New Frontiers in Physics 2014

11institutetext: Institute for Particle Physics, ETH Zürich, Rämistrasse 101,
8092 Zürich,Switzerland

# Search for a dileptonic edge with CMS

Konstantinos Theofilatos (on behalf of the CMS Collaboration) 11
<EMAIL_ADDRESS>
###### Abstract

We present a search for a kinematic edge in the invariant mass distribution of
two opposite-sign same-flavor leptons, in final states with jets and missing
transverse energy. The analysis makes use of $19.4$ fb-1 proton-proton
collision data at $\sqrt{s}=8$ TeV. The data have been recorded with the CMS
experiment. Complementary methods have been used for the background
estimation, which when combined achieve a total uncertainty of $5\%$ ($10\%$)
for leptons in the central (forward) rapidity of the detector. We do not
observe a statistically significant signal and the results are consistent with
the background-only hypothesis.

A search for new physics in final states with an opposite-sign same-flavor
lepton pair (dilepton), jets and missing transverse energy (MET) is presented
here. We search for evidence of production of a new heavy neutral particle,
which is produced in the cascade of a strongly interacting heavier resonance.
This report has been compiled from results that have been published here PAS .
We use proton-proton collision data at $\sqrt{s}=8$ TeV. The total accumulated
luminosity is $19.4$ fb-1. The data have been recorded with the CMS experiment
CMS .

The invariant mass distribution of the dilepton system can exhibit a rise that
increases with dilepton mass followed by a sharp cut-off, resembling the shape
of a triangle with an edge, if the two leptons originate from the decay of a
heavy neutral particle. In Supersymmetry (SUSY), such a signal is exemplified
by the decay of the heavy neutralino $(\tilde{\chi}^{2}_{0})$, which is
produced within the cascade of a squark ($\tilde{q}$) – the SUSY partner of a
quark. An example cascade is depicted in Fig. 1. The cascade can also start
from a gluino, the SUSY partner of the gluon, if that is heavier than the
squark resulting to an additional energetic jet in the final state. In either
cases, the location of the dileptonic mass endpoint is given by Eq. 1.

$m_{{\ell}{\ell}}^{max}=m_{\tilde{\chi}^{0}_{2}}\sqrt{\left(1-\left(\frac{m_{\tilde{\ell}}}{m_{\tilde{\chi}^{0}_{2}}}\right)^{2}\right)\left(1-\left(\frac{m_{\tilde{\chi}^{0}_{1}}}{m_{\tilde{\ell}}}\right)^{2}\right)}$
(1)

$q$$\ell^{\pm}$$\ell^{\mp}$$\tilde{q}$$\tilde{\chi}^{0}_{2}$$\tilde{\ell}^{\mp}$$\tilde{\chi}^{0}_{1}$invisible
Figure 1: An example SUSY process that will lead to an excess of events in the
invariant mass distribution of the two leptons $m_{{\ell}{\ell}}$. The
invariant mass distribution of the two leptons will exhibit a threshold effect
reaching a maximum endpoint. The location of the endpoint solely depends on
the masses of the particles involved and can be used to constrain different
physics models.

If the mass difference between the two neutralinos is small, or the mass of
the slepton $(\tilde{\ell})$ is large, the $\tilde{\chi}^{2}_{0}$ decay can
proceed through a virtual $\mathrm{Z}^{0*}$ boson exchange, resulting to a
direct 3-body decay. The endpoint in that case is given by the mass difference
of the two neutralinos,
$m_{{\ell}{\ell}}^{max}=m_{\tilde{\chi}^{0}_{2}}-m_{\tilde{\chi}^{0}_{1}}$ and
the invariant mass distribution of the dilepton system is not anymore an exact
triangle and it has more rounded shape. If there is sufficient energy to
produce a $\mathrm{Z}^{0}$ boson, i.e. when
$m_{\tilde{\chi}^{0}_{2}}-m_{\tilde{\chi}^{0}_{1}}\sim m_{\mathrm{Z}}$ this
will be preferred against a possible 3-body decay and the dilepton mass
distribution will exhibit an excess of events at $m_{\mathrm{Z}}\sim 91$ GeV.
Furthermore, similar cascades involving higher mass neutralino eigenstates
$m_{\tilde{\chi}^{0}_{x=2,3,4}}$ decaying to lighter neutralinos can occur
producing similar topologies.

Measuring the possible different endpoints and their $\sigma\times\mathrm{BR}$
is a way to constrain candidate new physics models with data edge1 . In
addition, since the above discussion was only based on kinematic
argumentation, any massive heavy neutral particle (SUSY or not) that decays
through the aforementioned cascades may give rise to similar phenomenology.

In SUSY, the lightest neutralino $\tilde{\chi}^{0}_{1}$ is massive, neutral,
weakly interacting and is most often assumed to be stable. This assumption is
not a necessity, but is experimentally well motivated and can explain the
evidence of dark matter in the universe. The stability of dark matter
particles in the universe implies the conservation of a new quantum number and
thus the dark matter particles are expected to be always pair produced in a
collider, if there is sufficient $\hat{s}$ available. The stability of
$\tilde{\chi}^{0}_{1}$ can be enforced in SUSY by the R-parity conservation
susyprimer . Being a carbon copy of the SM, we expect that SUSY particles will
be predominantly produced via the strong interaction
$pp\to\tilde{q}\tilde{q}$, $\tilde{g}\tilde{q}$ and $\tilde{g}\tilde{g}$. In
each signal event, two independent cascades will be formed, originating from
the decays of the squarks and gluinos that will decay producing jets and gauge
bosons until the lightest SUSY particle is produced $\tilde{\chi}^{0}_{1}$. We
therefore expect at least two $\tilde{\chi}^{0}_{1}$ in the final state. A
stable $\tilde{\chi}^{0}_{1}$ will escape from detection like a heavy
neutrino, producing missing transverse energy in the detector.

