#
Fluctuations and Correlations in Causal Diamonds

Tom Banks
Department of Physics and NHETC
Rutgers University, Piscataway, NJ 08854
E-mail<EMAIL_ADDRESS>

Willy Fischler
Department of Physics and Texas Cosmology Center
University of Texas, Austin, TX 78712
E-mail<EMAIL_ADDRESS>
###### Abstract

We argue that the correlation between the modular Hamiltonians of two causal
diamonds related by a finite time translation is equal to the uncertainty in
the modular Hamiltonian of the overlap diamond in empty Minkowski space-time.
In a separate section we investigate the derivation of length fluctuations in
Minkowski diamonds from the ’t Hooft commutation relations, and the connection
of those fluctuations to entropy fluctuations.

##  1 Introduction

In[1], Verlinde and Zurek made the fascinating suggestion that quantum
gravitational fluctuations might be measurable in modern interferometer
experiments. A variety of subsequent papers[2] have demonstrated that large
fluctuations of the sort suggested in[1] indeed occur in AdS/CFT, or follow
from other plausible theoretical conjectures. At present we do not have a
calculation, based on some sort of fundamental principles, of the actual
observables relevant to interferometer experiments. The aim of the present
paper is to make incremental progress towards that goal. The basic
interferometer observable is the time correlation function of arrival time
fluctuations in single passes of the interferometer separated by a proper time
interval $\tau$ at the beam splitter. We will instead calculate the time
correlation function of the modular Hamiltonians of two causal diamonds along
the same geodesic in Minkowski space, with proper time separation $\tau$ . We
will argue that when there is no overlap between the diamonds, the correlation
function is essentially zero, and that it is otherwise equal to the
fluctuation of the modular Hamiltonian of the overlap diamond .

In the second part of the paper we will try to relate the modular fluctuation
to the ’t Hooft commutation relations[3] which are also related to the
fluctuation of lengths along individual interferometer trajectories studied
in[1]. Our formulae differ from those found in[4]. We are unable to calculate
precise numerical coefficients for either the modular or the length
fluctuations using these methods. In particular, we cannot check the
relation[5][6][7]

$(\Delta K)^{2}=\langle K\rangle.$ (1)

However we argue that if the Planck scale regularization of the transverse
geometry of causal diamonds is ”fuzzy”, then the calculation of entropy
fluctuations from length fluctuations is compatible with this formula and the
normalization of length fluctuations ”at the same angle” is fixed by this
entropy fluctuation coefficient. This leads us to a conjecture about the time
correlation function of length fluctuations, which might have direct relevance
to interferometer experiments.

##  2 Time Correlations in terms of Equal Time Fluctuations

The basic observable in an interferometer experiment is the correlation
between arrival times of signals emitted from the beam splitter in different
passes of the apparatus, separated by proper time $\tau$ at the beam splitter.
In the interpretation of these signals in terms of changes in the space-time
metric, these are interpreted as distortions of the length of the arms of the
interferometer. We are interested in possible quantum gravitational
fluctuations of these lengths in an empty causal diamond in Minkowski space.
As usual, we neglect the back-reaction of the measuring apparatus on the
causal diamond in which it resides. A related, and more theoretically
tractable quantity to calculate is the time dependent correlation of modular
Hamiltonians for two causal diamonds along the same Minkowski geodesic,
separated by proper time $\tau$. A pictorial description of the setup is shown
in Fig.1 . There are two causal diamonds, $D_{1,2}$ which may or may not
overlap, and a large diamond $D_{L}$ which exactly contains them both. The
modular Hamiltonians for those three diamonds are denoted $K_{1},K_{2},K_{L}$
respectively. The quantity we wish to calculate is

$G(\tau)\equiv{\rm Tr}\ [e^{-K_{L}}(K_{1}K_{2})]-{\rm Tr}\
[e^{-K_{L}}K_{1}]{\rm Tr}\ [e^{-K_{L}}K_{2}],$ (2)

as a function of the proper time difference $\tau$ between the past tips of
$D_{1}$ and $D_{2}$.

Figure 1: Shifted Diamond Correlation Function.

We will assume some basic principles in order to perform this calculation.
These principles have been described in[11], but in fact we will need much
less than the full set of assumptions of that paper in order to establish our
result. Any causal diamond can be covered by both a past and future directed
nested set of diamonds Fig.2.

Figure 2: Future and past directed nested coverings of a causal diamond.

The time differences between the future (past) tips of the future (past)
directed nesting are all $L_{P}$. If we consider the operator
$e^{iK(t)}e^{-iK(t\pm L_{P})}$, for each pair of sub-diamonds in a future/past
directed nesting, this defines a sequence of unitary embeddings of the Hilbert
space of the smaller diamond into the larger diamond. We can view this as a
choice of time slicing in Lorentzian signature Minkowski space. Here, we’re
implicitly assuming all Hilbert spaces are finite dimensional and that the log
of their dimension is roughly given by the Bekenstein-Hawking-Jacobson-Bousso
area law. This implies in particular that a single Planck step in time gives
rise to an exponentially large change in dimension, once the dimension is
macroscopic.

The other important assumption we will make is the Milne red-shift: we assume
all of the quantum degrees of freedom in an empty causal diamond reside on the
stretched horizon of the diamond. The time evolution we are describing is
proper time along the geodesic traversed by the beam splitter. Dynamics on the
stretched horizon of a diamond of size $L$, has a time scale of order $L$ in
geodesic proper time.

We can now proceed to calculate $G(\tau)$. We first consider the case where
$D_{1}$ and $D_{2}$ have no overlap. In this case, the Hilbert space of
$D_{L}$ is exponentially larger than either of the subsystem Hilbert spaces.
Page’s theorem[10] says that, for a Haar random state on the Hilbert space of
$D_{L}$ the states on the subsystem Hilbert spaces are exponentially close to
maximally uncertain, which implies vanishing of $G(\tau)$: there is no
correlation between the two subsystems. The modular Hamiltonian on the Hilbert
space of $D_{L}$ is, according to[5][6][7][11], the $L_{0}$ generator of an
interacting two dimensional CFT, with interactions that mix up the individual
degrees of freedom describing the subsystems $D_{i}$ in a complicated way.
While we have no simple proof of the analog of Page’s theorem in this case it
seems extremely likely that $G(\tau)$ will vanish in this case as well.

Another limiting case is $\tau=0$. In this case $D_{L}=D_{1}=D_{2}$ and
$G(0)=(\Delta K_{1})^{2}$ the uncertainty, or fluctuation in the modular
Hamiltonian of either of the two subsystems. These two limits motivate the
following general conjecture:

$G(\tau)={\rm Tr}\ [e^{-K_{12}(\tau)}(K_{12}(\tau)-\langle
K_{12}(\tau)\rangle)^{2}],$ (3)

where $K_{12}(\tau)$ is the modular Hamiltonian of the overlap diamond, and
the angle brackets mean expectation value in the density matrix
$e^{-K_{12}(\tau)}$. We will ”prove” this conjecture by a sort of ”continuous
mathematical induction”. We assume that it is true for some value of $\tau$
and then argue that it continues to hold for $\tau+\delta\tau$ for small
enough $\delta\tau$. Since it holds for $\tau=0$ and for $\tau>2T$, the total
proper time in the diamonds $D_{1,2}$, this establishes the conjecture.

