[a]Ho Hsiao

# Chimera baryon spectrum in the Sp(4) completion of composite Higgs models

Ed Bennett  Deog Ki Hong  Jong-Wan Lee  C.-J. David Lin  Biagio Lucini
Maurizio Piai  Davide Vadacchino

###### Abstract

In strongly coupled gauge theories that serve as completions of composite
Higgs models, the fermionic bound states formed by fermions (hyperquarks)
transforming in different representations, called chimera baryons, could serve
as top partners, by embedding of the Standard Model appropriately. We report
our results on the spectrum of chimera baryons in the $Sp(4)$ gauge theory
with hyperquarks transforming in fundamental and two-index antisymmetric
representations. For this study, we adopt the quenched approximation. We
investigate the mass hierarchy between the lightest chimera baryons with
different quantum numbers, as a function of the lattice parameters. Inspired
by baryon chiral effective field theory, and the Akaike Information Criterion,
we perform a first extrapolation to the continuum and massless-hyperquark
limit.

##  1 Introduction

The Higgs boson discovery [1, 2] intensifies the quest for a deeper
understanding of nature. While experimental evidence supports the standard
model (SM), the triviality of its scalar sector suggests it is an effective
field theory (EFT) with a finite ultraviolet (UV) cut-off. Composite Higgs
models (CHMs) stand out as potential theories, as they provide a natural
framework for accommodating a light Higgs boson [3, 4, 5, 6]. These models
introduce a novel strongly coupled sector with an asymptotically-free gauge
theory coupled to fermions (hyperquarks), where the SM Higgs boson emerges as
a pseudo-Nambu-Goldstone boson (PNGB) associated with a global symmetry of the
new strong interaction. Furthermore, CHMs can tackle the flavor problem by
incorporating partial compositeness for the top quark [7]. By coupling the
theory to hyperquarks in two gauge group representations and embedding the SM
gauge group appropriately, bound states formed by hyperquarks in different
representations can share the quantum numbers as the top quark. These bound
states, known as top partners, contribute to the mass of the top quark through
mixing.

Our collaboration has been developing an extensive programme of lattice
studies [8, 9, 10, 11, 12, 13, 14] focused on the $Sp(4)$ gauge theory coupled
to $N_{f}=2$ Dirac fermions in the fundamental, $(f)$, and $n_{f}=3$ Dirac
fermions in the two-index antisymmetric, $(as)$, representations of the gauge
group [15]. In the $(f)$ sector, the $SU(4)/Sp(4)$ coset, due to the
pseudoreality [16], provides candidates for the SM Higgs doublet with only one
additional Goldstone mode. As the $(as)$ representation is real, the global
symmetry is $SU(6)$, broken to $SO(6)$ [16]. The $SU(3)$ subgroup of the
unbroken $SO(6)$ can be identified with the QCD gauge group [15, 6]. In this
framework, the top partners, dubbed chimera baryons, are composed of two $(f)$
and one $(as)$ hyperquarks.

In this contribution, we report the spectrum of the low-lying chimera baryons,
sourced by the following operators:

$\displaystyle{\mathcal{O}}_{\rho}^{ijk,5}$ $\displaystyle\equiv$
$\displaystyle
Q^{i\,a}_{\alpha}(C\gamma^{5})_{\alpha\beta}Q^{j\,b}_{\beta}\Omega^{ad}\Omega^{bc}\Psi^{k\,cd}_{\rho}\,,$
(1) $\displaystyle{\mathcal{O}}_{\rho}^{ijk,\mu}$ $\displaystyle\equiv$
$\displaystyle
Q^{i\,a}_{\alpha}(C\gamma^{\mu})_{\alpha\beta}Q^{j\,b}_{\,\beta}\Omega^{ad}\Omega^{bc}\Psi^{k\,cd}_{\rho}\,,$
(2)

where $Q$ and $\Psi$ are $(f)$ and $(as)$ hyperquarks, respectively, $a,b,c,d$
are hypercolor indices, $\alpha,\beta$, $\rho$ are spinor indices, $i,j,k$ are
flavor indices, $\gamma^{5}$ and $\gamma^{\mu}$ are $4\times 4$ Dirac
matrices, $C$ is the charge conjugation matrix and $\Omega$ is the symplectic
matrix. Given both operators in Eqs. (1) and (2) overlap with even- and odd-
parity states, and the operator ${\mathcal{O}}^{\mu}$ couples to both spin-1/2
and 3/2 states, we perform parity and spin projections to isolate the states
with designed quantum numbers, see Section IIIA in Ref. [17] for detailed
discussions. The lightest state sourced by ${\mathcal{O}}^{5}$, denoted as
$\Lambda_{\rm CB}$, is of spin-1/2 and even-parity. In the case of
${\mathcal{O}}^{\mu}$, the lightest spin-1/2 even-parity state and spin-3/2
even-parity state are $\Sigma_{\rm CB}$ and $\Sigma^{\ast}_{\rm CB}$,
respectively. Both $\Lambda_{\rm CB}$ and $\Sigma_{\rm CB}$ are top-partner
candidates.

Considering this is the first systematic lattice calculation for the chimera
baryon mass spectrum in the $Sp(4)$ gauge theory, we perform simulations in
the quenched approximation. The standard Wilson plaquette is applied for
gauge-field action, while we employ valence fermions described by the Wilson-
Dirac lattice action in calculations of the hyperquark propagators. We
generate five ensembles, listed in Tab. 1, with different lattice spacing
allowing for the continuum extrapolation. The gradient flow method is applied
to set the scale [18], and following the notation in Ref. [13] we express the
masses in unit of the gradient flow $w_{0}$ [19], denoting $\hat{m}=w_{0}m$.
The square of the pseudoscalar meson mass is adopted as a reference scale for
the hyperquark masses [10]. The masses of pseudoscalar mesons with $(f)$ and
$(as)$ hyperquarks are denoted as $m_{\rm PS}$ and $m_{\rm ps}$, respectively.
Regrading the chimera baryons, $m_{\Lambda_{\rm CB}}$, $m_{\Sigma_{\rm CB}}$
and $m_{\Sigma^{\ast}_{\rm CB}}$ denote the mass of even-parity $\Lambda_{\rm
CB}$, $\Sigma_{\rm CB}$, and $\Sigma^{\ast}_{\rm CB}$, respectively.