Both CMS and ATLAS Collaborations have extensively studied the dilepton
signature as a search strategy before the LHC start-up ptdr2 atlasptdr .
Previous results of a search for kinematic edge have been published by CMS at
$\sqrt{s}=7$ TeV edge2011 . In addition, a search for a $\mathrm{Z}^{0}$ boson
accompanied by jets and MET has been also published by CMS at $\sqrt{s}=7$ TeV
jzbpaper . This update, uses $\sqrt{s}=8$ TeV data and unifies the
experimental methods deployed earlier in the corresponding CMS $7$ TeV papers
edge2011 jzbpaper .

We select events with two leptons having $p_{\mathrm{T}}>20$ GeV and
$|\eta|<2.4$, with $p_{\mathrm{T}}$ denoting the transverse momentum and
$\eta$ the pseudorapidity. We also count the number of jets produced in the
event having $p_{\mathrm{T}}>40$ GeV and $|\eta|<3.0$. In the jet counting, we
exclude jets that are within $\Delta
R=\sqrt{\delta\phi^{2}+\delta\eta^{2}}<0.4$ away from the leptons. We require
that the two leptons are well separated ($\Delta R>0.3$). The signal region is
defined as ($N_{\mathrm{jets}}=2$ $\cup$ $\mathrm{MET}>150$ GeV) $\cap$
($N_{\mathrm{jets}}\geq 3$ $\cup$ $\mathrm{MET}>100$ GeV), and the phase space
is further subdivided to “central” region (Fig. 2) where both leptons have
$|\eta|<1.4$ and “forward” region when at least one lepton lays in the forward
region $1.6<|\eta|<2.4$ (Fig. 3).

Figure 2:  Comparison between the observed and estimated SM background
dilepton mass distributions in the central region, where the SM backgrounds
are evaluated from control samples in data.  Figure 3:  Comparison between the
observed and estimated SM background dilepton mass distributions in the
forward region, where the SM backgrounds are evaluated from control samples in
data.

A priori these two regions of the phase space, have different $S/B$. We expect
that the leptons that will originate from the decay of heavy resonance, will
be more centrally distributed in rapidity with respect to the standard model
(SM) background processes that mimic the same topology. The main SM background
is top pair production,

$t\bar{t}\to W^{+}W^{-}b\bar{b}\to\ell^{+}\ell^{-}b\bar{b}\nu\bar{\nu}$ (2)

which results in equal yields of same-flavor (SF = $ee$ and $\mu\mu$) and
opposite-flavor (OF = $e\mu$) events. This is due to the lepton universality
in the decays of the two $\mathrm{W}$ bosons, which decay with equal
probability to all lepton flavors. We use a control sample of OF data that
satisfies all signal selection criteria, but the flavor composition which is
SF for the signal and OF for the control sample. The OF data are corrected for
triggering and lepton reconstruction efficiency effects such as to match the
SF selection efficiency. The measured multiplicative correction factor
$\text{R(SF/OF)}=N_{\mathrm{SF}}/N_{\mathrm{OF}}$ is consistent with unit
within uncertainties and is very robust against changes in the muon over
electron efficiency ratio $(r=\epsilon_{\mu}/\epsilon_{e})$. This can be
understood by considering that $\text{R(SF/OF)}=0.5\times(r+r^{-1})\times
R_{\mathrm{T}}$ with $R_{\mathrm{T}}$ a constant factor related to the
triggering efficiencies of $ee$, $\mu\mu$ and $e\mu$ paths when they reach
their plateau. It is easy to verify that $0.5\times(r+r^{-1})$ stays close to
$1.0$ even when $r$ varies significantly, due to cancellation efficiency
effects when combining electron and muons channels to form the SF sample. We
measure the efficiency correction factors and their uncertainty, by two
complementary methods, which are detailed in PAS . The results of the two
methods are consistent and the two methods are finally combined. The final
uncertainty of R(SF/OF) is 4% (7%) for the central (forward) region. Another
important source of background comes from Drel–Yan (DY) events, which could
exhibit large apparent missing energy due to detector resolution and
reconstruction effects. This background is not flavor symmetric and therefore
the OF control sample is not alone sufficient to cope with it. We deploy two
independent methods to estimate the DY background, the MET templates and the
Jet-Z-Balance (JZB) method. The former uses photon+MET data reweighed to match
the DY kinematics while the latter exploits the sign symmetry of the JZB
variable, which is an estimate of the momentum imbalance in DY events jzbpaper
. The total uncertainty on the background is 5% (10%) for the central
(forward) region. The region of phase space with moderate MET and hadronic
activity HT111HT is defined as the scalar sum of $p_{\mathrm{T}}$ of all jets
that is probed by this analysis, is not easily accessible to inclusive
searches for jets+MET that use MET and HT triggers. The achieved level of
precision allows to search for soft signals that would have been missed in
more inclusive “all-hadronic” final states, which typically suffer from much
higher SM background uncertainties.

