# Interferometry of non-Abelian band singularities and Euler class topology

Ollie Breach  TCM Group, Cavendish Laboratory, University of Cambridge, JJ
Thomson Avenue, Cambridge CB3 0HE, United Kingdom  Rudolf Peierls Centre for
Theoretical Physics, Parks Road, Oxford, OX1 3PU, United Kingdom  Robert-Jan
Slager  TCM Group, Cavendish Laboratory, University of Cambridge, JJ Thomson
Avenue, Cambridge CB3 0HE, United Kingdom  F. Nur Ünal<EMAIL_ADDRESS>TCM
Group, Cavendish Laboratory, University of Cambridge, JJ Thomson Avenue,
Cambridge CB3 0HE, United Kingdom

###### Abstract

In systems with a real Bloch Hamiltonian band nodes can be characterised by a
non-Abelian frame-rotation charge. The ability of these band nodes to
annihilate pairwise is path dependent, since by braiding nodes in adjacent
gaps the sign of their charges can be changed. Here, we theoretically
construct and numerically confirm two concrete methods to experimentally probe
these non-Abelian braiding processes and charges in ultracold atomic systems.
We consider a coherent superposition of two bands that can be created by
moving atoms through the band singularities at some angle in momentum space.
Analyzing the dependency on the frame charges, we demonstrate an
interferometry scheme passing through two band nodes, which reveals the
relative frame charges and allows for measuring the multi-gap topological
invariant. The second method relies on a single wavepacket probing two nodes
sequentially, where the frame charges can be determined from the band
populations. Our results present a feasible avenue for measuring non-Abelian
charges of band nodes and the experimental verification of braiding procedures
directly, which can be applied in a variety of settings including the recently
discovered anomalous non-Abelian phases arising under periodic driving.

Introduction.— Within the active field of topological insulators and
semimetals [1, 2, 3], the role of on-site ten-fold way symmetries, such as
time reversal, particle-hole and chiral, as well as crystalline symmetries are
by now rather uniformly understood in and out of equilibrium [4, 5, 6, 7, 8,
9, 10, 11, 12, 13, 14, 15, 16]. Recently there has been new progress in terms
of multi-gap topological phases [17, 18, 19, 20, 21], where band subspaces
(sets of isolated bands) can attain non-trivial invariants that do not a
priori depend on the symmetry eigenvalues of the bands at high symmetry points
and, hence, fall outside of all hitherto-known classifications [7, 8, 9, 10].
When a system can be represented in terms of a real-valued Hamiltonian due to
the presence of $\mathcal{C}_{2}\mathcal{T}$ or $\mathcal{P}\mathcal{T}$ (two-
fold rotations or parity and time-reversal) symmetry, band degeneracies can
carry non-Abelian frame charges [22]. The sign of these frame charges can be
changed upon braiding band nodes in the momentum space, where the obstruction
to annihilate the similarly-valued charges is quantified by a multi-gap
invariant, the Euler class [18, 17, 19]. Multi-gap considerations away from
equilibrium have revealed even more exotic phenomena, such as novel quench
signatures [23, 24], optical responses [25, 26, 27, 28] and anomalous non-
Abelian phases that can exclusively arise under periodic driving [29].

Band nodes also play an important role in single-gap topological phases.
Conventional Weyl nodes act as sources of Berry flux [30], while higher order
degeneracies require stabilization of crystalline symmetries and are often
associated with topological invariants. Individually, their properties can be
probed in different experiments [31, 32, 33, 34, 35, 36]. For example, the
$\pi$-Berry flux of a Dirac cone has been measured through atomic momentum-
space interferometry in optical lattices [32], and the winding of isolated
linear or quadratic band touchings has been observed via exciting atoms by
moving them through the node [34]. Upon being elevated to multi-gap
topologies, the hallmark gained by band singularities is non-Abelian frame
charges that can induce a nontrivial Euler class. Consequently, a fundamental
question arises whether there exists observable signatures of these non-
Abelian charges, their braiding and this invariant.

We here address this question, providing an essential link to experiments.
Whilst frame charges correspond to $\pm\pi$ vortices within a gap,
distinguishing them is difficult because it amounts to discriminating between
zero and $2\pi$ phase windings of the frame after braiding. We demonstrate
that these two cases impose certain constructive or destructive interference
of atoms passing through different Euler nodes. Secondly, we construct another
protocol for sequential excitation of atoms by moving them through two band
nodes consecutively where the frame charges can be detected in the phase shift
of resulting oscillations in band populations. Our results reveal a physical
manifestation of non-Abelian charges and their relative sign changes upon
braiding in momentum space, imposing a feasible route to directly observe
signatures of multi-gap band topology in experiments.

Non-Abelian Frame Charges and Euler Class.— For a real Hamiltonian, the
eigenstates $|u_{n}(\bm{k})\rangle$ at quasimomentum $\bm{k}$ span an
orthonormal triad or, in general, vielbein (’frame’). The space of
spectralized Hamiltonians is given by $O(N)/O(1)^{N}=O(N)/\mathbb{Z}_{2}^{N}$
for $n\in N$-bands, where $N>2$. The quotient ensures that flipping the signs
of the eigenvectors does not change the Hamiltonian. Band singularities
obstruct a unique assignment of the eigenstates and correspond to a
$\pi$-rotation between the eigenstates that host the node (see Fig. 1b).
Namely, the frame rotation accumulates a $\pi$ Berry phase, where band nodes
act as the analogues of $\pi$ disclination vortices in biaxial nematics [37,
38]. While within the two-band subspace (a single ‘gap’) the nodes act as
$\pi$-rotations, where it should be noted that $2\pi$ rotations in fact
correspond to $-1$ as $\pi_{1}(SO(N))=\mathbb{Z}_{2}$, accumulated frame
charges anticommute with those in the adjacent gaps. Focusing on the case of
three bands, the frame charges associated to the nodes
$\pi_{1}(SO(3)/D_{2})=Q$ take values in the quaternion group $Q=\\{\pm 1,\pm
i,\pm j,\pm k\\}$ [37]. As a result, braiding band nodes in momentum space
(which can be achieved e.g. using stress/strain [17, 39], temperature effects
[40] or periodic driving [29]), converts their charges and can ensure that a
specific band subspace hosts similarly valued charges. In two dimensions, the
resulting obstruction to annihilate these charges (nodes) between states $n$
and $(n+1)$ is quantified by the Euler class [17],

$\chi_{n,n+1}{[\mathcal{D}]}=\dfrac{1}{2\pi}\left[\int_{\mathcal{D}}\mathrm{Eu}~{}dk_{1}\wedge
dk_{2}-\oint_{\partial\mathcal{D}}\mathcal{A}\cdot
d\boldsymbol{k}\right]\in\mathbb{Z},$ (1)

evaluated over any patch $\mathcal{D}$ encapsulating these nodes. Here, we
define the Euler form
$\mathrm{Eu}=\langle\partial_{k_{1}}u_{n}(\boldsymbol{k})|\partial_{k_{2}}u_{n+1}(\boldsymbol{k})\rangle-\langle\partial_{k_{2}}u_{n}(\boldsymbol{k})|\partial_{k_{1}}u_{n+1}(\boldsymbol{k})\rangle$
and associated connection one-form, $\mathcal{A}=\langle
u_{n}(\boldsymbol{k})|\boldsymbol{\nabla}u_{n+1}(\boldsymbol{k})\rangle$.

The braiding and Euler class can effectively be captured by tracking Dirac
strings [29]. These are gauge objects connecting pairs of nodes in each gap
(see Fig. 1) and represent the line across which the sign of the eigenstates
change due to the $\pi$-Berry phase induced by the band nodes. Crossing a
Dirac string (DS) residing in an adjacent gap changes the sign of the frame
charge, effectively encoding braiding rules of the band node charges.