For a detailed discussion including lattice formulation, projections and
analysis approaches, we refer readers to Ref. [17] and references therein.

Table 1:  Gauge ensembles generated for this study. We display the bare coupling $\beta$, the lattice size, $N_{t}\times N^{3}_{s}$, the average plaquette $\left<P\right>$, and the gradient-flow scale $w_{0}/a$. The gradient-flow scales are taken from Ref. [10].  Ensemble | $\beta$ | $N_{t}\times N^{3}_{s}$ | $\left<P\right>$ | $w_{0}/a$
---|---|---|---|---
QB1 | $7.62$ | $48\times 24^{3}$ | 0.6018905(94) | 1.448(3)
QB2 | $7.7$ | $60\times 48^{3}$ | 0.6088003(35) | 1.6070(19)
QB3 | $7.85$ | $60\times 48^{3}$ | 0.6203811(28) | 1.944(3)
QB4 | $8.0$ | $60\times 48^{3}$ | 0.6307426(27) | 2.3149(12)
QB5 | $8.2$ | $60\times 48^{3}$ | 0.6432300(25) | 2.8812(21)

##  2 Mass hierarchy

To study the hyperquark-mass dependence of the chimera-baryon mass hierarchy,
we perform the calculations at various bare masses, see Appendix A in Ref.
[17]. We observe a decreasing ratio $m_{\Lambda_{\rm CB}}/m_{\Sigma_{\rm CB}}$
with increasing $\hat{m}^{2}_{\rm ps}$. This ratio eventually approaches unity
in the large-$\hat{m}^{2}_{\rm ps}$ regime. A similar pattern emerges with
varying $\hat{m}_{\rm PS}^{2}$ except that $m_{\Lambda_{\rm
CB}}/m_{\Sigma_{\rm CB}}$ never approaches unity in the region of large
$m_{\rm PS}^{2}$. When the $(as)$ hyperquark is heavy, the ratio shows a mild
dependence on $\hat{m}_{\rm PS}^{2}$ and is primarily influenced by
$\hat{m}^{2}_{\rm ps}$. Throughout our entire range of hyperquark masses,
$\Lambda_{\rm CB}$ is not lighter than $\Sigma_{\rm CB}$. Likewise, we find
that $\Sigma^{\ast}_{\rm CB}$ is consistently heavier than $\Sigma_{\rm CB}$
and $\Lambda_{\rm CB}$. The mass gap between them decreases with increasing
$\hat{m}^{2}_{\rm PS}$ and $\hat{m}^{2}_{\rm ps}$. This behavior aligns with
expectations from heavy-hyperquark spin symmetry [20], as increasing
hyperquark masses suppress the effects of spin that contribute to the mass
difference between $\Sigma_{\rm CB}$ and $\Sigma^{\ast}_{\rm CB}$.

##  3 Mass extrapolation

We extrapolate the chimera baryon mass to the continuum and massless-
hyperquark limit by drawing inspiration from baryon chiral perturbation theory
in QCD [21, 22], and from its lattice realization [23]. Considering the
following ansatz, we perform uncorrelated fits to the chimera-baryon masses
using polynomial functions of $\hat{m}_{\mathrm{PS}}$,
$\hat{m}_{\mathrm{ps}}$, and the lattice spacing, $\hat{a}$,

$\displaystyle\hat{m}_{\textrm{CB}}=\hat{m}_{\rm CB}^{\chi}$ $\displaystyle+$
$\displaystyle F_{2}\hat{m}_{\rm PS}^{2}+A_{2}\hat{m}_{\rm
ps}^{2}+L_{1}\hat{a}$ (3) $\displaystyle+$ $\displaystyle F_{3}\hat{m}_{\rm
PS}^{3}+A_{3}\hat{m}_{\rm ps}^{3}+L_{2F}\hat{m}_{\rm
PS}^{2}\hat{a}+L_{2A}\hat{m}_{\rm ps}^{2}\hat{a}$ $\displaystyle+$
$\displaystyle F_{4}\hat{m}_{\rm PS}^{4}+A_{4}\hat{m}_{\rm
ps}^{4}+C_{4}\hat{m}_{\rm PS}^{2}\hat{m}_{\rm ps}^{2}\,.$

Here ${\rm CB}=\Lambda_{\rm CB}$, $\Sigma_{\rm CB}$ or $\Sigma_{\rm
CB}^{\ast}$, and $\hat{m}_{\rm CB}^{\chi}$ denotes the mass of the chimera
baryon in the continuum and massless-hyperquark limit. The coefficients
$F_{j}$ and $A_{j}$ are the low energy constants (LECs) associated with
corrections to $\hat{m}_{\rm CB}$ at the $j$-th power in $\hat{m}_{\rm PS}$
and $\hat{m}_{\rm ps}$, respectively, while the coefficient $C_{4}$ governs
the cross-term proportional to
$\hat{m}^{2}_{\mathrm{PS}}\hat{m}^{2}_{\mathrm{ps}}$. The LECs $L_{1}$,
$L_{2F}$, and $L_{2A}$, correspond to corrections due to the finite lattice
spacing, $\hat{a}$. We only consider polynomial dependence of $\hat{m}_{\rm
CB}$ on pseudoscalar meson masses, omitting possible logarithmic terms, and
perform the fit independently for each chimera baryon.

Our first attempt, fitting to Eq. (3) with the entire dataset, results in a
large value of $\chi^{2}/N_{\rm d.o.f.}$—indicating a very poor fit. As a
result, we impose a set of cuts on pseudoscalar meson masses,
($\hat{m}_{\mathrm{PS,cut}},\hat{m}_{\mathrm{ps,cut}}$), to fit a subset
including data points that sit within the cuts. The initial set of cuts is
chosen as $(\hat{m}_{\mathrm{PS,cut}},\hat{m}_{\mathrm{ps,cut}})=(0.52,0.52)$,
which ensures the minimal subset contains $13$ data points. We then increase
the cut values, $\hat{m}_{\mathrm{PS,cut}}$ and $\hat{m}_{\mathrm{ps,cut}}$,
independently, in steps of $0.05$, and introduce the condition, $am_{\rm
PS}<1$ and $am_{\rm ps}<1$, on the data points within that set of cuts. The
maximum cut values stop at
$(\hat{m}_{\mathrm{PS,cut}},\hat{m}_{\mathrm{ps,cut}})=(1.07,1.87)$, where the
whole dataset is included. In this way, we construct $158$ distinct data sets.
However, the $\chi^{2}/N_{\rm d.o.f.}$ values of fitting each data set to Eq.
(3) are still large. Thus, we proceed the following fitting strategy.

