refined into
modern quantum mechanics by Bohr, Heisenberg and Schrodinger, Dirac, and
others. Precision measurements of hydrogen energy levels by Willis Lamb in
1947 disagreed with the theory, which stimulated the development of Quantum
Electrodynamics (QED)which included the effects of the vacuum fluctuations of
the quantized electromagnetic field. The hydrogen atom is the fundamental two-
body system and perhaps the most important tool of atomic physics and the
challenge is to calculate its properties to the highest accuracy possible. The
current QED theory is the most precise of any physical theory: ”The study of
the hydrogen atom has been at the heart of the development of modern
physics…theoretical calculations reach precision up to the 12th decimal
place…high resolution laser spectroscopy experiments…reach to the 15th decimal
place for the $1S-2S$ transition…The Rydberg constant is known to 6 parts in
$10^{12}$ beyer .

Stochastic electrodynamics (SED) represents an effort to explain quantum
phenomena through classical physics done in the presence of a real stochastic
electromagnetic field which is the sum of a portion for $T=0$ with spectral
energy density $\rho(\omega)=\hbar\omega^{3}/2\pi^{2}c^{3}$, identical to that
of the virtual zero-point vacuum fluctuations of the quantized electromagnetic
field in QED, plus a Planckian spectrum for a finite temperature $T>0$. In
this paper, we focus on the $T=0$ component. SED has had success modeling the
harmonic oscillator, and other linear systems. However, analytical efforts to
address the H atom with SED have not been successful, so researchers have
turned to numerical calculations to explore the H atom in SED and determine if
SED can model the ground state zhou and cole ,nie . The assertion of SED
originally made by Boyer boyer  is that the energy lost by radiation from the
accelerating electron bound to the H nucleus is statistically compensated by
the energy gained from the fluctuating vacuum field. In a variation of this
work, Puthoff has shown that particularly for circular orbits, if a stable
ground state of constant energy is achieved, then the energy provided by the
stochastic vacuum field exactly compensates for the energy lost by radiation
hal . Others have asserted that no thermodynamic stable equilibrium will be
achieved claverie . SED theory has been developed in several books by de la
Pena and his collaborators de la pena  in which they have formulated an
approach that claims to lead to the equations of standard quantum mechanics.
In this paper, we will contrast the approach of quantum mechanics with that of
SED, discussing the role of vacuum fluctuations in both theories. Emphasis is
placed on the symmetry of the H atom since this plays such a pivotal role. For
both theories we make some simplifying assumptions: non-relativistic
mechanics, no spin, infinitely heavy proton. We discuss the most recent SED
calculations which have shown stability for a limited number of orbits, but
ionization for longer timeszhou and cole  nie .

##  II Role of Vacuum Fluctuations in Quantum Mechanics and SED

In the nineteen-thirties, quantum theory had predicted zero-point vacuum
fluctuations. Oppenheimer had calculated that the vacuum fluctuations gave an
infinite and therefore unphysical energy shift to the free electron. The
theory also predicted that the hydrogen $2S^{1/2}$ and $2P^{1/2}$ levels were
degenerate. In 1947 Lamb and Retherford announced they had measured a shift of
$1058$ MHz at the Shelter Island Conference. After the conference, Hans Bethe,
on the train ride to Schenectady, where he was consulting for GE, developed a
new approach to include the vacuum fluctuations and QED was born. He, and
others after him, applied renormalization methods, originally developed by
Kramers and others, to deal successfully with the infinities predicted by
prior approaches. Renormalization asserted that the vacuum fluctuations lead
to the observed physical mass and charge of the electron.

In QED the vacuum field is typically expressed as a sum over an infinite
number of plane waves with all possible momenta $\hbar\mathbf{k}$ and
directions $\mathbf{k}/k$ with the restriction that the energy $E_{k}$ in each
mode is $\hbar\omega_{k}/2=\hbar k/2c$. The vector potential is milonni

$\mathbf{E}(\mathbf{r},t)=\sum_{\mathbf{k},\lambda}\sqrt{\frac{\hbar\omega_{k}}{2\epsilon_{0}V}}(A_{\mathbf{k}\lambda}\cos{(\mathbf{k}\cdot\mathbf{r}-\omega_{k}t)}\\\
-B_{\mathbf{k}\lambda}\sin{(\mathbf{k}\cdot\mathbf{r}-\omega_{k\\}t})})\
\mathbf{e}_{\mathbf{k},\lambda}$ (1)

where the raising and lowering operators obey the commutation rules

$[a_{\bf{k}\lambda},a^{\dagger}_{\bf{k^{\prime}}\lambda^{\prime}}]=\delta_{\bf{k}\bf{k^{\prime}}}\delta_{\lambda\lambda^{\prime}}$
(2)

and the two polarization vectors ($\lambda=1,2$) are orthogonal to
$\mathbf{k}$ so $\mathbf{k}\cdot\mathbf{e}_{\mathbf{k},\lambda}=0$, and

$\mathbf{e}_{\mathbf{k},\lambda}\cdot\mathbf{e}_{\mathbf{k},\lambda^{\prime}}=\delta_{\lambda\lambda^{\prime}}.$
(3)

The quantization volume V is an artifice to avoid infinite volumes. In this
box normalization $k_{x}=2\pi n_{x}/L_{x}$ , $k_{y}=2\pi n_{y}/L_{y}$ , and
$k_{z}=2\pi n_{z}/L_{z}$ , with $V=L_{x}L_{y}L_{z}$, and the integers $n_{x}$,
$n_{y}$, and $n_{z}$ go from -$\infty$ to $+\infty.$ The electric field is
$\mathbf{E(\mathbf{r},t)}=-\partial\mathbf{A}(\mathbf{r},t)/\partial r$ and
$\mathbf{B}(\mathbf{r},t)=\nabla\times\mathbf{A}(\mathbf{r},t)$.

To summarize the properties of the vacuum field in QED: no real photons are
present, only random virtual photons of energy $\hbar\omega_{k}/2$ and
momentum $h\mathbf{k}/2c$, with all possible momenta present consistent with
Eq 1. The expectation values of the fields vanish but the variances do not.
The fields are isotropic (invariant under rotations), invariant under space-
time translations (homogeneous), and under boosts (Lorentz invariant). The
energy density spectrum which is proportional to $\omega^{3}$ is also Lorentz
invariant.

In QED, the zero-point quantum fluctuations are responsible for the mass and
charge renormalization of particles. In this process one postulates the
following:

bare point electron + vacuum fluctuations + radiative reaction $\rightarrow$

electron with physical mass, charge and effective size about Compton
wavelength.

A similar process occurs for an atom, in which the atom undergoes allowed
virtual (energy conserving) transitions due to the vacuum field. These
transitions can be seen as shifting the average energy of the atom. This is
one picture (approximate) of radiative shifts, such as the Lamb shiftmaclay .
The Lamb shift is a small shift, about 1 part in $10^{6}$. It has been
predicted correctly to about 1 part in $10^{9}$ by QED. In QED, radiative
shifts are generally calculated using Feynmann diagrams, in which the atom is
depicted as propagating in time, and it absorbs or emits a virtual photon
changing its state correspondingly, then a short time later (consistent with
the time-energy uncertainty principle) emits or absorbs the same virtual
photon and returns to the initial state. This process arises from the
$\mathbf{p}\cdot\mathbf{A}$ term in the Hamiltonian and can involve all
possible photons, but only photons of the resonant frequency will induce
transitions. This model in a sense describes the interaction of the electron
with its own radiation field. In QED this is equivalent to interacting with
the ubiquitous virtual fluctuating vacuum field. The electron mass
renormalization in QED is the state independent shift in energy for a bare
electron given by the $\mathbf{p}\cdot\mathbf{A}$ and the
$\mathbf{A}\cdot\mathbf{A}$ terms.