Figure 4:  Fit results for the signal-plus-background hypothesis in comparison to the measured dilepton mass distributions, in the central region, projected on the SF event sample. The combined fit shape is shown as a blue, solid line. The individual fit components are indicated by dashed lines. The flavor-symmetric background is denoted as FS and is displayed with a black dashed line. The Drell–Yan contribution is denoted as DY and is displayed with a red dashed line. The extracted signal component is denoted as Signal and is displayed with a green dashed line. Table 1: Results of the counting experiment. | Central | Forward
---|---|---
Observed data events $(N_{\mathrm{data}})$ | 860 | 163
Estimated background $(N_{\mathrm{b}})$ | $730\pm 40$ | $157\pm 16$
$N_{\mathrm{s}}=N_{\mathrm{data}}-N_{\mathrm{b}}$ | $130^{+48}_{-49}$ | $6^{+20}_{-21}$
Significance [$\sigma$] | 2.6 | $0.3$

A counting experiment is performed for $20<m_{\ell\ell}<70$ GeV while the
search for kinematic edge is performed in $20<m_{\ell\ell}<300$ GeV by fitting
the signal and background shapes in data (Fig. 4). The background shape
normalization is performed with a Gaussian constraint taking into account the
measured value of R(SF/OF) and its uncertainty. For the signal, we use a
triangular shape assumption, which is the nominal shape expected from
kinematics for two sequential two-body decays as e.g. in SUSY when the decay
occurs via an intermediate slepton. The local significance of the counting
experiment is $2.6\sigma$ $(0.3\sigma)$ in the central (forward) signal
region. The best fitted value for the location of the kinematic edge, assuming
a triangular signal shape, is at $78.7\pm 1.4$ GeV (Fig. 4), with significance
of $2.4\sigma$. The fit is simultaneous to both central and forward signal
regions. The reduced significance of the fit can be understood by the mixing
with equal weight of the central and forward regions. The comparison of data
versus simulation of the background and three different benchmark models is
shown in Fig. 5. These reference signal models are based on sbottom pair
production with masses as indicated in the legend. The decay mode of each
sbottom quark is
$\tilde{b}\to\tilde{\chi}_{2}^{0}b\to\tilde{\chi}_{1}^{0}\mathrm{Z}^{0*}b$
with $\mathrm{Z}^{0*}$ boson being off-shell and decaying to leptons,
quark–antiquark and invisible with their SM branching fractions. As a result,
about 13% of the signal events contain at least one SF dilepton in the final
state. The mass difference $m_{\tilde{\chi}_{2}^{0}}-m_{\tilde{\chi}_{1}^{0}}$
has been fixed to $70$ GeV, producing the characteristic endpoint expected for
a 3-body decay.

The results of the counting experiment are summarized in Tab. 1. In the
forward signal region, which is experimentally more challenging, we record as
much data as expected from the SM background. In the central signal region,
although the yield of signal candidates $N_{\mathrm{s}}=130^{+48}_{-49}$ is
sizable, it is not sufficiently large to reject the background-only
hypothesis. The possibility that the observed excess is solely a background
fluctuation of statistical nature, can not be excluded. This analysis probes a
phase space that is not easily accessible to inclusive searches for jets+MET
and it does that with foremost precision. We will have to wait for its
repetition with 13 TeV LHC data in order to wipe out the possibility that this
is a first hint of new physics.

Figure 5: Data compared with SM simulation in the central signal region.
Example signal scenarios based on the pair production of bottom-squarks is
also overlaid.

## References

  * (1) CMS Collaboration, “Search for physics beyond the standard model in events with two opposite-sign same-flavor leptons, jets, and missing transverse energy in pp collisions at $\sqrt{s}$ = 8 TeV, CMS-PAS-SUS-12-019, 2014
  * (2) CMS Collaboration, “The CMS experiment at the CERN LHC”, JINST 3 S08004, 2008
  * (3) I. Hinchliffe et al., “Precision SUSY measurements at CERN LHC”, Phys. Rev. D 55 5520-5540 doi:10.1103/PhysRevD.55.5520, 1997
  * (4) CMS Collaboration, “CMS Physics: Technical Design Report Volume 2: Physics Performance, J. Phys. G 34”, no. CERN-LHCC-2006-021. CMS-TDR-8-2, 995-1579. 669 p., 2007
  * (5) ATLAS Collaboration, “ATLAS: Detector and physics performance technical design report”, Vol. 2, 1999
  * (6) S. P. Martin, “A Supersymmetry Primer”, Adv. Ser. Direct. High Energy Phys. 21, 1-153, hep-ph/9709356, 2010
  * (7) CMS Collaboration, “Search for new physics in events with opposite-sign leptons, jets, and missing transverse energy in pp collisions at $\sqrt{s}=7$ TeV”, Phys. Lett. B 718, no. 3, 815, doi:10.1016/j.physletb.2012.11.036, 2013
  * (8) CMS Collaboration, “Search for physics beyond the standard model in events with a Z boson, jets, and missing transverse energy in pp collisions at $\sqrt{s}=7$ TeV” Phys. Lett. B 716, no. 2, 260, doi:10.1016/j.physletb.2012.08.026, 2012


††thanks: Email<EMAIL_ADDRESS>
# Interfacial phase frustration stabilizes unconventional skyrmion crystals