The $158$ data sets are fitted to five fit ansatze based upon truncating Eq.
(3) to include a reduced number of fitting parameters. We first consider the
ansatz, dubbed M2, restricted to the first line of Eq. (3), which contains
$\hat{m}^{\chi}_{\mathrm{CB}}$ and corrections quadratic in pseudoscalar-meson
masses and linear in lattice spacing. The ansatz M3 is introduced by
incorporating also corrections up to cubic in the pseudoscalar-meson masses,
as well as the lattice-spacing corrections, $\hat{m}_{\textrm{PS}}^{2}\hat{a}$
and $\hat{m}_{\textrm{ps}}^{2}\hat{a}$. Finally, we incorporate the three
highest-order terms from Eq. (3) individually. Within ansatze MF4, MA4, and
MC4, these additions correspond to introducing
$F_{4}\hat{m}_{\textrm{PS}}^{4}$, $A_{4}\hat{m}_{\textrm{ps}}^{4}$, or
$C_{4}\hat{m}_{\textrm{PS}}^{2}\hat{m}_{\textrm{ps}}^{2}$, respectively.

Table 2: List of the terms in Eq. (3) that are associated to each fit ansatz used in our analysis.  Ansatz | $\hat{m}_{\rm CB}^{\chi}$ | $\hat{m}_{\rm PS}^{2}$ | $\hat{m}_{\rm ps}^{2}$ | $\hat{m}_{\rm PS}^{3}$ | $\hat{m}_{\rm ps}^{3}$ | $\hat{m}_{\rm PS}^{4}$ | $\hat{m}_{\rm ps}^{4}$ | $\hat{m}_{\rm PS}^{2}\hat{m}_{\rm ps}^{2}$ | $\hat{a}$ | $\hat{m}_{\rm PS}^{2}\hat{a}$ | $\hat{m}_{\rm ps}^{2}\hat{a}$
---|---|---|---|---|---|---|---|---|---|---|---
M2 | ✓ | ✓ | ✓ | - | - | - | - | - | ✓ | - | -
M3 | ✓ | ✓ | ✓ | ✓ | ✓ | - | - | - | ✓ | ✓ | ✓
MF4 | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | - | - | ✓ | ✓ | ✓
MA4 | ✓ | ✓ | ✓ | ✓ | ✓ | - | ✓ | - | ✓ | ✓ | ✓
MC4 | ✓ | ✓ | ✓ | ✓ | ✓ | - | - | ✓ | ✓ | ✓ | ✓

With these five ansatze, tabulated with their associated terms in Tab. 2, and
$158$ data sets, we obtain $790$ analysis procedures. In order to
quantitatively select the best procedure, we compute the Akaike information
criterion (AIC) [24, 25]. For each analysis procedure we compute

${\rm AIC}\equiv\chi^{2}+2k+2N_{\rm cut}\,,$ (4)

where $\chi^{2}$ represents the standard chi-square, $k$ is the number of fit
parameters, and $N_{\rm cut}$ is the number of data points removed by the
cuts, $(\hat{m}_{\mathrm{PS,cut}},\hat{m}_{\mathrm{ps,cut}})$. The
corresponding probability weight is defined as

$W\equiv\frac{1}{\mathcal{N}}\exp\left[{-\frac{1}{2}{\rm AIC}}\right]\,,$ (5)

where the normalization factor $\mathcal{N}$ assures the sum of $W$ over all
$790$ analysis procedures equals to one.

Table 3: Low-energy constants in Eq. (3) for each chimera baryon, as determined by the best analysis procedure in the ansatz at $(\hat{m}_{\mathrm{PS,cut}},\hat{m}_{\mathrm{ps,cut}})$. The missing coefficients are set to zero. CB | $\hat{m}_{\rm CB}^{\chi}$ | $F_{2}$ | $A_{2}$ | $F_{3}$ | $A_{3}$ | $C_{4}$
---|---|---|---|---|---|---
$\Lambda_{\rm CB}$ | 0.999(27) | 0.709(61) | 0.383(10) | -0.150(34) | -0.092(4) | -0.026(5)
$\Sigma_{\rm CB}$ | 0.841(21) | 0.815(77) | 0.558(13) | -0.252(63) | -0.161(7) | -0.078(7)
$\Sigma^{\ast}_{\rm CB}$ | 1.259(34) | 0.360(110) | 0.393(29) | -0.071(93) | -0.129(16) | -
CB | $L_{1}$ | $L_{2F}$ | $L_{2A}$ | Ansatz | $\hat{m}_{\rm PS,cut}$ | $\hat{m}_{\rm ps,cut}$
$\Lambda_{\rm CB}$ | -0.137(43) | 0.071(69) | 0.004(11) | MC4 | 1.07 | 1.87
$\Sigma_{\rm CB}$ | -0.141(33) | 0.187(65) | -0.021(16) | MC4 | 0.77 | 1.47
$\Sigma^{\ast}_{\rm CB}$ | -0.335(54) | 0.342(83) | 0.004(30) | M3 | 0.82 | 1.17

By applying this fitting strategy to the three chimera baryons independently,
we find that the optimal choice of analysis procedure for $\Lambda_{\rm CB}$
is MC4 at $(\hat{m}_{\mathrm{PS,cut}},\hat{m}_{\mathrm{ps,cut}})=(1.07,1.87)$.
Regrading $\Sigma_{\rm CB}$ and $\Sigma^{\ast}_{\rm CB}$, the best analysis
procedures are MC4 at
$(\hat{m}_{\mathrm{PS,cut}},\hat{m}_{\mathrm{ps,cut}})=(0.77,1.47)$ and M3 at
$(\hat{m}_{\mathrm{PS,cut}},\hat{m}_{\mathrm{ps,cut}})=(0.82,1.17)$,
respectively. We present corresponding LECs obtained by the best analysis
procedures in Tab. 3, see more details in Section IIIC of Ref. [17].