What makes Equation 1 for the vector potential unique to the quantized
electromagnetic field is the presence of the operators $a_{\mathbf{k}\lambda}$
and $a^{\dagger}_{\mathbf{k}\lambda}$. The expression for the vector potential
or the $\mathbf{E}(\mathbf{r},t)$ field can be recast with real (Hermitian)
operators into the form:

$\mathbf{E}(\mathbf{r},t)=\sum_{\mathbf{k},\lambda}\sqrt{\frac{\hbar\omega_{k}}{2\epsilon_{0}V}}(A_{\mathbf{k}\lambda}\cos{(\mathbf{k}\cdot\mathbf{r}-\omega_{k}t)}\\\
-B_{\mathbf{k}\lambda}\sin{(\mathbf{k}\cdot\mathbf{r}-\omega_{k\\}t})})\
\mathbf{e}_{\mathbf{k},\lambda}$ (4)

where

$[A_{\mathbf{k}\lambda},B_{\mathbf{k^{\prime}}\lambda^{\prime}}]=2\ i\
\delta_{\mathbf{kk^{\prime}}\lambda\lambda^{\prime}}$ (5)

and the expectation values with respect to the vacuum are $\langle
A_{\mathbf{k}\lambda}\rangle=\langle B_{\mathbf{k}\lambda}\rangle=0$ and the
variances are $\langle A^{2}_{\mathbf{k}\lambda}\rangle=\langle
B^{2}_{\mathbf{k}\lambda}\rangle=1$.

Equation 4 appears precisely the same as the equation often used in SED, with
the differences that $A_{\mathbf{k}\lambda}$ and $B_{\mathbf{k}\lambda}$ are
not operators but independent Gaussian variables with average $0$ and variance
$1$. Also for each term the energy taken as the integral
$\int_{V}d^{3}r(\frac{\epsilon_{0}}{2}\mathbf{E}^{2}+\frac{1}{2\mu_{o}}\mathbf{B}^{2})=\hbar\omega_{k}$.
In some cases a convergence factor for the energy may be used nie . The SED
field is real and is scaled so that the spectral energy density
$\rho(\omega)=\hbar\omega^{3}/2\pi^{2}c^{3}$ is identical to that of QED. The
vacuum field in SED is also invariant under Lorentz transformations, which
includes translations, rotations, and boosts.

Vacuum fluctuations are at the heart of SED. Because the expression for the
vacuum field has an infinite number of plane waves, in SED calculations
mathematical simplifications are employed to make calculations tractable. How
the vacuum field is treated mathematically is probably one of the most
critical aspects of SED modeling. A finite sum over discrete $\mathbf{k}$ with
a restricted range of $\omega_{k}$ is generally employed, which may alter the
symmetry properties of the vacuum field. In the SED H atom calculations, the
frequencies employed have been within four orders of magnitude of the
rotational frequency of the electron in the atomzhou and cole ,nie . The
physical mass and charge of the electron are used, whereas in QED the higher
frequency components of the vacuum fluctuations have the effect of giving a
bare particle the physical mass and charge of the electron. These high
frequency components are not explicitly considered in the SED calculations.
Even with such simplifications, millions or billions of waves are included
requiring extensive computing power. In SED, the vacuum fluctuations are the
source of behavior characteristic of quantum systems, for example, for the
harmonic oscillator. Indeed that is the basic contention of SED, that
classical physics plus the vacuum field leads to the correct predictions for
atomic systems, as does quantum mechanics:

classical mechanics + radiative reaction + point electron with physical mass +
real vacuum fluctuations(SED)

$\rightarrow$ stable atom with correct behavior.

In summary, the role of the vacuum field in SED is to stabilize the electron
orbit by providing the energy lost to radiation from the
$d^{3}\mathbf{r}/dt^{3}$ term. Since the electron follows a classical orbit in
SED, the orbit must be constantly changing to mimic wavelike behavior shown in
quantum mechanics. Indeed from Earnshaw’s theorem there is no stable static
arrangement of charges so fluctuations are to be expected. One interpretation
of the orbital behavior is that when the electron loses energy and moves
closer to the nucleus, its orbital frequency increases and it interacts more
strongly with the higher frequency, higher energy waves in the vacuum field,
causing it to accelerate, gain energy and move to a larger radius orbit. With
the larger radius, the orbital frequency slows down, the electron interacts
more strongly with the lower frequency waves, loses energy and moves to a
smaller orbit. Thus the electron oscillates from smaller to larger radii boyer
cole v .

##  III The Hydrogen atom in Quantum Mechanics and Classical Mechanics with No
Radiative Reaction

In the quantum mechanical H atom there is no radiation from the electron due
to the standing wave nature of the electron orbitals when the atom is in a
stationary state. The wavefunctions and energy levels are determined solely by
the symmetry of the Coulomb potential. The Hamiltonian is

$H=\frac{\mathbf{p}^{2}}{2m}-\frac{Ze^{2}}{r}.$ (6)

We discuss the symmetry of H and its implication in the classical case with no
radiation since SED is a classical theory. We also outline the very similar
effects of the symmetry in the quantum mechanical system. Classically, the
orbits about the Coulomb potential are stable orbits (assuming no radiation),
meaning that a perturbation to the position of the electron in the orbit will
lead only to small oscillations. In the classical system there are orbits, in
the quantum mechanical system there are wavefunctions. Solutions to
Schrodinger’s equation $H|\psi\rangle=E|\psi\rangle$ are stationary energy
eigenstates. The rotational symmetry of the atom implies that H is invariant
under rotation. Rotations are generated by $e^{i\mathbf{L}\phi/\hbar}$ where
the angular momentum $\mathbf{L}=\mathbf{r}\times\mathbf{p}$, so
$[H,\mathbf{L}]=0$. On the other hand $[H,\mathbf{L}]=-i\hbar d\mathbf{L}/dt$
so the angular momentum is conserved. For the classical orbit, the energy E
and $\mathbf{L}$ are also constant, and $\mathbf{L}$ would be normal to the
orbital plane. Symmetry operators of H are conserved in time. There is an
additional symmetry operator of H, the Runge-Lenz vector $\mathbf{A}$:

$\mathbf{A}=\frac{\mathbf{p}\times\mathbf{L}-\mathbf{L}\times\mathbf{p}}{2a}-\frac{mZe^{2}\mathbf{r}}{r}$
(7)

where $a=(-2mE)^{1/2}$, the effective momentum. The Runge-Lenz vector is
constant, pointing along what would classically be the principal axis of the
orbit, and is perpendicular to the angular momentum
$\mathbf{L}\cdot\mathbf{A}=0$ (see Fig.1). In the classical H atom
$e^{i\mathbf{A}\rho/\hbar}$ is a generator of changes in the eccentricity
$\epsilon$ of the orbit in such a way that the energy remains unchanged. In
quantum mechanics, the Runge-Lenz vector generates a transformation that takes
a state of given principal quantum number n and given L ($L\leq n-1$) into a
superposition of the $n^{2}$ degenerate states with the same n.

Figure 1: The classical limit of the orbital is a non-precessing ellipse with
constant $\mathbf{L}$ and $\mathbf{A}$. The semi-major axis is $r_{c}$ and
$L^{2}+A^{2}=a^{2}r^{2}=\rangle$ energy. The eccentricity $\epsilon=\sin{\nu}$
and $a=(-2mE)^{1/2}$.

From the definitions of the two conserved vectors, it is trivial to obtain the
equations for the classical orbits:

$\mathbf{r}\cdot\mathbf{A}=rA\cos{\phi_{r}=-r(mZe^{2}/a)+\mathbf{r}\cdot\mathbf{p}\times\mathbf{L}},$
(8)

and substituting $\mathbf{r}\cdot\mathbf{p}\times\mathbf{L}=L^{2}$ gives

$r=\frac{L^{2}/mZe^{2}}{(a/mZe^{2})A\cos{\phi_{r}}+1}$ (9)

which describes an ellipse with eccentricity $\epsilon=A(a/mZe^{2})$.
Computing $\mathbf{A}\cdot\mathbf{A}$ and substituting the value for A gives

$\epsilon=[2|E|L^{2}/m(Ze^{2})+1]^{1/2}.$ (10)

The semi-major axis is the average of the radii at the turning points. From
the orbit equation, we find

$r_{c}=\frac{L^{2}}{mZe^{2}}\frac{1}{1-\epsilon^{2}}=\frac{Ze^{2}}{2|E|}.$
(11)

The energy depends only on the length of the semi-major axis $r_{c}$ not on
the eccentricity. This makes sense because $\mathbf{A}$ changes the
eccentricity but not the energy. Using the equations for $A^{2}$ and $L^{2}$,
with $\epsilon=\sin{\nu}$, we find

$L=ar_{c}\cos{\nu}\hskip 25.0ptA=ar_{c}\sin{\nu}$ (12)

therefore,

$L^{2}+A^{2}=a^{2}r^{2}_{c}=-\frac{m(Ze^{2})^{2}}{2|E|}.$ (13)

