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# Manipulation and braiding of Weyl nodes using symmetry-constrained phase
transitions

Siyu Chen<EMAIL_ADDRESS>TCM Group, Cavendish Laboratory, University of
Cambridge, J. J. Thomson Avenue, Cambridge CB3 0HE, United Kingdom  Adrien
Bouhon<EMAIL_ADDRESS>Nordic Institute for Theoretical Physics
(NORDITA), Stockholm, Sweden  Robert-Jan Slager<EMAIL_ADDRESS>TCM Group,
Cavendish Laboratory, University of Cambridge, J. J. Thomson Avenue, Cambridge
CB3 0HE, United Kingdom  Bartomeu Monserrat<EMAIL_ADDRESS>TCM Group,
Cavendish Laboratory, University of Cambridge, J. J. Thomson Avenue, Cambridge
CB3 0HE, United Kingdom  Department of Materials Science and Metallurgy,
University of Cambridge, 27 Charles Babbage Road, Cambridge CB3 0FS, United
Kingdom  TCM Group, Cavendish Laboratory, University of Cambridge, J. J.
Thomson Avenue, Cambridge CB3 0HE, United Kingdom  Nordic Institute for
Theoretical Physics (NORDITA), Stockholm, Sweden  TCM Group, Cavendish
Laboratory, University of Cambridge, J. J. Thomson Avenue, Cambridge CB3 0HE,
United Kingdom  TCM Group, Cavendish Laboratory, University of Cambridge, J.
J. Thomson Avenue, Cambridge CB3 0HE, United Kingdom  Department of Materials
Science and Metallurgy, University of Cambridge, 27 Charles Babbage Road,
Cambridge CB3 0FS, United Kingdom

###### Abstract

Weyl semimetals are arguably the most paradigmatic form of a gapless
topological phase. While the stability of Weyl nodes, as quantified by their
topological charge, has been extensively investigated, recent interest has
shifted to the manipulation of the location of these Weyl nodes for non-
Abelian braiding. To accomplish this braiding it is necessary to drive
significant Weyl node motion using realistic experimental parameter changes.
We show that a family of phase transitions characterized by certain symmetry
constraints impose that the Weyl nodes have to reorganise by a large amount,
shifting from one high-symmetry plane to another. Additionally, for a subset
of pairs of nodes with non-trivial Euler class topology, this reorganization
can only occur through a braiding process with adjacent nodes. As a result,
the Weyl nodes are forced to move a large distance across the Brillouin zone
and to braid, all driven by small temperature changes, a process we illustrate
with Cd2Re2O7. Our work opens up routes to readily manipulate Weyl nodes using
only slight external parameter changes, paving the way for the practical
realization of reciprocal space braiding.

Introduction.— The discovery of topological insulators (TIs) Qi and Zhang
(2011); Hasan and Kane (2010) has reinvigorated interest in band theory over
the past decade. The initial ideas behind topological insulators have been
extended to include different crystalline symmetries, resulting in a plethora
of topological characterisations Fu (2011); Slager  _et al._ (2012); Kruthoff
_et al._ (2017); Bouhon and Black-Schaffer (2017); Po  _et al._ (2017);
Bradlyn  _et al._ (2017); Codefects2; Höller and Alexandradinata (2018);
Juričić  _et al._ (2012); Shiozaki  _et al._ (2017); Fang  _et al._ (2012);
Watanabe  _et al._ (2018); Elcoro  _et al._ (2020); Bouhon  _et al._ (2021);
Wieder  _et al._ (2016); Lange  _et al._ (2021); Po  _et al._ (2018); Wieder
and Bernevig (2018); Bouhon  _et al._ (2019), and also to topological
semimetals and superconductors Armitage  _et al._ (2018a, b); Kobayashi and
Sato (2015); Sato and Fujimoto (2009). Within these, Weyl semimetals represent
an ultimate consequence of translational symmetry, allowing for locally stable
band crossings that carry a topological charge. More recently, there has also
been interest in braiding Weyl nodes residing in different band gaps in
systems exhibiting reality conditions, as this process can result in
configurations in which nodes in the same gap have similar charges. The
resulting obstruction for the nodes to annihilate is characterized by a new
type of multi-gap invariant known as Euler class Bouhon  _et al._ (2019); Ahn
_et al._ (2019); Bouhon  _et al._ (2020a); Wu  _et al._ (2019); Bouhon  _et
al._ (2020b); Peng  _et al._ (2021); Jiang  _et al._ (2021); Ünal  _et al._
(2020).

The practical use of Weyl nodes requires their manipulation with external
parameters, and there is a growing body of literature in this direction.
Examples include the use of stress and strain Sie  _et al._ (2019); Slager
_et al._ (2016), coupling to electromagnetism and light Zhang  _et al._
(2017); de Juan  _et al._ (2017), and disorder Pixley  _et al._ (2016); Slager
_et al._ (2017); Pixley and Wilson (2021); Roy  _et al._ (2018). However, in
these proposals the Weyl nodes only move by a small fraction of the Brillouin
zone dimension with experimentally realistic parameter changes, making the
manipulation of Weyl nodes impractical. Here, we explore a route to
manipulating Weyl nodes that combines insights from materials science via
temperature-driven structural changes with more fundamental theoretical
insights that find their origin in symmetry. Namely, we show that in a
particular class of structural phase transitions, Weyl nodes need to travel a
long distance across the Brillouin zone to follow the relevant symmetry
changes, and these large displacements are driven by small temperature
changes. We first exemplify this concept using Cd2Re2O7 as a material example,
and we then provide a more fundamental perspective setting the stage for a
general mechanism. This allows us to predict this behavior for a whole class
of compounds and symmetries.