Model.— As a concrete example, we employ a
$\mathcal{C}_{2}\mathcal{T}$-symmetric Kagome lattice, while our results
indeed apply to any system admitting an Euler description. The Hamiltonian
diagonal in momentum space is written as

$H(\bm{k})=-2J\sum_{\beta\neq\beta^{\prime}}\cos{\left(\bm{k}\cdot\bm{d}_{\beta\beta^{\prime}}\right)}c^{\dagger}_{\beta}c^{\phantom{\dagger}}_{\beta^{\prime}}+\sum_{\beta}\Delta_{\beta}c^{\dagger}_{\beta}c^{\phantom{\dagger}}_{\beta},$
(2)

with nearest-neighbour hopping of amplitude $J$ along the three directions
$\bm{d}_{\beta\beta^{\prime}}$ connecting three sublattices $\beta\in(A,B,C)$,
with the annihilation (creation) operators $c^{(\dagger)}_{\beta}$ [29]. For
vanishing sublattice offsets $\Delta_{\beta}$, the spectrum features two
linear band touching points which carry opposite frame charges (e.g. $\pm i$,
depicted with empty/filled markers in Fig. 1b), hence a vanishing patch Euler
class. The quadratic band touching point in gap 2 harbours $\chi=1$,
corresponding to two same-valued frame charges which we can assign to be $+j$
111The frame-charge assignment is a gauge choice, where only the relative sign
per gap is physically meaningful.. While these quaternion frame charges act as
$\pm\pi$ Berry fluxes in a given gap, conventional interferometry methods are
insensitive to their signs and, hence, cannot a priori reveal the relative
frame charges or their change after braiding.

Figure 1: a) Kagome lattice band structure for $\Delta_{\beta}=0$. b) Linear
band nodes $(K,K^{\prime})$ carry opposite non-Abelian frame charges $(\pm
i)$, and hence can annihilate ($\chi_{1,2}=0$ in the shaded area). They act as
$\pm\pi$ fluxes in gap 1, captured by the DS connecting them. The $\Gamma$
node is formed by two similar-valued frame charges $(+j)$ that are obstructed
to annihilate $(\chi_{2,3}=1)$. The orthonormal frame of the eigenstates
(arrows) rotates around a band node, accumulating a $\pi$-phase jump at the
DS. c) A wavepacket moving through a node at an angle $\theta$, and finishing
in a coherent superposition [Eq. (Interferometry of non-Abelian band
singularities and Euler class topology)] due to the frame rotation by an angle
$\phi$.

Method 1: Interferometry.— We employ a coherent superposition in the bands
forming a topological singularity (i.e. the Euler subspace) to deduce the
frame rotation around it relative to other nodes in this gap, hence, their
relative frame charges. Consider moving a wavepacket in momentum space in band
1 $(\mathinner{|{u_{1,\mathrm{in}}(\bm{k})}\rangle})$, which enters and exits
a band node at some angle $\theta$ as depicted in Fig. 1c [34]. The frame of
band eigenstates is rotated by an angle $\phi=\pm\theta/2$ between the in-
going and out-going paths 222For a uniform and symmetric Hamiltonian, the
frame-rotation angle winds smoothly $\phi=\pm\theta/2\in[0,\pi]$ for the
interferometry angle completing a circle $\theta\in[0,2\pi]$. Here $\pm$
indicates the direction of winding, hence the sign of the frame charge in the
fixed gauge. In general, $\phi$ might change non-uniformly while still
rotating by $\pi$ in total, see SM [44] for more elaboration.. The atoms,
therefore, are in a coherent superposition after passing through the node
(adiabatically with respect to other bands) as,

$\displaystyle\mathinner{|{u_{1,\mathrm{in}}}\rangle}$
$\displaystyle\to\mathinner{|{u_{1,\mathrm{out}}}\rangle}\mathinner{\langle{u_{1,\mathrm{out}}|u_{1,\mathrm{in}}}\rangle}+\mathinner{|{u_{2,\mathrm{out}}}\rangle}\mathinner{\langle{u_{2,\mathrm{out}}|u_{1,\mathrm{in}}}\rangle}$
$\displaystyle=\cos{\phi}\mathinner{|{u_{1,\mathrm{out}}}\rangle}-\sin{\phi}\mathinner{|{u_{2,\mathrm{out}}}\rangle},$
(3)

where the resulting probability to be in bands
$\mathinner{|{u_{1,\mathrm{out}}}\rangle}$ and
$\mathinner{|{u_{2,\mathrm{out}}}\rangle}$ is expressed in terms of the frame
rotation $\phi$ corresponding to the overlap.

When the system is characterized by an Euler class within a fixed gauge, the
relative phase of the component excited into the second band depends on the
chirality of the frame winding, i.e. the full dreibein that is naturally
encoded by the Hamiltonian evolution. By analyzing two such excitation events,
we devise an interferometry of the Euler nodes to extract the relative non-
Abelian frame charges and, hence, patch Euler class (1). We shall first
present the idea in the simple setting of a Kagome model by considering the
$(K,K^{\prime})$ nodes in between bands 1 and 2, where the $\mathcal{C}_{6}$
symmetry eliminates the complicating effects of dynamic phases, isolating the
key physics at play. We will subsequently discuss more general settings,
including the effects of DSs and dynamic phases.

Figure 2:  a,b) Interferometry to distinguish relative frame charges of nodes
in Kagome lattice. Up- and down-spin atoms are moved along paths I and II,
respectively, and recombined. Opposite (a) [same (b)] frame charges result in
a single spin [mixture of both spins] as in Table. 1. In (b2), interferometry
paths enclose an additional node, crossing a DS (blue line). c) Illustration
of the general interferometry. Geometric phases
$\Delta\alpha_{n}-\Delta\alpha_{n+1}=0$ at the close of the loop indicates
oppositely charged nodes, which will annihilate if brought together within the
shaded area, i.e., vanishing patch Euler class.

We consider a wavepacket of a single spin state
$\mathinner{|{\uparrow}\rangle}$ localised in a single band
$(\mathinner{|{u_{1}(\bm{k}_{0})}\rangle})$, starting from $\bm{k}_{0}$ at
equal distance to the targeted Euler nodes. Employing a $\pi/2$-pulse, a
superposition of
$(\mathinner{|{\uparrow}\rangle}+\mathinner{|{\downarrow}\rangle})/\sqrt{2}$
can be created, where the pseudo-spin can be encoded by, e.g., hyperfine
states [31, 32]. As demonstrated in Fig. 2, we split the atoms such that
$\mathinner{|{\uparrow}\rangle}$ and $\mathinner{|{\downarrow}\rangle}$ atoms
follow symmetric paths I and II, passing through two nodes with an angle
$\theta$ and $-\theta$ respectively before recombining. In optical lattices,
this may be achieved by using a combination of an applied magnetic field
gradient and lattice acceleration [32, 31].

Assuming that the frame-rotation angle for the node I is $\phi$, the atoms
following this path finish in the state
$\big{(}c\mathinner{|{u_{1}}\rangle}-s\mathinner{|{u_{2}}\rangle}\big{)}\mathinner{|{\uparrow}\rangle}$
as in Eq. (Interferometry of non-Abelian band singularities and Euler class
topology), where $c\equiv\cos{\phi}$ and $s\equiv\sin{\phi}$ and the momentum
index is suppressed for simplicity. If the node on path II carries the same
charge (in the chosen gauge), the frame features the same chirality on both
nodes. However, since the path winds in the opposite direction ($-\theta$),
the frame-rotation angle experienced on path II is $-\phi$. Conversely, if the
nodes host opposite charges, then path II corresponds to a frame-rotation
angle $\phi$ 333We consider these interferometry angles to be such that the
magnitude of frame rotation angle is the same on both paths, which can be
controlled e.g. using the techniques in Ref. [34] even for reduced lattice
symmetries..

Therefore, considering the paths together, the final state becomes

$\frac{1}{\sqrt{2}}\bigg{[}\big{(}c\mathinner{|{u_{1}}\rangle}-s\mathinner{|{u_{2}}\rangle}\big{)}\mathinner{|{\uparrow}\rangle}+\big{(}c\mathinner{|{u_{1}}\rangle}\pm
s\mathinner{|{u_{2}}\rangle}\big{)}\mathinner{|{\downarrow}\rangle}\bigg{]},$
(4)

where the $+(-)$ for down-spins on path II corresponds to same (opposite)
charges. Applying another $\pi/2$ pulse before closing the interferometry then
yields
$c\mathinner{|{u_{1}}\rangle}\mathinner{|{\uparrow}\rangle}-s\mathinner{|{u_{2}}\rangle}\mathinner{|{\downarrow}\rangle}$
for similarly-charged nodes, resulting in a mixture of both spin populations,
$p_{\uparrow}=\cos^{2}{\left(\phi\right)}\mathrm{~{}and~{}}p_{\downarrow}=\sin^{2}{\left(\phi\right)}$.
On the other hand, for oppositely-charged nodes, the final state
$c\mathinner{|{u_{1}}\rangle}\mathinner{|{\uparrow}\rangle}-s\mathinner{|{u_{2}}\rangle}\mathinner{|{\uparrow}\rangle}$
consists only of up spins $p_{\uparrow}=1$. We demonstrate our interferometry
technique along different paths in Fig. 2(a,b) with resulting populations
given in Table 1, where the relative non-Abelian frame charges are indeed
distinguished.

