# Spin-orbital correlations from complex orbital order in MgV2O4

H. Lane  School of Physics and Astronomy, University of Edinburgh, Edinburgh
EH9 3JZ, United Kingdom  School of Chemistry and Centre for Science at Extreme
Conditions, University of Edinburgh, Edinburgh EH9 3FJ, United Kingdom  ISIS
Pulsed Neutron and Muon Source, STFC Rutherford Appleton Laboratory, Harwell
Campus, Didcot, Oxon, OX11 0QX, United Kingdom  School of Physics, Georgia
Institute of Technology, Atlanta, GA 30332, USA  P. M. Sarte  Materials
Department, University of California, Santa Barbara, CA 93106, USA  K.
Guratinder  School of Physics and Astronomy, University of Edinburgh,
Edinburgh EH9 3JZ, United Kingdom  A. M. Arevalo-Lopez  Univ. Lille, CNRS,
Centrale Lille, Univ. Artois, UMR 8181 - UCCS - Unite de Catalyse et Chimie du
Solide, F-59000 Lille, France  R. S. Perry  ISIS Pulsed Neutron and Muon
Source, STFC Rutherford Appleton Laboratory, Harwell Campus, Didcot, Oxon,
OX11 0QX, United Kingdom  London Centre for Nanotechnology and Department of
Physics and Astronomy, University College London, London WC1E 6BT, United
Kingdom  E. C. Hunter  School of Physics and Astronomy, University of
Edinburgh, Edinburgh EH9 3JZ, United Kingdom  T. Weber  Institute Laue-
Langevin, 71 Avenue des Martyrs, CS 20156, 38042 Grenoble cedex 9, France  B.
Roessli  Laboratory for Neutron Scattering and Imaging, Paul Scherrer Institut
(PSI), 5232 Villigen PSI, Switzerland  A. Stunault  Institute Laue-Langevin,
71 Avenue des Martyrs, CS 20156, 38042 Grenoble cedex 9, France  Y. Su  Jülich
Centre for Neutron Science (JCNS) at Heinz Maier-Leibnitz Zentrum (MLZ),
Forschungszentrum Jülich GmbH, Lichtenbergstraße 1, 85748 Garching, Germany
R. A. Ewings  ISIS Pulsed Neutron and Muon Source, STFC Rutherford Appleton
Laboratory, Harwell Campus, Didcot, Oxon, OX11 0QX, United Kingdom  S. D.
Wilson  Materials Department and California Nanosystems Institute, University
of California, Santa Barbara, CA 93106, USA  P. Böni  Physik-Department E21,
Technische Universität München (TUM), James-Franck-Str. 1, 85748 Garching,
Germany  J. P. Attfield  School of Chemistry and Centre for Science at Extreme
Conditions, University of Edinburgh, Edinburgh EH9 3FJ, United Kingdom  C.
Stock  School of Physics and Astronomy, University of Edinburgh, Edinburgh EH9
3JZ, United Kingdom

###### Abstract

MgV2O4 is a spinel based on magnetic V3+ ions which host both spin ($S=1$) and
orbital ($l_{eff}=1$) moments. Owing to the underlying pyrochlore coordination
of the magnetic sites, the spins in MgV2O4 only antiferromagnetically order
once the frustrating interactions imposed by the $Fd\overline{3}m$ lattice are
broken through an orbitally-driven structural distortion at TS $\simeq$ 60 K.
Consequently, a Néel transition occurs at TN $\simeq$ 40 K. Low temperature
spatial ordering of the electronic orbitals is fundamental to both the
structural and magnetic properties, however considerable discussion on whether
it can be described by complex or real orbital ordering is ambiguous. We apply
neutron spectroscopy to resolve the nature of the orbital ground state and
characterize hysteretic spin-orbital correlations using x-ray and neutron
diffraction. Neutron spectroscopy finds multiple excitation bands and we
parameterize these in terms of a multi-level (or excitonic) theory based on
the orbitally degenerate ground state. Meaningful for the orbital ground
state, we report an “optical-like” mode at high energies that we attribute to
a crystal-field-like excitation from the spin-orbital $j_{eff}$=2 ground state
manifold to an excited $j_{eff}$=1 energy level. We parameterize the magnetic
excitations in terms of a Hamiltonian with spin-orbit coupling and local
crystalline electric field distortions resulting from deviations from perfect
octahedra surrounding the V3+ ions. We suggest that this provides compelling
evidence for complex orbital order in MgV2O4. We then apply the consequences
of this model to understand hysteretic effects in the magnetic diffuse
scattering where we propose that MgV2O4 displays a high temperature orbital
memory of the low temperature spin order.

##  I Introduction

Atomic orbitals of magnetic ions provide a link between the crystallographic
structure and local magnetic moments and hence an avenue to couple structural
and magnetic degrees of freedom. Lee  _et al._ (2010); Khomskii (2022);
Khomskii and Streltsov (2021); Streltsov and Khomskii (2020, 2017, 2016);
Tokura and Nagaosa (2000) A central parameter to controlling orbital order in
materials is spin-orbit coupling which exists in magnetic ions with an inherit
single-ion orbital degeneracy. Given that spin-orbit coupling in multielectron
atoms scales with the atomic number squared ($\lambda\propto Z^{2}$)Landau and
Lifshitz (1977), first row transition metals provide an opportunity to study
magnetism when spin-orbit coupling is of a similar energy scale to the
magnetic exchange and also energy scales of local structural distortions away
from a perfect, for example, octahedral environment. These energy scales have
consequences to the nature of the real space magnetic orbitals which spatially
order at low temperatures, breaking any degeneracy. In particular is the
situation of complex orbitals and real orbitals illustrated in Fig. 1. In this
paper we investigate the interplay between spin-orbital physics in MgV2O4 and
apply neutron spectroscopy to obtain information on the underlying orbital
order pointing to ordering of complex orbitals.

Figure 1: Graphical representation of the real $\ket{d_{\alpha\beta}}$ and
imaginary $\ket{\hat{l}_{z}=\pm 1}$ orbital wavefunctions. The surface
represents the absolute magnitude of the wavefunction whereas the color
represents the phase. Figure 2: $(a)$ The low temperature nuclear and magnetic
structure of MgV2O4 obtained from our neutron diffraction results. $(b)$ Our
coaligned single crystals. $(c)$ Comparison of powder x-ray and neutron
results illustrating the structural and magnetic transitions. The shaded
region indicates where both tetragonal and cubic phases coexist. $(d)$ Single
crystal neutron diffraction refinements of the nuclear structure in the high
temperature T=140 K cubic phase and low temperature T=50 K tetragonal phases.
The calculated Bragg peak structure factor squared ($F^{2}_{calc}$) is plotted
against the measured structure factor squared ($F^{2}_{obs}$) as measured on
D9 (ILL).

Consisting of orbitally degenerate $d^{2}$ trivalent V3+ ($S$=1, $l_{eff}$=1)
decorating a pyrochlore Gardner  _et al._ (2010) sublattice of corner-sharing
tetrahedra (Fig. 2 $a$), the spinel vanadates $A$V2O4 with $A$ = divalent
cation, provide a platform for the interplay of the underlying geometric
constraints of a frustrated lattice Moessner (2001) populated with strongly
correlated electrons, each with charge, spin, and orbital degrees of freedom.
Lee  _et al._ (2010); Radaelli (2005) The structural, electronic, and magnetic
properties of the $A$V2O4 series are based on two broadly different classes
with the $A$-site either being magnetic (as is the case for (Mn,Fe,Co)V2O4) or
not (for example in (Mg,Zn)V2O4). Tsurkan  _et al._ (2021) These two classes
of materials display some noteable differences with the size of the ordered
moment on the V3+ reportedly stronger when the $A$-site of the spinel
structure is magnetic Garlea  _et al._ (2008). Also, magnetic $A$-site
vanadate spinels display noncollinear magnetic order in contrast to their
nonmagnetic counterparts. Krishna  _et al._ (2019) For simplicity towards
investigating the orbital ground state of the V3+ ion in this class of
compounds, we discuss in this paper the case where the $A$-site is
nonmagnetic. Given the orbital degeneracy on the V3+ site, (Mg,Zn)V2O4
compounds exhibit a Jahn-Teller distortion with a contraction along the
$c$-axis accompanying a cubic to tetragonal structural phase transition at
$T_{S}$. This distortion relieves the underlying geometric magnetic
frustration and hence is followed by the formation of long-range collinear
antiferromagnetic order at $T_{N}<T_{S}$. The ordered moments in the Néel
phase are reduced from expectations of a spin-only moment of $gS=2\ \mu_{B}$
with values of, for example, $\simeq$ 0.5 $\mu_{B}$ reported for MgV2O4.
Wheeler  _et al._ (2010)

The specific case of MgV2O4 exhibits an orbitally driven Jahn-Teller
structural transition at T${}_{S}\simeq$ 60 K and a Néel transition at
T${}_{N}\simeq$ 40 K. The exact nature of the low temperature orbital order is
ambiguous because two types of low-temperature orbital order have been
proposed for this compound. The first, termed Real Orbital Ordering (ROO),
corresponds to orbital ordering of the real $t_{g}$ and $e_{g}$ orbitals which
are eigenstates in the limit that the crystal field imposed on the V3+ ion
from the surrounding oxygen octahedra is large. This model has been advocated
based on x-ray and electron beam diffraction data. Niitaka  _et al._ (2013) An
alternate suggestion Tchernyshyov (2004) has been made for ordering of the
complex basis of the $\hat{L}_{z}$ observable which are complex linear
combinations of the real $d$-orbitals. This is denoted as Complex Orbital
Ordering (COO) and is the basis state used in the weak-intermediate crystal
field limit for transition metal ions. Based on powder neutron spectroscopy
and diffraction, it has also been suggested that the orbital ordering could
possibly be intermediate between the two ROO and COO extremes. Wheeler  _et
al._ (2010) A graphical representation of the two different extremes is
illustrated in Fig. 1.

A theoretical study outlined in Ref. Perkins and Sikora, 2007 suggested
neutron spectroscopy as a means of distinguishing between COO and ROO orders
at low temperatures. In particular the study noted the importance of spin-
orbit coupling with COO giving multiple magnetic branches in the neutron
response and also a larger magnetic zone-center gap than would be expected in
the ROO model. Motivated by this, we investigate the magnetic neutron response
in single crystals of MgV2O4. We study the hysteretic critical dynamics using
diffraction, measure the dynamic response, and develop a theory to describe
the magnetic excitations from the spin-orbital ground state. Given the
spatially diffuse and correlated nature of the spin-orbital states used for
our theory, we term this an excitonic approach.

We apply this excitonic approach to model our neutron spectroscopy results
using the single-ion states and then treating the single-ion Hamiltonian
(including spin-orbit coupling) and exchange energetics equally. Sarte  _et
al._ (2019, 2020a); Buyers  _et al._ (1975); Sarte  _et al._ (2020b) This
model combined with the data favors the ordering of complex orbitals (COO) in
MgV2O4 and illustrates the importance of using a complex orbital basis for
understanding the properties of the first-row transition metal ions.