Structural phase transitions.— Cd2Re2O7 is an intriguing material as it is the
only known superconductor in the pyrochlore family Jin  _et al._ (2001); Sakai
_et al._ (2001). Interest in its superconductivity has led to extensive
exploration of its phase diagram under different conditions Yamaura  _et al._
(2017). As summarized in Fig. 1(a), Cd2Re2O7 crystallizes in a structure of
cubic symmetry (phase I, Fd$\bar{3}$m) Donohue  _et al._ (1965) at ambient
temperature and pressure. The structure can be described as two interwoven
pyrochlore lattices assembled from Re4 and Cd4 tetrahedra, respectively, where
the centers of O6 octahedra coincide with the centers of Re4 tetrahedra. With
gradual cooling, two low-temperature phases with higher conductance appear
successively at Ts1 = $\sim$$200\text{\,}\mathrm{K}$ Hanawa  _et al._ (2001);
Castellan  _et al._ (2002) and Ts2 = $\sim$$120\text{\,}\mathrm{K}$ Yamaura
and Hiroi (2002); Hiroi  _et al._ (2002), both exhibiting tetragonal symmetry
but lacking a center of inversion (phase II, I$\bar{4}$m2 and phase III,
I$4_{1}$22). Very recently, a magnetic torque experiment has revealed that the
transition between phases II and III is mediated by an additional phase of
point group D4h or D2 Uji  _et al._ (2021). Given that this additional phase
was not detected in previous experiments with the exact same setting but a
relatively large temperature step of $2.5\text{\,}\mathrm{K}$ Matsubayashi
_et al._ (2020), the new phase should only exist in the small temperature
range between T${}_{s2}-\Delta$T and Ts2, where $\Delta$T is less than
$2.5\text{\,}\mathrm{K}$.

Figure 1: (a) Energy landscape of Cd2Re2O7. The space groups of the three
phases observed in experiments are shown on the energy surface. The solid
arrows indicate the path of the phase transition during cooling and the dash
arrow indicates the Goldstone mode fluctuating between phases II and III.
(b-c) Distortions of neighbouring Re4 tetrahedra and O6 octahedra in (b) phase
II and (c) phase III. The Re and O ions are represented by the green and red
vertices respectively. The displacement of each O ion is indicated by the
arrow attached to the ion. The subtle differences between edge length of Re4
tetrahedra responding to O ion movement are distinguished by the colors. The
shaded area shows the dihedral mirror plane falling along the “indigo” and
“purple” edge of Re4 tetrahedra and the curved arrow shows the twofold axis
through the connecting Re ion. (d) Saddle points on the energy surface from
phase II to III as a function of root sum squared (rss) distance of atoms,
where the energy of phase II has set to zero.

The non-centrosymmetric lattice, strong spin-orbit coupling and metallic
properties of the two well-characterized low-temperature phases of Cd2Re2O7
naturally satisfy the prerequisites for non-magnetic Weyl semimetals,
providing an ideal platform to study the movement of Weyl nodes constrained by
crystallographic symmetry changes. This inspires us to consider a smooth path
connecting phases II and III. The crystal structures of phases II and III are
obtained through a first-principles phonon calculation performed
relativistically (including spin-orbital coupling) with a revised Perdew-
Burke-Ernzerhof GGA functional for solids (PBEsol) Perdew  _et al._ (2008) on
phase I (see details in SM), and the intermediate structures are obtained
using the nudged elastic band (NEB) algorithm Henkelman and Jónsson (2000) to
optimize a number of linear interpolated structures between phases II and III.
The results show that all fully optimized intermediate structures belong to
the same space group, constituting a smooth transition path of F222 symmetry.
Including the start and end of the path (namely phases II and III), any
crystal structure on the path can be interpreted as a consequence of
structural distortions of phase I, which is dominated by Cd and O ion
displacements. Figs. 1(b-c) visualize the distortions of O ions occurring in
phases II and III. It is worth noting that two neighbouring O6 octahedra have
displacements with opposite directions, thus the inversion symmetry of the
system is broken. Interestingly, although the distortion does not have a Jahn-
Teller origin but instead is very likely driven by strong spin-orbital
coupling Fu (2015), it exhibits extremely similar behavior compared to the
Jahn-Teller effect occurring among $d$-orbital coordination complexes in which
$E_{g}$ electronic states are coupled to $E_{g}$ vibrational modes. In phase
II, the distortion conserves the symmetry of the horizontal square of the O6
octahedra and gives half of the O6 octahedra an elongation along the vertical
direction whereas the other half a compression. In phase III, the horizontal
square is distorted to a rhombus, which breaks the vertical mirror planes that
fall along the “indigo” and “purple” edges of Re4 tetrahedra shown in Fig.
1(b). It is also worth noting that in phase II the two neighbouring Re4
tetrahedra are non-congruent, however when it comes to phase III, they become
congruent via losing the mirror plane in phase II and adding a twofold axis
through the connecting Re ion, as shown in Fig. 1(c)). Such a change in
symmetry results in the conversion of the point group of the system from D2d
to D4. We show later that this transition from a fourfold rotoinversion
symmetry ($S_{4}=IC_{4z}^{-1}$ in D2d) to a proper fourfold rotation symmetry
($C_{4z}$ in D4) drives a major topological transformation of the band
structure in reciprocal space.