Paths | Final state | $p_{\uparrow}$ | $p_{\downarrow}$ | Charges
---|---|---|---|---
(a1) | $c\mathinner{|{u_{1}}\rangle}\mathinner{|{\uparrow}\rangle}-s\mathinner{|{u_{2}}\rangle}\mathinner{|{\uparrow}\rangle}$ | $1$ | $0$ | opposite
(a2) | $c\mathinner{|{u_{1}}\rangle}\mathinner{|{\uparrow}\rangle}-s\mathinner{|{u_{2}}\rangle}\mathinner{|{\uparrow}\rangle}$ | $1$ | $0$ | opposite
(b1) | $c\mathinner{|{u_{1}}\rangle}\mathinner{|{\uparrow}\rangle}-s\mathinner{|{u_{2}}\rangle}\mathinner{|{\downarrow}\rangle}$ | $\cos^{2}{\theta/2}$ | $\sin^{2}{\theta/2}$ | same
(b2) | $c\mathinner{|{u_{1}}\rangle}\mathinner{|{\downarrow}\rangle}-s\mathinner{|{u_{2}}\rangle}\mathinner{|{\uparrow}\rangle}$ | $\sin^{2}{\theta/2}$ | $\cos^{2}{\theta/2}$ | same
Table 1: Interferometry results for paths depicted in Fig. 2(a,b). While
opposite frame charges result in a single spin species, similar charges yield
a mixture.

General configurations.— Our interferometry technique readily caters to more
complex band structures. The dynamic phases along the two paths considered
above (Fig. 2) are equal under $\mathcal{C}_{6}$ symmetry and cancel each
other (see Supplemental Material (SM) [44]). In general when the nodes of
interest are not positioned at high-symmetry points, dynamic phases must be
accounted for. This can be achieved e.g. by measuring band energies along the
interferometry with standard techniques such as band mapping [45, 31], while
we present another method based on performing a reference interferometry loop
with twice the acceleration in the SM [44].

Furthermore, there can be Dirac strings traversed by the interferometry loop,
which change the sign of the relevant band eigenstates, leading to an
additional $\pi$ geometric phase. While the DS crossed twice in Fig. 2a yields
no net effect as in Table. 1 444Equivalently, the Dirac string can be deformed
past the node along path I and omitted, the single crossing on path I in Fig.
2b induces a $\pi$ phase shift between the spins and switches $p_{\uparrow}$
and $p_{\downarrow}$, but still entails a spin mixture for the similar-valued
frame charges probed. We here give a general proof for generic DS
configurations and show that our interferometry scheme determines whether the
nodes can annihilate if brought together along the paths that close the
interferometry loop (see Fig. 2c). Given the Euler class, this therefore
reveals the relative frame charges and patch invariant (1) within this region.

We consider two Euler nodes between bands $n$ and $n+1$, and focus on the
geometric phases $\alpha_{n(n+1)}^{\text{I/II}}$ in each band along the paths
I and II, where the dynamic phases can be accounted for as before if required.
We now analyze the effect of all possible DS configurations:

1\. Right before the atoms starting in band $n$ enter the nodes, the phase
difference between the two paths
$\Delta\alpha_{n}\equiv\alpha_{n}^{\text{II}}-\alpha_{n}^{\text{I}}$ can be
$0$ or $\pi$, depending on any DSs between the initial point and the nodes as
depicted in Fig. 2c. These have no effect as it will be carried over to the
contributions in both bands on that path below [44].

2\. Just after exiting the nodes, the phase of the excited component
$\alpha^{\text{I/II}}_{n+1}$ on the two arms depends on the frame charges,
where we emphasize that the gauge is naturally fixed by the Hamiltonian
evolution. If the charges are opposite as in Fig. 2c, the total phase
difference between the two paths are the same for both bands
$\Delta\alpha_{n+1}-\Delta\alpha_{n}=0$, while there will be a sign change
$|\Delta\alpha_{n+1}-\Delta\alpha_{n}|=\pi$ for same charges.

3\. When the wavepackets are brought together avoiding other nodes, there can
be three types of DSs crossed; within the Euler subspace and in the adjacent
gaps, as shown in Fig. 2c, which result in changes in $\Delta\alpha_{n(n+1)}$
summarized in Table 2. Most importantly, these DSs would also alter the
relative sign of the targeted frame charges if they were to be brought
together themselves along the closing paths. Indeed, we see that each time
$\Delta\alpha_{n+1}-\Delta\alpha_{n}$ changes by $\pi$, the relative charge of
the nodes changes.

4\. The nodes’ obstruction to annihilate within a region of the Brillouin zone
is captured by the patch Euler class (1). We therefore conclude that the patch
Euler class in the area enclosing the second half of the interferometry [i.e.
post-node branches] vanishes $\chi_{n,n+1}=0$ (is finite, $\chi_{n,n+1}=1$) if
$|\Delta\alpha_{n+1}-\Delta\alpha_{n}|=0\,(\pi)$. Corresponding spin
populations (after a $\pi/2$-pulse upon closing the interferometry [44]) are
given in Table 3.

We here note that evaluating the Euler class is superfluous for two nodes
separated by a reciprocal lattice vector. While we illustrate the relative
charges using the simplest Kagome setting in Fig. 2b, our scheme applies to
generic situations. We validate this by employing a Kagome model with next-
nearest-neighbour tunneling terms [20] which host several more singularities
of various charges, see SM [44]. Furthermore, under linear periodic driving,
this model features an anomalous Euler phase [29] where the non-trivial patch
invariant arises by virtue of a DS in the anomalous Floquet gap, which we
confirm to be successfully captured by our interferometry [44].

DS between bands | Effect on $\Delta\alpha_{n}$, $\Delta\alpha_{n+1}$ | Relative charge
---|---|---
$n$ and $n+1$ | Both change by $\pi$ | No change
$n$ and $n-1$ |  $\Delta\alpha_{n}$ by $\pi$ | Changes
$n+1$ and $n+2$ |  $\Delta\alpha_{n+1}$ by $\pi$ | Changes
Table 2: Possible DS crossings (Fig. 2c) after the targeted nodes, and corresponding effects on the phase difference in each band and frame charges if the nodes were to be moved.  $\Delta\alpha_{n}$ | $\Delta\alpha_{n+1}$ | $p_{\uparrow}$ | $p_{\downarrow}$ | Charges
---|---|---|---|---
$0$ | $0$ | $1$ | $0$ | opposite
$\pi$ | $\pi$ | $0$ | $1$ | opposite
$0$ | $\pi$ | $\cos^{2}{\phi}$ | $\sin^{2}{\phi}$ | same
$\pi$ | $0$ | $\sin^{2}{\phi}$ | $\cos^{2}{\phi}$ | same
Table 3: Spin populations resulting from phase differences between the two
paths in bands $n$ and $n+1$. Figure 3: a) Illustration of the consecutive
deflection method. Along the path joining the nodes, a relative dynamic phase
$\alpha^{\prime}$ is developed between the bands. b) Starting from band $n=1$,
the final population $p_{1}$ in this band oscillates as a function of
$\theta_{1}$, which shifts right (left) for increasing $\theta_{2}$ values if
the frame charges are same (opposite), here demonstrated in Kagome lattice for
$(K,K^{\prime})$ nodes (2). c) The phase shift direction is deduced from
Eqs.(5,6), after determining $\cos{\alpha^{\prime}}<0$ at double acceleration
with $\theta_{2}=\pi/2$.