Our argument is pictorially described in Fig.3 .

Figure 3: Infinitesimal Diamond Shift.

We are comparing the situation where the diamond $D_{2}$ is the orange diamond
in this figure, to that in which it is the blue diamond, shifted forward in
time by $\delta\tau$. Two things have occurred. The diamond $D_{L}$ has grown,
by the addition of degrees of freedom that are associated with the region
between the blue and orange dotted lines. The overlap diamond
$D_{12}(\tau+\delta\tau)$ has shrunk, losing degrees of freedom associated
with the region labeled $\delta$ in the figure. Our assumption about the
relation between modular Hamiltonians, time evolution and embedding maps,
combined with the Milne red shift, implies that the new degrees freedom in
$D_{L}$ cannot yet be correlated with those in $D_{1}$ and do not contribute
to $G(\tau)$. The degrees of freedom in region $\delta$ are causally connected
to the new degrees of freedom in $D_{L}$, though not to those in the shifted
(blue) $D_{2}$. Since they travel with the speed of light nothing prevents
them from being entangled with those new variables. Monogamy of entanglement
then implies that they are no longer entangled with the variables in $D_{1}$
and so cannot contribute to $G(\tau)$.

By assumption, $G(\tau)$ before the shift by $\delta\tau$ was just the modular
fluctuation of $K_{12}(\tau)$. We have just argued that the entire effect of
the shift is simply to disentangle a small fraction of the total number of
variables in the Hilbert space of $D_{12}(\tau)$ leaving only those variables
associated with the smaller overlap. In the explicit models of[11] we are just
removing a small fraction of the total number of fermion fields in a Thirring
model involving the product of two commuting $U(1)$ currents. $G(\tau)$ will
still be given by the modular fluctuation in the smaller model.

##  3 Modular Fluctuations and ’t Hooft Commutators

In 1998, Carlip and Solodukhin argued that the effective theory on black hole
horizons, for non-extremal black holes in space-times with non-negative
cosmological constant (c.c.) was a (cut-off) $1+1$ dimensional CFT on an
interval. In[7] this argument was generalized to a large class of causal
diamond boundaries. It is even a correct description of boundary anchored Ryu-
Takayanagi diamonds in AdS space, or of single nodes in a tensor network
description of large AdS black holes in CFTs with a large radius AdS dual. For
any such diamond boundary, the modular fluctuations satisfy

$(\delta K)^{2}=\langle K\rangle.$ (4)

Recently, Verlinde and Zurek[4] have presented a derivation of this result
based on ’t Hooft’s shock wave commutation relations. This is extremely
interesting because it points to a possible connection between the more easily
accessible entropy fluctuation calculations and the angular dependent
correlations necessary for making connection to actual experiments. The
present section is devoted to our attempts to forge similar connections.

We will first present a calculation of the equal time two point functions

$\langle X^{\pm}(t,\Omega)X^{\pm}(t),\Omega^{\prime})\rangle,$ (5)

where we will explain below the meaning of the angle brackets. We then use
these in two ways. First, following[1] we calculate the length fluctuations
along a single light ray trajectory in a causal diamond. Then, combining such
calculations for many angles, we calculate the entropy fluctuations. Both
calculations give finite answers but the entropy calculation has an ambiguous
normalizations having to do with how we discretize things at the Planck scale.
The normalization of the entropy calculation can be fixed by invoking the C-S
answer.

We begin from ’t Hooft’s commutation relations for the fluctuating null
coordinates near the boundary of a causal diamond in Minkowski space[3][12].
Near the diamond boundary the largest components of Einstein’s equations are
those in the two nearly null directions. They imply that the transverse metric
$G_{ij}$ is independent of those null coordinates and that the two dimensional
Lorentzian metric

$\hat{g}_{ab}=g_{ab}-g_{ai}G^{ij}g_{bj},$ (6)

is flat. The fluctuations away from flat metrics are parametrized by a
conformal factor and a coordinate transformation from the Minkowski form
$\eta_{ab}L_{P}^{2}$. Setting the conformal factor to zero we get

$\hat{g}_{ab}=\eta_{ab}L_{P}^{2}+L_{P}(\eta_{\alpha
b}\partial_{\beta}X^{b}+\eta_{a\beta}\partial_{\alpha}X^{a})\equiv\eta_{ab}+h_{ab}.$
(7)

As shown in[12], when this is plugged into the Einstein action, we get a
topological action which is a pure boundary term and leads to the commutation
relations

$[X^{+}(t,\Omega),X^{-}(t,\Omega^{\prime})]=i(J^{2}+1)^{-1}(\Omega,\Omega^{\prime})L_{P}^{2}.$
(8)

Here $J^{2}$ is the squared angular momentum operator on the two sphere. Note
that we are using conventions where metrics have dimensions and coordinates
are dimensionless. Also, the parameter $t$ is $x^{+}$ on the past boundary and
$x^{-}$ on the future boundary. Furthermore, there is always a gauge choice in
which $X^{+}$ vanishes on the past boundary, and $X^{-}$ on the future. Thus,
this commutation relation applies only on the holographic screen or
bifurcation surface of the diamond boundary. Note however that every point on
the diamond boundary lies on the bifurcation surface of some diamond in either
a past or future directed nested covering of the diamond by smaller diamonds.
Thus, we have an ’t Hooft commutation relation at every point on the diamond
boundary, but they refer to operators that operate in different Hilbert
spaces. We should also emphasize that these commutation relations are
incompatible with our fundamental assumption that the diamond Hilbert spaces
are finite dimensional. We view them as approximate statements, which need
regularization. In[11], one of us proposed a regularization of these operators
in terms of approximate $U(1)$ currents in cut-off CFTs. In the present paper
we will work with the infinite dimensional algebras, but use them only to
calculate the Gaussian fluctuations in the $X^{\pm}$ variables.

In order to calculate correlation functions, we must decide on the state
representing an empty diamond. Fortunately, symmetry considerations are enough
to determine the state. Every empty diamond in Minkowski space is time
reversal invariant. Time reversal takes the canonically conjugate variables
$X^{\pm}$ into each other. This implies that the uncertainties of the angular
momentum components of $X^{\pm}$ must be identical, but that implies that the
state is the minimal uncertainty state for both $X^{+}$ and $X^{-}$.