As a final remark, we emphasize that the path proposed in this work is not
just a fictitious computational construct. Its existence is strongly supported
by group theory analysis A. Sergienko and H. Curnoe (2003) and by the latest
magnetic torque experiment performed with extremely small temperature steps
Uji  _et al._ (2021). The phenomenological order parameter characterizing the
cubic-to-tetragonal phase transitions of Cd2Re2O7 corresponds to a twofold
degenerate $E_{u}$ representation of the cubic point group, so it makes sense
that the NEB calculation has found a series of orthorhombic intermediate
phases of space group F222 (the maximal common subgroup of phases II and III)
which are induced by a mixture of the two components of the representation.
The result from the NEB calculation shows that the energy difference of
intermediate phases on the path is anomalously small ($\Delta
E\leq$0.3\text{\,}\mathrm{m}\mathrm{eV}\mathrm{/}\mathrm{a}\mathrm{t}\mathrm{o}\mathrm{m}$$,
see Fig 1(d)), that is of the order of $\Delta T=$$0.4\text{\,}\mathrm{K}$ for
a structure traversing the whole path. This extremely flat path leads to a
quasi-continuous $U(1)$ symmetry of the energy landscape of Cd2Re2O7. Breaking
this symmetry was predicted to yield a Goldstone mode manifesting as a
structural fluctuation between phase II and III with vanishing frequency and
excitation energy, and this has been observed by polarized Raman scattering
and X-ray diffraction experiments Kendziora  _et al._ (2005); Venderley  _et
al._ (2020). In particular, we have found an energy local minimum on the path
which has the same energy as phase II, and we propose it as a candidate
structure for the newly discovered intermediate phase in Ref. Uji  _et al._
(2021).

Figure 2: (a) Weyl nodes in the Brillouin zone of phase II. The chirality of
the Weyl nodes is distinguished by the red and blue colors of the balls/dots.
The three $C_{2}T$-invariant planes are highlighted. The Brillouin zone
polyhedron is shown in the lower left corner for reference. (b-c) The movement
of the red and the blue Weyl nodes as the phase transition smoothly progresses
from phase II to phase III. The thick arrows indicate the movement of the Weyl
nodes of interest, and the dash arrows indicate the movement of their
symmetry-related partners. The bars below show the “timelines” of the
movement. See text for more details.

Mechanism of Weyl node movement.— Since the path connecting phases II and III
preserves the symmetry of space group F222, the key symmetry elements
constraining the phase transition are three twofold rotation axes along the
$k_{x}$, $k_{y}$ and $k_{z}$ directions respectively: $C_{2x}$, $C_{2y}$, and
$C_{2z}$. Time-reversal symmetry $T$ is also preserved along the transition
path as no magnetic ordering has been observed in experiments Vyaselev  _et
al._ (2002) or in our first-principles calculations. Hence, there are three
mutually perpendicular $C_{2}T$-invariant planes dividing the Brillouin zone
polyhedron into eight equivalent regions, on which state
$\left|\psi_{k}\right\rangle$ can be mapped to itself by a combination of
symmetry operators $C_{2i}T$ ($i=x$ or $y$ or $z$). Fig.2(a) shows the
distribution of the Weyl nodes of phase II formed by the two partially
occupied bands closest to the Fermi energy $E_{\mathrm{F}}$, whose main
contributions are Re $t_{2g}$ orbitals (see the band structure in SM). There
are a total of twenty pairs of Weyl nodes in the whole Brillouin zone located
within the energy window of $E_{\mathrm{F}}\pm$100\text{\,}\mathrm{meV}$$.
Considering the symmetry of the system, there are four independent octets and
two independent quartets of Weyl points. All of them are pinned exactly on the
$C_{2}T$-invariant planes, which is in line with the previous theoretical
prediction that $C_{2}T$ symmetry is able to stabilize Weyl nodes Bouhon and
Black-Schaffer (2017); Bouhon  _et al._ (2019); Sun  _et al._ (2018).
Additionally, due to the presence of two dihedral mirror planes
($k_{x}$=$k_{y}$ and $k_{x}$=$-k_{y}$ planes) the Weyl nodes flip chirality
when mapped from one vertical $C_{2}T$-invariant plane to another, thus
forming quartets of Weyl points at fixed $k_{z}$ with alternating chirality.
$C_{2z}T$ symmetry then maps each quartet of Weyl points at $k_{z}$ to a
quartet at $-k_{z}$ with no change of chirality, thus giving rise to each
independent octet of nodes in Fig. 2(a) where the $S_{4}$ symmetry maps each
Weyl node at a fixed $k_{z}$ to a Weyl node of opposite chirality at $-k_{z}$.
In contrast, the symmetries of phase III require that each quartet of nodes at
a fixed $k_{z}$ has a unique chirality as imposed by $C_{4z}$ symmetry, and
similarly for the quartet mapped to $-k_{z}$ by $C_{2z}T$, thus giving rise to
octets of nodes of the same chirality. While the transition from one phase to
another requires the splitting of the octets into D2 symmetric quartets of the
same chirality (each contained in one of the two vertical $C_{2}T$ planes),
the intrinsic incompatibility between the D2d-symmetric octets of alternating
chirality and the D4-symmetric octets of the same chirality enforces a
qualitative rearrangement of the Weyl nodes through the Brillouin zone across
the structural phase transition. It is worth noting that unlike the situation
in real space where each atom can move along three independent degrees of
freedom, Weyl nodes pinned on the $C_{2}T$-invariant planes in reciprocal
space only have two degrees of freedom, as a single Weyl node on a given
$C_{2}T$-invariant plane cannot get out of that plane unless it recombines
with another Weyl node. Therefore, the Weyl nodes are forced to move a large
distance across the Brillouin zone as the structural phase transition occurs,
despite the very small temperature change and associated small atomic
displacements in real space.