Method 2: Consecutive Deflection.— Our second method does not require an
interferometry loop and offers the advantage of not increasing in complexity
when symmetries are removed. This relies on a single wavepacket entering and
exiting the two targeted nodes between bands $n$ and $n+1$ consecutively as
depicted in Fig. 3a. Starting in band $n$, the final population $p_{n}$ in
this band depends on the frame-rotation angles $\phi_{1}$ and $\phi_{2}$, as
well as the relative dynamic phase $\alpha^{\prime}$ developed in between the
nodes, which a priori complicates deducing the frame charges 555Any relative
dynamic phases appearing along the pre/post-node sections are irrelevant since
they will not affect the populations.. We circumvent this by expressing it
after some algebra [44] as
$p_{n}=\left[A\cos{\left(2\phi_{1}-\beta\right)}+1\right]/2$, where

$\displaystyle
A^{2}=\cos^{2}{\left(2\phi_{2}\right)}+\cos^{2}{\left(\alpha^{\prime}\right)}\sin^{2}{\left(2\phi_{2}\right)},$
(5)
$\displaystyle\tan{\beta}=-\cos{\left(\alpha^{\prime}\right)}\tan(2\phi_{2}).$
(6)

Namely, the band population varies with $\phi_{1}$ sinusoidally with an
amplitude $A$ and phase $\beta$. Crucially, the phase shift in Eq. (6) depends
on whether the frame winds in the same/opposite direction around the second
node $(\phi_{2})$, corresponding to relative same/opposite frame charges,
provided that $\cos{\left(\alpha^{\prime}\right)}$ is known. The latter can be
extracted from the amplitude (5), which reveals only
$\cos^{2}(\alpha^{\prime})$, by performing the experiment at double
acceleration to halve the dynamic phase. Obtaining
$\cos^{2}{\left(\alpha^{\prime}/2\right)}$ thus yields
$\cos{\alpha^{\prime}}=2\cos^{2}{\left(\alpha^{\prime}/2\right)}-1$.
Alternatively, the dynamic phase can be also directly calculated from energy
measurements.

We numerically demonstrate this method by using the $(K,K^{\prime})$ nodes in
Fig. 3 and other nodes in more complicated settings in SM [44], where
$p_{n=1}$ oscillates as a function of the first turning angle
$\theta_{1}=2\phi_{1}$. Repeating the consecutive deflection for different
turning angles in node 2 induces a phase shift, direction of which reveals the
relative frame charges targeted. Here, for $\cos\alpha^{\prime}<0$ extracted
from the amplitude (5), oscillations shift to right (left) for similar
(opposite) frame charges 666For $\cos\alpha^{\prime}>0$, these shifts are
reversed.

Discussion.— We have presented two interferometric schemes to extract non-
Abelian frame charges of band degeneracies of real-valued Hamiltonians, which
require control on lattice acceleration and path precision that have been
already demonstrated in state-of-the-art ultracold atom experiments [34, 32].
While the first method involves only two measurements –the actual
interferometry and in general a reference loop if dynamic phases are unknown
[44] which is not necessary in the presence of lattice symmetries–, the second
method relies on several measurements for a range of $\theta_{1,2}$ values to
detect the phase shift but applies independently of underlying lattice
symmetries. We note that pseudospins in Method 1 can be employed for guiding
atoms along an interferometry loop [32]. This can be further combined with
band specific readout of atoms to extract individual terms in Table 1. Method
2 can be fine-tuned for a specific experiment as well. For example, if the
rate of frame winding around individual nodes is known (e.g. uniform under
$C_{6}$ symmetry [44]), one requires only three measurements: Two for
observing $p_{n}$ under double acceleration by choosing
$|\phi_{1}|=|\phi_{2}|=\pi/4$ and reversing the sign of $\phi_{1}$, which
yield $\sin^{2}{(\alpha^{\prime}/4)}$ and $1-\sin^{2}{(\alpha^{\prime}/4)}$ to
find $\cos^{2}{(\alpha^{\prime}/2)}$. Repeating the first measurement at
normal acceleration then discloses
$1-\cos^{2}{(\alpha^{\prime}/2)}~{}\,(\cos^{2}{(\alpha^{\prime}/2)})$ for
similar (opposite) charges.

The interferometry and consecutive deflection methods reveal the relative
frame charge along the path joining the nodes. Hence, by varying the route
connecting the nodes, our techniques pave the way for directly demonstrating
the non-Abelian and path dependent nature of braiding in experiment for the
first time. Given the essential role of such charges in constituting Euler
class and multi-gap topology [18, 17, 19], these results serve as a crucial
route towards experimentally probing multi-gap topologies in and out of
equilibrium [29].

###### Acknowledgements.

Acknowledgments— O.B. thanks James Walkling and Calvin Hooper for productive
discussions. We acknowledge discussions with Dan Stamper-Kurn and Adrien
Bouhon. O.B. acknowledges funding from the Clarendon Fund, Merton College, and
a Leverhulme Studentship. R.-J. S. acknowledges funding from a New
Investigator Award, EPSRC grant EP/W00187X/1, a EPSRC ERC underwrite grant
EP/X025829/1, and a Royal Society exchange grant IES/R1/221060 as well as
Trinity College, Cambridge. F.N.Ü. acknowledges funding from the Marie
Skłodowska-Curie programme of the European Commission Grant No 893915, Trinity
College Cambridge, and thanks the Aspen Center for Physics for their
hospitality, where this work was partially funded by a grant from the Sloan
Foundation.

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  * Note [1] The frame-charge assignment is a gauge choice, where only the relative sign per gap is physically meaningful.
  * Note [2] For a uniform and symmetric Hamiltonian, the frame-rotation angle winds smoothly $\phi=\pm\theta/2\in[0,\pi]$ for the interferometry angle completing a circle $\theta\in[0,2\pi]$. Here $\pm$ indicates the direction of winding, hence the sign of the frame charge in the fixed gauge. In general, $\phi$ might change non-uniformly while still rotating by $\pi$ in total, see SM [44] for more elaboration.
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*

## Supplemental Material for “Interferometry of non-Abelian band singularities
and Euler class topology”

###  .1 I. Frame Windings, Dirac Strings, and Non-Abelian Charges

In $\mathcal{C}_{2}\mathcal{T}$ symmetric systems, the eigenvectors are real
and orthogonal, and thus at each quasimomentum form an orthonormal frame
$\\{\mathinner{|{u_{n}(k)}\rangle}\\}_{n=1}^{N}$ [17, 22]. However, each
eigenvector is only defined up to an overall sign $\pm$ (a gauge degree of
freedom). Considering a closed path, since the initial and final Hamiltonians
will be identical, the accumlated angle defines a ‘frame charge’ associated
with the path. In particular, a frame charge may be nontrivial when nodes are
enclosed. For example, for a path enclosing a single linear node between two
bands (the ‘principal bands’), the eigenvectors corresponding to the principal
bands undergo a $\pi$ rotation as we circle the node [17]. Such a $\pi$
winding means that it is not possible to define a globally smooth gauge and
there is a line across which the gauge switches so the sign of these two bands
changes. This line, similar to a branch cut, is called a Dirac string [18],
and is demonstrated in Fig. 1. A Dirac string may either terminate on a node
(joining a pair of nodes) or may cross the entire Brillouin zone (e.g. as in
the anomalous Floquet phases introduced in Ref. [29]) after nodes are
recombined over the Brillouin zone edge.