This paper is divided into four sections. First, we outline the materials
preparation methodology and experimental techniques used to probe the magnetic
fluctuations and critical scattering. Second, we outline the spectroscopic
experimental results of both the low-energy spin-wave excitations and a higher
energy optic-like mode. Third, we present a theory for the magnetic
excitations including spin-orbit coupling and the orbitally degenerate ground
state. Finally, we investigate the hysteretic effects resulting from the
orbital degeneracy using energy integrated neutron diffuse scattering.

##  II Experimental Information

Materials preparation: Single crystals (Fig. 2 $b$) of MgV2O4 were grown using
the floating zone technique with details provided in the Appendix. Given the
extreme sensitivity of the magnetic properties to stoichiometry and chemical
order in MgV2O4 Islam  _et al._ (2012), we have characterized our single
crystals using both thermodynamic and scattering probes with neutrons and
x-rays. The diffraction results are summarized in Fig. 2 $(c)$, with high
temperature structural (TS $\simeq$ 60 K) and magnetic (TN $\simeq$ 40 K from
the magnetic $\vec{Q}$=(1,1,0) Bragg peak) transitions consistent with the
published literature Mamiya  _et al._ (1997); Wheeler  _et al._ (2010).

Characterization: Figure 2 $(c)$ illustrates powder diffraction measurements
using synchrotron x-rays (Diamond, I11) and neutrons (PSI, RITA-II). Powder
synchrotron data shows a first order transitions at TS $\simeq$ 60 K from a
high temperature cubic to a low temperature tetragonal phase. This is
confirmed on single crystal neutron diffraction data following the
$\vec{Q}$=(0,0,2) Bragg peak on warming and cooling. A large hysteresis in the
neutron intensity is observed between warming and cooling over the same
temperature range where synchrotron x-rays measure a coexistence of tetragonal
and cubic phases. This is further discussed in the Appendix where this region
was found to extend from $\sim$ 55-60 K. We have further confirmed that the
structural properties of single crystals are consistent with the published
structural phases through single crystal neutron diffraction data (D9, ILL).
The refinement results are summarized in Fig. 2 $(d)$ in both the cubic (T=140
K) and tetragonal (T=50 K) phases with the structural parameters listed in the
Appendix. Fig. 2 $(d)$ illustrates the calculated Bragg peak intensities
($\propto F^{2}$) as a function of the measured intensities with a straight
line indicative that the models describe the results well. The slope of the
line a calibration factor scaling experiment and calculated values. We have
confirmed the magnetic structure to be consistent with that outlined in the
literature for ZnV2O4 Reehuis  _et al._ (2003) using polarized neutrons (DNS,
MLZ). The results of this are schematically shown in Fig. 2 $(a)$ with the
magnetic V3+ magnetic moments pointing along the tetragonal $c$-axis.

Critical magnetic fluctuations: In this paper we discuss energy-integrated
polarized diffuse scattering measurements sensitive to the magnetic critical
scattering and neutron inelastic data probing the spin-orbital dynamics. To
investigate the magnetic critical scattering as a function of temperature, we
studied the polarized neutron cross section using an $xyz$ geometry provided
by the DNS diffractometer (MLZ, Munich) Su1 (2015). Further details of the
polarized beam measurements and analysis are provided in the Appendix.

Neutron Spectroscopy: To probe the magnetic dynamics sensitive to the spin-
orbital ground state, neutron spectroscopy was performed on four different
spectrometers. To study the high-energy excitations which represent spin-
orbital excitations we used the MAPS spectrometer. The sample was oriented so
that $\vec{k}_{i}$ was oriented along the $c^{*}$ axis and the intensity along
$L$ integrated as typically done with time of flight direct geometry
spectrometers to measure one or two dimensional spin excitations. To obtain an
overview of the low-energy and low dimensional dispersive dynamics, the multi
rep rate option on MERLIN Bewley  _et al._ (2006) was exploited combined with
rotating the crystal. Further data was taken on the EIGER triple-axis
spectrometer Stuhr  _et al._ (2017) to characterize weakly three dimensional
dispersive excitations sensitive to interactions between chains. Finally, to
probe the low-energy gapped excitations, sensitive to local single-ion
anisotropy, we used the cold neutrons on the RITA-II spectrometer Lefmann  _et
al._ (2000).

##  III Experimental results

###  III.1 Low-energy dispersive dynamics

Figure 3: $(a)$ illustrates the magnetic excitations, taken from MERLIN (Ei=49
meV) up to $\sim$ 40 meV showing a band of excitations. This cut is taken
along the chain direction. $(b)$ illustrates MERLIN (Ei=24 meV) displaying a
comparatively weaker dispersion perpendicular to the chain direction. $(c)$
displays high resolution data taken on RITA-II displaying an excitation gap
reflecting underlying anisotropy. $(d-f)$ shows data taken from EIGER
characterizing the dispersion perpendicular to the chains.

The magnetic excitations characterizing the underlying spin-orbital ground
state below the Néel temperature TN are summarized in Fig. 3. The overall
dispersion is displayed in Fig. 3 $(a)$ taken on the MERLIN spectrometer with
Ei=49 meV. This constant momentum slice illustrates a strong band of magnetic
excitations which extends up to $\sim$ 35 meV. Perpendicular to this
direction, a much more weakly dispersive mode is illustrated in Fig. 3 $(b)$.
This reflects the underlying strongly one dimensional coordination Lee  _et
al._ (2004) of the V3+ ions. The dispersion is further studied through a
series of constant momentum scans using the EIGER thermal triple-axis
spectrometer in Fig. 3 $(d-f)$. Analyzing the low-energy fluctuations with the
cold triple-axis spectrometer RITA-II in Fig. 3 $(c)$, illustrates an energy
gap of $\sim$ 6 meV in the fluctuation spectrum. These results demonstrate the
presence of strongly one-dimensional magnetic fluctuations with an energetic
gap due to crystalline anisotropy. We discuss the origin of these below when
we present an excitonic model for the spin-orbital excitations where we also
discuss the relative magnetic exchange energies.

Figure 4: $(a)$ displays a constant momentum slice at E=[50,55] meV taken
using the MAPS spectrometer. $(b)$ illustrates a background-corrected constant
momentum cut showing a well-defined peak in energy indicative of a second
higher energy magnetic band. The red line shows a fitted Gaussian on an
exponentially decaying background peaked at 51 meV with a FWHM of 3 meV.

###  III.2 Higher energy gapped mode

One of the distinctions between COO and ROO orbital ordering scenarios is the
presence of an optical-like excitation at higher energies Perkins and Sikora
(2007) corresponding to a transition between the $j_{eff}=2$ and $j_{eff}=1$
manifolds (as theoretically outlined below). A higher energy band of magnetic
excitations is investigated using the MAPS spectrometer. The results are
summarized in Fig. 4. Fig. 4 $(a)$ displays a constant energy slice showing
weakly correlated excitations near $\vec{Q}$=(2,0,0) and equivalent positions.
A background (taken from large momentum detectors) corrected constant momentum
cut is shown in Fig. 4 $(b)$ displaying a well-defined peak at $\sim$ 50 meV.
We discuss the origin of this additional excitation and connect it with the
low-energy response below by investigating the spin-orbital neutron response
theoretically.

Figure 5: $(a)$ Tanabe-Sugano diagram for a $d^{2}$ ion in an octahedral
crystal field. The Racah parameters have been chosen such that $C/B=4.43$
McClure (1959). For the discussion of the crystal field parameters, we have
followed Ref. Abragam and Bleaney, 1986 (Table 7.3) and used B=0.11 eV and
$Dq$=0.22 eV, indicated by the dashed line. The VO6 octahedron as viewed from
$(b)$ an isometric viewpoint and $(c)$ the $c$-axis. The octahedron is
tetragonally distorted, as evidenced by the shorted bond lengths along $c$. A
further trigonal distortion is present along [111] leading to two different
bond lengths in the $a$-$b$ plane. $(d)$ Eigenvalues of the single-ion
Hamiltonian for an octahedrally coordinated V3+ ion ($S=1$, $l=1$) subject to
a crystallographic distortion as described in Sect. IV.1. Red lines indicate
the ground state and excitated states for which dipole-allowed transitions
exist.

##  IV Theory

In MgV2O4 the magnetic V3+ (3$d^{2}$) ions form a pyrochlore lattice with the
V3+ ions on the spinel B sites surrounded by an octahedral coordination of
oxygen O2- (Fig. 5 $b,c$) which determines the orbital ground state.
Neighboring V3+ ions occupy edge-sharing octahedra with non-magnetic
tetrahedrally-coordinated Mg filling the voids between VO6 octahedra. The
magnetic inter-ion interactions are likely governed by a combination of direct
$d$-$d$ orbital overlap and oxygen mediated superexchange via the $\sim
90^{\circ}$ V-O-V bonds.

In the rare earth pyrochlores Rau and Gingras (2019), where the B site is
occupied by the heavier $4f$ ions, spin-orbit coupling ($\lambda\sim Z^{2}$)
Landau and Lifshitz (1977) is the dominant energy scale and motivates the
projection onto a ground state manifold $J\equiv L+S$. The crystalline
electric field further splits the ground state with the resulting energy
distribution dependent on the species of ion and local environment. In the
case that the ground state is a Kramers doublet, a projection onto an
effective $S=1/2$ pseudospin Ross  _et al._ (2011); Savary  _et al._ (2012);
Thompson  _et al._ (2017) has been utilized and provided the motivation for
seeking out quantum spin liquid Savary and Balents (2016) behavior in these
materials.

The physics of the $3d$ pyrochlores is unlike that of their rare earth cousins
owing to a different hierarchy of single-ion energy scales. Particularly, in
$3d$ ions, the spin-orbit coupling ($\lambda$) is smaller than the crystalline
electric field ($Dq$) Abragam and Bleaney (1986); Sarte  _et al._ (2018a, b).
With typical energy scales of $Dq$ $\sim$ 0.1 eV and $\lambda$ $\sim$ 10 meV,
spin-orbit coupling is a perturbation on the crystalline electric field
Hamiltonian and the ground state is defined by the orbital angular momentum
$L$. Within this ground state manifold, the spin-orbit coupling, magnetic
exchange and Jahn-Teller energy scales are comparable. In the $3d$ transition
metal compounds it is therefore necessary to consider both single-ion spin-
orbital energy scales ($\mathcal{H}_{SI}$) and the corresponding magnetic
exchange interactions ($\mathcal{H}_{exch}$)Sarte  _et al._ (2018a)

$\mathcal{H}=\mathcal{H}_{SI}+\mathcal{H}_{exch}.$

We will begin by discussing the single ion physics of the $3d^{2}$ ions to
understand the magnetic ground state before considering the magnetic inter-ion
exchange in vanadium spinels. Using a Green’s function formalism, the
dynamical structure factor measured with neutrons will then be calculated
using the random phase approximation (RPA) applied to MgV2O4, treating the
single-ion Hamiltonian which produces a quantized multilevel ground state
explicitly.

###  IV.1 $\mathcal{H}_{SI}$ \- Single-ion physics

The single ion physics discussed in this section is schematically shown in
Fig. 5 which determines the quantized spin-orbital ground state of the
magnetic V3+ ions. We consider first the free-ion V3+ ground state followed by
the effects of the crystalline electric field, spin-orbit coupling,
distortions from an octahedral environment, and finally the Zeeman-like
molecular field originating from magnetic ordering in the low temperature Néel
phase.