$(\Delta X^{\pm}_{kl})^{2}=\frac{(l(l+1)+1)^{-1}}{2}L_{P}^{2}.$ (9)

Furthermore, the mixed two point function between $X^{+}$ and $X^{-}$,
vanishes. We conclude that

$\langle
X^{\pm}(t,\Omega)X^{\pm}(t,\Omega^{\prime})\rangle=\frac{1}{2}(J^{2}+1)^{-1}(\Omega,\Omega^{\prime})L_{P}^{2}.$
(10)

It is convenient to work with the angular momentum decomposition of these
relations, because this allows us to incorporate the finite entropy assumption
that we have always used in the Holographic Space Time (HST) formalism. The
variables in HST are sections of the spinor bundle on the two sphere, with an
angular momentum cutoff. In the updated version of HST compatible with the
Carlip-Solodukhin ansatz[11], the expansion coefficient multiplying each
angular momentum component of the transverse spinor is a one plus one
dimensional massless fermion field, living on an interval (the stretched
horizon) near the boundary of the causal diamond, with a UV cutoff. The UV
cutoff is chosen so that the entropy of the smallest transverse angular
momentum field theory coincides with that of the smallest causal diamond to
which the semi-classical CS analysis applies. According to our present
understanding, only some far future experiment might tell us what that size
is. However, the use of Cardy’s theorem by CS allows us to put a plausible
lower bound on it. Cardy’s theorem gives a more or less correct count of the
number of states of a single Dirac fermion on an interval when it has between
$10-15$ momentum modes.

Larger causal diamonds are modeled by adding angular momentum multiplets,
according to the CS rule. We hypothesize that the $X^{\pm}$ variables are
bilinears in these fermion fields, and so carry a maximal angular momentum
equal to twice that of the spinors. Both these maximum values scale like the
proper time $2T$ in the diamond.

The length fluctuation along a light ray trajectory on the diamond boundary
has to be treated with great care. Formally, the length is the sum of
integrals of $h_{++}$ along the past portion of the trajectory and $h_{--}$
along the future portion, and the metric variables are derivatives of the
$X^{\mp}$ variables. However, we have just argued that the $X^{\pm}$ variables
at different spheres on the boundary, which are the holographic screens of
different diamonds in a nested cover, are independent random variables. Thus
$X^{\mp}$ are nowhere differentiable functions of their time arguments.
Instead of simply integrating a total derivative, we divide proper time along
the geodesic into steps of size $L_{P}$ and define the infinitesimal length
fluctuation on the corresponding segment of the boundary trajectory to be the
difference of the $X^{\pm}$ variables at those two points along the
trajectory. The length fluctuation thus has an angular momentum decomposition

$\langle L_{Jm}(i)L_{Jm}(i)\rangle=(J^{2}+1)^{-1}L_{P}^{2},$ (11)

twice as large as that of the individual ’t Hooft variables. Furthermore, the
length fluctuations for independent segments of the trajectory are independent
random variables, because they each contain an $X^{\pm}$ variable not shared
by any previous segment. Thus, the total length, which is the sum of these
individual randomly fluctuating interval lengths, satisfies

$\langle
L^{T}_{Jm}(i)L_{Jm}(i)\rangle=\frac{2T}{L_{P}}(J^{2}+1)^{-1}L_{P}^{2}.$ (12)

Note that there is an additional hidden $T$ dependence in this formula,
because the maximal angular momentum grows like $T$. If one were really
contemplating light rays that were delta function localized in angle, this
would lead to an extra logarithmic enhancement of the length fluctuation. As
we’ll see below, the constraint that the entropy of a finite area diamond is
finite leads to the conclusion that its holographic screen has a pixel of
minimal angular area . The length fluctuation of a light beam should be
averaged over this pixel in some way. Consistency with the Carlip-Solodukhin
formula for entropy fluctuations will imply that the logarithmic enhancement
of the naive formula has a cutoff at a fixed finite value, related to the UV
cutoff on the $1+1$ dimensional fermion fields. We’ll see that it is natural
to obtain a finite value because ”pixels” made of a finite number of angular
momentum modes have only ”fuzzy” localization, but that there is no obvious
”correct” way to define the pixelization.

###  3.1 Entropy fluctuations from the ’t Hooft Commutation relations

Carlip[5] and Solodukhin[6] (C-S) showed that the fluctuations around the
horizon of generic black holes contained the hydrodynamic equations of a $1+1$
dimensional conformal field theory (CFT) living on a strip (the stretched
horizon), with central charge proportional to the horizon area. Using Cardy’s
formula they derived the Bekenstein-Hawking area formula. The authors of[7]
pointed out that the C-S derivation worked for arbitrary causal diamonds111To
understand large AdS black holes one needs to invoke the tensor network
description of AdS space. For CFTs with an Einstein-Hilbert dual, he C-S
argument works at the level of a single node of the tensor network, but the
entropy of large black holes is dominated by excitations that couple many
nodes together. The fluctuation formula for the modular Hamiltonian of large
AdS black holes is that of a higher dimensional CFT. and implied the universal
fluctuation formula for the modular Hamiltonian

$\langle(K-\langle K\rangle)^{2}\rangle=\langle K\rangle.$ (13)

This had been shown[8][9] to hold for boundary anchored Ryu-Takayanagi
diamonds in arbitrary AdS/CFT models with Einstein-Hilbert duals.

The description of the geometry of the holographic screen of a diamond in
terms of a $1+1$ CFT arose independently in our model of holographic
cosmology[14]. There, the CFT was naturally realized in terms of $1+1$
dimensional fermion fields, which are the expansion coefficients of a spinor
field on the holoscreen in terms of eigenspinors of the Dirac operator. The
idea of describing Riemannian geometry in terms of the Dirac operator is due
to Connes[13]. Finite central charge implies a UV cutoff on the Dirac
eigenvalue. Finite entropy implies that one should also cut off the number of
momentum modes of the individual fermion fields on the stretched horizon.
Mathematically it’s natural to take this cutoff to be at the point where
Cardy’s formula begins to be accurate, which is about $10-15$ momentum modes
for a single fermion field. Physically this corresponds to the smallest
diamond in Planck units for which one can believe the semi-classical,
hydrodynamic arguments of C-S. Ultimately the correct value of this cutoff in
a theory applying to the real world would have to be taken from experiment.

In Minkowski space, the cutoff on the transverse Dirac eigenvalue is a cutoff
on angular momentum. The expectation value of the radius of the holographic
screen is proportional to this cutoff, implying a UV/IR connection like
Maldacena’s scale/radius duality. Fluctuations in the unimodular geometry of
the screen are described by fluctuations of the two dimensional fermion
fields, which are the expansion coefficients of the spinor field on the
holographic screen in terms of Dirac eigenspinors on the classical screen
geometry.

In this paper however, we want to make a more semi-classical estimate of
fluctuations, using the ’t Hooft commutation relations. The cutoff on angular
momentum implies that we have only $\sum_{1/2}^{L_{M}}(2L+1)$ angular momentum
modes of the spinor fields on the holographic screen. Using the Bekenstein-
Hawking-Jacobson-Bousso formula we have

$\frac{A}{4G}\approx 2K\sum_{1/2}^{L_{M}}(2L+1){\rm ln}\ 2.$ (14)

where $K$ is the number of momentum modes of each $1+1$ dimensional Dirac
field on the stretched horizon. In[11] it was suggested that the ’t Hooft
operators are constructed out of $U(1)$ currents bilinear in the underlying
fermions. These have angular momenta that range from $0$ to $2L_{M}$. Thus the
number of ”pixels” into which we can break up our screen is
$\sum_{L=0}^{2L_{M}}(2L+1)$. A pixel, by definition, is the best approximation
of the delta function on the smallest angular subset of the sphere that we can
make out of those spherical harmonics. There are of course many ways to define
”best approximation”. We could use any of the $L^{p}$ norms on the space of
continuous functions. Since the spherical harmonics are infinitely
differentiable, a wide variety of other norms are also available. However we
define ”best”, we can form a basis of the space of cutoff functions,
consisting of ”pixel $\delta$-functions localized” on the points of an anti-
prism distributed over the surface of the sphere.