The sign of (the quaternion charge of) the node refers to the direction in
which the eigenvectors wind. While these signs can be changed through a gauge
transformation, relative charge in a band subspace can be defined by smoothly
fixing the gauge between the nodes. For two similarly-charged nodes in a given
gap, the net frame winding around a path encircling them is $2\pi$, indicating
an obstruction to the ability of these nodes to annihilate. For oppositely
charged nodes, the net winding is $0$, and the nodes may annihilate. The
relative frame charges can be path dependent as demonstrated in Fig. S4, in
which two nodes between the principal bands
($\mathinner{|{u_{1}}\rangle},\mathinner{|{u_{2}}\rangle}$) are labelled by
$\pm i$, and one ‘adjacent’ node by $+k$. Suppose that within the enclosed
region in (a), the two $i$ nodes have opposite charge. We now ask what the
relative charge would be for the region on the other side of the $k$ node, as
in (c). In fixing the gauge to be smooth within that region, we must move the
red Dirac string outside of this region. Pulling this string through the $-i$
node reverses the signs of
$\mathinner{|{u_{2}}\rangle},\mathinner{|{u_{3}}\rangle}$, and thus the
winding of the $\mathinner{|{u_{1}}\rangle},\mathinner{|{u_{2}}\rangle}$ frame
changes. As a result, within this region in (c) the $i$ nodes have the same
charge. We stress that the relative stability is a gauge invariant physical
quantity and even motivates a topological invariant, the patch Euler class.
Indeed, considering region $\Gamma_{a}$ in Fig. S4c the $i$ nodes still can
annihilate in this patch as one transverses the Dirac string of the adjacent
gap, a similar analysis holds for considering region $\Gamma_{c}$ in Fig. S4a.
This motivates a puzzle approach [39, 49] in which one fixes the gauge at a
patch by referring to relative stability, e.g. on region $\Gamma_{a}$ and
build up the global topological configuration using that the frame charge
changes sign when crossing the Dirac strings residing in adjacent gaps as well
as combination rules for Dirac strings [29].

Figure S4: Demonstration of how the relative charge of two nodes depends on
the region considered. Nodes and Dirac strings between bands 1 and 2 are in
blue and labelled by $i$, those between bands 2 and 3 are in red and labelled
by $k$. a) In the region $\Gamma_{a}$ the charges have opposite charge, with a
patch Euler class $\chi=0$, or equivalently the winding of the frame along
$\Gamma_{a}$ is $0$. b) Pulling the Dirac strings through the node in the
other gap change their charge, resulting in a smoothly defined gauge within
$\Gamma_{c}$. c) In the region $\Gamma_{c}$ the nodes have the same charge and
thus $\chi=1$. The frame winds by $2\pi$ along $\Gamma_{c}$ as a manifestation
of the non-Abelian nature of the band nodes. We emphasise that each subfigure
is the same physical situation, but each with a different gauge choice.

Another closely-related perspective is that of braiding, see Refs. [17, 29].
By tuning parameters of the Hamiltonian, the positions of nodes can be
adjusted. Moving a principal node around an adjacent node changes the charges
of the nodes, which can be interpreted as due to each node crossing the Dirac
string associated with the other. If two nodes in a given gap have opposite
charge, they can annihilate; but if the nodes have the same charge they
cannot, and will typically ‘bounce’ [17]. Due to the braiding, this ability to
annihilate will be path dependent. If the nodes are brought together within
$\Gamma_{a}$ in Fig. S4 they can annihilate, but within $\Gamma_{c}$ they
cannot.

The above labelling of the nodes with quaternion charges $i,j,k$ is not
unique. Consider nodes only in a 3-band subspace, and suppose we circle a node
between bands 1-2 and another between bands 2-3. Composing a $\pi$ rotation of
$\mathinner{|{u_{1}}\rangle},\mathinner{|{u_{2}}\rangle}$ followed by a $\pi$
rotation of $\mathinner{|{u_{2}}\rangle},\mathinner{|{u_{3}}\rangle}$, is in
fact equivalent to a $\pi$ rotation of
$\mathinner{|{u_{3}}\rangle},\mathinner{|{u_{1}}\rangle}$ [29]. Considering
different pairs of bands to get equivalent relations, and noting that a
trivial or $2\pi$ winding in any gap is equivalent, we can see that the set of
charges can be described by the quaternion group with charge composition
described by quaternion multiplication rules. More explicitly, we define the
charges of nodes e.g. of bands 1-2 as $\pm i$ (sign depending on direction of
winding), bands 2-3 as $\pm k$, bands 3-1 as $\pm j$, the trivial winding as 1
and the $2\pi$ winding as -1. Then charge composition is reflected by

$i^{2}=j^{2}=k^{2}=-1,~{}ij=k=-ji,$ (S1)

since $\pi_{1}(SO(3)/D_{2})=Q$. We emphasise that both the assignment of
$i,j,k$ to each gap is a gauge choice, in addition to defining the signs of a
given charge, but their relative charge when compared with a consistent gauge
is a physically meaningful quantity. The ability to annihilate within a region
is captured by the patch Euler class over the region $\mathcal{D}$

$\chi_{n,n+1}{[\mathcal{D}]}=\dfrac{1}{2\pi}\left[\int_{\mathcal{D}}\mathrm{Eu}~{}dk_{1}\wedge
dk_{2}-\oint_{\partial\mathcal{D}}\mathcal{A}\cdot
d\boldsymbol{k}\right]\in\mathbb{Z},$ (S2)

where the Euler form
$\mathrm{Eu}=\langle\partial_{k_{1}}u_{n}(\boldsymbol{k})|\partial_{k_{2}}u_{n+1}(\boldsymbol{k})\rangle-\langle\partial_{k_{2}}u_{n}(\boldsymbol{k})|\partial_{k_{1}}u_{n+1}(\boldsymbol{k})\rangle$
and associated connection one-form $\mathcal{A}=\langle
u_{n}(\boldsymbol{k})|\boldsymbol{\nabla}u_{n+1}(\boldsymbol{k})\rangle$.

###  .2 II. Details for Method 1: Interferometry

Here, we demonstrate explicitly how the wavefunction evolves along the
interferometry paths and give the general form of the final wavefunction. We
then describe in detail how the dynamic phases may be accounted for, to
extract information about the geometric phases.

Along path I, the wavefunction evolves into

$\left(ce^{i\alpha_{n}^{I}}\mathinner{|{u_{n}}\rangle}-se^{i\alpha_{n+1}^{I}}\mathinner{|{u_{n+1}}\rangle}\right)\mathinner{|{\uparrow}\rangle},$
(S3)

where $c\equiv\cos{\phi_{I}},s\equiv\sin{\phi_{I}}$ and $\phi_{I}$ is the
angle by which the frame rotates between ingoing and outgoing paths (see Fig.
1(c)). The phase $\alpha_{n}^{I}$ consists of dynamic and geometric
contributions (which are changes by either $0$ or $\pi$) and from the pre- (i)
and post-node (ii) sections of the path,
$\alpha_{n}^{I}=\alpha_{n}^{I}(i)+\alpha_{n}^{I}(ii)$, as demonstrated in Fig.
S5. $\alpha_{n+1}^{I}$ consists of the phase developed in band $n$ before the
node, and the dynamic/geometric phases accumulated in band $n+1$ after the
node, $\alpha_{n+1}^{I}=\alpha_{n}^{I}(i)+\alpha_{n+1}^{I}(ii)$. Along path II
we similarly have the final wavefunction

$\left(ce^{i\alpha_{n}^{II}}\mathinner{|{u_{n}}\rangle}-se^{i\alpha_{n+1}^{II}}\mathinner{|{u_{n_{1}}}\rangle}\right)\mathinner{|{\downarrow}\rangle}.$
(S4)

Here, the relative directions of frame winding $\phi_{II}=\pm\phi_{I}$ leads
to a relative sign in the component excited at the node. Different than the
main text, we can here assume that this frame winding is encompassed in the
final phase $\alpha^{II}_{n+1}$ for compactness.