Having determined the LECs, we explore the dependence of the chimera-baryon
masses in the continuum limit by taking $\hat{a}=0$ in Eq. (3). In Fig. 1, the
left (right) panel shows the change of $\hat{m}_{\Lambda_{\rm CB}}$,
$\hat{m}_{\Sigma_{\rm CB}}$, and $\hat{m}_{\Sigma_{\rm CB}}$ with respect to
the variation of $\hat{m}_{\rm PS}$ ($\hat{m}_{\rm ps}$) in the limit where
$\hat{m}_{\rm ps}=0$ ($\hat{m}_{\rm PS}=0$). The mass hierarchy,
$\hat{m}_{\Sigma_{\rm CB}}\lesssim\hat{m}_{\Lambda_{\rm
CB}}<\hat{m}_{\Sigma^{\ast}_{\rm CB}}$, is observed across the entire range of
hyperquark masses explored in this study. Specifically, the masses
$\hat{m}_{\Lambda_{\rm CB}}$ and $\hat{m}_{\Sigma_{\rm CB}}$ are compatible
only in the regime where $(as)$ hyperquarks are substantially heavy. This
hierarchy carries non-trivial implications for the development of composite
Higgs models with top partial compositeness.

Figure 1:  Mass of chimera baryons ${\Lambda_{\rm CB}}$, ${\Sigma_{\rm CB}}$,
and ${\Sigma^{\ast}_{\rm CB}}$ as a function of $\hat{m}^{2}_{\rm PS}$ (left)
and $\hat{m}^{2}_{\rm ps}$ (right) in the limit of vanishing lattice spacing.
The plots are generated with $\hat{m}^{2}_{\rm ps}=0$ (left) and
$\hat{m}^{2}_{\rm PS}=0$ (right), based on best-fit LECs from Tab. 3, with
bands representing statistical errors. Bands extend horizontally between zero
and optimal choices of $\hat{m}^{2}_{\rm PS,cut}$ (left) and $\hat{m}^{2}_{\rm
ps,cut}$ (right).

In addition, we extend the plot of the mass spectrum by including mesons and
glueballs in the theory. In Fig. 2, meson and glueball masses are taken from
our previous studies in the quenched approximation [10, 11], while the
chimera-baryon masses are taken in the massless-hyperquark limit
($\hat{m}^{2}_{\rm ps}=\hat{m}^{2}_{\rm PS}=0$). Mesons labeled with capital
letters consist of $(f)$ hyperquarks, whereas those represented by lowercase
letters are made of $(as)$ hyperquarks. All masses have been extrapolated to
the continuum and massless-hyperquark limit. They are displayed in both
gradient-flow units (vertical axis on the left-hand side) and units of the
fundamental pseudoscalar meson decay constant (vertical axis on the right-hand
side). The masses of the top-partner candidates, $\Lambda_{\rm CB}$ and
$\Sigma_{\rm CB}$, closely align with those of the $(as)$ vector mesons.

##  4 Summary and Outlook

We presented our measurement, performed in the quenched approximation and in
the Sp(4) gauge theory, of the mass spectrum of chimera baryons, $\Lambda_{\rm
CB}$, $\Sigma_{\rm CB}$, and $\Sigma^{\ast}_{\rm CB}$. The first two such
states are top partner candidates in a class of composite Higgs model with top
partial compositeness. We examined their mass hierarchy and performed
continuum and massless-hyperquark extrapolations. This study serves as a
foundation for future lattice simulations with dynamical hyperquarks.

Figure 2: Quenched spectrum of the $Sp(4)$ gauge theory in the continuum and
massless-hyperquark limit. The pseudoscalar, vector, tensor, axial-vector,
axial-tensor and scalar mesons composed of fundamental (antisymmetric)
hyperquarks are denoted as PS (ps), V (v), T(t), AV (av), AT (at) and S (s),
respectively, while glueball states are labelled by $J^{P}$. The results of
mesons and glueballs are taken from our previous works in Refs. [10, 11].

## Acknowledgments

The work of HH and CJDL is supported by the Taiwanese MoST grant
109-2112-M-009-006-MY3 and NSTC grant 112-2112-M-A49-021-MY3. The work of EB
has been supported by the UKRI Science and Technology Facilities Council
(STFC) Research Software Engineering Fellowship EP/V052489/1, and by the
ExaTEPP project EP/X017168/1. The work of DKH was supported by Basic Science
Research Program through the National Research Foundation of Korea (NRF)
funded by the Ministry of Education (NRF-2017R1D1A1B06033701). The work of JWL
was supported in part by the National Research Foundation of Korea (NRF) grant
funded by the Korea government(MSIT) (NRF-2018R1C1B3001379) and by IBS under
the project code, IBS-R018-D1. The work of DKH and JWL was further supported
by the National Research Foundation of Korea (NRF) grant funded by the Korea
government (MSIT) (2021R1A4A5031460). The work of BL and MP has been supported
in part by the STFC Consolidated Grants No. ST/P00055X/1, ST/T000813/1, and
ST/X000648/1. BL and MP received funding from the European Research Council
(ERC) under the European Union’s Horizon 2020 research and innovation program
under Grant Agreement No. 813942. The work of BL is further supported in part
by the Royal Society Wolfson Research Merit Award WM170010 and by the
Leverhulme Trust Research Fellowship No. RF-2020-4619. The work of DV is
supported by STFC under Consolidated Grant No. ST/X000680/1. Numerical
simulations have been performed on the Swansea University SUNBIRD cluster
(part of the Supercomputing Wales project) and AccelerateAI A100 GPU system,
on the local HPC clusters in Pusan National University (PNU) and in National
Yang Ming Chiao Tung University (NYCU), and on the DiRAC Data Intensive
service at Leicester. The Swansea University SUNBIRD system and AccelerateAI
are part funded by the European Regional Development Fund (ERDF) via Welsh
Government. The DiRAC Data Intensive service at Leicester is operated by the
University of Leicester IT Services, which forms part of the STFC DiRAC HPC
Facility (www.dirac.ac.uk). The DiRAC Data Intensive service equipment at
Leicester was funded by BEIS capital funding via STFC capital grants
ST/K000373/1 and ST/R002363/1 and STFC DiRAC Operations grant ST/R001014/1.
DiRAC is part of the National e-Infrastructure.

Open Access Statement—For the purpose of open access, the authors have applied
a Creative Commons Attribution (CC BY) licence to any Author Accepted
Manuscript version arising.