Narayan Mohanta  Materials Science and Technology Division, Oak Ridge National
Laboratory, Oak Ridge, TN 37831, USA  Department of Physics and Astronomy, The
University of Tennessee, Knoxville, TN 37996, USA  Department of Physics,
Indian Institute of Technology Roorkee, Roorkee 247667, India  Elbio Dagotto
Materials Science and Technology Division, Oak Ridge National Laboratory, Oak
Ridge, TN 37831, USA  Department of Physics and Astronomy, The University of
Tennessee, Knoxville, TN 37996, USA

###### Abstract

Chiral magnetic phases with an unconventional topological twist in the
magnetization are of huge interest due to their potential in spintronics
applications. Here, we present a general method to induce such exotic magnetic
phases using interfacial phase frustration within artificially grown
superlattices. To demonstrate our method, we consider a multilayer with two
different chiral magnetic phases as the competing orders at the top and bottom
and show, using Monte Carlo calculations, that the interfacial phase
frustration is realized at the central layer. In particular, we obtain three
unconventional phases: a checkerboard skyrmion crystal, an incommensurate
skyrmion stripe, and a ferrimagnetic skyrmion crystal. In these frustration-
induced phases, the spin chirality driven topological Hall conductivity can be
largely enhanced. This method provides a playground to realize unconventional
magnetic phases in any family of materials that can be grown in superlattices.

INTRODUCTION
Magnetic frustration arises from the incompatibility of the interactions
between magnetic degrees of freedom in a lattice with the underlying crystal
geometry and often produces exotic ground states, such as spin ice and spin
liquids Bramwell_Science2001 ; Balents2010 . These frustration-stabilized
magnetic phases opened a window to understand the fundamentals of magnetism
and to use in technological applications, as exemplified by the search for
quantum spin liquids Kitaev2003 ; Savary_RPP2016 ; Batista_RPP2016 . The
collective behavior of the spins, influenced by the frustration, often leads
to chiral correlations in otherwise non-chiral ordered magnetic ground states
Taguchi_Science2001 ; Grohol_NatMat2005 . Specifically, chiral magnetic
textures are the focus of intense investigation because of their potential in
spintronic applications, and for their role in producing topological Hall
effects from the emergent magnetic field created by the magnetic texture
Schulz_NatPhys2012 ; Nagaosa_NatNano2013 ; Fert_NarRevMat2017 .

A prominent example of such chiral magnetic texture with a continuous winding
of magnetization is a skyrmion, stabilized commonly as a triangular crystal in
various bulk compounds and interfaces by competing Heisenberg ferromagnetic
exchange and relativistic Dzyaloshinskii-Moriya interaction (DMI)
Rossler_Nature2006 ; Muhlbauer_Science2009 ; Yu_Nature2010 ; Balents_PRL2014 ;
Das_Nature2019 ; Mohanta_PRB2019 . The skyrmion crystal phase was also
theoretically predicted to be stabilized without DMI in geometrically-
frustrated magnets or in highly-symmetric lattices with long-range Ruderman-
Kittel-Kasuya-Yosida interaction Okubo_PRL2012 ; Leonov_NatComm2015 ;
Ozawa_PRL2017 ; Wang_PRL2020 . Three dimensional variants of skyrmions, viz.,
skyrmion strings Park_NatNano2014 , chiral bobbers Zheng2018 and hopfions
Kent_NatCommun2021 have been experimentally observed. However, the variety of
these chiral magnetic textures beyond the skyrmion crystal is severely limited
by the available magnetic interactions in bulk compounds.

Fig. 1: Schematic illustration of a multilayer to realize “interfacial phase
frustration’. Shown is a magnetic multilayer, involving different $5d$
compounds, top and bottom, and LSMO layers. An emergent phase will appear in
the middle layer (green), between two LSMO layers each influenced by two
different $5d$ compounds with different DMI strengths or different spiral
orientations. The central “green” layer is frustrated by competing tendencies
above and below. Fig. 2: Checkerboard skyrmion crystal. a Plot of the
$z$-component of the spin $S_{z}$, obtained via MC annealing, in a five-layer
heterostructure. b Spin configuration of the frustration-stabilized
checkerboard skyrmion crystal (CSkX) at the central layer ($z\\!=\\!3$). c and
d are the canonical spin spirals at the top layer ($z\\!=\\!5$) and the bottom
layer ($z\\!=\\!1$), oriented along mutually-orthogonal diagonals. Inset in b
shows the central layer spin structure factor $S(\mathbf{q})$ within the first
Brillouin zone, revealing a double-q magnetic order. The parameters used are
$D\\!=\\!J$, $A\\!=\\!0$, $J\\!=\\!1$ (both top and bottom layers).