Applying a $\pi/2$-pulse which maps
$\mathinner{|{\uparrow}\rangle}\to\frac{1}{\sqrt{2}}(\mathinner{|{\uparrow}\rangle}+\mathinner{|{\downarrow}\rangle})$
and
$\mathinner{|{\downarrow}\rangle}\to\frac{1}{\sqrt{2}}(\mathinner{|{\uparrow}\rangle}-\mathinner{|{\downarrow}\rangle})$,
and upon recombining the wavepackets we obtain

$\frac{1}{2}\mathinner{|{\uparrow}\rangle}\left[c\left(e^{i\alpha_{n}^{I}}+e^{i\alpha_{n}^{II}}\right)\mathinner{|{u_{n}}\rangle}-s\left(e^{i\alpha_{n+1}^{I}}+e^{i\alpha_{n+1}^{II}}\right)\mathinner{|{u_{n+1}}\rangle}\right]+\frac{1}{2}\mathinner{|{\downarrow}\rangle}\left[c\left(e^{i\alpha_{n}^{I}}-e^{i\alpha_{n}^{II}}\right)\mathinner{|{u_{n}}\rangle}-s\left(e^{i\alpha_{n+1}^{I}}-e^{i\alpha_{n+1}^{II}}\right)\mathinner{|{u_{n+1}}\rangle}\right].$
(S5)

We find that the populations of each spin and each band are

$\displaystyle p_{n,\uparrow}$
$\displaystyle=c^{2}\cos^{2}{\frac{\Delta\alpha_{n}}{2}},$ (S6) $\displaystyle
p_{n+1,\uparrow}$ $\displaystyle=s^{2}\cos^{2}{\frac{\Delta\alpha_{n+1}}{2}},$
(S7) $\displaystyle p_{n,\downarrow}$
$\displaystyle=c^{2}\sin^{2}{\frac{\Delta\alpha_{n}}{2}},$ (S8) $\displaystyle
p_{n+1,\downarrow}$
$\displaystyle=s^{2}\sin^{2}{\frac{\Delta\alpha_{n+1}}{2}}.$ (S9)

In Fig. S5 we demonstrate how the wavefunction evolves along each path, both
in the general case and for a specific example (in which we do not display
dynamic phases for simplicity). In Fig. S5b, we find that
$\alpha_{n}^{I}=\pi$, $\alpha_{n+1}^{I}=0$, $\alpha_{n}^{II}=\pi$, and
$\alpha_{n_{1}}^{II}=\pi$. Therefore
$\Delta\alpha_{n+1}-\Delta\alpha_{n}=\pi-\pi=0$, indicating that along this
path joining the nodes, the singularities have opposite frame charges and can
annihilate.

In the wavefunctions above, we have assumed that the magnitude of the frame
rotation angle $\phi$ is the same at both nodes, $|\phi_{I}|=|\phi_{II}|$. By
studying the winding about a single node using e.g. the techniques of Ref.
[34], the turning angles $\theta$ at each node can be chosen such that this is
satisfied, making our interferometry generically applicable even under reduced
lattice symmetries.

Figure S5:  Evolution of the wavefunction along the paths taken during
interferometry (note that in the gauge chosen the nodes have opposite charge).
a) A general example demonstrating how phases accumulate, where dashed lines
indicate that there may be some set of complicated Dirac string crossings. In
particular, notice that the phases accumulated by band $n$ in the first half
of the path (labelled by (i) for the phases) carries over to the excited
component. b) A specific example configuration of DSs. In this case,
$\Delta\alpha_{n+1}-\Delta\alpha_{n}=0$, indicating opposite charges if they
were to be brought together along the closing paths. In (b), dynamic phases
are not included.

####  .2.1 Accounting for Dynamic Phases

Whilst the phases above consist of both a dynamic part $\alpha^{\prime}$ and a
geometric part $\alpha=\alpha^{geo}$,
$\Delta\alpha_{n}^{\mathrm{tot}}=\Delta\alpha_{n}+\Delta\alpha_{n}^{\prime}$,
our goal here is to extract the geometric part only. Here, we discuss how this
can be achieved. Upon performing the interferometry, the final spin-up
population is [Eqs. (S6,S7)]

$p_{\uparrow}=\cos^{2}{(\phi)}\cos^{2}{\left(\Delta\alpha_{n}^{\mathrm{tot}}/2\right)}+\sin^{2}{(\phi)}\cos^{2}{\left(\Delta\alpha_{n+1}^{\mathrm{tot}}/2\right)}.$
(S10)

By performing this interferometry at two (known) values of $\phi$, or by
measuring spin populations separately in bands $n$ and $n+1$ [45], this allows
for $\cos^{2}{\left(\Delta\alpha_{n}^{\mathrm{tot}}/2\right)}$ and
$\cos^{2}{\left(\Delta\alpha_{n+1}^{\mathrm{tot}}/2\right)}$ to be determined.
We now seek to determine the geometric contribution, which may be either $0$
or $\pi$, to these phases.

We here construct a ‘reference interferometry loop’ technique which relies on
repeating the interferometry with twice the acceleration, halving the time for
the total process. As a result, the dynamic phase contribution is halved,
whilst the geometric phase remains fixed, therefore determining
$\cos^{2}{\left(\Delta\alpha^{\prime}_{n(+1)}/4+\Delta\alpha_{n(+1)}/2\right)}$.

The problem is then reduced to determining whether $\alpha=0$ or $\pi$ when we
have $x=\cos^{2}\left(\alpha^{\prime}/2+\alpha/2\right)$ and
$y=\cos^{2}\left(\alpha^{\prime}/4+\alpha/2\right)$ (we here drop the $\Delta$
and subscripts for clarity). This can be achieved e.g. by calculating
$x-(2y^{2}-1)^{2}=\sin{\left(\alpha/2\right)}\sin{\left(\alpha^{\prime}+3\alpha/2\right)}$.
Therefore when $\alpha=0$, $x-(2y^{2}-1)^{2}=0$, and generally this quantity
will be non-zero when $\alpha=\pi$. We note that when
$\alpha^{\prime}=\pm\pi/2$, this will yield an ‘accidental zero’. This can be
resolved by performing the reference interferometry with four times the
acceleration, yielding $z=\cos^{2}\left(\alpha^{\prime}/8+\alpha/2\right)$,
and
$y-(2z^{2}-1)^{2}=\sin{\left(\alpha/2\right)}\sin{\left(\alpha^{\prime}/2+3\alpha/2\right)}$
can be calculated. If this is also $0$, then one can be certain that the
geometric phase $\alpha=0$; if we previously had an ‘accidental zero’, then we
expect $1/\sqrt{2}$. Indeed, this set of measurements provides a reasonably
robust answer. This is because
$\mathrm{min}(\mathrm{abs}(x-(2y^{2}-1)^{2})+\mathrm{abs}(y-(2z^{2}-1)^{2}))=1/\sqrt{2}$
when $\alpha=\pi$, providing a significant ‘gap’ to the value expected for
$\alpha=0$ (i.e. $0$).

In this manner, the geometric contributions $\Delta\alpha_{n}$ and
$\Delta\alpha_{n+1}$ can be determined, and used to find the relative charge
of the nodes.

###  .3 III. Details for Method 2: Consecutive Deflection

Suppose that we start in a state purely in band $\mathinner{|{u_{n}}\rangle}$.
When we evolve this in and out of a node at an angle such that the frame has
rotated by $\phi_{1}$, upon exiting the node the state is given by

$\cos{\phi_{1}}\mathinner{|{u_{n}}\rangle}-\sin{\phi_{1}}\mathinner{|{u_{n+1}}\rangle}.$
(S11)

If we now evolve this state adiabatically towards the second node, a relative
dynamic phase $\alpha^{\prime}$ will appear between the bands, such that the
state (up to some overall phase which can be omitted) just before entering the
second node is

$e^{i\alpha^{\prime}}\cos{\phi_{1}}\mathinner{|{u_{n}}\rangle}-\sin{\phi_{1}}\mathinner{|{u_{n+1}}\rangle}.$
(S12)

Upon entering and exiting this second node with a frame rotation angle
$\phi_{2}$, we have (denoting $\cos{\phi_{i}}=c_{i},\sin{\phi_{i}}=s_{i}$)

$\left[e^{i\alpha}c_{1}c_{2}-s_{1}s_{2}\right]\mathinner{|{u_{n}}\rangle}-\left[e^{i\alpha}c_{1}s_{2}+s_{1}c_{2}\right]\mathinner{|{u_{n+1}}\rangle}.$
(S13)

The final population in band $n$ is therefore

$n_{1}=|e^{i\alpha}c_{1}c_{2}-s_{1}s_{2}|^{2}=\cos^{2}{\left(\phi_{1}+\phi_{2}\right)}+\sin{2\phi_{1}}\sin{2\phi_{2}}\sin^{2}{\left(\alpha/2\right)}.$
(S14)

With some algebra we can rewrite this in a more transparent form

$\displaystyle n_{1}$
$\displaystyle=\frac{1}{2}\left[\cos{\left(2\phi_{1}+2\phi_{2}\right)}+1+2\sin{2\phi_{1}}\sin{2\phi_{2}}\sin^{2}{\left(\alpha/2\right)}\right]$
(S15)
$\displaystyle=\frac{1}{2}\left[\cos{2\phi_{1}}\cos{2\phi_{2}}-\cos{\alpha}\sin{2\phi_{1}}\sin{2\phi_{2}}+1\right]$
(S16)
$\displaystyle=\frac{1}{2}\left[A\cos{\left(2\phi_{1}-\beta\right)}+1\right]$
(S17)

where

$\displaystyle
A^{2}=\cos^{2}{\left(2\phi_{2}\right)}+\cos^{2}{\left(\alpha\right)}\sin^{2}{\left(2\phi_{2}\right)},$
(S18) $\displaystyle\tan{\beta}=-\cos{\left(\alpha\right)}\tan(2\phi_{2})$
(S19)

as given in the main text.