Research Data Access Statement—The results for Sp(4) are based on preliminary
analysis. Further analysis and the data generated for this manuscript will be
released together with an upcoming publication [17]. Alternatively, data and
code can be obtained from the authors upon request.

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# Dirac-like fermions anomalous magneto-transport in a spin-polarized oxide
two-dimensional electron system

Yu Chen1<EMAIL_ADDRESS>Maria D’Antuono2,1 Mattia Trama3 Daniele
Preziosi4 Benoit Jouault5 Frédéric Teppe5 Christophe Consejo5 Carmine A.
Perroni2,1 Roberta Citro3,6 Daniela Stornaiuolo2,1
<EMAIL_ADDRESS>Marco Salluzzo1<EMAIL_ADDRESS>1CNR-
SPIN, Complesso Univ. Monte S. Angelo, Naples, I-80126, Italy  2Department of
Physics, University of Naples Federico II, Complesso Univ. Monte S. Angelo,
Naples,I-80126, Italy  3Department of Physics, Universitá degli studi di
Salerno, Via Giovanni Paolo II, 132, Salerno, I-84084, Italy  4Institut de
Physique et Chimie des Matériaux de Strasbourg, UMR 7504 CNRS and Université
de Strasbourg, Strasbourg, F-67034, France  5Laboratoire Charles Coulomb,UMR
5221, CNRS, Université de Montpellier, Montpellier, F-34095, France  6CNR-
SPIN, Via Giovanni Paolo II, 132, Salerno, I-84084, Italy

###### Abstract

In two-dimensional electron systems (2DES) the breaking of the inversion,
time-reversal and bulk crystal-field symmetries is interlaced with the effects
of spin-orbit coupling (SOC) triggering exotic quantum phenomena. Here, we
used epitaxial engineering to design and realize a 2DES characterized
simultaneously by ferromagnetic order, large Rashba SOC and hexagonal band
warping at the (111) interfaces between LaAlO3, EuTiO3 and SrTiO3 insulators.
The 2DES displays anomalous quantum corrections to the magneto-conductance
driven by the time-reversal-symmetry breaking occurring below the magnetic
transition temperature. The results are explained by the emergence of a non-
trivial Berry phase and competing weak anti-localization / weak localization
back-scattering of Dirac-like fermions, mimicking the phenomenology of gapped
topological insulators. These findings open perspectives for the engineering
of novel spin-polarized functional 2DES holding promises in spin-orbitronics
and topological electronics.

Low dimensional electron systems can display exotic physical phenomena
governed by the emergence of a non-trivial Berry curvature. The latter is
linked to a band structure locally mimicking the dispersion relations of
relativistic particles. The effects of these Dirac-like fermions in a 2DES are
revealed when the chemical potential is tuned near the avoided crossing of
Rashba-like SOC split bands. In three-dimensional (3D)-topological insulators
(TIs) Dirac fermions drive intriguing quantum transport properties, such as
the quantum anomalous Hall effect [1, 2], topological phase transitions [3]
and changeovers from weak anti-localization (WAL) to weak localization (WL)
quantum corrections to the magneto-conductance (MC) [4, 5, 6]. In particular,
the hexagonal band warping of most 3D-TIs can trigger a net out-of-plane spin
polarization induced by in-plane magnetic fields, and the opening of
scattering channels between multiple pairs of stationary points with opposite
momenta [7]. When a time-reversal-symmetry (TRS) breaking is introduced in
such systems [4, 5, 6], the opening of a magnetic gap leads to the creation of
an extra WL scattering channel, beside the WAL spin-orbit scattering
associated to a $\pi$ Berry phase [8, 9]) (see Figures 1(a-c)).

Among low dimensional electron systems, 2DES at the oxide interfaces are
particularly appealing, as they can be opportunely engineered by epitaxy to
host novel quantum phenomena, from unconventional superconductivity [10, 11,
12] to exotic magnetism and multiferroicity [13]. In the 2DES at the (001)
interfaces between LaAlO3 (LAO), EuTiO3(ETO) and SrTiO3 (STO) (LAO/ETO/STO),
for instance, ferromagnetic correlations, due to the magnetic ordering of Eu2+
(4f7) ions, coexist with a relatively large Rashba-like SOC and unconventional
superconductivity [11, 14].

Recently, the study of (111) STO-based interfaces unveiled a large and
unexpected second order bi-linear magneto-resistance and an anomalous planar
Hall effect [15, 16, 17]. The results were interpreted as signatures of an
out-of-plane spin-polarization and non-trivial Berry curvature nearby the
Dirac-like point formed at crossing bands split, at finite momentum, by the
Rashba-SOC. The phenomenon is triggered by a large external in-plane magnetic
field and, in part, linked to the hexagonal band-warping of (111)
heterostructures [16].

Here, we show how epitaxial engineering of oxide heterostructures enables the
realization of a rare example of oxide 2DES characterized by a non-trivial
Berry phase ($\gamma\notin\\{0,\pi\\}$) without the application of an in-plane
magnetic field. This non-trivial Berry phase gives rise to competing WL/WAL
corrections to the MC due to Dirac-like fermions, with a phenomenology
analogous to that of gapped 3D-TIs [4, 5, 6].

Figure 1: Schematics of WL, WAL, and WL/WAL back-scattering, and crystal,
energy-bands, spin-texture and Berry curvature in (111) 2DES. (a) Sketches of
WL in a system without spin-momentum locking, e.g., disordered metals with
constructive interference of self-crossing loops and a zero Berry phase. (b)
WAL in SOC systems, e.g., topological insulators and oxide interfaces, due to
a destructive interference and a $\pi$ Berry phase. (c) Competing WL and WAL
with a non-trivial Berry phase at low doping (right) in gapped Dirac systems,
e.g., magnetically doped TIs [4]. (d) Sketch of the LAO/ETO/STO
heterostructure with a back-gate electrode. (e) The (Eu, Sr)TiO3 lattice
(left) and its projection (right) on the (111) interface/surface plane. Along
the [111] interface/surface normal, the system consists of three non-
equivalent layers of Ti atoms (red, green, blue), with the two perpendicular
$[\bar{1}\bar{1}2]$ and $[\bar{1}10]$ crystallographic axes. (f) Electronic
band structure, E(k) of (111) interfaces, showing bands with $a_{1g}$ (red)
and $e_{g\pm}^{\pi}$ (green, blue) orbital character, together with their
Rashba-like split counterparts (dark dotted lines). Throughout this letter,
$k\tilde{a}$ are in units of $\pi$, where $\tilde{a}=\sqrt{2/3}a_{0}$ with the
cubic STO lattice constant $a_{0}=3.905\AA$. (g) Spin arrangements of the
lowest spin-split bands composed of $a_{1g}$ orbitals, exhibiting the
emergence of an out-of-plane component and (h) corresponding Fermi contours
shown as continuous/dotted lines and Berry curvature color map of a sub-band
at EF = 60 meV.