The trajectory of the Weyl nodes with ‘red’ chirality on the
$C_{2x}T$-invariant plane (shown in Fig.2(b)) is a typical example reflecting
the above scenario. In the initial configuration the Weyl nodes on the two
vertical $C_{2i}T$-invariant planes ($i=x,y$) are mirror images of each other.
When the mirror symmetry breaks and the system gradually approaches phase III
with fourfold rotational symmetry, the ‘red’ chirality Weyl nodes on the
$C_{2x}T$-invariant plane with $k_{y}>0$ first move towards each other and
meet at the $C_{2z}T$-invariant plane. After the collision, they get out of
the $C_{2x}T$-invariant plane and continue moving on the $C_{2z}T$-invariant
plane in opposite directions, until each of them meets its previous
$C_{2z}$-rotational symmetry partner with $k_{y}<0$ at the intersection of the
$C_{2y}T$\- and $C_{2z}T$-invariant planes. Then they leave the horizontal
$C_{2z}T$-invariant plane and stop somewhere on the $C_{2y}T$-invariant plane.
To summarize, in order to migrate from one vertical $C_{2}T$-invariant plane
to another, the ‘red’ Weyl nodes have to travel nearly a quarter of the
Brillouin zone. Fig.2(c) shows another two examples of significant motion of
Weyl nodes. The two ‘blue’ chirality Weyl nodes located near the boundary of
the $C_{2z}T$-invariant plane meet at the $k_{y}$ axis and finally stop
somewhere on the $C_{2x}T$-invariant plane (indicated by the light blue
arrows). The two ‘blue’ chirality Weyl nodes that are located on the
$C_{2z}T$-invariant plane and initially very close to each other first slowly
meet at the $k_{x}$ axis, then shift over an arc on the $C_{2y}T$-invariant
plane approaching the $k_{z}$ axis (indicated by the dark blue arrows).

Due to the bulk-boundary correspondence, the movement of bulk Weyl nodes can
also be traced in terms of the surface Fermi arcs. Figs.3 (a-b) show the
surface states of phases II and III on the [001] surface. We note that phase
II exhibits an ideal Weyl semimetal feature — two thick and straight Fermi
arcs connecting the projection of $C_{2x}T$\- and $C_{2y}T$-invariant planes,
which should be easily detected by angle-resolved photoemission spectroscopy.
To reach the fourfold rotational symmetry in phase III, the Fermi arcs in
phase II must be broken, therefore the Fermi arcs in phase III are more local
and not as obvious as in phase II.

Figure 3: $\boldsymbol{k}$-dependent local density of states (LDOS) for phases
(a) II and (b) III on the [001] surface, where gold (olive) colour represents
high (low) LDOS. The diagonal dash lines highlight the projections of
$C_{2x}T$\- and $C_{2y}T$-invariant planes on the [001] surface.

$C_{2}$-indicated locking of Weyl nodes and unlocking through braiding.—
Crucially, we find that a subset of pairs of Weyl nodes are locked on their
$C_{2}T$ plane as a consequence of non-trivial Euler class topology Ahn and
Yang (2019); Bouhon  _et al._ (2020a, b). This implies that the braiding with
adjacent nodes, that is, nodes within the next energy gap below or above, must
take place before their transfer to another $C_{2}T$ plane. Calling the Weyl
nodes considered so far the principal nodes, we take as an example the
$C_{2x}$-symmetric pair of Weyl nodes with the same (‘blue’) chirality located
within the horizontal $C_{2z}T$ plane at $k_{x}>0$ (whose trajectories are
indicated by the dark blue arrows in Fig. 2(c)). We have computed the Euler
class $\chi_{C_{2z}T}$ on a patch containing this pair of principal Weyl nodes
in phase II, and find $\chi_{C_{2z}T}=1$ (see details in SM). The non-trivial
Euler class indicates an equal $C_{2z}T$-non-Abelian frame charge of the two
nodes Ahn and Yang (2019); Bouhon  _et al._ (2020a); Wu  _et al._ (2019);
Bouhon  _et al._ (2020b); Sun  _et al._ (2018), which implies a topological
obstruction to transferring these nodes to the vertical $C_{2y}T$ plane.
During the structural phase transition, we observe a band inversion happening
in the adjacent gap below, which creates a new pair of adjacent nodes on the
$\Gamma X$-line that crosses the patch of the principal nodes (see the
snapshots in SM), that is, the principal nodes are braided with the adjacent
ones. This leads to the trivialization of the $C_{2z}T$-Euler class for the
principal nodes on the patch, which then allows them to move to the vertical
$C_{2y}T$ plane after their merging on the $C_{2x}$-symmetric $\Gamma X$-line.
It is remarkable that the non-vanishing of the $C_{2z}T$-Euler class in phase
II readily follows from the opposite nature of the $C_{2x}$-eigenvalues of the
bands forming the principal nodes, see also Ref. Murakami  _et al._ (2017). In
this regard, the unlocking of the nodes from the $C_{2z}T$ plane is indicated
by the exchange of the $C_{2x}$-eigenvalue of the lower band, which is induced
by the braiding through an adjacent band inversion, so that both bands now
have equal $C_{2x}$-eigenvalue. We conclude that on top of inducing large
trajectories of the Weyl nodes across the Brillouin zone, the structural phase
transition in Cd2Re2O7 also requires multiple interesting braiding processes
of Weyl nodes that are necessary to unlock their transfer from one $C_{2}T$
plane to another.