####  IV.1.1 ${}^{3}F$ \- Free ion ground state

For the case of a free V3+ ions with 2 electrons in the five degenerate $d$
orbitals, the ground state is determined by Hund’s rules which defines $L=3$
and $S=1$. This fixes the orbital ground state of the free-ion V3+ state to be
${}^{3}F$ in spectroscopic notation.

####  IV.1.2 $\mathcal{H}_{CEF}$ \- Crystalline electric field

As discussed above, the dominant energy scale for magnetic $3d$ is the
crystalline electric field originating from the O2- ions forming an octahedron
around the V3+ ion. Application of an octahedral crystalline electric field in
terms of Steven’s operators $O_{l}^{m}$ gives the following

$\mathcal{H}_{CEF}=B_{4}\left(O_{4}^{0}+5O_{4}^{4}\right)$

where $B_{4}<0$ for $d^{2}$ ions Abragam and Bleaney (1986). The energy
spectrum resulting from this crystalline electric field Hamiltonian is
schematically displayed in Fig. 5 $(a)$ (Tanabe-Sugano diagram) and
parameterized in terms of the crystal field strength ($Dq$) and the Racah
parameter $B$ which physically corresponds to the energy cost associated with
the Coulomb repulsion. The limit $Dq/B\rightarrow 0$ corresponds to the weak
crystal field limit where Hunds rules across all $d$-orbitals applies. In the
large $Dq/B$ limit, the crystal field energy scale is dominant and in some
transition metal ions (such as Co2+) can result in spin transitions.

Figure 5 $(a)$ shows that the octahedral crystal field splits the orbital 3F
ground state into two triplets (${}^{3}T_{1,2}$) and a singlet
(${}^{3}A_{2}$), with the ${}^{3}T_{1}$ triplet being the orbital ground
state. The splitting between ${}^{3}T_{1}$ and the first excited triplet is
$8Dq=480B_{4}\approx$ 1.8 eV Abragam and Bleaney (1986) (in the limit of small
$Dq$). Each of the multiplets under the octahedral crystal field forms an
irreducible representation of the octahedral double group, $O^{\prime}$
Tinkham (2003). By the orthogonality of different irreps of the same group,
for an operator that is invariant under all octahedral symmetry operations,
matrix elements between multiplets are zero Tinkham (2003). Assuming further
symmetry lowering terms away from an octahedral field are small and the large
energy separation in comparison to the temperatures of interest, one can
justifiably work in the ground state ${}^{3}T_{1}$ multiplet, neglecting the
excited states.

Considering energy levels beyond the ${}^{3}F$ manifold in Fig. 5 $(a)$, the
octahedral field mixes in the excited ${}^{3}P$ state (Fig. 5 $a$ when
$Dq/B\rightarrow 0$). The lowest energy orbital state of the ${}^{3}P$
manifold has the same symmetry being a ${}^{3}T_{1}$ triplet Abragam and
Bleaney (1986) and therefore the overall groundstate is a linear combination
of the two triplet states from the ${}^{3}F$ and ${}^{3}P$ manifolds

$\ket{\psi(^{3}T_{1})}=\epsilon\ket{{}^{3}F(^{3}T_{1})}+\tau\ket{{}^{3}P(^{3}T_{1})}$
(1)

where $\epsilon^{2}+\tau^{2}=1$ Abragam and Bleaney (1986). One can represent
the ground state orbital triplet using the fictitious orbital angular momentum
operator, $\mathbf{\hat{l}}$ via the transformation
$\mathbf{\hat{L}}=\alpha\mathbf{\hat{l}}$, where the projection factor
$\alpha<0$ can be read off the block describing the ground state manifold
after projecting the matrices $\hat{L}_{\alpha}$ onto the space spanned by the
eigenvectors of $\mathcal{H}_{CEF}$. For the $\ket{{}^{3}F(^{3}T_{1})}$ block,
$\alpha=-\frac{3}{2}$ Lane  _et al._ (2021a), whilst for the
$\ket{{}^{3}P(^{3}T_{1})}$ state, $\alpha=+1$. The admixture of these two
states leads to an effective projection factor that varies between these two
extremal values, $\alpha=\frac{5}{2}\tau^{2}-\frac{3}{2}$.

It is instructive to discuss the orbital triplet ground state in terms of the
electronic orbital basis states which have the advantage over the eigenstates
of the observables $L^{2}$ and $L_{z}$ of being real and therefore give a
greater connection to the microscopic picture of electron hopping through
chemical bonding. In the $3d$ ions, the outer electrons partially fill
$d$-orbitals, which are split in an octahedral crystal field into the triply
degenerate $t_{2g}$ level ($d_{xy},d_{yz},d_{xz}$) and doubly degenerate
$e_{g}$ level ($d_{x^{2}-y^{2}},d_{3z^{2}-r^{2}}$). The energy splitting
between $t_{2g}$ and $e_{g}$ levels is set by $10Dq$, and therefore this
parameter is key in distinguishing the weak (small $Dq$) from the strong
(large $Dq$) crystal field limits. For weak crystal fields it is more natural
to use a complex orbital basis that are eigenstates of the observables $L^{2}$
and $L_{z}$. In the large crystal field limit, the real $t_{2g}$ and $e_{g}$
states are the convenient basis.

Neglecting the spin-orbit coupling and further distortions, the ground state
for 2 electrons in the $3d$ orbitals is triply degenerate with each member of
ground state manifold having two of the $t_{2g}$ levels occupied. In terms of
these real basis states one can represent the ground state triplet with the
admixture caused by the crystal field (Eq. 1) equivalently as

$\ket{\psi(^{3}T_{1})}=\mathrm{cos}\theta\ket{t_{2g},t_{2g}}-\mathrm{sin}\theta\ket{t_{2g},e_{g}}.$
(2)

The first of these two basis states has two occupied $t_{2g}$ levels as
expected in an octahedral crystal field, and the second has an occupied
$e_{g}$ level but has the same symmetry as $\ket{t_{2g},t_{2g}}$. By
diagonalizing the energy matrix for a $d^{2}$ ion with both a Coulomb term and
an octahedral crystal field Abragam and Bleaney (1986) (Fig. 5$a$) one finds
$\mathrm{tan}2\theta=12B/(9B+10Dq)$ and using the correspondence between the
two descriptions (Eqs. 1 and 2) one can quantify the fraction of the ${}^{3}P$
level in the ground state orbital triplet,
$\tau=\frac{1}{\sqrt{5}}(\mathrm{cos}\theta-2\mathrm{sin}\theta)$. With
$B=0.11$ eV and $Dq=0.22$ eV, as expected for a free V3+ ion, we have
$\tau\approx 0.27$. We therefore expect a projection factor of
$\alpha\approx-1.32$ in comparison to a value of -1.5 in the absence of
mixing. We discuss this parameter later in context of the reported ordered
magnetic moment and the possibility for quantum fluctuations.

####  IV.1.3 $\mathcal{H}_{SO}$ \- Spin-orbit coupling

The next largest term (illustrated in Fig. 5 $d$) in the single-ion
Hamiltonian for $3d$ ions is the spin-orbit coupling

$\mathcal{H}_{SO}=\lambda\mathbf{\hat{L}}\cdot\mathbf{\hat{S}}=\alpha\lambda\mathbf{\hat{l}}\cdot\mathbf{\hat{S}}$

where $\lambda>0$ for $d^{2}$ systemsYosida (1996). This term splits the
ground state ninefold spin-orbital manifold into three
$\mathbf{\hat{j}}=\mathbf{\hat{l}}+\mathbf{\hat{S}}$ levels according to the
Landé interval rule with $j_{eff}$=0, 1, 2 (Fig. 5 $d$). In $d^{2}$ ions, the
ground state is the $j_{eff}=2$, separated from the $j_{eff}=1$ state by
$|2\alpha\lambda|$ Abragam and Bleaney (1986). In the language of the strong
crystal field $t_{2g}$ levels, the effect of the inclusion of spin-orbit
coupling is to mix the $d_{xz}$ and $d_{yz}$ orbitals in complex combinations
such that three basis states in which the Hamiltonian is diagonal become
$\ket{l_{z}=0}=\ket{d_{xy}}$ and $\ket{l_{z}=\pm
1}=\frac{1}{\sqrt{2}}(\ket{d_{xz}}\pm i\ket{d_{yz}})$ Khomskii (2014).

####  IV.1.4 $\mathcal{H}_{dis}$-Distortion

In MgV2O4 at the low temperatures of interest here, the symmetry of the local
VO6 octahedra is not $O_{h}$, since a Jahn-Teller distortion, originating from
the orbital degeneracy of the $d^{2}$ vanadium ion, occurs on the transition
from the high temperature cubic phase to the low temperature tetragonal phase
Wheeler  _et al._ (2010). This distortion results in the octahedra being
subtly compressed along the fourfold $\hat{z}$-axis (illustrated in Fig. 5
$b$). The distortion can be modeled by the following term in the Hamiltonian

$\mathcal{H}_{dis}=\Gamma\left(\hat{l}_{z}^{2}-\frac{2}{3}\right)$ (3)

where $\Gamma<0$ for a compression. As displayed in Fig. 5 $(d)$ the effect of
this term is to break the five fold orbital groundstate $j_{eff}=2$ degeneracy
into three levels with a groundstate doublet, an excited state doublet, and an
excited state singlet.

Considering the effect of this distortion only on orbitals (without
considering spin or spin-orbit coupling and in the limit of
$|\Gamma|\gg|\lambda|$) and in terms of the strong crystal field basis of real
orbitals, the effect of this axial distortion is to break the ground state
$t_{2g}$ orbital triplet degeneracy, yielding a ground state orbital doublet
and excited singlet. The distortion lowers the energy of the $d_{xy}$ orbital
relative to the $d_{xz}$ and $d_{yz}$ orbital. If we populate these two levels
with two $d$ electrons applying Pauli’s exclusion principle, this results in a
doubly degenerate ground state with a hole in either the $d_{xz}$ or $d_{yz}$
orbital.

We note that in addition to the primary tetragonal compression driven by
orbital degeneracy, the VO6 octahedra are trigonal distorted (Fig. 5 $c$) – a
compression along the threefold $[111]$ axis (even within the high temperature
cubic phase). Within the manifold of the groundstate $t_{2g}$ orbitals, the
multiplet structure under this distortion is the same as the tetragonal
distortion (Eq. 3) Khomskii (2014). In this paper, we will take advantage of
the projection onto the ground state triplet,
$\mathbf{\hat{L}}=\alpha\mathbf{\hat{l}}$ and treat the mixing of the $e_{g}$
levels as a small perturbation and so we can collect all contributions to the
distortion Hamiltonian into a single distortion parameter, $\Gamma$. Under a
dominant trigonal distortion, the orbital spectrum has the form
$\ket{a_{1g}}=\frac{1}{\sqrt{3}}(\ket{d_{xy}}+\ket{d_{xz}}+\ket{d_{yz}})$ with
the excited doublet
$\ket{e_{g}^{\pi}}=\pm\frac{1}{\sqrt{3}}(\ket{d_{xy}}+e^{\pm 2\pi
i/3}\ket{d_{xz}}+e^{\mp 2\pi i/3}\ket{d_{yz}})$ Khomskii (2014). From
crystallographic considerations, the low temperature tetragonal distortion
discussed above is expected to be dominant, however one might expect a small
difference in orbital coefficients arising due to the subleading trigonal
distortion. Such a situation maybe supported by the small $\sim 8^{\circ}$
canting of the spin towards the apical oxygen reported in some diffraction
studies Wheeler  _et al._ (2010).