The 2DES was realized by sequential epitaxial deposition of 3 unit cells (uc)
delta-doping (111) ETO layer and 14 uc LAO on a Ti-terminated (111) STO single
crystal (see Methods and Supplementary Fig. S1a). Magneto-transport properties
were studied on Hall-bar devices patterned using photolithography and ion beam
etching at low temperature [18] (see Methods). Transport characterizations
[19] confirm the formation of a 2DES with a sheet resistance temperature
dependence similar to that of (001) and (111) LAO/STO heterostructures
(Supplementary Fig. S1b). Tuning of the chemical potential was achieved by
back-gating, as schematically shown in Fig. 1d.

(111) STO-based oxide 2DES can be described as repetition of three
inequivalent Ti-layers stacked along the [111] vector, i.e., perpendicular to
the interface, arranged in a hexagonal lattice [20, 21, 22, 23] (Fig. 1e).
Inversion symmetry breaking gives rise to a $D_{3d}$ trigonal symmetry,
resulting in hybridized $t_{2g}$ Ti-orbitals forming $a_{1g}$, $e_{g}^{\pi}$
states [24] (Fig. 1f and Supplementary Fig. S2(a,b)). The trigonal crystal
field has several consequences on the electronic band structure of the 2DES.
First, as shown by tight-binding calculations (Supplementary Note I), the
Fermi surface of the lowest-energy bands exhibits a snowflake shape and a
hexagonal band warping in a wide range of chemical potentials, affecting the
electronic and transport properties [22]. Second, the hexagonal warping brings
about an out-of-plane spin textures (see Fig. 1g and ref. [15, 23, 16]) and a
Berry curvature with alternating positive and negative values around the
Brillouin zone, as illustrated by numerical calculations for the lowest band
with $a_{1g}$ orbital character (Fig. 1h). This Berry curvature leads to an
overall $\pi$ Berry phase (Supplementary Note II). As a result, pure WAL
corrections to the low field MC are expected in these (111) systems that do
not have a TRS breaking [16, 12, 17].

Figure 2: Magneto-transport properties of the (111) LAO/ETO/STO 2DES. (a) Gate
voltage (at 1.8 K) and temperature (at Vg = -10 V) dependence of the Hall
effect on a Hall-bar oriented along the $[\bar{1}10]$ crystallographic
direction. The inset is a zoomed-in view. (b) Inverse Hall coefficient versus
the sheet conductance, tuned by the gate voltage as shown in the inset. (c) MC
as function of gate voltages in the range +30 V to -40 V (upper panel) and -40
V to -120 V (lower panel). MC data in full voltage range is also shown in
Supplementary Fig. S4. The color-code of each MC data corresponds to the sheet
conductance shown in the colorbar ($\sigma\approx 1\>mS-2.9\>\mu S$). Red
lines are the best fits using Equation (1). (d) Extracted MC data from the
data set and their fits, at 30, 0, -40, -80 and -120 V, to highlight the
evolution of the peak-shoulder feature as function of the gate voltage. The MC
at +30 V in the full range is displayed in the right axis.

On the other hand, (111) LAO/ETO/STO is characterized by ferromagnetically
ordered Eu-ions, as confirmed by x-ray magnetic circular dichroism and SQUID
magnetization measurements (see Supplementary Fig. S3 and ref. [19]). The
interaction between Eu-4f spins and Ti-3d moments induces a spin-polarization
in the 2DES. Moreover, the trigonal crystal field splitting in this system is
three times larger than the one of the (111) LAO/STO system [24], enhancing
the band splitting between the lowest $a_{1g}$ and $e_{g}^{\pi}$ derived
bands. The enhanced splitting and the TRS-breaking driven by the ferromagnetic
order are expected to induce novel quantum states, distinct from non-magnetic
2DES.

To study the low temperature physics of (111) LAO/ETO/STO, we performed an
extensive study of the gate-voltage dependent magneto-transport properties of
the system (Fig. 2). In Fig. 2a we show the Hall effect data measured as
function of the gate voltage and temperature. The Hall-effect resistivity is
always linear, suggesting single-band transport of electrons belonging to the
lowest $a_{1g}$ band. The inverse Hall coefficient decreases when increasing
the gate voltage (Fig. 2b), which is opposite to the expected accumulation of
electron-carriers observed for (001) oxide interfaces. This result is a direct
consequence of the band-warping, i.e., the non-parabolic nature of the
$a_{1g}$ band. The snowflake Fermi contours of this band give rise to regions
having positive and negative curvatures, thus opposite Fermi velocities, akin
to (111) LAO/STO interfaces in the low doping regime (see ref. [22] and
Supplementary Note III).

In Fig. 2(c, d) we show the gate voltage Vg dependence of the anomalous MC of
(111) LAO/ETO/STO. At large negative gate voltage, i.e. -120 V, the MC is
negative at low fields, similarly to what happens in the presence of WAL
corrections (Fig. 2(d) bottom panel). Increasing the gate voltage, the shape
changes drastically and the MC becomes positive with a cusp-like shape,
followed by a peak and a shoulder. The peak to shoulder feature increases with
Vg, reaches a maximum at Vg=-40 V, and then decreases again until it almost
(but not completely) disappears at +30 V (Fig. 2(d) upper panel). This
behavior, in particular the positive cusp-like shape, is remarkably different
from the positive parabolic WL or negative cusp-like WAL corrections observed
in non-magnetic 2DES at the (001) and (111) LAO/STO [25, 12, 16] and
LaTiO3/STO interfaces [17]. The latter were well explained by the models of
Hikami-Larkin-Nagaoka, Iordanskii-Lyanda-Geller-Pikus, or Maekaewa-Fukuyama
[26, 27, 28]. On the other hand, the MC data of (111)LAO/ETO/STO (Fig. 2(c,d))
are reminiscent of the phenomenology observed in the magneto-transport of
magnetically doped 3D-TIs [6], where it was attributed to competing WAL and WL
channels induced by the opening of a magnetic gap $\Delta$ (see Figure 1(c)),
and well reproduced by the formula derived in ref. [4]:

$\delta\sigma(B)=\sum_{i=0,1}\frac{\alpha_{i}e^{2}}{\pi
h}\left[\Psi\left(\frac{\ell_{B}^{2}}{\ell_{\phi
i}^{2}}+\frac{1}{2}\right)-\ln\left(\frac{\ell_{B}^{2}}{\ell_{\phi
i}^{2}}\right)\right],$ (1)

where $\Psi$ is the Digamma function, $\ell_{B}^{2}={\hbar}/{4eB}$ and
$\ell_{\phi i}$ are the effective phase coherence lengths. The pre-factors
$\alpha_{0}$ and $\alpha_{1}$, with opposite signs, represent the weights of
the WAL and WL channels contributions. As shown in Fig. 2(c), where the
fitting-curves are displayed as red-lines, Equation (1) captures well the data
in the full range of gate voltages. As for 3D-TIs [4], both $\alpha_{i}$ and
$\ell_{\phi i}$ are expected to be a function of the ratio between the
magnetic gap and chemical potential $\Delta$/EF (see Supplementary Fig. S5).
The good agreement between the data and Equation (1) shown in Fig. 2(c,d)
indicates that the low temperature anomalous MC is explained by two competing
WL and WAL scattering channels related to the magnetic gap $\Delta$ opening,
each contributing to the MC quantum corrections.

Figure 3:  Temperature dependence of MC, calculated band structures and Berry
curvatures in a 2DES with in plane ferromagnetic correlations. (a) MC data as
function of the temperature at -10 V, 0 V, 5 V gate voltages, and (b)
corresponding color-maps. The anomalous peak to shoulder feature disappears at
slightly different temperatures depending on the gate voltage. (c) Energy
bands in the presence of a planar exchange field with the direction along
$[\bar{1}\bar{1}2]$. Lowest Rashba-spin split energy bands and (d) Berry
curvature at EF = 60 meV of a spin-split band analytically evaluated from the
minimal model described in the main text. The calculations show the emergence
of a hot-spot, i.e. a non-trivial Berry phase, near the Dirac-like point.
Using tight-binding Hamiltonian (beyond the minimal model) band structure
calculations of E(k) along $[\bar{1}10]$ (e), and non-trivial Berry curvature
(f). The avoided crossing of the bands and the opening of a magnetic gap are
magnified in the insets of panels (c,e).

In this framework, a purely WAL MC is predicted upon closing of the magnetic
gap, which should take place approaching the FM transition of the system. In
order to verify the link between the anomalous MC and the magnetism of the
2DES, in Fig. 3(a,b) we show the peak-shoulder feature evolution as a function
of the temperature at different gate voltages. By increasing the temperature,
the MC-shape changes completely within few Kelvins and a negative MC, i.e.
WAL, is recovered at temperatures in the range between 5 and 8 K, compatible
with the fading away of magnetic correlations induced by the Eu2+ magnetic
order (Tc between 5 and 10 K) [11, 29]. Thus the experimental results show
that the anomalous MC is linked to the magnetic gap, that is, the band
topology character.

As mentioned above, in magnetically doped 3D-TIs this band topology gives rise
to a non-trivial Berry phase (see Fig. 1c and ref. [4]). Now we introduce a
minimal model including Rashba-split bands derived from the lowest $a_{1g}$
orbital, a hexagonal warping term, and a magnetic exchange interaction related
to the magnetic correlations in the 2DES. It is worth noting that within this
minimal model, the hexagonal warping approximates well the band-structure of
(111) 2DES only at low values of the momentum (Supplementary Fig. S2a).
However, it has the advantage to provide analytical expressions for most of
the quantities, which helps in understanding the main consequences of the
interplay between the Rashba-SOC, the TRS-breaking, and the breaking of the
bulk ($D_{4h}$) crystal-field symmetry into the $D_{3d}$ symmetry.

Magnetic correlations in the 2DES are included by adding an in-plane
ferromagnetic magnetization of the form $H_{FM}={\bf
h}_{||}\cdot\vec{\sigma}=\epsilon_{||}(\sigma_{x}\cos\psi+\sigma_{y}\sin\psi)$
(Supplementary Note IV). Here $\psi$ is the angle between the magnetization
direction and the in-plane axis $\hat{x}$ ($[\bar{1}10]$), and ${\bf h}_{||}$
is the effective parallel field associated to the in-plane magnetic exchange
$\epsilon_{||}$. The energy band dispersion assumes the form
$E_{\pm}=\hbar^{2}\mathbf{k}^{2}/2m^{*}\pm\sqrt{\alpha^{2}k^{6}\cos^{2}(3\phi)+\lambda^{2}\tilde{k}^{2}}$,
where tan($\phi$)=$k_{y}/k_{x}$,
$\tilde{k}^{2}=(k_{x}-\epsilon_{||}\sin\psi/\lambda)^{2}+(k_{y}+\epsilon_{||}\cos\psi/\lambda)^{2}$,
$m^{*}$ is effective mass, $\lambda$ is Rashba-SOC strength, and $\alpha$ is
the warping parameter (Supplementary Note II). In the presence of the internal
in-plane magnetic exchange, a magnetic gap
$\Delta=2\alpha(\epsilon_{||}/\lambda)^{3}\sin(3\psi)$ opens up because the
warping term couples the in-plane magnetization to the out-of-plane spin (Fig.
3(c,f)), except for $\psi=2\pi n/6$, i.e. along $[\bar{1}10]$ (Supplementary
Note IV and Supplementary Fig. S6). The Dirac-like point is then shifted from
the center of Brillouin zone to the
$\bf{k^{*}}=\hat{z}\times{\bf{h}}_{||}/\lambda$ position, where $\hat{z}$ is
the unit vector along the [111] direction, leading to a non-trivial Berry
curvature (Fig. 3d) and a hot-spot at the avoided crossing between the two
Rashba-SOC spin-split bands [16].