Generalizations.— We emphasize that the physical manifestations addressed in
this work are controlled by symmetry, and not by the details of the specific
material involved. Other materials have also been shown to undergo a
structural phase transition mediated by a Goldstone mode, such as the
manganites Skjærvø  _et al._ (2019); Meier  _et al._ (2020); Juraschek  _et
al._ (2020); Marthinsen  _et al._ (2018). A common feature of these systems is
the breaking of the $U(1)$ symmetry of their energy landscape (as an effect
the discrete point group symmetry of the crystal) with discrete minima
separated by small potential barriers. While in our case this facilitates an
unconventional phase transition between two isomorphic but distinct crystal
structures, in the case of the manganites, the high-temperature phase
$P6_{3}/mmc$ ($D_{6h}$) gives rise to two non-isomorphic lower-temperature
phases $P6_{3}cm$ ($C_{6v}$) and $P\bar{3}c1$ ($D_{3d}$), that both exhibit a
sixfold degenerate ground state. Focusing on the non-centrosymmetric phase
($P6_{3}cm$) which is the only one permitting Weyl nodes from a symmetry
perspective, each discrete minimum corresponds to a distinct polar lattice
distortion (combining three Mn-trimerizations and two ferroelectric
polarizations) Skjærvø  _et al._ (2019); Meier  _et al._ (2020); Huang  _et
al._ (2014); Kumagai  _et al._ (2012). This suggests that our framework would
also apply in the case of a transition between two distinct polar phases.
These questions, as well as studies into other symmetry groups and materials,
provide intriguing future research directions.

Conclusions and discussion.— In conclusion, we have shown that phase
transitions in the materials science sense can result in relatively large
movements of Weyl nodes due to symmetry changes across the transition. This
offers an interesting route to significantly manipulate Weyl nodes and their
associated Fermi arcs using only moderate temperature changes. Although the
principles behind the mechanism are generic, we have also discussed a specific
material example in the form of Cd2Re2O7. We note a recent high-throughput
screening for Weyl semimetals with fourfold rotoinversion symmetry Gao  _et
al._ (2021), and based on our findings it is worthwhile to explore whether
these candidates may have any phase transition from $S_{4}$ to $C_{4}$, which
would suggest similar physics to that reported here. Overall, our work sets
the stage for the practical manipulation Weyl nodes, a prerequisite for their
ultimate technological use.

Acknowledgements.— S.C. acknowledges financial support from the Cambridge
Trust and from the Winton Programme for the Physics of Sustainability. R.-J.
S. acknowledges funding from the Marie Skłodowska-Curie Programme under EC
Grant No. 842901, the Winton Programme for the Physics of Sustainability, and
Trinity College at the University of Cambridge. B.M. acknowledges support from
the Gianna Angelopoulos Programme for Science, Technology, and Innovation and
from the Winton Programme for the Physics of Sustainability. The calculations
in this work have been performed using resources provided by the Cambridge
Tier-2 system (operated by the University of Cambridge Research Computing
Service and funded by EPSRC [EP/P020259/1]), as well as by the UK Materials
and Molecular Modelling Hub (partially funded by EPSRC [EP/P020194]), Thomas,
and by the UK National Supercomputing Service, ARCHER. Access to Thomas and
ARCHER was obtained via the UKCP consortium and funded by EPSRC
[EP/P022561/1]).

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Supplemental material for Manipulation and braiding of Weyl nodes using
symmetry-constrained phase transitions

Supplemental Material for
Manipulation and braiding of Weyl nodes using symmetry-constrained phase
transitions

Siyu Chen Adrien Bouhon Robert-Jan Slager Bartomeu Monserrat

##  I Phonons in phase I

Figure S1: (a) Phonon dispersions of phase I of Cd2Re2O7 with and without SOC.
(b) The energy of phase I as a function of the frozen soft phonon amplitudes,
where the energy of phase I has been set to zero.