The orbital state for each electron can be written in general as

$\ket{\psi}=\alpha\ket{d_{xy}}+\beta e^{i\theta}\ket{d_{xz}}+\gamma
e^{i\phi}\ket{d_{yz}}$ (4)

where $\alpha^{2}+\beta^{2}+\gamma^{2}=1$ ensures normalization . Based on the
single-ion physics of the V3+ ion ($3d^{2}$) alone, the orbital order in
MgV2O4 is expected to be intermediate between the regimes of validity of real
orbital order (ROO), which occurs when the tetragonal distortion dominates and
complex orbital order (COO) where the spin-orbit coupling dominates, since the
energy scales of the distortion and the spin-orbit coupling in the $3d$ ions
are typically nearly comparable Lane  _et al._ (2021a); Sarte  _et al._
(2019); Cowley  _et al._ (1973). The weightings of the $d_{\alpha\beta}$
orbitals for each electron within each regime are summarized in Table 1. We
will discuss the nature of the orbital order further later in the paper based
on the insight gained through our analysis.

Table 1:  Orbital wavefunction coefficients in common orbital ordering regimes for electron I, electron II. Coefficients correspond to Eq. 4. Column two states the dominant energy scale in each case.  OO | Dom. | $\alpha$ | $\beta$ | $\gamma$ | $\theta$ | $\phi$
---|---|---|---|---|---|---
COO | $\lambda$ | 1,0 | 0,$\frac{1}{\sqrt{2}}$ | 0,$\frac{1}{\sqrt{2}}$ | 0,0 | 0,$\frac{\pi}{2}$
ROO | $\Gamma_{[001]}>0$ | 1,0 | 0,$\frac{1}{\sqrt{2}}$ | 0,$\frac{1}{\sqrt{2}}$ | 0,0 | 0,0
Trig. COO | $\Gamma_{[111]}$ |  $\frac{1}{\sqrt{3}}$,$\frac{1}{\sqrt{3}}$ |  $\frac{1}{\sqrt{3}}$,$\frac{1}{\sqrt{3}}$ |  $\frac{1}{\sqrt{3}}$,$\frac{1}{\sqrt{3}}$ | 0,$\frac{2\pi}{3}$ | 0,$\frac{-2\pi}{3}$

####  IV.1.5 $\mathcal{H}_{MF}$ \- Molecular field below TN

The final term present in the single-ion Hamiltonian is the molecular field.
This results from a mean field decoupling of the exchange interaction between
coupled ions, and is required such that the single ion ground state about
which one expands is correct. This term breaks time reversal symmetry

$\mathcal{H}_{MF}=h_{MF}\hat{S}_{z}$

as is consistent with the establishment of long-range magnetic order. The
magnitude of $h_{MF}$ will be discussed further once the inter-ion spin
Hamiltonian, $\mathcal{H}_{exch}$, has been introduced.

In terms of the single-ion Hamiltonian, the ground state order is expected to
be $(S=1,l=+1)$ with positive $+l$ (instead of $-l$) selected due to the
negative spin-orbit coupling constant in $d^{2}$ ions which promotes alignment
of the spin and orbital moments. This expected ground state gives rise to a
magnetic moment
$\mu=\mu_{B}(\mathbf{\hat{L}}+2\mathbf{\hat{S}})=\mu_{B}(-1.32+2)=0.68\mu_{B}$
which agrees well with the observed (reduced) magnetic moment,
$\mu=0.47\mu_{B}$. Wheeler  _et al._ (2010) The discrepancy between the
observed and calculated magnetic moment suggests the presence of quantum
fluctuations, $S-\Delta S\approx 0.9$, as is to be expected given the small
value of $S$ and reduced dimensionality originating from the frustrated
lattice. This reduction in the ordered moment in turn contributes to a
reduction in the molecular mean field over what would be expected for the full
spin value.

###  IV.2 Inter-ion coupling

We now discuss the inter-ion interactions present in MgV2O4. In the high
temperature cubic phase, the V3+ ions lie on an ideal pyrochlore lattice, with
each nearest neighbor V-V bond equivalent by symmetry. In the case of
antiferromagnetic nearest-neighbor interactions, the spin arrangement on a
pyrochlore lattice is geometrically frustrated Bramwell and Harris (1998),
opening the possibility of spin liquid states Moessner and Chalker (1998);
Savary and Balents (2016) resulting from the large ground state degeneracy.
Pyrochlore systems with $S=1$ have been studied both theoretically Iqbal  _et
al._ (2019); Zhang  _et al._ (2019) and experimentally Gardner  _et al._
(1999); Krizan and Cava (2015); Plumb  _et al._ (2019), particularly in search
of an understanding of the crossover from quantum to classical spin liquid
behavior Plumb  _et al._ (2019); Zhang  _et al._ (2019). The vanadium spinels
have also attracted great interest Tchernyshyov (2004); Di Matteo  _et al._
(2005); Perkins and Sikora (2007); Motome and Tsunetsugu (2005); Chern  _et
al._ (2010) since the orbital degree of freedom allows for Jahn-Teller
distortions which break the ground state degeneracy and permit magnetic order.
This behavior is observed in MgV2O4 at TS $\simeq$ 60 K, where the system
becomes tetragonal before ordering antiferromagnetically at T${}_{N}\simeq$ 40
K Wheeler  _et al._ (2010). The cubic to tetragonal structural transition
renders the nearest neighbor bonds along $[HH0]$ and $[H0H]$ inequivalent.

Figure 6: Magnetic structure of MgV2O4 as viewed from an isometric viewpoint.
The structure comprises chains in the $a$-$b$ plane with the nearest-neighbor
bond, $J_{2}$. Inter-chain bond $J_{1}$ is frustrated and couples chains along
$[H0H]$ and $[0HH]$. A longer range bond, $J_{3}$ couples perpendicular
chains.

The orbital order plays an important role in the magnetic inter-ion exchange
since both the $d$-$d$ and $d$-$p$ orbital overlap are dependent on the V3+
orbital ground state, and ultimately where the electron hole lies. This
provides a mechanism for a breaking of the equivalence of magnetic bonds both
in terms of the amplitude of the direct and super-exchange interactions
Khomskii (2014). The question of the magnetic exchange in the vanadium spinels
has been discussed at length by Di Matteo et al Di Matteo  _et al._ (2005) and
Perkins et al Perkins and Sikora (2007). We will now briefly summarize the
findings of the aforementioned references Di Matteo  _et al._ (2005); Perkins
and Sikora (2007) insofar as they are relevant to the magnetic exchange in
MgV2O4.

Di Matteo et al considered the limit where the tetragonal distortion is
dominant (ROO - real electronic orbitals discussed above) and where the spin-
orbit coupling is dominant (COO - complex basis described above when the
crystalline electric field $Dq$ is weak) Di Matteo  _et al._ (2005). Both
limits give rise to multiple magnetic and orbital configurations dependent on
the underlying physical parameters Di Matteo  _et al._ (2005) and described by
the general Heisenberg spin Hamiltonian

$\mathcal{H}_{exch}=\frac{1}{2}\sum_{ij}\mathcal{J}_{ij}\mathbf{\hat{S}}_{i}\cdot\mathbf{\hat{S}}_{j},$

where the factor of $\frac{1}{2}$ accounts for double counting. In the case of
ROO, only one possible orbital ground state is consistent with the
experimentally observed magnetic structure Wheeler  _et al._ (2010). This ROO
ground state comprises strong antiferromagnetic (AFM) bonds in the $xy$ plane
and a weaker ferromagnetic coupling along $[H0H]$ and $[0HH]$. The ROO model
thus suggests that MgV2O4 is comprised of strongly-coupled AFM chains in the
$xy$ plane with weak FM coupling between chains. In each tetrahedron, two FM
bonds are satisfied and two FM bonds are frustrated.

In contrast, when the spin-orbit coupling is dominant (COO), there are two
ground states consistent with the observed magnetic structure. Both possible
orbital ground states have strongly coupled AFM chains in the $xy$ plane,
however one of these states has two weak AFM and two weak FM bonds per
tetrahedron – thus no bond is frustrated. In the other possible COO ground
state, the inter-chain bonds are weak AFM bonds, two of which are frustrated.
Based on the parameters obtained from spectroscopy data for vanadium
perovskites Tsunetsugu and Motome (2003); Mizokawa and Fujimori (1996) and
typical strengths of the spin-orbit coupling for V3+ reported in the
literature Tchernyshyov (2004); Abragam and Bleaney (1986), Di Matteo et al.
concluded that the latter of the two COO states is likely the ground state Di
Matteo  _et al._ (2005).

Finally we note that the predicted magnetic ground states in the ROO and COO
limits differ only in the sign of the inter-chain coupling, with both inter-
chain couplings of a similar strength Di Matteo  _et al._ (2005). It is
interesting to speculate as to whether these two limits are continuously
connected via a phase with zero inter-chain coupling and hence might be tuned
through with the application of strain, using the tetragonal distortion,
$\Gamma$, as a tuning parameter. It is also interesting to note that a metal-
insulator has been predicted in MgV2O4 based on ab initio calculations.
Baldomir  _et al._ (2008)

###  IV.3 Neutron scattering intensity calculation

Table 2:  Summary of labeling convention for indices. Index | Description
---|---
$\gamma$, $\gamma^{\prime}$ | V3+ sites
$p$, $q$ | spin-orbital crystal field states
$\alpha$, $\beta$, $\mu$, $\nu$ | Cartesian coordinates

In order to calculate the neutron scattering response, we employ a spin-
orbital exciton model in terms of Green’s functions as applied previously to
describe multi-level systems Buyers  _et al._ (1975); Sarte  _et al._ (2019,
2020a); Lane  _et al._ (2021a, b, 2022). A full derivation for a general
single-$\mathbf{Q}$ multi-level magnetic system is presented in Ref. Lane  _et
al._ , 2022, here we quote only the key results. The neutron scattering
intensity is proportional to the structure factor

$S({\bf{q}},\omega)=g_{L}^{2}f^{2}({\bf{q}})\sum_{\alpha\beta}(\delta_{\alpha\beta}-\hat{q}_{\alpha}\hat{q}_{\beta})S^{\alpha\beta}({\bf{q}},\omega),$

where $g_{L}$ is the Landé g-factor, the V3+ form factor is $f(\mathbf{q})$
and $S^{\alpha\beta}(\mathbf{q},\omega)$ is the partial dynamical structure
factor (with cartesian indices $\alpha,\beta$, Table 2), defined as

$S^{\alpha\beta}({\bf{q}},\omega)=\frac{1}{2\pi}\int dte^{i\omega
t}\langle\hat{S}^{\alpha}({\bf{q}},t)\hat{S}^{\beta}(-{\bf{q}},0)\rangle.$

The factor preceeding the partial dynamical structure factor is the
polarization factor which picks out the component of the structure factor
perpendicular to the scattering wavevector, to which the scattered neutrons
are sensitive. We note that neutron scattering is sensitive to magnetic
correlations through interacting with the spin. It is through the presence of
spin-orbit coupling in the magnetic Hamiltonian described above, neutrons are
sensitive to orbital effects. The structure factor can be related to the
Green’s function by way of the fluctuation-dissipation theorem