In this work, we propose a method to create unconventional magnetic phases by
using interfacial magnetic phase frustration as a driving mechanism. In the
conventional geometrical frustration, as in spins in a triangular lattice,
couplings at various distances lead to unconventional phases due to competing
tendencies. Here, in a multilayer geometry the competition arises at the
center of a multilayer ensemble where dominant top and bottom states are not
compatible with one another, thus stabilizing exotic central magnetic phases
through interfacial phase frustration (IPF). In Fig. 1, we show a sketch of
our proposed platform, involving two $5d$ compounds at top and bottom, such as
SrIrO3 and CaIrO3 with robust DMI coupling, as well as metallic lanthanum
manganites La1-xSrxMnO3 (LSMO). The interface-induced DMI from the iridate
generates, e.g., chiral magnetic phases in thin LSMO regions Mohanta_PRB2019 ;
Mohanta_CommPhys2020 . In this set up, the two LSMO layers can acquire a
variety of magnetic phases including a spiral or a skyrmion crystal with
various radii, depending on the doping level in the LSMO and the DMI strength
at the iridate/LSMO interfaces. When two different magnetic phases are
realized in the two LSMO layers, the interface between them will suffer the
IPF that will produce an unconventional magnetic phase. We verified our
proposal by performing Monte Carlo (MC) calculations using a spin Hamiltonian
for a magnetic multilayer. We consider three cases and obtain three
unconventional magnetic phases: (i) a checkerboard skyrmion crystal (CSkX)
that appears from the competition between two orthogonally-aligned spin
spirals, (ii) an incommensurate skyrmion stripe (ISkS) that arises from the
competition between a spin spiral and a triangular skyrmion crystal, and (iii)
a ferrimagnetic skyrmion crystal (FSkX) that arises from the competition
between a triangular skyrmion crystal and a standard antiferromagnet. The
emergent phases in the middle layer are unconventional, multi-q magnetic
textures and are predicted to stabilize in future experimental investigations.
The CSkX is particularly interesting as it is created without any external
magnetic field and is, therefore, promising for spintronic application.
Furthermore, the CSkX produces a large topological Hall effect, induced by a
nonzero scalar spin chirality, which is absent in the competing spin spirals.
Thus, chirality arises from non-chiral components.

IPF paves the way to study artificial electrodynamics and unconventional
magnetic excitations hosted by the frustration-stabilized magnetic phase in
magnetic multilayers. It also broadens the class of chiral magnetic phases
hosting skyrmions. This IPF can be realized generically using any two
different competing magnetic phases. The primary advantage of the generation
of unconventional magnetic orders via IPF is that the periodicity of the
generated textures can be controlled by changing the constituent magnetic
phases above and below. Additional potential material platforms for the
experimental realization of the proposed IPF include superlattices of
transition-metal oxide interfaces involving manganites in which a plethora of
magnetic phases are already available for both Sr- and Ca-doped compounds
Yunoki_PRL2000 ; Moreo_PRL2000 ; Dagotto_PhyRep2001 ; Hotta_PRL2003 , chiral
magnets e.g. MnSi and MnGe, as well as quasi-two-dimensional magnets, such as
CrI3 or Fe3GeTe2. Similar frustration-induced patterns can also be found at
the domain walls between two or more magnetic phases Schoenherr_NPhys2018 .

Fig. 3: Incommensurate skyrmion stripe. a Plot of the $z$-component of the
spin $S_{z}$, obtained via MC annealing, in a five-layer heterostructure. b
Spin configuration of the IPF frustration-stabilized incommensurate skyrmion
stripe (ISkS), obtained via MC annealing at the central layer ($z\\!=\\!3$). c
and d are the spin configurations of a skyrmion crystal and a spin spiral at
the top layer ($z\\!=\\!5$) and the bottom layer ($z\\!=\\!1$), respectively.
The inset in b shows the central-layer spin structure factor $S(\mathbf{q})$
within the first Brillouin zone, revealing a multi-q magnetic order. The
parameters used are $D\\!=\\!1.5J$, $A_{z}\\!=\\!0.3J$, $J\\!=\\!1$ (top
layer), and $D\\!=\\!J$, $A_{z}\\!=\\!0$, $J\\!=\\!1$ (bottom layer).

RESULTS
Model
Consider a multilayer heterostructure, with the top and the bottom layers
hosting two different stable magnetic phases, such as the LSMO 1 and LSMO 2
layers in Fig. 1. The magnetic phases of LSMO, above and below our focus – the
central layer, are influenced by the iridate/LSMO(1,2) interfaces. They can be
described by the following Hamiltonian, with a ferromagnetic Heisenberg
exchange coupling (because LSMO has a metallic ferromagnetic phase in a broad
Sr-doping region), a DMI term from the iridates, and a single-ion uniaxial
anisotropy

$\displaystyle{\cal H}\\!=$ $\displaystyle\\!-J\\!\sum_{\langle
ij\rangle}\\!\mathbf{S}_{i}\\!\cdot\\!\mathbf{S}_{j}\\!-\\!D\\!\sum_{\langle
ij\rangle}\\!\hat{e}_{ij}\\!\cdot\\!\Big{(}\\!\mathbf{S}_{i}\\!\times\\!\mathbf{S}_{j}\\!\Big{)}-A_{z}\\!\sum_{i}\\!S_{zi}^{2}.$
(1)

$\mathbf{S}_{i}$ represents a classical spin vector at site $i$, $J$ is the
ferromagnetic exchange between neighboring spins, $D$ is the DMI coupling,
$\hat{e}_{ij}$ is the unit vector connecting sites $i$ and $j$, and $A_{z}$ is
the strength of uniaxial anisotropy along the $z$ direction. The values of
these parameters define separately the magnetic ground states in each layer.
We keep the spin amplitude $S\\!=\\!1$ and exchange coupling amplitude
$J\\!=\\!1$ throughout this paper, for simplicity. The individual layers are
coupled via a ferromagnetic Heisenberg exchange coupling ${\cal
H}\\!=\\!-J\\!\sum_{\langle
ij\rangle}\\!\mathbf{S}_{i}\\!\cdot\\!\mathbf{S}_{j}$. We consider a five-
layer heterostructure and perform MC annealing to obtain the spin
configurations in all layers.