We now hypothesize that we can compute the entropy fluctuation by computing
the length fluctuation in each of these pixels as independent Gaussian random
variables, with covariance computed from the ’t Hooft uncertainty relations
described above, averaged over the pixel. Indeed, the area of a pixel is just
its angular area, fixed by the number of pixels, times the radius of the
screen. While the number of independent pixels is the same as the number of
independent angular momentum components, the form of the fuzzy pixel
characteristic functions is not determined by any principle we have been able
to invent. What is clear is that there will be many choices for which

$\int
d^{2}nd^{2}n^{\prime}\delta_{LM}(\Omega)\delta_{LM}(\Omega^{\prime})(J^{2}+1)^{-1}(\Omega,\Omega^{\prime})=F,$
(15)

is finite as $L_{M}\rightarrow\infty$. Here $\delta_{LM}$ is the cutoff
$\delta$-function.

If we now view the fluctuation in area of a large diamond as built up from the
fluctuation in area of individual nested diamonds, we can write an area
diffusion equation

$\frac{dP(A)}{dt}=FAL_{P}\frac{d^{2}P}{dA^{2}},$ (16)

for the probability distribution of areas. The $L_{P}$ in this equation
appears because the time step in our nested sequence of diamonds is the Planck
time, and the constant $F$ represents the ambiguity coming from our lack of
knowledge of how to regularize the transverse geometry at the Planck scale.

If we now use the Bekenstein-Hawking-Jacobson-Bousso law for the entropy of
the diamond we find that the ’t Hooft commutation relations predict area
fluctuations that are proportional to the entropy of the diamond, with a
proportionality constant that depends on $F$. Note that in terms of the number
of pixels defined above, both the ’t Hooft calculation of area fluctuations
and the C-S calculation of entropy fluctuations are proportional to the number
of pixels, but the C-S calculation contains the ambiguous factor related to
the UV cutoff on $1+1$ dimensional fermion fields on the stretched horizon.
Recall that this cutoff was also related to UV ambiguities in the Planck scale
transverse geometry.

Both the ’t Hooft commutation relations and the C-S ansatz of $1+1$
dimensional CFT on the stretched horizon are bold attempts to extract the
quantum dynamics of gravity from its classical hydrodynamics, the Einstein
field equations. What we have shown is that, at the level of quadratic
fluctuations of entropy, these two approaches are consistent with each other,
up to Planck scale ambiguities which we believe are inevitable. We do not
think that these ambiguities can be resolved without experimental input. The
arguments presented in[15] suggest that neither perturbative string theory nor
AdS/CFT can resolve them. The C-S approach makes much more far reach
assumptions about dynamics on the stretched horizon of causal diamonds, and
consequently it makes a more precise prediction for the coefficient in the
entropy fluctuations. We can use this to fix the coefficient $F$ in the length
fluctuation calculation.

The connection between length and entropy fluctuations leads us to a
conjecture about the time correlation function of length fluctuations, namely
that it will be given by the length fluctuation in the overlap diamond. On a
very naive level, the argument we gave in the first section for the modular
Hamiltonian seems valid also for the length fluctuation, but we do not find
this convincing.

We conclude this section by noting that our approach appears to disagree with
that of[4]. Those authors claim to calculate the modular Hamiltonian of a
diamond directly in terms of the ’t Hooft operators. They then argue that,
given their formula, both the expectation value and the fluctuation of the
modular Hamiltonian are infinite but that the infinities are the same, so that
the ratio of the two quantities is $1$.

##  4 Conclusions

We have shown that the time dependent correlation of modular Hamiltonians in
two shifted causal diamonds can be computed in terms of the uncertainty in the
modular Hamiltonian of the overlap between the two diamonds. We then used the
’t Hooft commutation relations, together with a regulator scheme inherited
from the HST formalism to calculate the length fluctuation in a single
diamond, verifying the large value suggested in[1] and consistent with[2].
Finally, we showed how the length and entropy fluctuation calculations could
be related, though here our results were not as precise as one might have
hoped. Nonetheless, taken together they suggest that the length correlation in
two shifted diamonds is equal to the length fluctuation in the overlap of the
two diamonds and that the logarithmic divergence in the latter is rendered
finite in a way that can be calculated from the C-S fluctuation formula for
entropy. This should lead to predictions for the Power Spectral Density in
interferometer experiments.

Acknowledgements

We thank Kathryn Zurek for sharing her insights about fluctuations in causal
diamonds. The work of TB is supported in part by the DOE under grant DE-
SC0010008. The work of WF is supported by the NSF under grant PHY-1914679

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# Exploration of the spin-phonon coupling in the noncollinear
antiferromagnetic antiperovskite Mn3NiN

L. Florez-Gomez1,2 1School of Physics, ficomaco Research Group, Universidad
Industrial de Santander, Carrera 27 Calle 09, 680002, Bucaramanga, Colombia.
2Centro de Investigación y Estudios Avanzados del IPN - cinvestav-Querétaro,
MX-76230, Querétaro, México.  W. Ibarra-Hernandez3 0000-0002-5045-4575
3Facultad de Ingeniería, Benemérita Universidad Autónoma de Puebla, Apartado
Postal J-39, Puebla, Pue. 72570, México.  A. C. Garcia-Castro4
0000-0003-3379-4495<EMAIL_ADDRESS>4School of Physics, cimbios Research
Group, Universidad Industrial de Santander, Carrera 27 Calle 09, 680002,
Bucaramanga, Colombia.

###### Abstract

Antiferromagnetic antiperovskites, of the form Mn${}_{3}B$N ($B$ = Ni, Cu, Zn,
Sn, Ir, and Pt), have shown an outstanding behavior in which, giant negative
thermal expansion and chiral magnetic structures are intertwined. As such,
aiming to shed light on the magnetostructural behavior, related to the
magnetic ordering and structure of this type of materials, we studied by
theoretical first-principles calculations the spin-phonon coupling in the
Manganese-based antiperovskite Mn3NiN, as a prototype in the Mn${}_{3}B$N
family. We found a strong spin-lattice coupling by means of the understanding
of the phonon-dispersion curves obtained when including the chiral
noncollinear magnetic structure. To do so, we highlight the importance of the
exchange-correlation scheme selected and its influence on the electronic,
structural, and vibrational degrees of freedom. Finally, we found two unstable
vibrational modes at the $M$\- and $R$-points when the magnetic ordering is
switched to ferromagnetic from the chiral $\Gamma_{4g}$ and $\Gamma_{5g}$. The
latter coupling was observed as a key signature of the strong spin-lattice
interaction.