$S^{\alpha\beta}({\bf{q}},\omega)=-\frac{1}{\pi}\frac{1}{1-\exp(-\omega/k{\rm{{}_{B}}}T)}\Im{G^{\alpha\beta}(\bf{q},\omega)}.$

The Green’s function for multi-level spin-systems can be written as a Dyson
equation where the propagator describes the dynamics of the single ion at
mean-field level with the inter-ion interaction treated at one-loop level with
the self energy $\underline{\underline{\mathcal{J}}}(\mathbf{q})$. The single
ion Green’s function can be written as

$g_{\tilde{\gamma}\tilde{\gamma}^{\prime}}^{\alpha\beta}(\omega)=\sum_{qp}\frac{S^{\tilde{\gamma}}_{\alpha
qp}S^{\tilde{\gamma}^{\prime}}_{\beta
pq}\phi_{qp}}{\omega-(\omega_{p}-\omega_{q})},$ (5)

Figure 7: Calculation of the component of the structure factor $a)$ parallel
and $b)$ perpendicular to the chain direction. Overplotted is the extracted
dispersion from neutron scattering. The green data points are from the EIGER
data set and the red and blue points are from the MERLIN $E_{i}=$ 24 meV and
49 meV data sets respectively. The white solid line indicates the approximate
peak position of the weak in intensity high energy mode and the dashed lines
indicate the approximate width.

where $\omega_{p}$ is the single-ion eigenvalue of state $\ket{p}$ and
$S^{\tilde{\gamma}}_{\alpha
qp}=\bra{p}\hat{S}_{\alpha}^{\tilde{\gamma}}\ket{q}$.
$\phi_{qp}=(f_{q}-f_{p})$ where $f_{q}$ is the Bose occupation factor of level
$q$. The site indices are denoted as $\tilde{\gamma},\tilde{\gamma}^{\prime}$
(Table 2). We perform a coordinate transformation onto the rotating frame so
that the molecular mean field is non-oscillatory. In the rotating frame, the
single-ion of each of the magnetic ions is independent of the moment direction
and varies only between crystallographically inequivalent sites. In the
rotating frame the Green’s function is defined as

$\begin{split}&\tilde{G}_{\tilde{\gamma}\tilde{\gamma}^{\prime}}^{\alpha\beta}(\mathbf{q},\omega)=g_{\tilde{\gamma}\tilde{\gamma}^{\prime}}^{\alpha\beta}(\omega)\delta_{\tilde{\gamma}\tilde{\gamma}^{\prime}}\\\
&\qquad+\sum_{\gamma^{\prime}}^{\mu\nu}\tilde{\mathcal{J}}^{\mu\nu}_{\tilde{\gamma}\gamma^{\prime}}(\mathbf{q})g_{\tilde{\gamma}\tilde{\gamma}}^{\alpha\mu}(\omega)\tilde{G}_{\gamma^{\prime}\tilde{\gamma}^{\prime}}^{\nu\beta}(\mathbf{q},\omega)\end{split}$
(6)

which can be solved as a matrix equation. Eq. 6 contains the Fourier
transformed exchange coupling in the rotating frame,
$\tilde{\mathcal{J}}^{\mu\nu}_{\tilde{\gamma}\gamma^{\prime}}(\mathbf{q})$,
which takes the form of a matrix of dimension $3N\times 3N$, where $N$ is the
number of sites in the unit cell. At this point it is useful to examine the
structure of MgV2O4. Below T${}_{N}\simeq$ 40 K, long-range antiferromagnetic
magnetic order is established, with $\mathbf{Q}=(0,0,1)$ Wheeler  _et al._
(2010). The full crystallographic and magnetic structure can be described by
an eight site unit cell as considered in Ref Perkins and Sikora, 2007.

Table 3:  V3+ ion positions in the unit cell. Index | Moment orientation | Position vector
---|---|---
1 | $\uparrow$ | (0.125,0.625,0.125)
2 | $\downarrow$ | (0.375,0.875,0.125)
3 | $\uparrow$ | (0.375,0.625,0.375)
4 | $\downarrow$ | (0.125,0.875,0.375)
5 | $\uparrow$ | (0.375,0.375,0.625)
6 | $\downarrow$ | (0.625,0.625,0.625)
7 | $\downarrow$ | (0.375,0.125,0.875)
8 | $\uparrow$ | (0.125,0.375,0.875)

In terms of this unit cell (Table 3), no rotation between neighboring unit
cells is required and, assuming Heisenberg coupling, in the rotating frame one
has

$\underline{\underline{\tilde{\mathcal{J}}}}^{\mu\nu}_{\gamma\gamma^{\prime}}(\mathbf{q})=X^{\prime}\left(\underline{\underline{\tilde{\mathcal{J}}}}_{\gamma\gamma^{\prime}}(\mathbf{q})\otimes\mathbb{I}_{3}\right)X$
(7)

where $X=\mathrm{diag}(R_{1},...,R_{n})$ rotates the spins within the unit
cell. The matrix $R_{n}$ is the $3\times 3$ rotation matrix which rotates the
spin on site $n$ onto the $\hat{z}$-axis. Similarly
$X^{\prime}=\mathrm{diag}(R^{T}_{1},...,R^{T}_{n})$. Finally, the Green’s
function in the laboratory frame is found by rotating the Green function back
using the matrices, $X,X^{\prime}$

$G_{\tilde{\gamma}\tilde{\gamma}^{\prime}}^{\alpha\beta}(\mathbf{q},\omega)=X\tilde{G}_{\tilde{\gamma}\tilde{\gamma}^{\prime}}^{\alpha\beta}(\mathbf{q},\omega)X^{\prime}.$

We now apply this theory to the neutron scattering data presented in Sect.
III.

###  IV.4 Application to MgV2O4

The excitonic dispersion relation was found according to the model presented
in the previous section by finding the poles of the Green’s function,
$G_{\tilde{\gamma}\tilde{\gamma}^{\prime}}^{\alpha\beta}(\mathbf{q},\omega)$.
In Fig. 5 $(d)$, we illustrate dipole allowed transitions through the red
lines when investigating the effect of $\mathcal{H}_{MF}$. Based on this, we
would expect two branches with a lower strongly dispersive branch
corresponding to transitions within the groundstate $j_{eff}$=2 manifold and
another transition from the ground state to a $j_{eff}$=1 level. We might
expect that given that this is a “crystal-field-like” transition, this second
higher energy and dipole allowed transition will be weakly dispersive in
momentum akin to crystal field transitions commonly observed for rare earth
ions.

The experimentally observed dispersion was found both along the V-V chain
direction and perpendicular to the chain by fitting Gaussian peaks to one-
dimensional constant energy cuts through the neutron scattering data. These
data points were then fitted using the dispersion derived according to the
excitonic model in order to extract physical parameters. The data in
comparison to theory is summarized through constant momentum and energy slices
in Figs. 7-9. The excitonic theory qualitatively shows two sets of modes with
low-energy fluctuations corresponding to transitions within the ground state
$j_{eff}=2$ manifold and another very weak, and comparatively disperisonless,
high energy mode corresponding to transitions to the excited $j_{eff}=1$ spin-
orbit manifold. We have plotted the weak mode on the same linear intensity
scale in Fig. 7 to illustrate the strong difference in intensity between these
two branches of excitations. We note that the excitonic theory predicts a very
weak and comparatively flat mode at $\sim$ 10 meV. Due to the presence of
overlapping domains and a comparable magnetic zone center excitation gap, we
were not able to reliably resolve this mode from our data.

Figure 8: $a)$ Constant energy slice through the inter-multiplet mode at
$E\sim 50$ meV. $b)$ One-dimensional cut through $Q=(2,0)$, with multi-level
spin wave calculation plotted in red. The Lorentzian half-width has been
chosen to approximately match the energy resolution on MAPS, $\epsilon=1.5$.
The calculation has been integrated along the $c^{*}$ direction across the
full width of the Brillouin zone. The presence of domain coexistence and the
coupling of energy transfer and the wavevector along $c^{*}$ have been
neglected. The differences between calculations and data at low energies
originates from an over subtraction of the data with the chosen background.
Figure 9: Constant energy slice at $10\pm 1$ meV, showing the comparison
between the measured data from MERLIN and the excitonic model with all three
domains summed as described in the text. The intensities have been integrated
along the $c^{*}$ direction ($L=-1\pm 0.2$ r.l.u).

The inter-chain bond, $J_{1}$, was determined to be negligible since the
inclusion of this bond (in the presence of finite $J_{3}$) gives rise to a
splitting of the lower dispersive mode at approximately $Q\geq\pm 0.1$ (r.l.u)
which is not seen in the data. We therefore determine the magnitude of $J_{1}$
to be negligible given the absence of any of these signatures, but note that
frustrated nature of this bond makes its determination difficult
experimentally. The neglect of $J_{1}$ is further justified given that
previous calculations have suggested that this bond is weakly ferromagnetic or
weakly antiferromagnetic in the limits of ROO and COO respectively Di Matteo
_et al._ (2005). In fact, the splitting caused by $J_{1}$ is a hybridization
with the lifted zero energy mode (which acquires some dispersion due to finite
$J_{3}$ ) and the low energy mode due to the intra-chain coupling. As pointed
out by Perkins and Sikora (2007), the presence of a zero energy mode
corresponds to the degeneracy associated with the rotation of the chains in
the $xy$-plane. This degeneracy is lifted by the anisotropy provided by spin
orbit coupling and crystallographic distortions. A small, but finite, $J_{3}$
is expected based on the dispersive excitation seen in Fig. 3 $(b)$ which is
consistent with coupling of parallel chains. We note that there exists a
further neighbor bond $J_{4}$ with bond length marginally longer than $J_{3}$.
Inclusion of this bond again splits the lower dispersive mode but does not
appreciably change the bandwidth or gap, provided that it is small. In the
absence of any observed splitting, and in the spirit of writing down a minimal
model, we neglect this term.