The crystal structure of Cd2Re2O7 at room temperature has been reported in
entry 28445 of the Inorganic Crystal Structure Database sm-Mariette2004 with
space group Fd$\bar{3}$m. Starting from this experimentally reported
structure, we have first performed a symmetry-constrained geometry
optimization using the Vienna Ab initio Simulation Package (vasp) sm-VASP1,sm-
VASP2, followed by a phonon calculation with and without considering spin-
orbital coupling (SOC) using the finite displacement method in conjunction
with nondiagonal supercells sm-Lloyd2015. The results taking SOC into account
(blue curves in Fig. S1(a)) show two soft modes at the center of the Brillouin
zone, a twofold degenerate $E_{u}$ mode and a threefold degenerate $T_{1u}$
mode. Following the soft phonons, we have found that the two-fold degenerate
$E_{u}$ phonon mode drives the structure to the I$\bar{4}$m2 and I4122 phases
(which are phases II and III observed in experiments), and the three-fold
degenerate $T_{1u}$ mode drives the structure to R3m, Ima2, and Cm phases,
which have relatively higher energy compared to phases II and III (see Fig.
S1(b)). Also, by comparing the phonon dispersions calculated with and without
SOC, we have noticed that SOC suppresses the distortions caused by the soft
modes to a certain extent. The mechanism behind this is the same as what has
been found in In5Bi3, where the SOC interaction reduces the density of states
near the Fermi energy and stabilizes the structure sm-Chen2019.

##  II Band structure of phase II

Figure S2: Band structure of phase II of Cd2Re2O7 with SOC. The shaded region
is zoomed in to show the details of the band crossings, where the ‘$+$’ sign
and ‘$-$’ sign mark the $C_{2x}$ eigenvalues of the bands. The corresponding
high-symmetry path is shown in the Brillouin zone polyhedron.

The band structure of phase II is shown in Fig. S2, where four partially
occupied bands closest to the Fermi energy (which has been set to zero) are
highlighted. The red and blue solid (dashed) bands are the $\mathcal{B}_{\pm
1}$ ($\mathcal{B}_{\pm 2}$) bands defined in the main text. Since bands
$\mathcal{B}_{\pm 1}$ have opposite $C_{2x}$ eigenvalues on the $\Gamma
X$-line, their crossings (marked in Fig. S2), which correspond the pair of the
Weyl node of opposite chirality located on the $k_{x}$ axis shown in Fig.
2(a)), are protected by symmetry.

|  |
---|---|---
|  |
Figure S3: (a-f) Snapshots of the band structure along the $\Gamma X$-line
during the structural phase transition. The ‘$+$’ sign and ‘$-$’ sign mark the
$C_{2x}$ eigenvalues of the bands with the same color as the symbol.

##  III Locking and unlocking of Weyl nodes through braiding

We further elaborate on the Weyl node locking on $C_{2}T$-planes. As stated in
the main text, we find that a subset of Weyl node pairs are locked on their
$C_{2}T$ plane as a consequence of non-trivial Euler class topology sm-
Ahn2019, sm-bouhon2019nonabelian, sm-bouhonGeometric2020. We call the Weyl
nodes discussed in the main text the principal nodes (i.e. formed by bands
$\mathcal{B}_{-1}$ and $\mathcal{B}_{+1}$, where we label them taking 0 at the
Fermi energy), and the nodes between bands $\mathcal{B}_{-1}$ and
$\mathcal{B}_{-2}$ ($\mathcal{B}_{+1}$ and $\mathcal{B}_{+2}$), the lower
adjacent nodes (the higher adjacent nodes). Taking as an example the same
(blue) chirality and $C_{2x}$-symmetric pair of principal Weyl nodes located
within the horizontal $C_{2z}T$ plane at $k_{x}>0$ whose trajectories are
indicated by the dark blue arrows in Fig. 2(c) of the main text, we find that
a braiding with Weyl nodes of one of the adjacent gaps is necessary to allow
their transfer to the $C_{2y}T$ vertical plane. We show below that this is a
consequence of the opposite $C_{2x}$-eigenvalues of the bands forming the two
principal nodes.

Let us first consider what would happen without braiding. In phase II, the
bands forming the principal nodes have opposite $C_{2x}$-eigenvalues with
$\xi_{C_{2x}}[\mathcal{B}_{-1}]=-1$ and $\xi_{C_{2x}}[\mathcal{B}_{+1}]=1$,
see Fig. S3(a). In the absence of braiding, the merging of the principal nodes
on the $\Gamma X$-line would induce a band inversion along the $C_{2x}$-axis
between two bands of opposite $C_{2x}$-eigenvalues, forming two unavoided band
crossings on the $\Gamma X$-line that are protected by $C_{2x}$, see also Ref.
sm-Murakamie1602680. As a consequence, the two principal nodes would scatter
on the $\Gamma X$-line, thus remaining on the $C_{2z}T$ plane after their
merging. Such a scattering of node pairs confined on a $C_{2i}T$ plane
indicates a non-trivial $C_{2i}T$-Euler class topology sm-
bouhon2019nonabelian. We have computed the Euler class on a patch of the
$C_{2z}T$ plane that contains the two principal nodes under consideration
while avoiding all other nodes, see Fig. S4, for which we obtain
$\chi_{C_{2z}T}=1$ (see below for details). This confirms the prediction based
on the $C_{2x}$-eigenvalues.