DOI:

##  I Introduction

Perovskite oxides, and perovskite structures in general ($i.e.$ $ABX_{3}$),
have been in the center of debate and development in the condensed matter
field with phenomena ranging from multiferroism and magnetoelectricity [1, 2,
3, 4, 5] to applications and development in solar cells with hybrid-
perovskites [6, 7, 8]. More recently, an associated family of materials has
caught the attention of the scientific community due to the multifunctional
character of its properties that cover, for example, anomalous Hall effect
[9], superconductivity [10, 11, 12], novel batteries [13, 14],
ferroelectricity [15], among other functionalities. This family of materials
is known as antiperovskite, or inversed-perovskite ($i.e.$ $A_{3}BX$) [16,
17], in which, the electrostatic balance and the oxidation site occupation is
reversed with respect to the known perovskites [18]. In the antiperovskites,
the anionic sites occupy the octahedral center, instead of the corners, and
the cationic metal transition occupies the corners, instead of the octahedral
center, forming the $XA_{6}$ octahedra. This change in coordination and atomic
occupations give rise to astonishing novel properties. For instance, when
magnetically active cations are placed at the $A$-sites to form the octahedra,
added to the triangular geometric coordination of the magnetic sites, a
magnetic frustration is induced resulting in chiral noncollinear
antiferromagnetic orderings [19]. This fascinating magnetic phenomenon is
associated with the observed barocaloric response [20], large piezomagnetic
behavior [21, 22, 23, 24], topological anomalous Hall effect that can be
controlled electrically [25, 26], and spin-torque switching [27]. Undoubtedly,
the control of all of the latter effects is highly desirable when designing
novel and efficient noncollinear spintronics controlled by external electric
fields [28]. Some efforts have demonstrated the possible control of the
magnetic ordering by the application of biaxial strain in thin films [29].
Moreover, intertwined magnetic and structural transitions are expected to
occur as demonstrated by M. Wu _et al._ [30]. Additionally, the negative
thermal expansion observed in the Mn${}_{3}B$N family [31, 32, 33] is a
signature of the strong magnetostructural coupling in this class of materials.
For example, in their perovskite oxides counterparts, the magnetostructural
coupling [34, 35] can induce tangible electric polarization, as in the case of
the SrMnO3. Thus, such materials are driven to a multiferroic state thanks to
their strong spin-phonon coupling assisted by epitaxial strain [36].
Consequently, when all the phenomena above are considered, the investigation
of the phonon spectra and their coupling with the magnetic orderings can shed
light on the physics hidden in these antiferromagnetic antiperovskites.

In this paper, we theoretically study, from first-principles calculations, the
magnetostructural coupling by exploring the spin and vibrational degrees of
freedom in the Mn3NiN antiferromagnetic antiperovskite, as a prototype among
its family. Moreover, we performed a careful analysis of the influence of the
exchange-correlation on-site Coulomb $U$ term into the physics behind the
spin-phonon coupling and the electronic structure in this antiperovskite. This
paper is organized as follows: In Section II are presented all the
computational details and the theoretical approaches used in the analysis of
this work. In Section III are shown the obtained results and the associated
analysis related to the vibrational landscape of the Mn3NiN noncollinear
antiferromagnetic antiperovskite. Finally, in Section IV we highlight our
conclusions and perspectives in regards to the strong magnetostructural
response observed in Mn3NiN.

##  II Computational Details

We performed first-principles calculations within the density-functional
theory (DFT) [37, 38] approach by using the vasp code (version 5.4.4) [39,
40]. The projected-augmented waves, PAW [41, 42] scheme was used to represent
the valence and core electrons. The electronic configurations considered in
the pseudo-potentials as valence electrons are Mn: (3$p^{6}$3$d^{5}$4$s^{2}$,
version 02Aug2007), Ni: (3$p^{6}$3$d^{8}$4$s^{2}$, version 06Sep2000), and N:
(2$s^{2}$2$p^{3}$, version 08Apr2002). The exchange-correlation, $E_{xc}$, was
represented within the generalized gradient approximation, GGA-PBEsol
parametrization [43], and the $E_{xc}$ of the $d$-electrons was corrected
through the DFT$+U$ approximation within the Liechtenstein formalism [44]. We
used a Coulomb on-site value of $U$ = 2.0 eV parameter. The latter optimized
to reproduce the experimentally observed lattice parameter. Also, a metaGGA
formalism [45], within the SCAN implementation [46], was adopted to correlate
with the Hubbard correction within the PBEsol$+U$ calculations. The dependence
of the lattice parameter on the electronic correlations was also explored
considering the LDA [47] and PBE [48, 49] exchange correlations. The periodic
solution of the crystal was represented by using Bloch states with a
Monkhorst-Pack [50] _k_ -point mesh of 12$\times$12$\times$12 and 800 eV
energy cut-off to give forces convergence of less than 0.001 eV$\cdot$Å-1 and
an error in the energy less than 0.5 meV. The spin-orbit coupling (SOC) was
included to consider noncollinear magnetic configurations [51]. The phonon
calculations were performed within the finite-differences methodology [52, 53]
and analyzed through the phonopy interface [54]. The modes symmetry was
studied with the support of the amplimodes toolkit [55]. Finally, the atomic
structure figures were elaborated with the vesta code [56].

Figure 1: (Color online) In (a) is presented the Mn3NiN $Pm\bar{3}m$ structure
in which, the $\Gamma_{5g}$ and $\Gamma_{4g}$ noncollinear ordering are shown.
(b) Lattice parameter as a function of the Hubbard $U$ parameter computed for
PBE, PBEsol, and LDA pseudopotentials, respectively. The experimentally
included lattice parameter was measured at TN = 262 K below the magnetic
transition temperature [30]. (c) Phonon dispersion curves were obtained for
the Mn3NiN at a fixed $U$ value of 2.0 eV and the FM, $\Gamma_{4g}$, and
$\Gamma_{5g}$ orderings. The unstable modes are shown at negative frequency
values by notation.

##  III Results and Discussion

As commented before, in the antiferromagnetic antiperovskites where
magnetically active (for example Mn and Ni) cations are placed in the
octahedral corner sites, a magnetic frustration is produced as a response of
the triangular coordination resulting in the octahedral faces. Thus, from the
structural point of view, the [111] family of planes form structural Kagome
lattices that couples to the resulting magnetic structure after the magnetic
frustration is resolved into the symmetry allowed magnetic orderings ($i.e.$
$\Gamma_{4g}$ and $\Gamma_{5g}$ in our case [19], see Fig. 1(a)). Therefore,
by choosing the $\langle$111$\rangle$ direction as the symmetry axis, the
resulting noncollinear structure lies in the [111] plane. Then, a strong
magnetostructural coupling is expected to be observed in the vibrational
landscape of the Mn3NiN when phonon modes that are entangled to the structural
Kagome lattices are involved. The latter interaction is mediated by the
magnetic exchange coupling constants and Dzyaloshiinski-Moriya interaction
that in turn, depends on the bond’s length and bonding angles [57, 58].