Table 4:  Spin-orbit exciton parameters Parameter | Bond Distance (Å) | Value (meV)
---|---|---
$J_{1}$ | 2.885 | $\approx 0$
$J_{2}$ | 2.885 | 19.04 ($\pm 2.86$)
$J_{3}$ | 4.997 | -0.173 ($\pm 0.026$)
$\alpha\lambda$ | - | -8.8 ($\pm 1.3$)
$\Gamma$ | - | -42.2 ($\pm 6.33$)
$\Theta$ | - | 0.8334 ($\pm 0.1250$)

For the lower energy dispersive mode, the data were fitted by taking one-
dimensional constant energy cuts through the neutron data to extract a
dispersion curve to which the poles of the Green’s function were fit as the
parameters were varied (Fig. 7). To fit the higher energy mode at $\sim 50$
meV (Fig. 8), a constant $\mathbf{q}$ cut was made at (2,0,0) and subtracted
as a background. This method is justified at high energy transfers hear the
$\sim$ 50 meV mode, however it cuts through magnetic intensity at lower
energies and therefore is an overestimate of the background. This is reflected
in Fig. 7 which shows an overestimate of the intensity by the exciton model.
The fitted parameters are listed in Table 4. Stated uncertainties represent a
nominal 15% error to reflect the inherent uncertainties associated with the
experimental measurements (such as spectrometer resolution) rather than the
error bars associated with the numerical fitting procedure which are
unphysically small. The approximate magnitude of the intra-chain coupling
$J_{2}$ is consistent with that theoretically predicted in Ref. Perkins and
Sikora, 2007 using a Kugel-Khomskii model Kugel' and Khomskiĭ (1982). A
reduction of the molecular mean field strength from that expected for an $S=1$
antiferromagnet

$\mathcal{H}_{MF}=h_{MF}\hat{S}_{z}=\Theta\left(-2J_{2}+4J_{3}\right),$ (8)

where $\Theta<1$, is present due to quantum fluctuations. A reduction factor
of $\Theta\approx 0.9$ is expected based on the observed magnetic moment
Wheeler  _et al._ (2010) and the orbital projection factor based on the free
ion values Abragam and Bleaney (1986). This parameter is fixed through a
comparison between the calculated magnetic moment (discussed above) and the
measured magnetic moment. The calculated magnetic moment is sensitive to the
projection factor $\alpha$ which is tied to the single-ion parameters $Dq$ and
Racah parameter $B$. There is, however, a degree of uncertainty about the
single-ion parameters, $B$, $C$ and $Dq$ in the crystal environment, hence
$\Theta$ was refined in our analysis of the neutron spectroscopy results. The
refined value of $\Theta$ implies that the true value of the orbital
projection factor is reduced from that expected in a free ion. One possible
origin of this is the presence of a nephelauxetic effect which reduces the
electron Coulomb repulsion and hence reduces the value of the Racah $B$
parameter. We note that reductions of $\sim$ 10-20 % have been reported
Housecroft and Sharpe (2012), consistent with the reduction that would be
required to explain the value of $\Theta$ fitted here.

The calculated structure factor is plotted in Fig. 7 $(a-c)$. Overplotted are
the extracted peak positions from the neutron scattering data taken on EIGER
and MERLIN. The calculation well describes both the dispersion and the
intensity variation throughout the Brillouin zone.

In the experimental section above it was noted that single crystal samples
should be expected to exhibit domain coexistence as was observed in Ref.
Niitaka  _et al._ , 2013. This occurs below the structural transition and
results from the overall crystal structure retaining the average high
temperature cubic symmetry in the absence of strain. We now briefly note the
consequences of domain coexistence on the neutron scattering spectrum. A
rotation matrix can be defined for each domain which transforms between the
crystallographic basis vectors of each domain, $d$,

$\underline{\underline{R}}_{d}\mathbf{\hat{Q}}_{0}=\mathbf{\hat{Q}}_{d}$ (9)

with the $\mathbf{\hat{Q}}_{0}$ defining the nominal reference basis plotted
in Fig. 5. The crystallographic structure of MgV2O4 comprises chains along
$[110]$ and so one can define the rotation matrices

$\displaystyle\underline{\underline{R}}_{0}=\begin{pmatrix}1&0&0\\\ 0&1&0\\\
0&0&1\end{pmatrix}$ (10a)
$\displaystyle\underline{\underline{R}}_{1}=\begin{pmatrix}1&0&0\\\ 0&0&1\\\
0&1&0\end{pmatrix}$ (10b)
$\displaystyle\underline{\underline{R}}_{2}=\begin{pmatrix}0&0&1\\\ 0&1&0\\\
1&0&0\end{pmatrix}$ (10c)

corresponding to three inequivalent domains. Fig. 9 shows a constant energy
slice with the scattering for all three domains superposed. The diagonal modes
originate from the domains with basis vectors $\mathbf{\hat{Q}}_{1}$ and
$\mathbf{\hat{Q}}_{2}$ with the horizontal and vertical lines originating from
the $\mathbf{\hat{Q}}_{0}$ domain. A small difference in the position of the
diagonal modes likely originates from the fact that the unit cell is
tetragonal with $c/a=0.994$ Wheeler  _et al._ (2010). In the calculation, the
domain population has been assumed to be equal however the relative
intensities of the streaks in the data indicate that this is not the case.
This is in agreement with previous diffraction data Niitaka  _et al._ (2013).

###  IV.5 Orbital order

There has been a great deal of intrigue in the nature of the orbital order in
MgV2O4. Much of this has centered around whether the system exhibits real or
complex orbital order. We will briefly summarize both orbital ordering
schemes, remarking on how our results influence this discussion.

We begin by considering the isolated single ion in an octahedral environment,
neglecting the inter-ion coupling. An orbitally degenerate crystal field
ground state can lower its energy via an octahedral distortion Jahn and Teller
(1937) (the Jahn-Teller effect). An elongation of the octahedron lowers the
$d_{xz}$ and $d_{yz}$ orbitals by $\frac{1}{2}E_{JT}$ and raises $d_{xy}$ by
$E_{JT}$ Khomskii (2014). The lower energy doublet is then split by spin-
orbital coupling into $\ket{l_{z}=\pm 1}=\frac{1}{\sqrt{2}}(d_{xz}\pm
id_{yz})$ by $\pm\frac{1}{2}\alpha\lambda$. For a $d^{2}$ ion, assuming the
spin-orbit coupling does not lift $\ket{l_{z}=-1}$ above the $d_{xy}$ orbital,
then both $\ket{l_{z}=-1}$ and $\ket{l_{z}=1}$ are occupied and the resultant
ground state has quenched orbital angular momentum.

Conversely, if the octahedron is compressed, the Jahn-Teller spectrum is
inverted and the $d_{xy}$ level is lowered, with the $d_{xz}$ and $d_{yz}$
levels lying at $+\frac{1}{2}E_{JT}$. The ground state thus has one electron
in the $d_{xy}$ orbital and one in the $\ket{l_{z}=+1}$ level. In this case
the orbital angular momentum is unquenched owing to the orbital degeneracy
which is broken by spin-orbit coupling, lowering the ground state energy.
Based on single-ion physics alone, an elongation is favored if
$|\alpha\lambda|>E_{JT}=\frac{2}{3}\Gamma$. From the fitted values in Table 4
it is clear that the compressed structure is energetically favored by the
comparatively large value of $\Gamma$, hence the parameters extracted from the
neutron scattering measurements are consistent with the refined
crystallographic structure.

We now turn to the effect of orbital order on the inter-ion exchange in
MgV2O4. Di Matteo et al. and Perkins et al. discussed this in Refs. Di Matteo
_et al._ , 2005; Perkins and Sikora, 2007 by way of a Kugel-Khomskii model
Kugel' and Khomskiĭ (1982) to model the superexchange. Here we only quote the
key results. Two states were found to be consistent with the observed magnetic
and crystallographic structure. The energy of the COO state is
$E_{COO}=-\tilde{J}_{1}-\frac{1}{2}\tilde{J}_{2}(3+2S^{2})-\lambda-\frac{1}{2}E_{JT}$,
where $\tilde{J}_{1}=J(1-\eta)/(1-3\eta)$, $\tilde{J}_{2}=J(1+\eta)(1+2\eta)$,
$J=t^{2}/U_{1}$ and $\eta=J_{H}/U_{1}$. In terms of these variables,
$J_{2}$=$\tilde{J}_{2}$ and, assuming we have COO,
$J_{1}=\tilde{J}_{2}-2\tilde{J}_{0}$, with $\tilde{J_{0}}=\eta J/(1-3\eta)$.
We thus find that our extracted exchange parameters suggest $\eta\approx 0.19$
and $J\approx 22.1$ meV which are close to values suggested by photoemission
spectroscopy ($\eta_{exp}=0.11$, $J_{exp}=20.4$ meV) Perkins and Sikora
(2007); Takubo  _et al._ (2006). In contrast, for ROO one has
$J_{1}=-\tilde{J}_{0}$, such that for $J_{1}\approx 0$ one expects
$\eta\approx 0$. Such a situation is clearly unphysical as it suggests either
a vanishing Hund’s coupling or extremely large Coulomb term. For reasonable
values of $\eta\approx 0.1$ and with $J\approx 20$ meV, the ROO model predicts
$|J_{2}|\approx$ 3 meV, which would give rise to a splitting of the low energy
dispersive mode that would likely be resolvable in neutron scattering
measurements.

Figure 10: The energy integrated magnetic cross section extracted from the
scattering of polarized neutrons using the DNS diffractometer (FRM2). The
arrows at the top and bottom of the figure indicate the cooling and heating
history. Upon cooling to 3 K and reheating the diffuse scattering evolves
hysteretically with no return to the short range ordered phase at T =75 K. No
such hysteresis is found if the sample is only cooled to 50 K and reheated
(black arrow).

As shown by Perkins et al in Ref. Perkins and Sikora, 2007, qualitative
differences exist between the spectra expected for ROO and COO, namely an
optical mode is present in the case of COO whereas ROO gives rise only to an
acoustic mode. This optical mode finds its origin in the unquenched orbital
angular momentum which permits inter-multiplet transitions between the
$j_{eff}=2$ and $j_{eff}=1$ levels (Fig. 5 $d$). Our neutron scattering
experiments confirm the existence of this optical mode at $E\sim 50$ meV which
we were able to model within a multi-level excitonic formalism. We find that
the resultant single ion ground state, consistent based on the fitted
parameters (Table 4), is the $\ket{l_{z}=1,S_{z}=1}$ state presented in Ref.
Di Matteo  _et al._ , 2005. In this state, the effective orbital angular
momentum aligns with the spin moment on all V3+ sites. It should be noted that
due to the negative projection factor, this corresponds to the antialignment
of the spin and true orbital moments which gives rise to the reduction in the
magnetic moment measured in diffraction Wheeler  _et al._ (2010). By treating
the spin-orbit coupling, tetragonal distortion and molecular field on the same
level using an excitonic model, we have demonstrated that MgV2O4 can be
understood to behave according to the the COO picture presented in Refs. Di
Matteo  _et al._ , 2005; Perkins and Sikora, 2007 though the distortion, spin-
orbit coupling and molecular mean field strengths are comparable. This
analysis helps to explain the apparent discrepancy between the previous
neutron scattering results Wheeler  _et al._ (2010) and the predicted spectrum
for COO Perkins and Sikora (2007) which considered $|\Gamma|\ll|\lambda|$.

We note that Di Matteo et al. Di Matteo  _et al._ (2005) propose several types
of COO ranging from all V3+ being in a complex orbital state and a mixture
between real and complex orbital orders on different sites. In our theoretical
analysis discussed above, we have considered the case that all V3+ ions are in
a complex orbital state. Qualitatively, a mixture of real and complex orbital
states would decrease and likely damp the high energy spin-orbit exciton
corresponding to excitations from the ground state $j_{eff}=2$ manifold to a
higher energy $j_{eff}=1$ spin-orbit manifold. Given how weak this mode is
this would likely not be observable. Another point against a mixture of
complex and real orbitals is crystal lattice symmetry with the ground state
being inconsistent with such a situation. We therefore consider here the case
of full complex orbital order (COO) in MgV2O4.