Thus the values of A and L determine the energy E. A similar equation can be
derived in quantum mechanics, and using the quantization of the operators
$\mathbf{A}$ and $\mathbf{L}$, we obtain the quantized energy levels as a
function of the principal quantum number n. The commutators of $\mathbf{A}$
and $\mathbf{L}$ are

$\displaystyle[L_{i},L_{j}]$ $\displaystyle=i\epsilon_{ijk}L_{k}$
$\displaystyle[A_{i},A_{j}]$ $\displaystyle=i\epsilon_{ijk}L_{k}$ (14)
$\displaystyle[L_{i},A_{j}]$ $\displaystyle=i\epsilon_{ijk}A_{k}.$ (15)

The generators ${\mathbf{A},\mathbf{L}}$ form a Lie algebra that closes. To
determine the corresponding symmetry group define the generators
$\mathbf{M}=1/2(\mathbf{L}-\mathbf{A})$ and
$\mathbf{N}=1/2(\mathbf{L}+\mathbf{A})$ which have the commutation properties
of two disjoint O(3) groups $O(3)\times O(3)\approx O(4)$:

$\displaystyle[M_{i},N_{j}]$ $\displaystyle=0$
$\displaystyle[M_{i},M_{j}]=i\epsilon_{ijk}M_{k}$ (16)
$\displaystyle[N_{i},N_{j}]$ $\displaystyle=i\epsilon_{ijk}N_{k}.$ (17)

The Lie algebra for M or N is the same as the familiar commutation relations
for angular momentum $[J_{i},J_{j}]=i\epsilon_{ijk}J_{k}$. For angular
momentum the allowed representations are characterized by the scalar
$J^{2}=J(J+1)$, where $J=0,1/2,1,...$ The $2J+1$ angular moment kets
$|Jm\rangle$ for this representation are generally chosen so
$J_{z}|jm\rangle=m|Jm\rangle$, where $m=-J,-J+1,...,0,1,2,...,J$. In analogy
with the usual O(3) results
$\mathbf{M}^{2}=\mathbf{N}^{2}=1/4(\mathbf{L}^{2}+\mathbf{A}^{2})=j(j+1)$ for
$j+0,1/2,1,...$ Thus we obtain

$\mathbf{L}^{2}+\mathbf{A}^{2}+1=\frac{(mZe^{2})^{2}}{-2mE}=(2j+1)^{2}=n^{2}$
(18)

where n=1,2,3,…is the principal quantum number and $n^{2}$ is the degree of
degeneracy for the representation. The wavefunctions are eigenfunction of
$L_{z}$ and $A_{z}$. The O(4) symmetry of the wavefunctions is especially
manifest in momentum space, where the wavefunctions are spherical harmonics in
4 dimensions. It is possible to enlarge the symmetry group from O(4) to
SO(4,2) in such a way that all states, scattering and bound, are included in
the representation of the groupmaclay .

In summary, the symmetry of the Hamiltonian for the H atom determines the
energy levels, the orbits or wavefunctions, the degeneracy, the constants of
the motion. For the ground state n=1 and L=0 and there is no degeneracy.

##  IV The Hydrogen Atom in SED

In SED, because of the radiative reaction, the energy of the electron is not
conserved and the equation of motion generally used is the Abraham-Lorentz
force equationzhou and cole cole fop nie

$m\frac{d^{2}\mathbf{r}}{dt^{2}}=-\frac{Ze^{2}\mathbf{r}}{r^{3}}+\frac{2e^{2}}{3c^{2}}\frac{d^{3}\mathbf{r}}{dt^{3}}\\\
-e(\mathbf{E}(\mathbf{r},t)+\mathbf{v}\times\mathbf{B}(\mathbf{r},t)).$ (19)

The first term on the right of the equal sign is the Coulomb force, the next
term is the radiative reaction, followed by the Lorentz force due to the
vacuum fields $\mathbf{E}(\mathbf{r},t)$ and $\mathbf{B}(\mathbf{r},t)$. The
presence of the radiative reaction and the Lorentz force eliminates the
symmetry of equation: angular momentum is not conserved, the Runge-Lenz vector
is not constant and the electron energy is not constant. It should be
mentioned that there is a fully relativistic form of Eq. 19, the Abraham-
Lorentz-Dirac equation, used, for example, in the study of charges accelerated
uniformly through the vacuum coleP .

The Abraham-Lorentz force $(2e^{2}/3c^{2})d\mathbf{a}/dt$ is the non-
relativistic recoil force for an accelerating charged particle caused by the
particle emitting radiation which carries momentum, angular momentum and
energyteit . The radiation field from the particle is essentially exerting a
force on itself, sometimes called a “self-field”, a phenomena which can leads
to renormalization and a host of unsolved complications ultimately dealing
with the nature of elementary particlesteit jackson . Radiative reaction and
vacuum fluctuations are intimately intertwinedmilonni . For example, in
quantum mechanics, both are required to maintain the canonical commutation
relations. The third order derivative for the Lorentz force requires the
vacuum energy density to go as $\omega^{3}$. Renormalization methods in
classical physics and in QED have been developed to deal with these issues in
an effective if not fundamental way. The Abraham-Lorentz equation is is a
third order differential equation in time, which requires three initial
conditions (the initial position, velocity and acceleration), and which leads
to mathematical problems such as runaway solutions and acausal behavior and
divergences (see milonniqv  for a discussion).

In most calculations the magnetic field is dropped and a dipole approximation
is used so the electric field is a function of time alone
$\mathbf{E}(\mathbf{r},t)\approx\mathbf{E}(t)$. The electron is assumed to
move in a plane. Also often the third derivative is approximated by using the
equation obtained when there is no radiative reaction or vacuum field
$md^{2}\mathbf{r}/dt^{2}=-\frac{Ze^{2}\mathbf{r}}{r^{3}}$ so
$(2e^{2}/3c^{2})d\mathbf{a}/dt\approx(2Ze^{4}/3m^{2}c^{2})d\mathbf{r}/dt$.

##  V SED Simulations of the Ground State of the H Atom

Cole and Zou performed the first simulations of the hydrogen atomzhou and cole
cole v . They retained the effect of the magnetic field, approximated the
radiative term as discussed above, and did not use a dipole approximation.
They used an adaptive time step Runge-Kutta 4th order algorithm, roughly
equivalent to a net 5th order algorithm in dt to simulate 2 d orbits in a box
27 Å by 27 Å by 0.41 cm, where the normal to the plane of the orbit was in the
larger Z direction. The advantage of choosing this geometry was that it
reduced the number of plane waves they needed to consider to $2.2\times
10^{6}$, with wavelengths from 0.1Å to 900Å and waves only in the Z direction,
yet was able to capture the most important physical featureszhou and cole .
They did 11 simulations, each starting at a radius of 0.53 Å but with
different random amplitudes for the waves. They ran the simulations for about
$10^{-11}$ sec or roughly 100,000 orbits. In total these calculations took
about 55 CPU days.

Figure 2: Typical plot of r vs time for one trajectory. The inset shows the
probability density P(r) vs r for this trajectory. A lower curve of r vs. t
shows that if no zero-point radiation was present, then atomic collapse would
occur in about $1.3\times 10^{-11}$ sec. The starting trajectories began at
0.53 Å zhou and cole .

Fig. 2 shows a typical trajectory as a function of time. It shows fluctuations
in the radius by a factor of about 3 for roughly 100,000 orbits. As the inset
shows, without the ZP field, the electron would have collapsed after about
$1.3\times 10^{-11}$ sec, so clearly the ZP field is preventing collapse
during the interval shown.

Fig. 3 shows the radial probability distribution P(r) vs. radius, computed
from a time average of the 11 simulations, up to the time indicated for each
plot. Also shown is the P(r) computed from the ground state Schrodinger
wavefunction. As time increases, the distribution approaches that predicted by
quantum mechanics.

Figure 3: Plots of the radial probability density vs. radius. The solid line
was calculated from the ground state Schrodinger wavefunction. The dotted
curves are simulation results, with starting trajectories began at 0.53Å
calculated as a time average for all 11 simulation runs from t=0 to the
indicated time: left side, $1.417\times 10{{}^{-}12}$ sec, right side,
$7.25\times 10^{-12}sec$ zhou and cole .