Fig. 4: Ferrimagnetic skyrmion crystal. a Plot of the $z$-component of the
spin $S_{z}$, obtained via MC annealing, in a five-layer heterostructure. b
Spin configuration of the IPF frustration-stabilized ferrimagnetic skyrmion
crystal (FSkX), obtained via MC annealing at the central layer ($z\\!=\\!3$).
c and d are skyrmion crystal and antiferromagnet configurations at the top
layer ($z\\!=\\!5$) and the bottom layer ($z\\!=\\!1$), respectively. The
inset in b shows the spin structure factor $S(\mathbf{q})$ within the first
Brillouin zone (denoted by the dashed lines), revealing a multi-q magnetic
order. The parameters used are $D\\!=\\!J$, $A_{z}\\!=\\!0.24J$, $J\\!=\\!1$
(top layer), and $D\\!=\\!0$, $A_{z}\\!=\\!0$, $J\\!=\\!-1$ (bottom layer).

Checkerboard skyrmion crystal. We first consider two spin spirals as the top
and bottom layers. The spin spiral phase, which appears with a finite DMI in
the absence of any magnetic field or uniaxial anisotropy, has two degenerate
spiral solutions, aligned mutually orthogonal to each other. One of these two
preferred directions can be realized by directional tuning of a magnetic
anisotropy Garnier_JPM2020 or strain mismatching Bishop_PRL2016 . In our MC
calculations, these two spin spirals were spontaneously generated along the
preferred directions by applying a small inplane anisotropy, described by
${\cal H}_{\rm in}^{\rm aniso}\\!=\\!-A_{\rm
in}\\!\sum_{i}\\!(S_{xi}^{2}\cos{\theta}+S_{yi}^{2}\sin{\theta})$ along
$\theta\\!=\\!45$∘ and $\theta\\!=\\!135$∘ at the top and the bottom layers,
with $A_{\rm in}\\!=\\!0.05J$. Figures 2a-d show the spin configurations at
different layers of the considered five-layer heterostructure. The competition
between the two spin spirals produces an unconventional checkerboard skyrmion
crystal (CSkX). The spin structure factor
$S(\mathbf{q})\\!=\\!(1/N)\sum_{ij}\langle\mathbf{S}_{i}\\!\cdot\\!\mathbf{S}_{j}\rangle
e^{-i\mathbf{q}\cdot\mathbf{r}_{ij}}$ – where $\mathbf{r}_{ij}$ is the
position vector connecting spins $\mathbf{S}_{i}$ and $\mathbf{S}_{j}$ and $N$
is the total number of lattice sites – is shown in the inset of Fig. 2b and
characterizes the double characteristic-momenta q of the CSkX. Skyrmions are
known to arise in non-centrosymmetric systems with a DMI and in
centrosymmetric magnets with a geometrically-frustrated triangular lattice. It
was unclear if skyrmions could appear without inversion symmetry breaking or
geometrical frustration. Recent experiments reported a square skyrmion crystal
in GdRu2Si2, which is not explained by any of these two known mechanisms
Khanh_NatNano2020 . Our results may shed light on the origin of the reported
square arrangement of skyrmions in centrosymmetric itinerant magnets.

Incommensurate skyrmion stripe. Next we focus on the second exotic state that
we found, by considering a triangular skyrmion crystal at the top layer and a
spin spiral at the bottom layer, as depicted in Figs. 3b and c, respectively.
The lower critical magnetic field to obtain the skyrmion crystal depends on
the DMI strength Mohanta_PRB2020 , and therefore, in a heterostructure setting
with two different values of the DMI strengths, the top and the bottom layers
can host skyrmion crystal and spin spiral phases at a finite magnetic field.
More interestingly, a skyrmion crystal can be stabilized by a uniaxial
anisotropy without the need of any magnetic field, as observed, for example,
in the quasi-two-dimensional van der Waals magnet Fe3GeTe2 Yang_SciAdv2020 .
In our MC calculation, the skyrmion crystal is realized in the presence of a
perpendicular magnetic anisotropy at the top layer, at zero magnetic field. As
shown in Fig. 4, the spin configuration stabilized at the central layer by the
IPF, due to the top skyrmion crystal and bottom spin spiral, is an
unconventional multi-q magnetic phase which we call incommensurate skyrmion
stripe (ISkS) since the diagonal stripe order consists of four types of
incommensurate skyrmionic patterns. Note the appearance of a skyrmionic
diagonal pattern induced by the bottom layer which does not have skyrmions.
Specifically, the spin structure factor $S(\mathbf{q})$, shown in the inset of
Fig. 3b, reveals multiple peaks. A careful inspection of this $S(\mathbf{q})$
profile indicates that among the nine peaks there are five characteristic
momenta q for this ISkS—the peak at zero momentum is for the ferromagnetic
correlation, one of the two bright peaks inclined along 45∘ is for the
diagonal stripe order, and three out of the six hexagon-shaped peaks are for
the skyrmionic correlations.

Ferrimagnetic skyrmion crystal. In our third case, we consider a magnetic
trilayer with a skyrmion crystal at the top layer and staggered
antiferromagnetic order at the bottom layer. This situation can appear in our
considered multilayer with different amount of doping in the top and bottom
LSMO layers. As in the previous case, the skyrmion crystal is stabilized in
our MC calculations by a finite DMI and an uniaxial anisotropy. On the other
hand, the antiferromagnetic order is achieved in our spin Hamiltonian by a
negative Heisenberg exchange constant $J\\!=\\!-1$, without DMI and uniaxial
anisotropy. The magnetic configuration, generated at the central layer due to
the IPF, is shown in Fig. 4a-b, while the skyrmion crystal at the top layer
and the antiferromagnetic configuration at the bottom layer are shown in Figs.
4c and d. The IPF-stabilized texture contains remnants of the skyrmion crystal
in the background of a ferrimagnetic order, thus the name ferrimagnetic
skyrmion crystal (FSkX). The spin structure factor $S(\mathbf{q})$, shown in
the inset of Fig. 4b, reveals a multi-q pattern which has fingerprints of both
the skyrmion crystal and the antiferromagnetic order.