###  .4 IV. Numerical calculations

Here we describe the numerical validation for the time-evolution along paths
in the BZ.

When an external force $\vec{F}$ is applied to particle which otherwise has
Hamiltonian $H$, the Schrödinger equation is

$i\partial_{t}\mathinner{|{\psi}\rangle}=(H-\vec{F}\cdot\vec{r})\mathinner{|{\psi}\rangle}.$
(S20)

We now seek to write this in the basis of Bloch wavefunctions. Suppose the
initial state has definite (quasi) momentum $\bm{k}$

$\mathinner{|{\psi}\rangle}=e^{i\bm{k}\cdot\vec{r}}\mathinner{|{u_{k}}\rangle},$
(S21)

where $\mathinner{|{u_{k}}\rangle}$ is a superposition of bands

$\mathinner{|{u_{k}}\rangle}=\sum_{n}\alpha_{n}\mathinner{|{u_{n,k}}\rangle}.$
(S22)

The effect of the force is that $\dot{\bm{k}}=\vec{F}$ [50], such that through
an applied force we can choose the path through momentum-space $\bm{k}(t)$.
Therefore, the wavefunction will evolve as

$\mathinner{|{\psi(t)}\rangle}=e^{i\bm{k}(t)\cdot\vec{r}}\mathinner{|{u_{k(t)}}\rangle},$
(S23)

with

$\mathinner{|{u_{k(t)}}\rangle}=\sum_{n}\alpha_{n}(t)\mathinner{|{u_{n,k(t)}}\rangle}.$
(S24)

Substituting this into Eqn. (S20), we obtain a form of Schrödinger’s equation
which acts within the Hilbert space of Bloch states

$i\partial_{t}\mathinner{|{u(t)}\rangle}=H_{k(t)}\mathinner{|{u(t)}\rangle},$
(S25)

where

$H_{k}=e^{-i\bm{k}\cdot\vec{r}}He^{i\bm{k}\cdot\vec{r}}$ (S26)

is the Bloch Hamiltonian, and $\bm{k}(t)$ now simply acts as a varying
external parameter.

Defining the path of interest $\bm{k}(t)$, we solve this equation numerically
using Runge-Kutta [51], for a given starting state. The initial state is
typically a pure band eigenvector.

To confirm the results of the interferometry, we must find the phase
difference along two paths for a given band. If the final states along paths
$I$ and $II$ are given by $\mathinner{|{u_{I}}\rangle}$ and
$\mathinner{|{u_{II}}\rangle}$, and we would like to find the phase difference
of the component in band $\mathinner{|{u_{n}}\rangle}$, we evaluate the
argument of
$\mathinner{\langle{u_{I}|u_{n}}\rangle}\mathinner{\langle{u_{n}|u_{II}}\rangle}$.
For example,
$\Delta\alpha_{n+1}\Delta\alpha_{n}=\mathinner{\langle{u_{I}|u_{n}}\rangle}\mathinner{\langle{u_{n}|u_{II}}\rangle}-\mathinner{\langle{u_{I}|u_{n+1}}\rangle}\mathinner{\langle{u_{n+1}|u_{II}}\rangle}$.
The populations were calculated from these using Eqns. S6-S9. We worked
backwards from these populations, using the techniques to deal with dynamic
phases that were described in both the main text (for Method 2) and
supplementary materials (for Method 1) to recover the geometric phase
contributions. For both methods, to achieve adiabaticity we assumed timescales
of $t\gtrapprox 10^{2}J^{-1}$, which typically corresponded to a total phase
evolution $\mathcal{O}(10^{2})$.

###  .5 IV A. Results and Examples for Method 1

We performed numerical calculations to verify that the phases and band
populations are as expected from the theory. The results, in addition to
detailed explanations to clarify the physics, are presented here.

####  .5.1 Static Kagome with NN and NNN tunneling terms

First consider the paths shown in Fig. 2, for the nearest-neighbour Kagome
model introduced in the main text. We numerically find phase differences
between the paths in band 1 and band 2 as shown in Table S4, which are as
expected analytically.

Path | $\Delta\alpha_{1}$ | $\Delta\alpha_{2}$
---|---|---
(a1) | $0$ | $0$
(a2) | $0$ | $0$
(b1) | $0$ | $\pi$
(b1) | $\pi$ | $0$
Table S4: Geometric phase differences between the paths for the components in
the starting band 1, $\Delta\alpha_{1}$, and band 2, $\Delta\alpha_{2}$, for
the static nearest-neighbour Kagome model along the paths shown in Fig. 2.
Here, due to the $\mathcal{C}_{6}$ symmetry there is no difference in dynamic
phases. All phases have been confirmed numerically, and are in-line with
analytic predictions.  Figure S6: a) Location of nodes in the Brillouin zone
for the Kagome lattice with next-nearest-neighbour hopping. Blue (triangular)
nodes are between bands 1 and 2, whilst red (circular) nodes are between bands
2 and 3. The central $\Gamma$ node is also between bands 2 and 3. Here, the
next-nearest neighbour hopping strength $t^{\prime}$ is related to the
nearest-neighbour hopping $J$ through $t^{\prime}=0.71875J$. b) and c)
Possible paths for interferometry of the nodes between bands 1 and 2. d)
Possible paths for a consecutive deflection approach to determining relative
charge.

We further studied interferometry of nodes for a Kagome lattice including
next-nearest-neighbour (NNN) hopping terms of amplitude $t^{\prime}=0.71875J$
(see supplementary information of Ref. [20]). In Fig. S6, we demonstrate the
location of the nodes in the Brillouin zone for these parameters, with many
linear nodes in both gaps. We perform interferometry along paths between the
points labelled in Fig. S6, and tabulate the results (confirmed numerically)
in Table S5. The paths were also numerically verified in the reverse
direction, but results are not included in the table. Note that, similar to
the nearest-neighbour-only case, the $\mathcal{C}_{6}$-symmetry means that
dynamic phases along each of the chosen paths cancel.

Path | $\Delta\alpha_{1}$ | $\Delta\alpha_{2}$
---|---|---
$\mathrm{A}\to\mathrm{C}$ | $0$ | $0$
$\mathrm{A}\to\mathrm{D}$ | $0$ | $\pi$
$\mathrm{B}\to\mathrm{C}$ | $0$ | $0$
$\mathrm{B}\to\mathrm{D}$ | $0$ | $\pi$
$\mathrm{E}\to\mathrm{F}$ | $0$ | $0$
$\mathrm{E}\to\mathrm{G}$ | $0$ | $\pi$
$\mathrm{E}\to\mathrm{H}$ | $\pi$ | $0$
Table S5: Geometric phase differences between the paths for the components in
the starting band 1, $\Delta\alpha_{1}$, and band 2, $\Delta\alpha_{2}$, for
the Kagome model including next-nearest-neighbour hopping between the points
shown in Fig. S6. These results, alongside those for the reversed paths, have
been numerically confirmed.

It is worth lingering on these results briefly, since they demonstrate
important points. First (for points shown in Fig. S6), notice that starting at
A compared to B makes no difference - here, this is because the Dirac string
is between bands 2 and 3 whereas we only have a component of the wavefunction
in band 1 at this point. However, had we started in band 2, the Dirac string
would have led to a $\pi$ phase shift along one of the paths, changing
$\Delta\alpha_{1}$ and $\Delta\alpha_{2}$ by $\pi$ (since this phase shift
would carry through to the component excited in band 1). This wouldn’t change
the value of $\Delta\alpha_{2}-\Delta\alpha_{1}$, as we would expect by the
arguments in the main text. Physically, this is because the technique measures
the patch Euler class along the legs of the interferometry after the node.