We thus conclude that a braiding with adjacent nodes must take place sm-
bouhon2019nonabelian in order to unlock the pair of principal nodes from the
$C_{2z}T$-plane, thus allowing its transfer on the vertical $C_{2y}T$ plane.
Indeed, as shown in Fig. S3(a-f) with successive snapshots of the band
structure during the phase transition, we find that a band inversion within
the lower adjacent gap is taking place along the $\Gamma X$-line, leading to
the exchange of the $C_{2x}$-eigenvalue of band $\mathcal{B}_{-1}$ and giving
rise to two lower adjacent nodes along the $C_{2x}$-line. We then observe that
the two principal nodes merge on the $\Gamma X$-line, which now involves two
bands of equal $C_{2x}$-eigenvalue, i.e.
$\xi_{C_{2x}}[\mathcal{B}_{-1}]=\xi_{C_{2x}}[\mathcal{B}_{+1}]=1$. As a
result, the band inversion in the principal gap along the $\Gamma X$-axis now
induces avoided band crossings, indicating that the two principal nodes have
moved from the horizontal $C_{2z}T$ plane to the vertical $C_{2y}T$ plane.
This implies that their $C_{2z}T$\- and $C_{2x}T$-Euler classes on patches
that avoid the adjacent nodes are both zero.

Figure S4: Patch (marked in gray shade) of the $C_{2z}T$-invariant plane over
which we compute the Euler class $\chi_{C_{2z}T}$ in phase II. The inset zooms
in on the details around the $C_{2x}$-symmetric pair of Weyl nodes. Using the
Bloch eigenvectors obtained through the symmetrized Wannierization, we get
$\chi_{C_{2z}T}=1$, in agreement with the $C_{2x}$-eigenvalues indicator. See
text for more details.

##  IV Real Bloch eigenvectors

Performing the Wannierization of the bands around the Fermi energy over a
24-orbital-spin basis with wannier90 sm-wannier90 and symmetrizing the tight-
binding Hamiltonian sm-PhysRevMaterials.2.103805, we obtain the Bloch
eigenvectors of the two bands forming the two principal nodes over a discrete
mesh of the patch (highlighted in Fig. S4) in momentum space. We compute the
Euler class using the code provided in Ref. sm-Bzdusek:Mathematica-Euler, for
which the Bloch eigenvectors must be rotated into the basis that makes them
real sm-bouhon2019nonabelian. To accomplish this, we first need to specify the
effective atomic degrees of freedom of the wannierized tight-binding model,
and then perform a Takagi factorization of the $C_{2}T$ operator acting on
these.

Let us write the primitive cell lattice vectors for space group I$\bar{4}$m2
as

$\boldsymbol{a}_{1}=\dfrac{1}{2}(-a,a,c),~{}\boldsymbol{a}_{2}=\dfrac{1}{2}(a,-a,c),~{}\boldsymbol{a}_{3}=\dfrac{1}{2}(a,a,-c).$
(S1)

We obtain a Wannierized tight-binding model in the Wannier basis that spans
four Re inequivalent atomic sites at the Wyckoff position $8\mathrm{i}$, with
the three $d$-orbitals $\\{d_{xz},d_{yz},d_{xy}\\}$ and the two spin-$1/2$
components. The position vectors for four inequivalent Re sites at Wyckoff
position $8\mathrm{i}$ are given through (the other four equivalent sites of
Wyckoff position $8\mathrm{i}$ are obtained through a shift by
$\boldsymbol{\tau}=\dfrac{1}{2}(a,a,c)$)

$\boldsymbol{r}_{1}=(0,x,z),~{}\boldsymbol{r}_{2}=C_{2z}\boldsymbol{r}_{1}=(0,-x,z),~{}\boldsymbol{r}_{3}=S_{4z}\boldsymbol{r}_{1}=(-x,0,-z),~{}\boldsymbol{r}_{4}=S^{-1}_{4z}\boldsymbol{r}_{1}=(x,0,-z),$
(S2)

with the fourfold rotoinversion $S_{4z}=C_{4z}^{-1}I$. We write the Wannier-
Löwdin type basis of the tight-binding model as

$|\boldsymbol{\varphi}_{i},\boldsymbol{R}+\boldsymbol{r}_{i}\rangle=\left(|\varphi_{i,d_{xz},\uparrow}\rangle~{}|\varphi_{i,d_{yz},\uparrow}\rangle~{}|\varphi_{i,d_{xy},\uparrow}\rangle~{}|\varphi_{i,d_{xz},\downarrow}\rangle~{}|\varphi_{i,d_{yz},\downarrow}\rangle~{}|\varphi_{i,d_{xy},\downarrow}\rangle\right),~{}\mathrm{for}~{}i=1,2,3,4,$
(S3)

from which we obtain the Bloch basis per inequivalent lattice site,

$|\boldsymbol{\phi}_{i},\boldsymbol{k}\rangle=\sum\limits_{\boldsymbol{R}\in\boldsymbol{T}}\mathrm{e}^{\mathrm{i}\boldsymbol{k}\cdot(\boldsymbol{R}+\boldsymbol{r}_{i})}|\boldsymbol{\varphi}_{i},\boldsymbol{R}+\boldsymbol{r}_{i}\rangle,~{}\mathrm{for}~{}i=1,2,3,4,$
(S4)

with the tetragonal body-centered Bravais lattice
$\boldsymbol{T}=\\{n_{1}\boldsymbol{a}_{1},n_{2}\boldsymbol{a}_{2},n_{3}\boldsymbol{a}_{3}\\}_{(n_{1},n_{2},n_{3})\in\mathbb{Z}^{3}}$.
The total Bloch basis is then written