The simulations of Cole and Zou typically ran for about $10^{-11}$sec or
roughly $100,000$ orbits and showed that as time increased, the radial
distribution in $r$ tended to look more like that predicted by quantum
mechanics. In subsequent simulations using 30 PCs, Cole increased the span of
wavelengths used for the applied zero-point field from $10^{4}$ to $10^{6}$
and found ionization occurred unless they increased the precision of the
computation cole v . These preliminary results were only for about $10^{-12}$
sec, about one tenth the time of the previous work and showed general
agreement with the previous work. Cole and Zou also explored fundamental
interactions between circularly polarized (CP) electromagnetic radiation and
the electron initially in orbit about the proton, demonstrating that a stable
circular orbit could be obtained by judicious choice of the amplitude and
phase of a single CP normally incident wave with the same frequency as the
orbital motion, and that a stable elliptical orbit could be obtained with a
superposition of CP waves consisting of many harmonics of the orbital
motioncolejsc . Cole and Zoucolesub  and Cole10  did an ensemble of
simulations with different initial positions of the electrons, and explored
resonances that arise for certain orbital conditions, deepening understanding
about the details of SED simulations and the classical H atom, and possibly
paving the way for understanding transitions in H atom energy levels with SED.

The simulations by Nieuwenhuizen and Liska nie , done over a decade after the
pioneering work of Cole and Zouzhou and cole  used the vastly improved
computing power ($5.6$ TFLOPS single precision floating point). They did a 3d
simulation that included about $10^{7}$ plane waves. To simplify the
computation they replaced the 3d sums over all $\mathbf{k}$ by 1d sums over
frequency with Gaussian amplitudes chosen such that they reproduce the same
correlation function in the limit where the frequency mesh vanishes
$(N\rightarrow\infty)$. They also computed orbits for times that were twice as
long, about $3\times 10^{-11}$ sec, with about $10^{6}$ orbits.

A 4th order Runge-Kutta algorithm was used with about 4000 iterations/orbit.
The electric field was updated as a function of time ten times each orbit,
with interpolation in between. Numerous methods were employed to check the
computational accuracy of the results.

Fig. 4 shows the energy in Bohr units and the eccentricity as a function of
the time with units of $t_{o}=1/\omega_{0}$, the Bohr time
$t_{0}=\hbar/Z^{2}\alpha^{2}mc^{2}$, which is $(1/2\pi)$ of the QM orbital
period. The energy of the ground state is $\hbar\omega_{0}/2$. These
simulations employed a moving cutoff equal to 2.5 times the orbital frequency.
The energy varies from about 0.1 to -1.6 Bohr units (without radiative
reaction the energy would be -.5). The eccentricity $\epsilon$ varies from
about 0.05 to 0.95 (without radiative reaction $\epsilon$ would be 0 for a
circular orbit).

Figure 4: Plots of the energy E (left side) and eccentricity $\epsilon$ (right
side) as a function of time in units of $t_{0}$ for Z=3, for $1.5\times
10^{6}$ revolutions. The energy for the ground state in quantum mechanics
would be 0.5 with the scale used (Bohr units)nie .

Fig. 5 shows the variation in the radius with time in units of $t_{0}$ and a
corresponding histogram of the radius. The radius varies very rapidly from 0.1
to 8 Bohr radii , while the eccentricity and energy remain somewhat more
stable on longer timescales. Unlike Cole and Zou zhou and cole , they did not
see a trend to reproduce the wavefunction, although their simulations
suggested this might occur for shorter times scales.

Figure 5: Plots of the radius r of the electron orbit as a function of time in
units of $t_{0}$ (left side) and a histogram of r (right side), for the same
simulation as Fig. 4. The smooth curve is the probability distribution
expected from quantum mechanics.nie

These plots suggest the possibility that the orbits may stabilize. However,
when the simulations are done for longer times of the order of $10^{7}t_{0}$,
instabilities developed which eventually lead to ionization Fig.6. When higher
harmonics were included (4.5 and 6.5), ionization occurred earlier. Ionization
was defined as the moment when the electron stayed above $E=-0.05$ for at
least $10^{7}t_{0}$. The moment of ionization is not shown in the subsequent
plotsnie .

With an upgrade in computing power, allowing double precision and a fixed
cutoff of the frequency spectrum of the random fields, ionization occurred at
an even earlier time. The energy of the electron tended to decrease to zero
while the eccentricity increased leading to ionization. It is not clear
precisely why these computational upgrades led to ionization at significantly
earlier times.

Figure 6: Plots of the energy E (left side) and the eccentricity $\epsilon$ as
a function of time for Z=1 with a fixed cutoff exposing the trend towards
ionization at $E=0,\ \epsilon=1$. The time window in 45 times shorter than in
Fig. 4 and Fig. 5 nie .

In a second set of simulations, Nieuwenhuizen and Liska included relativistic
corrections and the effects of the magnetic fieldsnierel . Nevertheless, they
obtained essentially the same results, ionization at early times. Niewenhuizen
did a theoretical model of the H atom with the stochastic field, and computed
the change in energy with each orbit of the electronnieent . If the energy
gets very low and the angular momenta goes below $0.588\hbar$, then the orbits
are very eccentric, and the electron gains energy with each rotation, leading
to self-ionization. In a simulation, as shown in Fig. 7 they observed that the
angular momentum seldom went below this critical value, in agreement with the
predicted result.

Figure 7: Distribution of the angular momentum L (in units of $\hbar$) from
the numerical simulation of the SED dynamics reported in nie . Not much weight
lies below $L=0.588$ , confirming that when such a value is reached at near-
zero energy, self-ionisation may occur rapidly and the run is ended. Full
curve: the distribution of L from the conjecture for the would-be stable
ground state distribution from nieaip .

##  VI Discussion

SED researchers have made significant progress in the very complex task of
modeling the ground state of the H atom, and shown that for short times
corresponding to tens of thousands of orbits the real stochastic zero-point
field can provide the energy lost to radiation and maintain the orbital
motion. However, as time passes and more regions of phase space are explored,
the orbits tend to become highly elliptical, with lower angular momentum, and
self-ionization occurs. The question arises: is this instability due: 1) to
the specific implementations of SED, including the approximations and
numerical methods used, or 2) is this instability inherent in the SED approach
to the H atom, or 3) is it more fundamental in nature, and perhaps reflects a
characteristic chaos arising from non-linear potentials cole fop 10 , as
observed, for example, in comparisons of classical and quantum models of
photo-dissociation with the Morse potential for diatomic molecules
milonnimorse . For a variety of non-linear systems there are regions of
classical phase-space characterized by chaos.

Contrasting the approach of quantum mechanics and quantum electrodynamics to
that of SED is illuminating. In QED, the various physical phenomena, including
the zero-point field, the fundamental properties of the electron, the presence
of radiation, the properties of the Coulomb field, and the stability of atoms,
are all parsed in such a way that the system is mathematically tractable and
well behaved solutions, reflecting the symmetry of the 1/r potential, are
found. On the other hand it appears that the current SED approach lumps all
these phenomena together in the Abraham-Lorentz equation with the zero-point
field, which leads to very challenging mathematical entanglements that mask
the symmetry of the solutions. For example, in QED, the self interaction leads
to a big effect on the electron: mass renormalization. The remaining radiative
shift for the atom is very small. In SED, the radiative effects in the
Abraham-Lorentz equation lead to weak but persistent forces, whereas the
vacuum field forces are much larger and fluctuate greatly from positive to
negative values, making it difficult to obtain a stable orbit.

There are many more issues that SED needs to address beyond the attainment of
a stable orbit. Can SED develop a set of states for the H atom that are
isomorphic to those predicted by QM, with characteristic quantized values of
angular momentum and energy? with the same degree of degeneracy? Can SED
predict the transitions between energy levels? Can SED predict the results of
measurements on the atom? It is very difficult to see a path for attaining
these objectives with the current SED approach. For a first step, for example,
SED needs not only to obtain stable ground state orbit, but, to probably
predict an orbit that is similar to the quantum ground state which has a
spherically symmetric probability distribution with quantized angular momentum
of zero. Perhaps this might be obtained with a superposition of suitable
initial conditions. But considering superpositions leads to ideas
characteristic of quantum mechanics. This raises the question: does making SED
predict the H atom correctly require the modification of SED so that it
becomes essentially a reformulation of quantum mechanics? Perhaps SED, which
predicts particles with definite orbits, has a relationship to Bohmian
mechanics, which also describes definite orbits, has a non-local potential,
and predicts measurements that agree with those of quantum mechanicsholland .