##  V Hysteretic Magnetic correlations

There are three different spin-orbital phases in MgV2O4 on cooling from high
temperature. $(1)$ At high temperatures exceeding Tc, the nuclear unit cell is
cubic and the magnetism is paramagnetic. $(2)$ In the temperature regime TN
$<$ T $<$ TS, the structural unit cell distorts to tetragonal. $(3)$ For low
temperature T $<$ TN, antiferromagnetic order sets in. As displayed in Fig. 5
$(d)$, the single-ion spin-orbital ground state is different in each one of
these phases transitioning from a degenerate $j_{eff}=2$ state in the high
temperature cubic phase to an orbital doublet below TS, and then finally this
degeneracy being split by a molecular field in the low temperature
antiferromagnetic phase. This temperature dependence opens up the possibility
of hysteretic effects on cooling and warming through the intermediate phase
given the orbital degeneracy present, which is broken when cooling through the
antiferromagnetic transition by a Zeeman like molecular field. We now
investigate these spin-orbital ground states using energy-integrated magnetic
diffuse scattering.

To investigate the low-energy critical magnetic fluctuations as a function of
temperature in MgV2O4, we studied the magnetic cross section using the DNS
polarized diffractometer. The combined magnetic intensities from the diffuse
scattering measurements are displayed in Fig. 10. We note the these
measurements display a hysteresis depending on if the sample is cooled below
the magnetic ordering temperature, TN $\simeq$ 40 K. Cooling to T= 75 K from
120 K (Fig. 10 $a\rightarrow b$) one sees the development of magnetic diffuse
scattering, consistent with a spin-frustrated pyrochlore system above the
structural transition. On further cooling to T=50 K (Fig. 10 $b\rightarrow
c$), the magnetic scattering changes into chain-like rods below the structural
transition. Upon heating from T = 50 K back to 75 K (Fig. 10 $c\rightarrow b$)
one recovers these short-ranged magnetic correlations. However, if the sample
is cooled below the long-range magnetic transition, TN and then warmed up to
T=75 K (Fig. 10 $a\rightarrow d$) a different behavior is observed. Whilst at
T = 50 K (Fig. 10 $d\rightarrow e$) one still observes chain-like scattering,
above the structural transition (Fig. 10 $e\rightarrow f$), TS, the short-
ranged correlations are absent and no magnetic scattering is observable. This
indicates that either the magnetic fluctuations have moved outside our energy
window determined by kinematics on DNS with fixed Ei=4.64 meV, or the critical
fluctuations are highly extended in momentum and energy.

We suggest that the explanation for this hysteresis tied to cooling through
the magnetic ordering transition at TN lies in the orbital order present in
MgV2O4. At T=75 K, above the structural transition, in terms of a real orbital
basis the ground state has equal occupation of $\ket{xy}$, $\ket{xz}$ and
$\ket{yz}$ orbitals (Fig. 5). In this state the strength of the superexchange
is the same for both inter and intra-chain bonds, $J_{1}=J_{2}$ and MgV2O4 is
a canonical frustrated pyrochlore. As one cools below the structural
transition, the local octahedra tetragonally compress leading to a doubly
degenerate state with $l_{z}=\pm 1$. The average occupancy of both the
$\ket{xz}$ and $\ket{yz}$ orbitals is $\frac{1}{2}$ and so the strength of the
inter-chain coupling $J_{1}$ is reduced, rendering the system quasi-one-
dimensional which gives rise to rods in the magnetic diffuse scattering. At
this point, long-range magnetic order has yet to be established and so the
orbital ground state is doubly degenerate. Below the antiferromagnetic
ordering temperature, three-dimensional order is established thanks to the
cooperative effect of the third-nearest-neighbor coupling $J_{3}$ and the
anisotropy provided by the distortion and spin-orbit coupling.

One key effect of longer-range coupling (beyond $J_{1}$ and $J_{2}$) is to
reduce the degree to which the spin moment is suppressed due to fluctuations,
both thermal and due to the logarithmic divergence of spin fluctuations in the
one-dimensional quantum antiferromagnet (as per the Mermin-Wagner theorem
Mermin and Wagner (1966)). Upon establishment of long-range order, the orbital
doublet ground state is split by the molecular mean field, establishing the
COO ground state of Refs. Di Matteo  _et al._ , 2005 and Perkins and Sikora,
2007. As the sample is then reheated, at T $>$ TN long-range magnetic order is
lost as fluctuations build, but we speculate that the orbital moments remain
frozen in their COO configuration as the tetragonal symmetry prevents them
from reorienting. Finally as one reaches the structural transition from below,
it should be expected that the orbital moments reorient. However, given the
frozen nature of the orbital moments in the ground state established by
magnetic order, this is prevented by a potential barrier established by the
low temperature molecular fields. To overcome this requires a thermal energy
equivalent to the molecular field of order $\sim$ 10 meV, or above $\sim$ 100
K. We speculate that through the Zeeman energy applied by the molecular field
induced through magnetic ordering, MgV2O4 displays an orbital memory of the
low temperature spin order.

##  VI Conclusions

In conclusion, we have mapped out the spin-orbital fluctuations in MgV2O4
using neutron spectroscopy where we have observed two different branches. We
have parameterized these in terms of a spin-orbital excitonic theory where the
single-ion Hamiltonian is used to determine the quantized ground state which
is then coupled on a lattice using RPA. The results are strongly supportive of
a COO ground state. We then used this model to understand hysteretic magnetic
fluctuations through the three different spin-orbital phases.

###### Acknowledgements.

The authors thank E. Chan and M. Mourigal for useful discussions. H. L. was
co-funded by the ISIS facility development studentship programme and the
EPSRC. C. S. acknowledges support from the Carnegie Trust for the Universities
of Scotland, EPSRC, and the STFC. P.M.S. acknowledges support from the
California NanoSystems Institute through the Elings Fellowship program. S.D.W.
acknowledges financial support from the US Department of Energy (DOE), Office
of Basic Energy Sciences, Division of Materials Sciences and Engineering under
Grant No. DE-SC0017752. The authors gratefully acknowledge the financial
support provided by the Jülich Centre for Neutron Science (JCNS) to perform
the neutron scattering measurements at the Heinz Maier-Leibnitz Zentrum (MLZ),
Garching, Germany.

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  * Shirane  _et al._ (2002) Gen Shirane, Stephen M. Shapiro, and John M. Tranquada,  _Neutron Scattering with a Triple-Axis Spectrometer: Basic Techniques_ (Cambridge University Press, 2002).

##  VII Appendix - sample characterization

In this section we outline additional experimental information regarding
crystal growth and characterization of the nuclear and magnetic properties of
our MgV2O4 samples.

###  VII.1 Additional experimental information

Materials Preparation- Single crystals of MgV2O4 were grown using a four
mirror image furnace (Crystal Systems Inc.). Dry magnesium oxide, MgO (Sigma
Aldrich, 99.995 %) and vanadium (III) oxide, V2O3 (Sigma Aldrich, 99.99 %)
were ground in an agate mortar and pestle with an excess of 1.5 % V2O3. The
excess V2O3 was added to combat the evaporation of vanadium oxide during the
growth. For single crystal growth, the ground powders were pressed in a rod
and sintered in the image furnace on a lower power in a flowing argon
atmosphere (10 bar pressure, 0.1 L/min). The crystal growth conditions were 5
mm/hr growth speed, 35 rpm rotation, 10 bar argon with flow rate $\sim$ 0.1
L/min. The partial oxygen pressure was monitored on the input and output of
the gas supply (Cambridge Sensotec) and the flow rate was adjusted to keep the
partial pressure below $\sim$10-15 ppm. A radiation-heated molybdenum getter
was installed in the furnace to help reduce the oxygen partial pressure and
keep it stable over the growth. Powders of MgV2O4 were synthesized using the
same procedure (and samples) outlined in Ref. Browne and Attfield, 2020.

Polarized neutron diffraction- To investigate the momentum-broadened magnetic
correlations characterizing the critical properties in the paramagnetic
phases, polarized diffuse scattering measurements were performed on the cold
neutron diffractometer DNS at the Heinz Maier-Leibnitz Zentrum (MLZ) with an
incident neutron wavelength of 4.2 Å selected with a double-focusing pyrolytic
graphite monochromator. While DNS has time-of-flight capabilities, data
presented was collected in non time-of-flight mode, providing the integrated
scattering intensity with energy transfers up to an Ei = 4.64 meV. The
flipping ratio was measured to be 20 and a weak 50 Gauss guide field was
applied at the sample for the preservation of neutron polarization. A vanadium
standard was used to account for variations of the detector response and the
solid angle coverage. Corrections for the polarization efficiency of the
supermirror analyzers were performed by using the scattering from a NiCr
alloy. Each measurement was performed over a period of 8-12 hours after a
sample thermalization period of 1 hour.

Unpolarized neutron diffraction- To characterize the structural properties of
our single crystals in both the cubic and tetragonal phases, unpolarized
single crystal neutron diffraction data were taken on the D9 diffractometer
(ILL, France) with a fixed incident wavelength of $\lambda$=0.842 Å.

Synchrotron x-ray diffraction- Temperature dependent x-ray diffraction on
powders were performed to confirm structural properties and to search for any
hysteretic effects in the lattice properties. Experiments were performed on
the I11 high resolution beamline Thompson  _et al._ (2009) at the Diamond
Light Source (Didcot, UK) using the MAC detector system. A wavelength of
$\lambda$ = 0.8258Å was used.

###  VII.2 Thermodynamics

Figure 11: The heat capacity and magnetic susceptibility measured on a piece
of the single crystals used for our neutron studies and displayed in Fig. 1.
The data clearly show the presence of two transitions which we will show below
are a cubic to tetragonal structural transition ($\simeq 60K$) and a lower
antiferromagnetic transition ($40K$) based on neutron and x-ray diffraction.

Heat capacity and magnetic susceptibility measurements (taken using Quantum
Design MPMS and PPMS systems) on a small piece cut from the crystals (Fig. 2
$(b)$) used for our neutron studies are displayed in Fig. 11. The data display
two transitions at $\simeq$ 60 K and 40 K. We discuss below based on neutron
diffraction on pieces of these crystals that these two transitions represent a
high temperature structural transition from cubic to tetragonal unit cells
(T${}_{S}\simeq$60 K) and a lower temperature transition to an
antiferromagnetically ordered phase (T${}_{N}\simeq$ 40 K).

###  VII.3 Temperature dependent diffraction

Figure 12: Representative refinements of MgV2O4 powders used to characterize
the structural transition in the $(a)$ cubic phase at T=100 K and in the $(b)$
tetragonal phase at T=6 K. Figure 13: Representative scans of the structural
cubic (4,0,0) Bragg peak near the structural transition temperature. Different
temperature cycles are displayed. The temperature cycle is $(a)\rightarrow$
$(b)\rightarrow$ $(c)\rightarrow$ $(d)$.

We discuss the two phase transitions present in our single crystals using
temperature dependent powder and single crystal diffraction. In terms of
single crystal neutron diffraction, Fig. 2$(c)$ in the main paper illustrates
the (0,0,2) intensity on cooling and warming displaying a structural change at
$\simeq$ 60 K with a significant difference in the intensity on heating and
cooling. We speculate this originates from domains and strain in the sample
resulting in changes in extinction on cooling/warming. We further discuss this
point with regards to synchrotron x-ray measurements below. Fig. 2 $(c)$ also
plots the onset of magnetic order at T$\rm{{}_{N}}\simeq$ 40 K probed with
neutron diffraction by monitoring the (1,1,0) magnetic Bragg peak. The results
are consistent with published data and illustrative of the sample quality.
Watanabe  _et al._ (2014); Islam  _et al._ (2012); Mamiya  _et al._ (1997);
Wheeler  _et al._ (2010); Watanabe  _et al._ (2014); Islam  _et al._ (2012)

We further studied the structural transition using synchrotron x-ray
diffraction and Fig. 2$(c)$ displays the lattice parameters as a function of
temperature. This further demonstrates that the high temperature transition is
structural. We have confirmed the unit cells and structural parameters in the
literature in both the high and low temperature phases and representative
diffraction patterns are shown in Fig. 12.