As a first step into our analysis, we considered the volume effects, widely
reported in the literature [59, 30, 23, 60], in prototype antiperovskite
materials such as Mn3NiN. Moreover, the relationship of the lattice parameter,
and associated volume, with the electronic structure in this family of
compounds highlights the importance of an appropriate approach to study the
spin-phonon coupling (see Fig. 1S in the Supplement Material). As such, aiming
to properly map the spin-phonon coupling present in this material, we started
by exploring the exchange and correlation effects by studying the influence of
different pseudopotentials formalisms on the lattice parameter. This
structural parameters directly dependent on the electronic RKKY interaction
between the Mn-sites forming the octahedra. Then, we considered LDA, PBE, and
PBEsol formalisms in this study. We have fixed the magnetic orderings to the
noncollinear $\Gamma_{4g}$ and $\Gamma_{5g}$, shown in Fig. 1(a), and then, we
proceeded with the full optimization of the electronic structure and the
lattice parameters. In both orderings, we obtained, at $U$ = 0.0 eV, lattice
parameters of 3.648 Å, 3.830 Å, and 3.749 Å for the LDA, PBE, and PBEsol,
respectively, as included in Fig. 1(b). As expected, the value obtained for
PBEsol sits in the middle of LDA and PBE formalisms due to their
underestimation and overestimation of the lattice parameters, respectively,
acknowledged in the field [61].

Experimentally, the observed lattice parameter is $a$ = 3.886 Å at low
temperature, below $T_{N}$ = 262 K [30] where the frustrated non-collinear
magnetism is present. Interestingly, although the PBEsol approach is expected
to be well-reproduce in comparison to the experimentally observed value, the
obtained parameter is well below the expected by $\Delta a$ = 0.137 Å,
equivalent to an error of 3.5 %. This disagreement is explained in terms of
the strong electronic correlation and the entanglement between the magnetic
structure and the volume in this type of compounds. Surprisingly, at the
relaxed volume obtained within the LDA approach, the magnetism vanishes and no
trace of the magnetic moments is obtained, nor for the Mn-sites, nor for the
Ni cell corner sites. Moreover, this relaxed lattice parameter is well below
the experimentally obtained above the magnetic transition temperature of $a$ =
3.888 Å. In contrast, the net magnetic moments per Mn-atom and noncollinear
ordering are obtained when using PBE and PBEsol approaches at the same volume
obtained with LDA. Here again, suggesting a strong coupling between the
magnetic, electronic, and lattice structure. Therefore, to improve the
electronic exchange-correlation misrepresentation, we introduced the DFT+$U$
approach, in which, the $U$ value was considered for the magnetically active
Mn-site that is the main responsible for the magnetostructural response. In
Fig. 1(b) we present the lattice parameter values, for the LDA, PBE, and
PBEsol, obtained as a function of the on-site Coulomb $U$ value. We observed a
rather large expansion for the three considered schemes. For example, for the
PBEsol approach, the lattice parameter varies from 3.749 Å to 4.092 Å when the
$U$ value is varied from 0.0 to 6.0 eV. This giant expansion of the lattice is
due to the strong effect of the electronic structure, and the associated Mn–Mn
magnetic exchange, into the structural degrees of freedom possibly influencing
the phononic structure. It is worth noticing that, in the LDA calculation for
values above $U$ = 1.0 eV, the expansion and the increase in the correlation
correction favor the appearance of magnetism, and then, a development of a net
magnetic moment per Mn atom is clearly observed within the noncollinear
antiferromagnetic structure. The latter is along the lines of the expected
negative thermal expansion explained in terms of the existence and
annihilation of the frustrated magnetic structure as a function of temperature
[30]. The observed contraction in the equilibrium volume, in absence of the
Hubbard correction, is due to the Mn–Mn overbinding produced by LDA and GGA
schemes in metallic Mn-rich compounds [62, 63, 64], which in turn, produces a
substantial contraction of the lattice parameter, as observed at $U$ = 0.0 eV
in Fig. 1(b).

After considering a careful characterization of the electronic structure and
based-on our previous results aiming to better reproduce the magnetostructural
phenomenon and the associated spin-phonon coupling, we have defined the
pseudopotential to the PBEsol approach and the on-site correction $U$ value to
$U$ = 2.0 eV in the PBEsol+$U$ scheme. Moreover, to correlate and corroborate
our theoretical strategy with a more advanced approach, we employed the
metaGGA formalism within the SCAN representation [46]. This approach has been
successfully applied in the description of metal-transition oxides [65], Mn-
rich compounds [64], and metallic Heusler compounds [66]. As such, within the
SCAN approach we fully optimized the electronic and structural parameters
without the Hubbard correction ($i.e.$ $U$ = 0.0 eV) and conserving a
$\Gamma_{4g}$ noncollinear magnetic ordering. Under the SCAN representation,
we obtained a fully relaxed lattice parameter of $a$ = 3.863 Å representing an
error of 0.59 % with respect to the experimentally obtained value and the
computed lattice parameter for PBEsol$+U$, $U$ = 2.0 eV. Additionally, we
computed the band-structure for the LDA, PBE, PBEsol, and SCAN representation
of the exchange-correlation with and without the $U$ correction, shown in Fig.
2S in the Supplement Material. After comparing the electronic structure with
the obtained with the SCAN scheme, it can be observed that the electronic
states close to the Fermi energy are considerably improved, for the three
schemes, when the on-site Coulomb energy is considered and correctly
introduced into the calculations. Nonetheless, the PBEsol$+U$ offers the
advantage of being designed to better reproduce the lattice parameter avoiding
computational pressure/strain that could alter the electronic structure. Then,
we conclude that the PBEsol$+U$ methodology fairly reproduces both, the
electronic properties and the structural degrees of freedom that might be
associated with the vibrational landscape. Interestingly, most of the reports
in the literature [67, 21, 29, 9] devoted to this type of compounds do not
consider the on-site Coulomb $U$ correction in the exchange-correlation and
the cell volume is fixed to the experimentally obtained parameter. The latter
approach might be introducing a fictional negative external pressure, which,
in turn, could induce differences in the observed properties due to the strong
coupling between the magnetic and the atomic structure. Besides, as suggested
by Fig. 2S, the electronic structure strongly differs from the one obtained
with more advanced methodologies, such as meta-GGA. For this reason, the
exchange-correlation correction is needed to better reproduce the electronic
structure close to the Fermi energy, and with it, the magnetic response and
structural behavior. Thus, the computational and theoretical methodology
proposed here is highly recommended when treating this class of compounds.

Figure 2: (Color online) (a) Phonon dispersion for the $\Gamma_{5g}$ ordering
in the Mn3NiN computed at different $U$ values in which, the lattice volume
was fixed to the experimentally obtained value. The latter is in order to
explore the effect of the correlation into the vibrational modes. Here, the
unstable modes are shown at negative frequency values by notation. (b)
Obtained change in the frequencies, $\Delta\omega$ = $\omega_{FM}$ $-$
$\omega_{\Gamma_{5g}}$, when the magnetic transitions in induced. In the
latter, the extracted phonon frequencies of the lowest vibrational modes at
the high-symmetry points in the Brillouin zone. (c) Representation of the
structural distortions associated with the $M_{2}^{+}$ and $R_{5}^{-}$ modes
where the octahedral rotation and tilting can be appreciated, respectively.