Topological Hall response. Magnetic skyrmions are known to produce a
topological Hall effect due to the emergent magnetic field
$B_{z}\\!=\\!\frac{1}{2}\mathbf{n}\\!\cdot\\!\Big{(}\\!\partial_{x}\mathbf{n}\\!\times\\!\partial_{y}\mathbf{n}\Big{)}$
generated from the magnetic texture, where $\mathbf{n}$ represents the spin
vector on the unit sphere created by the spin angles Schulz_NatPhys2012 ;
Nagaosa_NatNano2013 . Even though the proposed IPF mechanism is applicable to
any type of magnetic phases—irrespective of its chiral nature — because the
three cases considered in this study are related to skyrmions – either as top
and bottom components or induced by IPF – we analyzed the topological Hall
response for the IPF-stabilized magnetic phases and compare with that of the
constituent phases.

Fig. 5: Topological Hall effect. Real-space profile of the scalar spin
chirality $\chi(\mathbf{r}_{i})$ for the three IPF-stabilized textures that
appear in our MC simulations at the central layer of the considered
multilayer: a checkerboard skyrmion crystal (CSkX), b incommensurate skyrmion
stripe (ISkS), and c ferrimagnetic skyrmion crystal (FSkX). d Comparison of
the topological Hall conductivity $\sigma_{{}_{\rm TH}}$ for these three
unconventional magnetic phases and the constituent competing phases.

To considering the impact that a chiral magnetic phase, such as those
predicted, would have on the transverse Hall conductivity, it is useful to
investigate the local scalar spin chirality $\chi(\mathbf{r}_{i})$, which is a
real-space equivalent of the Berry curvature that produces the Anomalous Hall
Effect and is defined as

$\displaystyle\chi(\mathbf{r}_{i})\\!=\\!\sum_{\langle
ijk\rangle}\\!\mathbf{S}_{i}\\!\cdot\\!(\mathbf{S}_{j}\\!\times\\!\mathbf{S}_{k}),$
(2)

where $\langle ijk\rangle$ represents three neighboring lattice sites on a
triangule of spins within the considered square lattice grid. The profile of
$\chi(\mathbf{r}_{i})$ for the three IPF-stabilized magnetic phases are
depicted in Figs. 5a-c, revealing a finite scalar spin chirality in all three
phases. In order to compute the transverse Hall conductivity of these magnetic
phases, we consider the following mobile fermions double-exchange Hamiltonian

$\displaystyle{\cal H}_{\rm DE}\\!=\\!-t\\!\sum_{\langle
ij\rangle}\\!c_{i\sigma}^{\dagger}c_{j\sigma}-J_{\rm
DE}\sum_{i,\sigma,\sigma^{\prime}}(\mathbf{S}_{i}\\!\cdot\bm{\sigma}_{\sigma\sigma^{\prime}})c_{i\sigma}^{\dagger}c_{i\sigma^{\prime}}$
(3)

where $t$ is the nearest-neighbor electron hopping energy and $J_{\rm DE}$ is
the Hund exchange coupling strength between the itinerant electron spin
$\bm{\sigma}$ and the localized spin $\mathbf{S}_{i}$ at a particular layer.
In other words, we consider mobile electrons in the background of the fixed
IPF textures. The topological Hall conductivity is obtained via the canonical
Kubo formula

$\displaystyle\sigma_{{}_{\rm
TH}}\\!=\\!\frac{e^{2}}{h}\frac{2\pi}{N}\\!\sum_{\epsilon_{m}\neq\epsilon_{n}}\\!\frac{f_{m}-f_{n}}{(\epsilon_{m}\\!-\\!\epsilon_{n})^{2}\\!+\\!\eta^{2}}{\rm
Im}\Big{(}\langle m|\hat{j}_{x}|n\rangle\langle
n|\hat{j}_{y}|m\rangle\Big{)},$ (4)

where $f_{m}$ is the Fermi function at temperature $T$ and energy eigenvalue
$\epsilon_{m}$, $\hat{j}_{x}$ is the current operator along the $x$ direction,
$|m\rangle$ is the $m^{\rm th}$ eigenvector of ${\cal H}_{\rm DE}$, and $\eta$
is the relaxation rate. We used $t\\!=\\!1$, $J_{\rm DE}\\!=\\!1$, and
$\eta\\!=\\!0.1$ for the present analysis, with no qualitative difference in
the description for other choices, as discussed before Mohanta_PRB2019 . In
Fig. 5d, we present the topological Hall conductivity for the magnetic
textures that we consider in the three cases discussed above. Clearly,
$\sigma_{{}_{\rm TH}}$ is finite for all three chiral magnetic phases
stabilized by the IPF. Notably, $\sigma_{{}_{\rm TH}}$ acquires a finite value
for the CSkX phase, even though the contribution from the spin spirals on the
top and the bottom layers are nearly zero. This finding, therefore, could be
"fingerprints" useful in deciphering topological Hall signals from magnetic
multilayers where the top and bottom spin spirals are oriented along different
directions. We also note that $\sigma_{{}_{\rm TH}}$ from the other two IPF-
stabilized magnetic phases, viz., ISkS and FSkX, are comparable to that from
the constituent skyrmion crystal, which implies that the resultant topological
Hall effect will be nearly doubled than that of the expected value from only
the skyrmion crystal. The appearance of topological Hall effect from non-
skyrmionic magnetic phases is an interesting topic of recent discussion
Mohanta_PRB2020 ; Nakane_PRB2020 . The current findings support these
theoretical predictions, albeit via a totally different IPF platform, and seek
further experimental studies in this direction—exploring topological Hall
effect from non-skyrmionic chiral magnetic textures.