Secondly, compare the effect of ending at C to ending at D. When ending at D,
a Dirac string between bands 2 and 3 is crossed, leading to a $\pi$ phase
shift in band 2. This corresponds to the fact that the patch Euler class of
the nodes is $0$ in a region enclosing the path to C, but has magnitude $1$ in
a region enclosing the path to D. This is a directly equivalent to the non-
Abelian braiding of a node in the gap $1-2$ around a node in gap $2-3$, and
physically means that the nodes would annihilate if brought together at C, but
would not annihilate if brought together at D. These arguments also apply to
ending the path at points F compared to G.

Finally, compare ending at G to ending at H. When we end at H a Dirac string
between bands 1 and 2 is crossed, giving a $\pi$ phase shift to the components
in both bands. Importantly, this does not affect
$\Delta\alpha_{2}-\Delta\alpha_{1}$, which physically corresponds to the fact
that braiding around a node in the same gap has no effect.

####  .5.2 Anomalous Floquet Phases

As a final numerical check of the interferometry technique, we look at
anomalous Floquet phases recently introduced in Ref. [29]. While out-of-
equilibrium settings in general entail intriguing topological phenomena such
as topologically protected linking structures [52, 53, 24, 54], periodically
driven systems have been attracting a lot of attention in recent years with
the anomalous Floquet topological phases that does not exist at equilibrium
[13, 14, 55, 56]. With multi-gap topological considerations, periodic driving
has been recently shown as a versatile tool to move the band singularities
within the BZ and braid them [29]. Most interestingly, it has been also
predicted that by braiding with band nodes appearing in the anomalous gap at
the edge Floquet BZ (FBZ) one can obtain an anomalous Euler phase and a
anomalous Dirac string phase where all gaps in the Floquet spectrum including
the anomalous one feature either obstructed Euler nodes or Dirac strings [29].
These novel phases therefore can be obtained solely under periodic driving and
by braiding band singularities in this out-of-equilibrium setting.

We here first consider a weak linear periodic driving that breaks the
$\mathcal{C}_{6}$ symmetry of the Kagome lattice and splits the $\Gamma$ node
into two linear nodes, which we dub (i) the ‘Split Gamma’ phase. Secondly, we
consider (ii) the ‘Anomalous Euler’ phase, whereby intricate braiding of nodes
through the anomalous Floquet gap results in (K,K’) nodes carrying a patch
Euler class of magnitude $1$ [29]. The DSs in the BZ are shown in Fig. S7,
including interferometry paths to study relative charges. Importantly, the
fact that these phases no longer possess $\mathcal{C}_{6}$ symmetry means that
the dynamic phases do not cancel in the the chosen paths, and we must take
account of these in studying the results.

To deal with the dynamic phases numerically, we employ two approaches. First,
we calculate directly the dynamic phase by integrating the band energies along
the paths and subtract it at the end to leave only the geometric contribution.
We also work directly from the calculated populations, using the reference
interferometry technique described in Sec.II ”Accounting for Dynamic Phases”
chapter, to extract the geometric phase difference. Both methods are
numerically confirmed to allow for determining the correct geometric phase
difference.

Figure S7: Interferometry paths in momentum space for phases induced by
Floquet driving. (a) is the ‘Split Gamma’ phase whereby the driving has split
the $\Gamma$ node into two linear nodes. (b) is the anomalous Euler phase,
where braiding means that the K-nodes now have non-zero patch Euler class.

For the ‘Split Gamma’ phase, if we start in band 2 and follow the path drawn
in Fig. S7(a), we obtain $\Delta\alpha_{2}=0$ and $\Delta\alpha_{3}=\pi$,
indicating a patch Euler class of magnitude 1, as expected. For the ‘Anomalous
Euler’ phase, starting in band 1 and following the path in Fig. S7(b), we
obtain $\Delta\alpha_{1}=0$ and $\Delta\alpha_{2}=\pi$. This latter result can
be inferred from Fig. S7(ii) in two possible ways. In the gauge that has been
chosen for the diagram, the nodes have opposite charge. However, if we pull
the Dirac string between bands 2 and 3 through the node on (say) the left, the
left node will change sign. In this new gauge it is transparent that the nodes
have opposite sign, and the phase differences follow directly. On the other
hand, thinking about the interferometry in the gauge in the diagram is also
illuminating. The lower path crosses the Dirac string before reaching the
node, and since we are in band 1 this therefore has no effect (similarly to
the next-nearest-neighbour Kagome, we can also reason that if we had started
in band 2, this would like to a $\pi$ phase shift in both components). The
upper path, however, crosses the Dirac string after the node, and so the
component in band 2 experiences a $\pi$ phase shift here. We therefore have
$\Delta\alpha_{2}-\Delta\alpha_{1}=\pi$, indicating a patch Euler class of
magnitude 1.

###  .6 IV B: Results and Examples for Method 2

In this section we present corroborating numerical results for Method 2. In
particular, we take the opportunity to step-by-step demonstrate how our
techniques work when the winding is not uniform, and emphasise the distinction
between the frame-rotation angle $\phi$ and the path rotation angle $\theta$.

We have confirmed the technique for nearest-neighbour Kagome in Fig. 3 for
similarly- and opposite-charged nodes (K and K’). We first note that the value
of $\cos{\alpha^{\prime}}$ extracted from the ‘double acceleration’ plots
(Fig. 3c) matches up with dynamic phases found by direct integration of
energies along the paths, as well as the shifts in the oscillations (Fig. 3b)
are quantitatively as expected. In this case, the frames wind uniformly
($\phi=\pm\theta$) and thus choosing $\theta=\pi/2$ fixes $|\phi|=\pi/4$ as
required.

We subsequently confirmed the consecutive deflection technique with the next-
nearest-neighbour Kagome model, shown in Fig. S6. The nodes between bands 2
and 3 wind uniformly, and thus are simple to investigate (e.g. along paths
such as Fig. S6(d)(I)). However, the nodes between bands 1 and 2 do not wind
uniformly and therefore more care must be taken. We now demonstrate step by
step how this can be accounted for in studying path Fig. S6(d)(II).

Step 1: We first must determine the path angles $\theta$ that lead to frame
rotation angles $|\phi|=\pi/4$. To do this, we use the methods of Ref. [34].
Taking the path which joins the nodes as the fixed point, study the population
changes upon entering and exiting each node at an angle. This yields the
winding shown in Fig. S8. From this data, we can read off the values of
$\theta$ for which the population is $0.5$, corresponding to $|\phi|=\pi/4$.
In this case, we obtain $\theta_{1}=2.43$ or $\theta_{1}=-0.71$. Repeating
this for the other node, we have $\theta_{2}=2.43$ (we do not need the
negative option here).

Figure S8: The population in band 1 after entering and exiting the node at an
angle $\theta_{1}$ relative to the path joining the nodes, which corresponds
to the frame rotation angle $\phi_{1}$ through $p_{1}=\cos^{2}\phi_{1}$. This
allows the value of $\theta_{1}$ to be chosen such that the frame rotation
angle $|\phi|=\pi/4$. Notice that the winding is not uniform, but the net
winding is $\pi$ as it is a topologically protected value.

Step 2: For both these values of $\theta_{1}$, and the fixed $\theta_{2}$, we
prepare a wavepacket in band 1 and accelerate it along the required path with
an acceleration $2a$ (where $a$ must be chosen such that the evolution is
adiabatic), before measuring the final population in band 1. For the two
values of $\theta_{1}$ we obtain $p_{1}=0.214$ and $p_{1}=0.786$. The square
of the difference of these values gives us the dynamic phase
$\cos^{2}{\alpha/2}=0.326$.

Step 3: Finally, we set $\theta_{1}=2.43$ and $\theta_{2}=2.43$ (i.e. the
$\theta$’s have the same sign), and accelerate the wavepacket along the path
with an acceleration $a$. We find that the final population in band 1 is
$0.326$, which is the same as $\cos^{2}{\alpha/2}$, and therefore we know that
along this line joining the nodes, the nodes are oppositely charged, and the
patch Euler class along this line is 0.

If we instead accelerate the wavepacket along path Fig. S6(d)(III), then we
find that the nodes have the same charge. This directly demonstrates the non-
Abelian nature of the nodes.