Once the noncollinear magnetic structure is considered into the symmetry
operations, the group symmetry is reduced from the $Pm\bar{3}m$ (SG. 221)
space group to the $R\bar{3}m$ (SG. 166) although the point group is conserved
to the $\bar{3}m$. The magnetic symmetry groups are defined as
$R\bar{3}m^{\prime}$ (MSG. 166.101) and $R\bar{3}m$ (MSG. 166.97) for the
$\Gamma_{4g}$, and $\Gamma_{5g}$ magnetic orderings, respectively, see Fig.
1(a). Next, after carefully having defined the exchange-correlation and the
Hubbard correction effects into the electronic and atomic structure, we
proceeded by obtaining the full phonon-dispersion curves for the noncollinear
fm (111), $\Gamma_{4g}$, and $\Gamma_{5g}$ magnetic orderings, as shown in
Fig. 1(c). Due to the metallic character of the Mn3NiN antiperovskite and the
frustrated magnetic structure that is resolved in the non-collinear chiral
magnetic $\Gamma_{4g}$ and $\Gamma_{5g}$ orderings, we adopted the finite-
differences method in which, we considered the non-collinear magnetic ordering
at each atomic displacements configuration. The latter displacements are
obtained by symmetry into the 2$\times$2$\times$2 supercell in which the
interatomic forces are computed and later used to reconstruct the dynamical
matrix tensor and with it, the full phonon-dispersions. In Fig. 1(c) we
present the phonon-dispersion curves for the 3 magnetic orderings at the fully
relaxed lattice parameter of $a$ = 3.885 Å with $U$ = 2.0 eV. It is relevant
mentioning that despite a symmetry lowering is observed when the noncollinear
magnetic orderings are introduced, within the margin error of 2.0 cm-1, the
$Pm\bar{3}m$ and $R\bar{3}m$ phonons are indistinguishable. For this reason,
the high-symmetry points are selected with respect to the $Pm\bar{3}m$
Brillouin zone, BZ. As it can be appreciated, for the chiral $\Gamma_{4g}$ and
$\Gamma_{5g}$ noncollinear magnetic orderings, the phonons and vibrations are
fully stable along the entire path in the BZ with non-observable tangible
differences in the vibrational landscape between them. In contrast, for the fm
(111) ordering, a negative ($i.e.$ unstable) branch is observed at the
$R$-point. This unstable phonon has a frequency value of $\omega$ = $-$59
$i$cm-1. Clearly, this unstable phonon observed, as a result of the magnetic
transition between the fm to $\Gamma_{4g}$, or $\Gamma_{5g}$, is proof of the
strong spin-lattice coupling present in this antiperovskite.

Table 1: Phonon frequencies, in cm-1, at the high-symmetry points computed for the $\Gamma_{4g}$, $\Gamma_{5g}$, and fm orderings and keeping the on-site Coulomb value of $U$ = 2.0 eV. Ord. / Ph. mode | $\Gamma_{5}^{-}$ | $\Gamma_{4}^{-}$ | $\Gamma_{4}^{-}$ |  $\Gamma_{4}^{-}$ |  |
---|---|---|---|---|---|---
$\Gamma_{4g}$ | 149 | 155 | 226 | 631 |  |
$\Gamma_{5g}$ | 148 | 155 | 226 | 631 |  |
fm | 151 | 162 | 282 | 662 |  |
| $X_{1}^{+}$ | $X_{1}^{+}$ | $X_{5}^{+}$ | $X_{3}^{-}$ | $X_{1}^{+}$ |  $X_{5}^{+}$
$\Gamma_{4g}$ | 127 | 155 | 214 | 354 | 600 | 637
$\Gamma_{5g}$ | 127 | 155 | 214 | 354 | 600 | 637
fm | 131 | 162 | 241 | 363 | 603 | 659
| $M_{2}^{+}$ | $M_{5}^{-}$ | $M_{2}^{-}$ | $M_{4}^{+}$ | $M_{5}^{-}$ |  $M_{2}^{-}$
$\Gamma_{4g}$ | 45 | 141 | 193 | 392 | 590 | 636
$\Gamma_{5g}$ | 45 | 140 | 193 | 392 | 590 | 636
fm | 31 | 152 | 216 | 389 | 592 | 657
| $R_{5}^{-}$ | $R_{4}^{-}$ | $R_{4}^{-}$ | $R_{3}^{-}$ | $R_{2}^{-}$ |  $R_{5}^{+}$
$\Gamma_{4g}$ | 43 | 133 | 199 | 324 | 428 | 591
$\Gamma_{5g}$ | 44 | 133 | 199 | 323 | 428 | 592
fm |  $-$59 | 126 | 234 | 319 | 440 | 576

In Table 1 we present the extracted frequency values for the phonons in the
Mn3NiN at the $\Gamma$, $X$, $M$, and $R$ high-symmetry points. We observed
that the frequency values computed for the $\Gamma_{4g}$ and $\Gamma_{5g}$ are
almost the same in agreement with the degenerated magnetic states. Now, when
compared to the fm (111) ordering, we found that several specific modes are
highly affected due to the different magnetic orderings. This is the case of
the $\Gamma_{4}^{-}$, $X_{5}^{+}$, $M_{2}^{+}$, and $R_{5}^{-}$ where the
computed |$\Delta\omega$| values are 56 cm-1, 27 cm-1, 14 cm-1, and 103 cm-1,
respectively. After analyzing these modes, we observed that the atomic
displacements associated with each of them are related to the Mn–Mn
interaction responsible for the magnetic response. In the $\Gamma_{4}^{-}$
case, this mode involves an in-phase octahedral stretching accompanied with a
displacement of the nitrogen atoms present in the center of the octahedron.
The $X_{5}^{+}$ mode is related with an in-phase stretching involving only the
Mn atoms positioned in the equatorial sites of the octahedron alongside with
movements of the nitrogen atoms. The case of the $M_{2}^{+}$ mode is also
interesting because involves a rigid in-phase octahedral rotation. The most
affected mode, the $R_{5}^{-}$, involves a rigid out-of-phase octahedral
rotation that conserves the Mn–Mn distance unaffected, as in the $M_{2}^{+}$
case, but, alters the bonding angles inducing a distortion within the [111]
Kagome lattice.

For our next step in the analysis, we studied only the effect of the
electronic correlations in the vibrations and the magnetic structure. Then, we
fixed the lattice parameters to the relaxed structure within the PBEsol$+U$
($U$ = 2.0 eV), and then, we vary the $U$ value from 0.0 to 6.0 eV also
keeping the $\Gamma_{5g}$ and fm magnetic orderings fixed. In Fig. 2(a) we
present the phonon-dispersion curves obtained at the fixed volume for $U$ =
0.0, 2.0, 4.0, and 6.0 eV considering the $\Gamma_{5g}$ ordering. Despite the
fixed volume, we found that the vibrational modes are strongly affected by the
exchange-correlation correction where, in general, a hardening of the branches
is observed as the on-site Coulomb term is increased. Interestingly, for
values of $U$ = 6.0 eV, there is an appearance of two unstable phonons at the
$M$\- and $R$-points. Here, for $U$ values above 4.0 eV, the $M_{2}^{+}$ and
$R_{5}^{-}$ modes become unstable as a consequence of the large on-site
Coulomb repulsion that is affecting the exchange coupling, and in turn,
modifying the modes’ frequencies. The latter modes are identified as
$M_{2}^{+}$ and $R_{5}^{-}$ and are representing the in-phase octahedral
rotation and the out-of-phase octahedral tilting, respectively, as shown in
Fig. 2(c).