###### Acknowledgements.

The author would like to thank Peter Milonni for his encouragement and his
comments on this paper, and to thank Dan Cole for sharing references and
research I was unaware of, and for his extraordinarily careful reading and
numerous insightful suggestions and comments.

## References

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# Kagome KMn3Sb5 metal: Magnetism, lattice dynamics, and anomalous Hall
conductivity

Sobhit Singh  0000-0002-5292-4235<EMAIL_ADDRESS>Department of
Mechanical Engineering, University of Rochester, Rochester, New York 14627,
USA  A. C. Garcia-Castro  0000-0003-3379-4495<EMAIL_ADDRESS>School of
Physics, Universidad Industrial de Santander, Carrera 27 Calle 09, 680002,
Bucaramanga, Colombia

###### Abstract

Kagome metals are reported to exhibit remarkable properties, including
superconductivity, charge density wave order, and a large anomalous Hall
conductivity, which facilitate the implementation of spintronic devices. In
this work, we study a novel kagome metal based on Mn magnetic sites in a
KMn3Sb5 stoichiometry. By means of first-principles density functional theory
calculations, we demonstrate that the studied compound is dynamically stable,
locking the ferromagnetic order as the ground state configuration, thus
preventing the charge-density-wave state as reported in its vanadium-based
counterpart KV3Sb5. Our calculations predict that KMn3Sb5 exhibits an out-of-
plane (001) ferromagnetic response as the ground state, allowing for the
emergence of topologically protected Weyl nodes near the Fermi level and
nonzero anomalous Hall conductivity ($\sigma_{ij}$) in this centrosymmetric
system. We obtain a tangible $\sigma_{xy}=314$ S$\cdot$cm-1 component, which
is comparable to that of other kagome metals. Finally, we explore the effect
of the on-site Coulomb repulsion ($+U$) on the structural and electronic
properties and find that, although the lattice parameters and $\sigma_{xy}$
moderately vary with increasing $+U$, KMn3Sb5 stands as an ideal stable
ferromagnetic kagome metal with a large anomalous Hall conductivity response.

DOI:

##  I Introduction

Kagome lattices [1], such as those observed in FeSn [2], Co3Sn2S2 [3], ScV6Sn6
[4], and the ones present in Mn${}_{3}B$N ($B$ = Ni, Ga, Pt, Pd, and Sn) and
V3AuN antiperovskites [5, 6, 7, 8, 9, 10], exhibit remarkable electronic,
phononic, topological, and magnetic entangled properties, mainly owing to
their particular star-shaped hexagonal symmetry [11]. In such symmetry,
triangularly-coordinated magnetically active cations present substantial
magnetic and electronic frustration that leads, for example, to charge-density
waves (CDW) phases [12], superconductivity [13] and chiral noncollinear
magnetic orderings [11]. Belonging to the kagome materials, the $A$V3Sb5 ($A$
= K, Rb, and Cs) family has recently attracted tremendous attention due to the
richness of their charge-density wave states, giant anomalous Hall response,
frustrated electronic structure, and associated superconductivity [14, 15, 16,
17, 18]. The reported CDW phase in $A$V3Sb5 metals is originated by the
electronic frustration in the in-plane vanadium sites, due to the unbalanced
charge based on the expected nominal charges, $i.e.$, K:$1+$,
V:(5-$\delta$)$+$ and Sb:$3-$, to enforce the charge neutrality with
K1+V${}_{3}^{(5-\delta)+}$Sb${}_{5}^{3-}$ [14]. Moreover, ferromagnetic and
noncollinear antiferromagnetic states can be expected to stabilize in this
family of materials, leading to tangible topological features related to Dirac
and Weyl fermions in the vicinity of the Fermi level, resulting in non-
vanishing Berry curvature induced observables such as anomalous Hall
conductivity (AHC) and nonlinear Hall effects [19, 20, 21, 22, 23].
Importantly, the electrical manipulation of Berry curvature-induced anomalous
Hall effect at room temperature offers unimaginable capabilities in future
spintronic devices [24, 25, 26, 27]. In ferromagnetic kagome metals,
reversible magnetization can be used as an external parameter to control and
switch the AHC response. Therefore, the search for novel ferromagnetic kagome
materials, belonging to the $A$$M$3Sb5 (with $A$ = K, Rb, Cs. $M$ = V and Mn)
stoichiometry, might lead to the discovery of novel quantum phases driven by
nontrivial electronic, topological, magnetic, and superconducting properties.

In this study, we employ first-principles density functional theory (DFT)
calculations to investigate the structural, vibrational, magnetic, and
topological electronic properties of the kagome metal KMn3Sb5, which was
recently predicted by Jiang _et al._ [28]. Our results reveal that, despite
the inherent magnetic frustration in the kagome Mn plane, KMn3Sb5 exhibits a
stable ferromagnetic ground state, with the magnetic easy axis oriented along
the (001) direction. This ferromagnetic ground state breaks time-reversal
symmetry and allows the emergence of topological Weyl nodes in this
centrosymmetric crystal system, resulting in large concentrations of Berry
curvature near the Weyl nodes. As a result, KMn3Sb5 exhibits a substantial
anomalous Hall conductivity response ($\sigma_{xy}$ = 314 S$\cdot$cm-1).
Incorporating the on-site Coulomb parameter $+U$ in our DFT calculations
moderately affects the optimized lattice parameters, magnetic moments, phonon
frequencies, and $\sigma_{xy}$ values, but no unstable phonon modes are
observed within the studied range of $+U$ parameter. Our findings provide
valuable insights for experimentalists in synthesizing and confirming the
predicted properties of the ferromagnetic kagome metal KMn3Sb5.

Figure 1: (Color online) (a) KMn3Sb5 _P6/mmm_ (SG. 191) hexagonal structure
obtained for the ferromagnetic, (001) FM, ground state. In this structure, the
K, Mn, and Sb sites are drawn in dark blue, violet, and pink colors,
respectively. (b) Chiral noncollinear antiferromagnetic orderings allowed in
KMn3Sb5. Here, the $\Gamma_{4g}$ and $\Gamma_{5g}$ AFM orders hold $+1$
magnetic chirality whereas the $\Gamma_{6g,x}$ and $\Gamma_{6g,y}$ AFM orders
inherit $-1$ magnetic chirality. (c) Phonon dispersion calculated for the
ground state FM order (DFT-PBEsol). (d) Atom-projected phonon density of
states (PDOS).

This paper is organized as follows: In Section II, we provide details of the
computational and theoretical methods used in this study. In Section III, we
present the results, starting with a discussion of the crystal structure and
its vibrational and dynamical stability. We then analyze different magnetic
configurations and explore the electronic structure, including intriguing
features associated with Berry curvature and anomalous Hall conductivity.
Furthermore, we examine the role of electron-correlation effects on the
properties of KMn3Sb5. Finally, in Section IV, we draw our conclusions and
discuss future perspectives.

##  II Computational Details

We performed first-principles DFT calculations [29, 30] using the projected-
augmented wave (PAW) [31] method as implemented in the vasp code (version
5.4.4) [32, 33]. The valence electron configurations considered in the PAW
pseudopotentials are as follows: K: (3$p^{6}$4$s^{1}$, version 02Aug2007), Mn:
(3$p^{6}$3$d^{5}$4$s^{2}$, version 02Aug2007), and Sb: (5$s^{2}$5$p^{3}$,
version 06Sep2000). The exchange-correlation functional was computed using the
generalized-gradient approximation as parameterized by Perdew-Burke-Ernzerhof
for solids (GGA-PBEsol) [34] and the on-site correlation effects for Mn-3$d$
electrons were corrected using the rotationally-invariant Liechtenstein
(DFT$+U$) formalism [35]. We used $+U$ values ranging from 0.0 to 3.0 eV,
which were optimized to explore the behavior of the lattice parameters,
magnetic moment, and anomalous Hall conductivity. The reciprocal space was
sampled using a $\Gamma$-centered Monkhorst-Pack $k$-mesh [36] of size
11$\times$11$\times$9, and a kinetic energy cutoff of 600 eV was used for the
plane wave basis set. These values resulted in the convergence of residual
forces and total energy to better than 0.001 eV$\cdot$Å-1 and 0.1 meV. Spin-
orbit coupling (SOC) was included to consider noncollinear magnetic
configurations [37]. Phonon calculations were performed within the finite-
differences approach [38, 39] and post-processed using the Phonopy code [40].
To compute the anomalous Hall conductivity and Berry curvature, we utilized
the Wannier functions methodology, for which the wannierization was performed
using the Wannier90 code [41, 42] and post-processed with the WannierBerri
package [43]. For the wannierization process, $s$ and $p$ orbitals were
considered for K and Sb atoms, while $s$, $p$, and $d$ were considered for Mn
atoms. To plot the Berry curvature projected on the Fermi surface, we utilized
the FermiSurfer software [44]. Crystal structures were visualized using the
vesta software [45], and electronic structure data were post-processed using
the PyProcar [46] software.