An interesting result in the context of our hysteretic studies outlined in the
main text is shown in Fig. 13 where we plot representative powder data of the
(4,0,0) Bragg peak (in the cubic setting). At high temperatures, a single peak
is displayed representative of the cubic unit cell and a splitting is seen at
low temperatures indicative of a structural distortion to a tetragonal unit
cell. However, for a significant temperature region below the structural
transition, a coexistance is observed. The relative fractions of the two
structural phases, indicated by the relative peak intensities, is different
dependent on the temperature history. This supports that the hysteresis in
intensity displayed in Fig. 2 originates from relative domain populations.

###  VII.4 Single Crystal Nuclear Structure

To characterize the structural properties in our single crystals we applied
neutron diffraction on a small piece from our crystals used for spectroscopy.
Owing to the large uncertainty related to possible mixing between the Mg and V
sites in single crystalline MgV2O4, it was important to measure the nuclear
structure above and below TS=60 K. As demonstrated in Ref. Islam  _et al._ ,
2012, the site mixing has strong effects on the magnetic and structural
transitions.

In agreement with previous studies (Ref. Niitaka  _et al._ , 2013), the
refinements of the T=140 K and 5 K temperature (Fig. 2) data sets confirm the
high-temperature $Fd\overline{3}m$ and low temperature $I4_{1}/amd$ crystal
structures for MgV2O4, with $a$=8.42 Å (cubic phase), and $a$=5.77 Å and
c=8.16 Å in the tetragonal phase. The Mg and O sites were refined to be fully
occupied within error, demonstrating the quality of the single crystal. The
refined structural parameters (including domains which are further discussed
below) are listed in Table LABEL:table_absolute_a.

There has been some discussion in the literature about the possibility of
different structures based on the observation of a predicted extinct
$\vec{Q}$=(0,0,2) Bragg peak. Wheeler  _et al._ (2010) We have observed this
Bragg peak in our D9 unpolarized experiment. However, an azimuthal scan
keeping the momentum transfer fixed found significant intensity variations
indicating that this peak is likely due to multiple scattering processes. We
therefore consider the low-temperature structure to be $I4_{1}/amd$ for the
purposes of this paper.

The high temperature cubic $Fd\overline{3}m$ space groups has a fourfold
symmetry along each crystallographic axis. On cooling a macroscopic crystal
through the Jahn-Teller transition to a $I4_{1}/amd$ tetragonal space group
with a twofold and a fourfold symmetry axis. In the absence of an external
symmetry breaking field such as strain this symmetry must, on average, be
preserved, resulting in structural domains at low temperatures. There are
therefore three possible low temperature domains associated with a distortion
along one of the axes of the high temperature unit cell. These domains were
refined not to be of equal population as indicated by the results in Table
LABEL:table_absolute. This unequal domain population is also consistent with
the findings of Ref. Niitaka  _et al._ , 2013 and may be domain and
environment dependent.

Table 5: Atomic parameters for MgV2O4 at T=140 K (cubic) and 50 K (tetragonal) from single crystal (neutron diffraction. Positions in the cubic phase are $8a$ (${1\over 8}$, ${1\over 8}$, ${1\over 8}$), $16d$ (${1\over 2}$, ${1\over 2}$, ${1\over 2}$), and $32e$ ($x$, $x$, $x$) for Mg, V, and O respectively, and in the $4a$ (0, ${3\over 4}$, ${1\over 8}$), $8d$ (0, 0, ${1\over 2}$), and $16h$ (0, $y$ , $z$) for the tetragonal phase. V was fixed to full occupancy. $*$ Denotes parameters fixed from the T=140 K refinement. Three different domains were observed and fractions included in the T=50 K refinement yield 88(1)%, 8(1)%, and 4(1)% populations. | T=140 K | T=50 K*
---|---|---
Space Group | $Fd\overline{3}m$ | $I4_{1}/amd$
$x$(O) | 0.26000(6) | 0.0
$y$(O) | - | 0.47929(14)
$z$(O) | - | 0.26002(8)
Mg $B_{iso}$ (Å2) | 0.51(4) | 0.16(3)
O $B_{iso}$ (Å2) | 0.45(4) | 0.24(2)
Mg occupancy (%) | 97(7) | 97*
O occupancy (%) | 94(6) | 94*
No. reflections | 741 | 827
No. indep. ref. | 74 | 188
Rf % | 2.55 | 4.63
Rw % | 4.83 | 6.29
$\chi^{2}$ | 2.15 | 3.14
Table 6: Atomic parameters for MgV2O4 at T=100 K (cubic) and 7 K (tetragonal) from the powder sample (synchrotron). Positions in the cubic phase are $8a$ (${1\over 8}$, ${1\over 8}$, ${1\over 8}$), $16d$ (${1\over 2}$, ${1\over 2}$, ${1\over 2}$), and $32e$ ($x$, $x$, $x$) for Mg, V, and O respectively, and in the $4a$ (0, ${3\over 4}$, ${1\over 8}$), $8d$ (0, 0, ${1\over 2}$), and $16h$ (0, $y$ , $z$) for the tetragonal phase. | T=100 K | T=7 K
---|---|---
Space Group | $Fd\overline{3}m$ | $I4_{1}/amd$
$x$(O) | 0.25985(4) | -
$y$(O) | - | 0.01833(2)
$z$(O) | - | 0.26108(11)
Mg $B_{iso}$ (Å2) | 0.386(5) | 0.240(11)
O $B_{iso}$ (Å2) | 0.578(6) | 0.360(11)
V $B_{iso}$ (Å2) | 0.366(1) | 0.234(3)
Rp % | 5.17 | 8.96
Rwp % | 7.19 | 12.7
RB % | 2.28 | 5.44
Rf % | 1.57 | 4.15

###  VII.5 Magnetic Structure

To confirm the magnetic structure, we have used polarized neutron scattering
measurements at DNS (MLZ) in the low temperature ordered phase with the sample
aligned in the (HHL) scattering plane. The measured magnetic Bragg peaks and
intensity differences are consistent with the magnetic structure reported in
ZnV2O4 (Ref. Reehuis  _et al._ , 2003) and is illustrated in Fig. 2 $(a)$. To
guide the analysis of the magnetic structure, we have used the program SARAh
Wills (2000) to calculate the irreducible representations and basis vectors
for the space group. There are 2 common irreducible representations for the
space group $I4_{1}/amd$ with propagation vector (001). The polarized data at
low temperatures shows a uniaxial structure within experimental error
consistent with only one irreducible representation being involved in the
magnetic transition. The involvement of basis functions from only one
irreducible representation is consistent with the TN $\simeq$ 40 K magnetic
transition being second order (Fig. 2 $c$) as predicted from Landau theory.
The magnetic moment was estimated by comparing magnetic $\vec{Q}$=(110) and
nuclear (111) Bragg peaks to be $\mu$=0.70 $\pm$ 0.15 $\mu_{B}$. This is in
reasonable agreement with the refined magnetic moment value found by neutron
diffraction in Ref. Wheeler  _et al._ , 2010, $\mu_{z}=0.47\mu_{B}$ (with an
$\sim$ 8∘ canting from the $z$-axis). The magnetic structure in Fig. 2 $(a)$
and Fig. 6 is based upon interpenetrating chains of magnetic V3+ which carry a
magnetic moment. The chain-like structure originates from the tetragonal
structural distortion which occurs at Ts that breaks the frustrating magnetic
interactions present in the high temperature cubic unit cell as discussed in
the main text.

###  VII.6 Polarization analysis

Polarized neutron spin-flip (SF) and non-spin-flip (NSF) scattering using an
$xyz$ geometry was performed on the DNS diffractometer. Magnetic neutron
scattering obeys the selection rule that it is sensitive only to components of
the magnetic vector perpendicular to the momentum transfer. In Ref. Shirane
_et al._ , 2002, this projected vector is denoted as $\vec{S}_{\perp}$ and
polarized neutron scattering affords a means of separating out the various
components of $\vec{S}_{\perp}$. The components of $\vec{S}_{\perp}$ which are
perpendicular to the neutron polarization appear in the SF channel, while the
component parallel to the neutron polarization will produce NSF scattering.

A problem with performing polarized neutron experiments on vanadium based
materials is the fact that vanadium has a strong incoherent cross section
which translates into an overall momentum independent background. In polarized
experiments, the incoherent cross section in the SF channel is twice that in
the NSF channel, which complicates comparing the channels directly. To
circumvent this, we considered the $xyz$ channels in the SF and NSF channels
separately. In our experiment, the $x$ channel was defined as when the neutron
polarization was parallel to the average momentum transfer $\vec{Q}$; $y$ was
set to be perpendicular to $x$, but within the horizontal scattering plane;
and $z$ was defined as vertical being perpendicular to $x$, $y$, and $z$ is
normal to the horizontal scattering plane.

Given the selection rules by neutron scattering $S_{\perp,x}$ is always
absent. The problem is then to extract $S_{\perp,y}$ and $S_{\perp,z}$
reliably. Using these combined selection rules we note the following for the
magnetic neutron intensities ($I$) in the SF channel including incoherent
background ($B_{incoh}$):

$\displaystyle I_{SF,x}=S_{\perp,y}+S_{\perp,z}+B_{incoh,SF}$ $\displaystyle
I_{SF,y}=S_{\perp,z}+B_{incoh,SF}$ $\displaystyle
I_{SF,z}=S_{\perp,y}+B_{incoh,SF}$ (11)

and in the NSF channel,

$\displaystyle I_{NSF,x}=B_{incoh,NSF}$ $\displaystyle
I_{NSF,y}=S_{\perp,y}+B_{incoh,NSF}$ $\displaystyle
I_{NSF,z}=S_{\perp,z}+B_{incoh,NSF}$ (12)

Since the magnetic diffuse scattering does not overlap with the nuclear
scattering we have not considered feedthrough from one channel to another due
to the incomplete flipping ratio or misalignment of the spin with respect to
the momentum transfer. We have removed the scattering around the nuclear Bragg
peaks in this analysis. In an ideal experimental setup, $2\times
B_{incoh,NSF}=B_{incoh,SF}$, however we have treated SF and NSF scattering
separately to extract the individual cross section for $S_{\perp,y}$ and
$S_{\perp,z}$. For example, from the SF channel,
$S_{\perp,y}=I_{SF,x}-I_{SF,y}$ and $S_{\perp,z}=I_{SF,x}-I_{SF,z}$. From the
the NSF channel, $S_{\perp,y}=I_{NSF,y}-I_{NSF,x}$ and
$S_{\perp,z}=I_{NSF,z}-I_{NSF,x}$. Therefore, the required components for
$S_{\perp,y}$ and $S_{\perp,z}$ can be extracted independently from both the
SF and NSF channels and combined. This method provides us a means of
subtracting off the incoherent background from vanadium.