In our five-layer structures, the extreme top and the bottom layers represent
the stable constituent magnetic phases, while the central layer reveals the
IPF-induced magnetic phase. The second and fourth layers show deformed
intermediate magnetic configurations, as shown in Figs. 2a, 3a, and 4a, and
these layers can also possess a finite scalar spin chirality or topological
charge. We also noted that despite the little imperfections in the top and
bottom spin configurations that we obtained via MC annealing, the IPF-induced
pattern at the central layer defines an unconventional state.

DISCUSSION
The interfacial DMI, that is primarily responsible for the chiral magnetic
interactions, can be controlled in $3d$–$5d$ transition-metal-oxide-based
superlattices, such as those made of manganite/iridate interfaces by
engineering the interfacial inversion symmetry Yi_PNAS2016 ;
Skoropata_SciAdv2020 . With advancements in the deposition techniques, the
layer thickness can also tune the Rashba-type interfacial DMI, Dresselhaus-
type bulk DMI, and the anisotropy in the B20 structures, such as
CrGe/MnGe/FeGe and their superlattices Kawakami_JCG2017 . Additional strain
control in these B20 structures can, furthermore, create complex spin textures
from the superposition of different helical domains Repicky_Science2021 .
Therefore, the CrGe/MnGe/FeGe compounds could be used for the experimental
realization of the discussed IPF and generate unconventional magnetic phases.
Besides transition metal oxide interfaces and chiral magnets hosting
skyrmions, the IPF may be explored in quasi-two-dimensional magnets, such as
CrI3, or Fe-based van der Waals magnets, such as Fe3GeTe2.

The appearance of a multi-q order is a generic property of the IPF-induced
phases. These multi-q phases can harbor soft magnetic modes that can be
controlled by an electric field, leading to a characteristic magnetoelectric
effect Leonov_NatComm2015 . Furthermore, due to the frustration, these multi-q
phases may allow skyrmions and anti-skyrmions to be present together in an
exotic thermodynamic phase that can host magnetic monopoles Okubo_PRL2012 .

The IPF-induced phases are, in general, expected to appear below the critical
temperatures of the constituent magnetic phases in the multilayer, whichever
is lesser. The chiral magnetic phases including the SkX commonly appear below
100 K, as revealed by topological Hall effect measurements
PhysRevLett.102.186602 ; Matsuno_sciadv ; Wang2018 ; Vistoli2019 . Therefore,
the three unconventional SkX phases that we discuss here are also expected to
appear within a similar temperature range. Nonetheless, by choosing
constituent phases that persist at higher temperatures, it is possible to
realize unconventional magnetic phases at higher temperatures.

There is a growing interest in three-dimensional chiral magnetic phases for
the urge to understand the complexity arising from topology and frustration in
three dimensions, and also for their potential spintronic applications
Pacheco_NatComm2017 ; Gao_Nature2020 ; Kent_NatCommun2021 . With the layer
control that is predicted here, it would be possible to achieve unconventional
magnetic phases with potential functionalities. The synthetic spin-orbit
coupling available in these exotic skyrmion crystals may also be useful in
fault-tolerant topological quantum computing Mohanta_CommPhys2021 ;
Mascot_NPJQM2021 .

To summarize, we proposed a method to stabilize never-observed-before magnetic
phases by using interfacial phase frustration in multilayers. As example, we
obtained three exotic magnetic phases in our MC calculations, generated by
this interfacial phase frustration. We hope that the proposed frustration
mechanism will open a path, aiding the search of unconventional magnetism in
three dimensions.

METHODS
Monte Carlo simulations
We perform Monte Carlo (MC) annealing to obtain the ground state spin
configuration in the discussed magnetic multilayer. We consider periodic
boundary conditions in plane and use the Metropolis spin update procedure. The
ground state magnetic configurations at the top two and the bottom two layers
in our five-layer structure were first stabilized by performing MC annealing,
without any communication between the top two and bottom two layers. Next, we
switch on the interlayer coupling between all the layers and perform MC
annealing in the entire multilayer. During this process, we start from a
completely random spin configuration at the middle layer. In the MC annealing
processes, we start at a high temperature $T\\!=\\!5J$ and very slowly reduce
the temperature down to $T\\!=\\!0.001J$. At each temperature, a large number
of MC spin updates, typically of the order of $10^{11}$, were performed to
avoid being trapped in metastable states.

Data availability
Data are available from the corresponding author upon reasonable request.

Code availability
Simulation codes are available from the corresponding author upon reasonable
request.

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Acknowledgements
This work was supported by the U.S. Department of Energy, Office of Science,
Basic Energy Sciences, Materials Sciences and Engineering Division.

Author contributions
All authors contributed to all aspects of this work.

Competing interests
The authors declare no competing interest.

Additional information
Correspondence should be addressed to N. Mohanta.