##  III Results and Discussion

###  III.1 Structural and Magnetic Configurations:

Similar to $A$V3Sb5, KMn3Sb5 adopts the $P6/mmm$ (SG. 191) phase in its
primitive unit cell. It consists of an Mn kagome lattice situated in the (0,
0, 1/2) plane, with several Sb sites embedded into the central kagome hexagon,
as shown in Fig. 1(a). Additionally, the Mn kagome lattice is packed with
graphite-like Sb layers in positions close to (0, 0, 1/4) and (0, 0, 3/4)
atomic planes. Lastly, the hexagonally coordinated K atoms are located at the
(0, 0, 0) sites. The DFT-optimized lattice parameters of KMn3Sb5 are provided
below in Table 2.

As a first step in the magnetic analysis, we define all the symmetry-allowed
noncollinear magnetic states that can be associated with a q propagation
vector of (0, 0, 0). We find that, similar to the case of antiperovskites
[10], four $xy$ in-plane noncollinear chiral antiferromagnetic (AFM) orderings
are allowed, as shown in Fig. 1(b), along with a $z$-axis out-of-plane
ferromagnetic (FM) ordering.

We then calculated the total energies (PBEsol) for all the considered
noncollinear magnetic orderings and obtained $E_{\Gamma_{4g}}$ = $-$54.0587
eV$\cdot$f.u.-1 and $E_{\Gamma_{5g}}$ = $-$54.0614 eV$\cdot$f.u.-1 for the
chiral $+1$, and $E_{\Gamma_{6g,x}}$ = $-$54.0453 eV$\cdot$f.u.-1 and
$E_{\Gamma_{6g,y}}$ = $-$54.0452 eV$\cdot$f.u.-1 for the chiral $-1$ orderings
[47]. See note 111The magnetic vector chirality is defined as $\kappa$ =
$\frac{2}{3\sqrt{3}}\sum_{i,j}$(Si$\times$Sj) =
$\frac{2}{3\sqrt{3}}$(S1$\times$S2+S2$\times$S3+S3$\times$S1) for the kagome
lattice where the $i$ and $j$ index run over the magnetic moments in the unit
cell for more details regarding magnetic chirality. Interestingly, we find
that the (001) FM state, with a total energy of $E_{FM}$ = $-$54.4928
eV$\cdot$f.u.-1 is the lowest energy magnetic ground state with an average
difference energy around 500 meV$\cdot$f.u.-1. This is in contrast with the
observed ground state magnetic configuration in other similar materials such
as the antiperovskites. In antiperovskites, the 3D kagome symmetry resolves
magnetic frustration by stabilizing chiral noncollinear antiferromagnetic
states instead of ferromagnetic states.

The ground state FM ordering in KMn3Sb5 can be explained in terms of the out-
of-plane broken magnetic frustration which tends to lower the total energy
compared to the strong frustrations present in the kagome lattice of
antiperovskites. In antiperovskites, the kagome lattices are formed, by
symmetry, in the complete ensemble of [111] family of planes, leading to the
stabilization of chiral noncollinear antiferromagnetic states due to the 3D
kagome symmetry. However, in KMn3Sb5, the out-of-plane magnetic frustration
breaks this symmetry and favors the ferromagnetic ordering as the ground state
magnetic configuration.

To gain a better understanding of the magnetic easy axis, we investigate the
preferred direction in which the magnetic moments tend to align in the kagome
plane of Mn. Specifically, we examine whether the moments lie in the
$xy$-plane (within the plane) or if they are oriented perpendicular to it
(along the $z$-axis). Our calculations (PBEsol) reveal that $E_{xy-plane}$ =
$-$54.4920 eV$\cdot$f.u.-1 whereas $E_{z-axis}$ = $-$54.4928 eV$\cdot$f.u.-1,
thus implying $z$-axis as the easy axis for the ground state FM order in
KMn3Sb5.

From this point on, all the reported calculations and analyses are for the
(001) ferromagnetic ground state of KMn3Sb5. The DFT (PBEsol) optimized
lattice parameters in this magnetic ground state are $a=b$ = 5.337 Å and $c$ =
9.028 Å. With increasing $+U$ value (PBEsol$+U$) from 0 to 3 eV on Mn-$3d$
orbitals, primitive cell moderately expands within the $a$-$b$ plane while
shrinking along the $c$ axis (see Table 2) by approximately $\pm 5\%$.

###  III.2 Lattice Dynamics:

To test the dynamical stability of the ground state noncollinear (001) FM
order in KMn3Sb5, we calculated the full phonon spectrum along the high-
symmetry directions of the Brillouin zone, as shown in Fig. 1(c). We observe a
fully stable vibrational landscape, with no unstable modes at imaginary
frequencies appearing along the entire Brillouin zone. The absence of unstable
phonons suggests the suppression of the charge-density-wave phenomenon induced
by the unstable phonon modes, as reported in other kagome systems such as
$A$V3Sb5 [15, 17, 18]. Thus, it is important to note that the kagome compound
KMn3Sb5 is added to the set of materials with this symmetry, presenting
potential opportunities for exhibiting a tangible ferromagnetic response.
According to group theory, the irreducible representation of all allowed
vibrational modes for KMn3Sb5 ($P6/mmm$) at Brillouin zone center is defined
as follows:

$\displaystyle\Gamma\textsubscript{vib}=\emph{A}\textsubscript{1g}\oplus
4\emph{A}\textsubscript{2u}\oplus\emph{B}\textsubscript{1g}\oplus\emph{B}\textsubscript{1u}\oplus
2\emph{B}\textsubscript{2u}\oplus 2\emph{E}\textsubscript{2u}$ (1)
$\displaystyle\oplus\,\emph{E}\textsubscript{2g}\oplus
5\emph{E}\textsubscript{1u}\oplus\emph{E}\textsubscript{1g}.$

Out of the total 27 allowed phonon modes (9 atoms per cell), the three
acoustic modes are $\Gamma\textsubscript{acoustic}$ =
$\emph{A}\textsubscript{2u}\oplus\emph{E}\textsubscript{1u}$, and 24 optic
modes are $\Gamma\textsubscript{optic}$ = $\emph{A}\textsubscript{1g}\oplus
3\emph{A}\textsubscript{2u}\oplus\emph{B}\textsubscript{1g}\oplus\emph{B}\textsubscript{1u}\oplus
2\emph{B}\textsubscript{2u}\oplus
2\emph{E}\textsubscript{2u}\oplus\emph{E}\textsubscript{2g}\oplus
4\emph{E}\textsubscript{1u}\oplus\emph{E}\textsubscript{1g}$. Here, A1g, E2g,
and E1g modes are Raman active, whereas A2u and E1u modes are infrared (IR)
active. All other modes are silent.

Table 1 presents the zone center phonon frequencies calculated using PBEsol
($U$ = 0.0 eV) for the ground state (001) FM ordering in KMn3Sb5. As $U$ is
increased, the phonon frequencies vary, but no phonon instability was observed
within the range of studied $U$ values (see for example the full-phonon
dispersion and the phonon DOS at $U$ = 3.0 eV in Fig. 4 in Appendix A). To
fully provide the behavior of the active Raman and IR frequencies, as a
function of $U$, we present the $A_{1g}$, $A_{2g}$, and $E_{1g}$ modes in Fig.
5(a), (in Appendix A). The $A_{2u}$ and $E_{1u}$ modes are presented in Fig.
5(b). In the overall trend, we can appreciate a softening of the modes when
the $U$ value is increased.