$|\boldsymbol{\Phi},\boldsymbol{k}\rangle=\left(|\boldsymbol{\phi}_{1},\boldsymbol{k}\rangle~{}|\boldsymbol{\phi}_{2},\boldsymbol{k}\rangle~{}|\boldsymbol{\phi}_{3},\boldsymbol{k}\rangle~{}|\boldsymbol{\phi}_{4},\boldsymbol{k}\rangle\right).$
(S5)

The representation of $C_{2z}T$ on the total Bloch basis is thus

${C_{2z}T}|\boldsymbol{\Phi},\boldsymbol{k}\rangle=|\boldsymbol{\Phi},m_{z}\boldsymbol{k}\rangle\cdot
U_{C_{2z}T}\mathcal{K}$ (S6)

with the mirror operation $m_{z}(k_{x},k_{y},k_{z})=(k_{x},k_{y},-k_{z})$, the
complex conjugation $\mathcal{K}$ and the $U_{C_{2z}T}$ unitary matrix given
by

$U_{C_{2z}T}=\mathrm{i}\sigma_{x}\otimes U_{2,\mathrm{site}}\otimes
U_{2,\mathrm{orb}},$ (S7)

where $U_{2,\mathrm{site}}=\mathbb{1}_{2}\otimes\sigma_{x}$ and
$U_{2,\mathrm{orb}}=\mathrm{diag}(-1,-1,1)$. Writing the tight-binding Bloch
Hamiltonian as

$\mathcal{H}=\sum\limits_{\boldsymbol{k}}|\boldsymbol{\Phi},\boldsymbol{k}\rangle\cdot
H(\boldsymbol{k})\cdot\langle\boldsymbol{\Phi},\boldsymbol{k}|=\sum\limits_{\boldsymbol{k}}|\boldsymbol{\Phi},\boldsymbol{k}\rangle\cdot\mathcal{V}(\boldsymbol{k})\cdot\mathcal{E}(\boldsymbol{k})\cdot\mathcal{V}^{\dagger}(\boldsymbol{k})\cdot\langle\boldsymbol{\Phi},\boldsymbol{k}|,$
(S8)

with the diagonal matrix of energy eigenvalues
$\mathcal{E}(\boldsymbol{k})=\mathrm{diag}(E_{1},\dots,E_{24})$ and the
corresponding matrix of column eigenvectors
$\mathcal{V}(\boldsymbol{k})=(v_{1}\dots v_{24})$, we obtain the real Bloch
eigenvectors through a change of basis,

$\displaystyle|\boldsymbol{\Phi},\boldsymbol{k}\rangle$
$\displaystyle=|\widetilde{\boldsymbol{\Phi}},\boldsymbol{k}\rangle\cdot W,$
(S9)

that is

$\widetilde{\mathcal{V}}(\boldsymbol{k})=W\cdot\mathcal{V}(\boldsymbol{k}).$
(S10)

We now show that the unitary matrix $W$ is obtained from the Takagi
factorization of the $U_{C_{2z}T}$ unitary matrix. Indeed, we find
$[C_{2z}T]^{2}=U_{C_{2z}T}\cdot U_{C_{2z}T}^{*}=\mathbb{1}_{24}$, which
implies $U_{C_{2z}T}=U_{C_{2z}T}^{\intercal}$. As for any symmetric matrix,
there is a Takagi factorization given by
$U_{C_{2z}T}=U_{\mathrm{tf}}\cdot\Lambda\cdot U_{\mathrm{tf}}^{\intercal}$,
from which we define $W=\sqrt{\Lambda^{*}}\cdot U_{\mathrm{tf}}^{\dagger}$.
The representation of $C_{2z}T$ in the new basis then gives

$\displaystyle{C_{2z}T}|\widetilde{\boldsymbol{\Phi}},\boldsymbol{k}\rangle$
$\displaystyle=|\widetilde{\boldsymbol{\Phi}},m_{z}\boldsymbol{k}\rangle\cdot\left(W\cdot
U_{C_{2z}T}\cdot
W^{\intercal}\right)\mathcal{K}=|\widetilde{\boldsymbol{\Phi}},m_{z}\boldsymbol{k}\rangle\mathcal{K}.$
(S11)

In the case of a unitary and symmetric matrix, as it is the case for
$U_{C_{2z}T}$, the Takagi factorization can be readily obtained from the
singular value decomposition $U_{C_{2z}T}=U_{\mathrm{svd}}\cdot\Lambda\cdot
V_{\mathrm{svd}}$ where $\Lambda=\mathbb{1}$ for unitary matrices, through
$U_{\mathrm{tf}}=U_{\mathrm{svd}}\cdot\sqrt{U_{\mathrm{svd}}^{\dagger}\cdot
V_{\mathrm{svd}}^{*}}$ sm-CHEBOTAREV2014380, from which we find
$W=U_{\mathrm{tf}}^{\dagger}$.

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