In order to confirm the overall picture provided by the minimal model, we
performed calculations also using the full Hamiltonian tight binding approach,
which better reproduces the snowflake shape of the Fermi surfaces. As shown in
Fig. 3(e,f), the full Hamiltonian tight binding calculations confirm the
emergence of a non-trivial Berry curvature and give an even stronger evidence
of a hot-spot at the avoided crossing between the two Rashba-SOC spin-split
bands (related spin textures shown in Supplementary Fig. S5(b-d)).

The evolution with the gate-voltage of the anomalous MC-data in the (111)
LAO/ETO/STO 2DES can be, then, directly related to the tuning of the chemical
potential: a maximum of the peak-shoulder feature associated to the competing
WL and WAL terms is attained when the chemical potential approaches the hot-
spot, while departing from it drives the gradual reduction. Thus, the MC is
strongly affected by the back-scattering of the Dirac-like fermions,
experiencing a non-trivial Berry phase along self-crossing loops. This
behavior, already established in gapped 3D-TIs [4, 5, 6], was rarely found in
other 2DES, since it requires the simultaneous presence of a large Rashba-like
SOC, a sizable gap $\Delta$ due to TRS-breaking, and very low chemical
potential [4].

Our work establishes a method to create oxide 2DES characterized by transport
properties mimicking those of systems hosting Dirac fermions, as in gapped
3D-TIs. The simultaneous presence of Rashba-SOC, magnetic correlations, and
the hexagonal symmetry of the system with enhanced trigonal crystal field
splitting, allows the tuning of the 2DES chemical potential near a Dirac-like
point generated by the spin-split lowest energy bands. This triggers Berry-
curvature hot-spot and topological charges in the 2DES without the need of
external planar magnetic field, which are on the other hand needed in the case
of non-magnetic 2DES, e.g., in (111) LAO/STO and LaTiO3/STO [16, 17]. Similar
approaches can be envisaged in the case of other novel interfaces, including
2D-atomic non-magnetic/magnetic bi-layers and other oxide 2DES, alike
superconducting EuO/KTaO3 (111) based-2DES [30], opening a vast space for
exploration at the intersection between topology and correlations of broad
interest in the field of spin-orbitronics and topological electronics.

## Acknowledgments

This project has received funding from the European Union’s Horizon Europe
research and innovation programme project IQARO, under grant agreement n.
101115190, by the Ministry of University and Research project PRIN 2022
SONATA, No. P2022SB73K, funded by the EU - Next Generation EU, and PRIN 2022
STIMO, No. 2022TWZ9NR. M. S. and Y. C. acknowledge financial support from PNRR
MUR project PE0000023-NQSTI. M.T. acknowledges financial support from
”Fondazione Angelo Della Riccia”. DP thanks the French National Research
Agency (ANR) through the ANR-JCJC FOXIES ANR-21-CE08-0021.

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## Methods

Sample growth: Epitaxial LAO/ETO heterostructures were deposited on Ti-
terminated STO(111) single crystals using pulsed laser deposition (PLD)
assisted by reflection high energy electron diffraction (RHEED) [19] at 720
${}^{\circ}C$ and in a background oxygen pressure of $7.5\times 10^{-5}$ mbar.
A KrF excimer laser with 248 nm wavelength and 1 Hz repetition was focused on
sintered Eu2Ti2O7 target and on a crystalline LAO target at a fluence of 1.3
Jcm-2. The sequential deposition of ETO (3 uc) and LAO (14 uc) films on Ti-
terminated (111) STO single crystal was monitored by the oscillations of the
specular RHEED intensity (Supplementary Fig. S1a).

Device fabrication: Different from the pattern technique of lithography -
amorphous LAO deposition - liftoff - epitaxial film growth [31], here 300 $\mu
m$ $\times$ 100 $\mu m$ Hall bars were realized by a combination of optical
lithography and cold ion milling on the film LAO/ETO/STO in order to maintain
the interface cleanliness as much as possible, following the procedure
described in Ref. [18].

Magneto-transport measurements: Magneto-transport measurements were performed
in a He4 flow cryostat using standard lock-in amplifiers (f = 10 Hz and jRMS =
100 nA). The longitudinal and transverse voltages were measured
simultaneously. The magnetic field was swept in the $\pm$ 12 T range. Data as
function of the back-gate were acquired in a +30 V to -120 V range. The
leakage current across the (111) STO gate-oxide was below 1 nA. The sheet
resistance saturates above +30 V. MC and Hall effect data were also acquired
at different temperatures in the 1.8 K to 8 K range.

XMCD and SQUID magnetization measurements: Eu-M4,5 edge XAS spectra were
acquired at the Extreme beamline of the PSI-SLS and at the ID32 ESRF
synchrotron facilities using the total electron yield (TEY) method (see ref.
[19]). The magnetic-field-dependent magnetization loops were obtained by
measuring, at each field, the difference between the TEY intensities at the
M5-Eu edge peaks obtained with two different helicities (combination of
polarization and field direction), and normalized to the background acquired
at an energy below the absorption edge. SQUID data were collected by using a
Quantum Design MPMS3 magnetometer at the Université de Strasbourg, IPCMS. The
data were corrected for the STO diamagnetism subtracting the high field linear
contribution.

## Declarations

  * •

Author contribution
Conceptualization: YC, DS and MS. Sample preparation: YC, MS, MDA, DS.
Transport measurements and analysis: YC, MDA, DS, BJ, DP, CC, FT and MS. SQUID
experiment: DP. XMCD experiment: MS and DP. Theory modeling: YC, MT, CAP and
RC. Supervision: MS and DS. Writing: YC and MS. Writing-review and editing:
all the authors participated in the editing of the paper.

  * •

Data availability Data needed to evaluate the conclusions of this paper are
present in the Letter and its Supplementary Information. Additional data are
available from the corresponding authors upon request.

  * •

Code availability
The codes used to evaluate the conclusions of this paper are available from
the corresponding authors upon request.

  * •

Competing interests
The authors declare no competing interests.

  * •

Current address Maria D’Antuono is currently employed at Instituto de
Microelectronica de Barcelona (IMB-CNM, CSIC), Campus UAB, 08193 Bellaterra,
Spain. Mattia Trama is currently working at Institute for Theoretical Solid
State Physics, IFW Dresden, Helmholtzstr. 20, 01069 Dresden, Germany.

See pages ,1,,2,,3,,4,,5,,6,,7,,8,,9,,10,,11 of
Supplementary_Anomalous_MC_111LETO2024.pdf