Table 1: DFT-PBEsol calculated zone center phonon frequencies for the (001) FM ground state of KMn3Sb5. Data for different values of $+U$ parameter are provided in Appendix A. Mode |  $\omega$ (cm-1) | Mode |  $\omega$ (cm-1)
---|---|---|---
$A_{1g}$ | 106.8 | $A_{2u}$ | 53.9, 82.5, 232.8
$B_{1g}$ | 135.3 | $B_{1u}$ | 112.5, 251.1
$E_{2g}$ | 131.3 | $B_{2u}$ | 216.7
$E_{1g}$ | 65.5 | $E_{2u}$ | 110.7, 227.0
— | — | $E_{1u}$ | 61.6, 94.1, 184.7, 250.2

Interestingly, we observe that, based on the atomically-project phononic
density-of-states, in Fig. 1(d), the Mn sites contribute strongly at the high-
frequency modes, in between 150 to 250 cm-1. On the contrary, the strongest Sb
contribution can be noted between 0 cm-1 to 150 cm-1. Finally, the K atom’s
contribution to the lattice dynamics is strongly localized around 60 cm-1.

###  III.3 Electronic Structure:

Figure 2: (Color online) Electronic band structure computed in the
ferromagnetic ground state for (a) without SOC and (b) with SOC cases. For the
SOC case, a (001) noncollinear ferromagnetic configuration was used. Here, in
orange ovals, are marked the gapped nodes along the high-symmetry $k$
directions whereas, in red ovals, are shown the symmetry-protected Weyl nodes.
The latter are located along the $M$–$K$, and $\Gamma$–$K$ paths at the
equivalent points in the (0, 0, 1/2) plane.

Moving forward, in Fig. 2 we present the PBEsol calculated electronic bands
obtained with and without considering SOC effects. Here, it is worth recalling
that noncollinear magnetic moments of Mn are aligned along the $z$-axis. As
expected, our band structure calculations reveal metallic features. We note
that the inclusion of $+U$ in our DFT calculations does not result in the
opening of any bandgap; it only moderately modifies the details of the bands
near the Fermi level (see, for example, the electronic band dispersion
obtained for $U$ = 3.0 eV Fig. 6 in Appendix B.).

At first glance, in the absence of SOC, we notice multiple band crossings near
the Fermi level that might be associated with potential topological nodes, in
Fig. 2(a). Some of these band crossings become gapped once the SOC effects are
included, leading to the emergence of a multitude of topologically protected
Weyl nodes in the vicinity of these gapped band crossings and away from the
high-symmetry k-path considered in Fig. 2(b) [49, 50, 51]. Additionally,
several crystal symmetry-protected nodes occur along the high-symmetry
directions of the Brillouin zone, as expected for the kagome symmetry [3, 52].
These crystal symmetry-protected nodes are located, for example, along the
$M$–$K$ and $\Gamma$–$K$ k-paths at energies close to 150 meV above the Fermi
level. There are symmetry-protected nodes also located at the $L$–$H$ and
$H$–$A$ in the (0, 0, 1/2) $k$-plane. All these Weyl nodes serve as sources
and sinks of Berry curvature in the momentum space yielding a large AHC
response in the FM kagome metal KMn3Sb5, as we discuss below. As expected, the
spin-projection of the Weyl nodes is reversed, as can be appreciated from the
spin-polarized electronic band structure shown in Fig. 7 at Appendix C. Here,
only the $S_{z}$ component is observed in the vicinity of the Fermi energy,
which is in agreement with the underlying magnetic structure.

###  III.4 Anomalous Hall Conductivity:

By analyzing the symmetry-allowed properties and electronic structure, we
notice that the ferromagnetic kagome metal KMn3Sb5 may exhibit a tangible
anomalous Hall conductivity response. Consequently, in the _P6/mm’m’_ magnetic
space group (MSG. 191.240), the AHC tensor ($\sigma_{ij}$) is extracted and
presented in Eq. 2. Most of the AHC tensor elements are forced to vanish due
to symmetry, except for $\sigma_{xy}$ and $\sigma_{yx}$ which are allowed to
have nonzero values. However, it is important to note that $\sigma_{yx}$ =
$-\sigma_{xy}$. Thus, the nonzero AHC components lie within the kagome plane
and perpendicular to the orientation of Mn magnetic moments breaking the time-
reversal symmetry.

$\displaystyle\sigma_{6/mm^{\prime}m^{\prime}}=\begin{pmatrix}0&\sigma_{xy}&0\\\
-\sigma_{xy}&0&0\\\ 0&0&0\end{pmatrix}$ (2)

Fig. 3(a) shows the variation of $\sigma_{xy}$ as a function of the chemical
potential near the Fermi level, calculated using the Kubo formula [53, 54].
See note 222the anomalous Hall conductivity component, $\sigma_{xy}$, has been
computed by following the relationship:
$\sigma_{xy}=-\frac{e^{2}}{\hbar}\sum_{n}^{occ}\int_{BZ}\frac{d^{3}k}{(2\pi)^{3}}f_{n}({\bf{k}})\Omega_{n,xy}(\bf{k}),$
(3) where
$\Omega_{xy}(\bf{k})$=$\sum_{n}^{occ}f_{n}(\bf{k})$$\Omega_{n,xy}(\bf{k})$
corresponds to the Berry curvature in the $xy$-plane and it is the result of
the summation of all the occupied $n$-bands and $f_{n}\bf(k)$ represents the
Fermi distribution. In this calculations, the $\Omega_{n,xy}(\bf{k})$ Berry
curvature is estimated such as:
$\Omega_{n,xy}(\textbf{k})=-2i\hbar^{2}\sum_{m\neq
n}\frac{\left\langle\psi_{n,\textbf{k}}\left|v_{x}\right|\psi_{m,\textbf{k}}\right\rangle\left\langle\psi_{m,\textbf{k}}\left|v_{y}\right|\psi_{n,\textbf{k}}\right\rangle}{\left[E_{m}(\textbf{k})-E_{n}(\textbf{k})\right]^{2}}$
(4) In the latter equation, $\psi_{n,\textbf{k}}$ and $v_{x}$, $v_{y}$ are the
Bloch functions and the velocity operators, respectively. In this step, an
320$\times$320$\times$320 $k$-mesh was used. for more details. As it can be
observed, the AHC at the Fermi level is $\sigma_{xy}$ = 314 S$\cdot$cm-1. This
is comparable to the values obtained in similar kagome compounds with large,
or even giant, AHC such as $\sigma_{xy}$ = 380 S$\cdot$cm-1 in LiMn6Sn6 [56],
$\sigma_{xy}$ = 1130 S$\cdot$cm-1 in Co3Sn2S2 [3], and $\sigma_{xy}$ = $-$400
S$\cdot$cm-1 in Fe3Sn2 [57]. Notably, by tuning the Fermi level in kagome
metal KMn3Sb5, it is possible to achieve a $\sigma_{xy}$ value close to or
even larger than 1000 S$\cdot$cm-1.

In Fig. 3(b), we present the Berry curvature components $\Omega_{x}$,
$\Omega_{y}$, and $\Omega_{z}$ calculated along the high-symmetry directions
in the Brillouin zone. The largest contributions to the Berry curvature are
obtained for $\Omega_{z}$ at the $H$-point and along the $H$–$K$ path. The
values extend up to $-$7000 Å2, though in Fig. 3(b), the $y$-axis is displayed
only up to $-$1000 Å2 to allow for the observation of smaller contributions
along the other Brillouin zone paths. By correlating the Berry curvature with
the electronic band structure calculated with SOC, we notice that there is a
SOC-induced gapped node at the $H$ point near the Fermi energy (see Fig. 2),
suggesting the presence of potential gapless Weyl nodes in the vicinity of the
$H$ point. This explains the divergent Berry curvature at the Weyl nodes [58]
which are the main source of the large AHC response in KMn3Sb5. Similar
behavior has also been reported in antiperovskite compounds such as Mn3NiN and
V3AuN [8, 10]. Fig. 3(b) also displays the hexagonal Brillouin zone obtained
for $P6/mmm$ space group in which, the high-symmetry points and relevant
$k$-paths are marked. Fig. 3(c) shows the calculated Berry curvature
$\Omega_{z}$(k) projected onto the Fermi surface of KMn3Sb5. Here, the red and
blue colors denote the positive and negative $\Omega_{z}$(k) values.
Throughout the entire Brillouin zone, multiple strong concentrations of Berry
curvature can be observed.