In Fig. 2(b) we present the differencies, $\Delta\omega$ =
$\omega_{\textsc{fm}}$ $-$ $\omega_{\Gamma_{5g}}$, in the phonon frequency,
for the most relevant modes close to 0 cm-1. We observe that most of the modes
show a tangible influence on the $U$ value. Remarkably, the $R_{5}^{-}$ mode
experiences a change of 113 cm-1. Interestingly, when the $U$ correction
approaches 6.0 eV, the difference in the frequencies between the FM and
$\Gamma_{5g}$ magnetic orderings, starts to decrease considerably, possibly
due to an overestimation of the correlation.

Figure 3: (Color online) In (a) are presented the $M_{2}^{+}$ and $R_{5}^{-}$
phonon frequencies computed at the $\Gamma_{5g}$ and FM orderings in the
Mn3NiN for different $U$ values in which, the volume relaxation is allowed in
the optimization process. Evolution of the $M_{2}^{+}$ and $R_{5}^{-}$
frequencies, in (c) and (c) respectively as a function of the Hubbard $U$
value. The $M_{2}^{+}$ ($R_{5}^{-}$) mode is unstable above $U$= 4.0 eV ($U$ =
6.0 eV) for both, relaxed and fixed volume, cases.

Aiming to explore the effect of the on-site Coulomb correction into the
volume, and consequently, into the vibrational landscape of Mn3NiN, we
proceeded with the analysis by allowing the volume to relax for the different
$U$ values and extracting the phonon-dispersion curves at each relaxed volume,
following the procedure explained above. Similarly, the phonons are fully
stable for a small regime of $U$ values in the $\Gamma_{5g}$ and only the
$M_{2}^{+}$ and $R_{5}^{-}$ modes are strongly affected by the correlation and
volume increase. In Fig. 3(a) we present the extracted phonon frequencies for
the $R_{5}^{-}$ and $M_{2}^{+}$ modes computed within the fm and $\Gamma_{5g}$
magnetic orderings. As it can be observed, the $R_{5}^{-}$ is unstable for the
entire regime of $U$ values when the ferromagnetic ordering is fixed. The
latter is corroborating that the fm ordering is unstable, independently of the
correlation value adopted, and confirms the strong AFM exchange interaction
within the Kagome lattice. Interestingly, the $M_{2}^{+}$ becomes unstable, in
the FM magnetic ordering for $U$ values above 4.0 eV. In Fig. 3(b), for
$M_{2}^{+}$, and Fig. 3(c), for $R_{5}^{-}$, we present the frequencies
obtained for the fixed and relaxed volumes as a function of $U$ by keeping the
$\Gamma_{5g}$ ordering. Here, we noticed that despite the volume changes, the
phonon modes hold a similar trend showing an unstable behavior, for both
$M_{2}^{+}$ and $R_{5}^{-}$ modes, when the $U$ values are above $U$ = 4.0 eV.

After analyzing the antiperovskite structure, it becomes clear that the
vibrational modes related to rigid octahedral rotations and tilting will be
strongly coupled to the magnetic behavior thanks to the magnetic exchange
interactions that, in the antiperovskite case, takes place in the Mn–Mn
corner-to-corner interaction. As such, the strong magnetic frustration will
tend to keep the triangular structure unaffected. However, the hypothetical fm
and/or external constrains such as strain, could introduce into the system
changes in the electronic structure taht could have as a consequence,
dislocations of the Kagome lattices formed along with the [111] family of
planes thanks to the strong magnetoestructural coupling. Therefore, we could
expect that induced rotations and tiltings at experimentally growth thin-
films, onto substrates with such octahedral patterns, might trigger the
ferromagnetic state in the Mn3NiN antiperovskite.

As observed in the literature for the perovskite oxides counterparts, the
strongest spin-lattice coupling phenomenon is observed in manganite-based
oxides, such as SrMnO3 [34, 35, 36]. In this manganite oxide, it is possible
to achieve a switching in the magnetic structure and to induce phase
transitions by external constraints, as it might be obtained in the Mn3NiN.
Thus, in order to disentangle the frustrated lattice contribution and the
manganese cation intrinsic tendency to exhibit a large spin-phonon coupling,
we performed a computational experiment by computing the lattice parameter
dependency on the on-site Coulomb correction in the Fe3NiN antiperovskite
compound. From Fig. 3S, in the Supplement Material, we observe that, although
the Fe3NiN is affected, the influence of the $U$ parameter is smaller,
suggesting that the magnetic frustrations in the $A_{3}B$N ($A$ = Fe, Mn, and
Ni) family are crucial ingredients for the magnetostructural behavior that it
is enhanced by the manganese contribution.

##  IV Conclusions

We have investigated, by means of first-principles calculations, the
magnetostructural behavior in the Mn-based nitride Mn3NiN atiferromagnetic
antiperovskite by exploring the spin-phonon coupling. According to our
results, we found a rather strong correlation between the magnetic,
electronic, and vibrational degrees of freedom. As such, a variation of the
phonon frequencies, as well as the cell volume, was observed as a function of
the $U$ correlation. Interestingly, phonon modes associated with in-phase and
out-of-phase rotations, represented by the $M_{2}^{+}$ and $R_{5}^{-}$
respectively, can be tuned by the magnetic ordering in the Mn3NiN. Remarkably,
we highlight the importance of the exchange-correlation treatment in the
calculations showing its influence on the electronic, magnetic, and phononic
structures. We thus hope that our work will motivate further studies leading
to a deeper exploration of the spin-phonon coupling in this antiferromagnetic
antiperovskite compound and then, potentially achieving the control of the
magnetic features by external constraints, such as epitaxial strain and/or
applied magnetic field.

## Acknowledgements

We thank prof. Aldo H. Romero for helpful comments and discussions. The
calculations presented in this paper were carried out using the GridUIS-2
experimental testbed, being developed under the Universidad Industrial de
Santander (SC3-UIS) High Performance and Scientific Computing Centre,
development action with support from UIS Vicerrectoría de Investigación y
Extension (VIE-UIS) and several UIS research groups as well as other funding
resources. Additionally, we acknowledge the computational support extended to
us by Laboratorio de Supercomputo del Sureste (LNS), Benemérita Universidad
Autónoma de Puebla, BUAP, for performing heavy theoretical calculations.
A.C.G.C. acknowledge the grant No. 2677 entitled “Quiralidad y Ordenamiento
Magnético en Sistemas Cristalinos: Estudio Teórico desde Primeros Principios”
supported by the VIE – UIS.

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