Figure 3: (Color online) (a) Anomalous Hall conductivity, $\sigma_{xy}$
component, calculated (PBEsol) as a function of the chemical potential near
the Fermi energy ($E_{F}$) for the ferromagnetic (001) KMn3Sb5 within the
_P6/mm’m’_ magnetic space group (MSG. 191.240). (b) Berry curvature,
$\Omega_{x}(k)$, $\Omega_{y}(k)$, and $\Omega_{z}(k)$ components, calculated
using DFT-PBEsol and integrated along the Brillouin zone $k$-path. In the
inset, we display the hexagonal Brillouin zone for the SG. 191 space group in
which, the high-symmetry points and relevant $k$-paths are marked. (c)
$\Omega_{z}$(k) component of Berry curvature projected onto the Fermi surface
of ferromagnetic (001) KMn3Sb5 using a color map. Blue and red colors denote
the negative and positive components of $\Omega_{z}$(k), respectively.

###  III.5 Role of Electronic Correlations:

As shown in previous reports [59, 60, 61, 9], the kagome lattices with
magnetic cations, especially cations having partially-filled $3d$ valence
shell, exhibit strong electronic-correlation phenomenon that leads to the
emergence of exciting electronic properties. Therefore, to investigate the
effects of the onsite Coulomb interaction on Mn-$3d$ electrons and its impact
on the structural, phononic, and electronic properties, we employed DFT+$U$
calculations which capture the electronic-correlations effects at the mean-
field level [35]. This analysis is particularly important due to the lack of
experimental reports on KMn3Sb5, and it may help experimentalists in
identifying suitable observables for future investigations.

In Table 2, we present the evolution of the lattice parameters, the magnetic
moment per Mn atom, and anomalous Hall conductivity ($\sigma_{xy}$) values as
a function of the $+U$ Coulomb correction parameter applied to Mn-$3d$
electrons. Our results reveal an increase in the in-plane lattice parameters
($a=b$) while decrease in the out-of-plane ($c$) lattice parameter with
increasing $U$. This demonstrates the dominant role of the Mn–Mn bondings and
interactions within the kagome lattice. The observed expansion in the in-plane
lattice parameters is in agreement with the previous observation of the strong
spin-lattice coupling related to the negative thermal expansion in the kagome
antiperovskite Mn3NiN [9]. It is worth noting that with increasing $U$ value,
some phonon modes (especially the in-plane vibrational modes) soften due to
the increased in-plane lattice parameters. However, no unstable phonon modes
are observed within the range of considered $U$ values. As expected, the
magnetic moment per Mn-site is increased due to the increased electron
localization governed by the Coulomb term. Interestingly, the $\sigma_{xy}$
component remains substantial at all the considered $U$ values.

Finally, we fully relax the KMn3Sb5 FM (001) structure using the meta-GGA scan
[62] and r2scan [63] functionals, aiming to obtain more precise values of the
interesting observables. We find that the magnetic moment is 3.022
$\mu_{B}$$\cdot$Mn-1 and 3.229 $\mu_{B}$$\cdot$Mn-1 for the scan and r2scan
functionals, respectively. Moreover, the lattice parameters are $a$ = 5.435 Å
and $c$ = 9.153 Å for the scan functional whereas $a$ = 5.512 Å and $c$ =
9.042 Å for the r2scan functional. Using these meta-GGA values as observables,
we suggest that a $U$ value close to 2 eV could reasonably reproduce the
structural and electronic properties of KMn3Sb5. However, it is important to
note that more systematic studies are needed to fully comprehend the electron-
correlation effects in this studied kagome metal.

Table 2: Lattice parameters ($a=b$, and $c$), magnetic moment, and $\sigma_{xy}$ AHC component as a function of different $U$ values in our PBEsol+$U$ calculations. Values obtained using the meta-GGA scan, and $r^{2}$scan functionals are also listed. All the structures were fully relaxed considering the ground state (001) noncollinear ferromagnetic ordering.  XC-type |  $a$ (Å) |  $c$ (Å) |  $m$ ($\mu_{B}$$\cdot$Mn-1) |  $\sigma_{xy}$ (S$\cdot$cm-1)
---|---|---|---|---
PS+$U$=0.0 eV | 5.337 | 9.028 | 2.458 | 314
PS+$U$=1.0 eV | 5.372 | 9.014 | 2.758 | 217
PS+$U$=2.0 eV | 5.470 | 8.875 | 3.312 | 241
PS+$U$=3.0 eV | 5.602 | 8.606 | 3.743 | 172
scan | 5.435 | 9.153 | 3.022 | —
r2scan | 5.512 | 9.042 | 3.229 | —

##  IV Conclusions

In this study, we performed comprehensive first-principles DFT calculations to
investigate the properties of the novel Mn-based kagome metal KMn3Sb5. Our
results indicate that KMn3Sb5 is both structurally and vibrationally stable,
with no observed unstable phonon modes. This is in contrast to the KV3Sb5
compound in which, phonon instability leads to the appearance of CDW phase. A
detailed investigation of the possible candidate magnetic configurations
(noncollinear FM as well as AFM) reveal the (001) ferromagnetic ordering as
the ground state in KMn3Sb5, although various other magnetically-frustrated
kagome metals favor noncollinear chiral antiferromagnetic orderings
$\Gamma_{4g}$ and $\Gamma_{5g}$, similar to the case of Mn3NiN antiperovskite,
due to the inherent magnetic frustration. The electronic structure shows
multiple nodal crossings associated with Weyl nodes near the Fermi energy,
indicating topologically nontrivial features present in the studied material.
Notably, the ferromagnetic order breaks the time-reversal symmetry, resulting
in a tangible anomalous Hall conductivity response. We find a value of
$\sigma_{xy}$ = 314 S$\cdot$cm-1, which moderately varies with increasing
onsite Coloumb parameter $U$ in our PBEsol+$U$ calculations. We find that the
predicted $\sigma_{xy}$ in KMn3Sb5 to be considerably large when compared to
other similar kagome compounds. We suggest a $U$ value in the range of 2-3 eV
to reasonably reproduce the properties of the ferromagnetic kagome metal
KMn3Sb5. Our findings are expected to motivate experimentalists to pursue the
synthesis, realization, and subsequent confirmation of the predicted
properties in KMn3Sb5.

## Acknowledgments

Calculations presented in this article were carried out using the GridUIS-2
experimental testbed and Center for Integrated Research Computing (CIRC)
facilities at the University of Rochester. The GridUIS-2 testbed was developed
under the Universidad Industrial de Santander (SC3-UIS) High Performance and
Scientific Computing Centre with support from UIS Vicerrectoría de
Investigación y Extensión (VIE-UIS) and several UIS research groups. We also
acknowledge the computational resources awarded by XSEDE, a project supported
by National Science Foundation grant number ACI-1053575. The authors also
acknowledge the support from the Texas Advances Computer Center (with the
Stampede2 and Bridges supercomputers).

##  Appendix A Phonons under the $+U$ correction.

Figure 4: (Color online) (a) Full phonon-dispersion curve and (b) Atom-
projected phonon density of states obtained at the Coulomb $+U$ value of 3.0
eV as a correction in the 3$d$:Mn states at the KMn3Sb5 Kagome compound.
Figure 5: (Color online) BZ zone center Raman and IR active modes as a
function of the $+U$ parameter in the KMn3Sb5.

##  Appendix B Electronic bands under the $+U$ correction.

Figure 6: (Color online) (a) Full dispersion bands obtained at the Coulomb
$+U$ value of 3.0 eV as a correction in the 3$d$:Mn states at the KMn3Sb5
Kagome compound.

##  Appendix C Spin-polarized band structure.

Figure 7: (Color online) Spin-polarized electronic band structure showing the
$S_{x}$, $S_{y}$, and $S_{z}$ spin components, respectively. Here, it can be
noted that the contributions from the $S_{x}$ and $S_{y}$ spin projections are
negligible, and only, the $S_{z}$ component is appreciated in the EF
proximity. Additionally, the Weyl nodes can be identified in the path between
the M–K and K–$\Gamma$ points and consequently, in the associated point, L–H
and H–A in the (0, 0, 1/2) plane. Here, the blue and red colors denote the up-
and down-spin projections, respectively.

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