# Quantum geometry of bosonic Bogoliubov quasiparticles

Isaac Tesfaye<EMAIL_ADDRESS>André Eckardt<EMAIL_ADDRESS>Institut für Theoretische Physik, Technische Universität Berlin,
Hardenbergstr. 36, D-10623 Berlin, Germany

###### Abstract

Topological and geometrical features arising in bosonic Bogoliubov-de Gennes
(BdG) systems have mainly been studied by utilizing a generalized symplectic
version of the Berry curvature and related Chern numbers. Here, we propose a
symplectic quantum geometric tensor (SQGT), whose imaginary part leads to the
previously studied symplectic Berry curvature, while the real part gives rise
to a symplectic quantum metric, providing a natural distance measure in the
space of bosonic Bogoliubov modes. We propose how to measure all components of
the SQGT by extracting excitation rates in response to periodic modulations of
the systems’ parameters. Moreover, we connect the symplectic Berry curvature
to a generalized symplectic anomalous velocity term for Bogoliubov Bloch wave
packets. We test our results for a bosonic Bogoliubov-Haldane model.

_Introduction._ —Bosonic Bogoliubov-de Gennes (BBdG) Hamiltonians naturally
arise in systems of weakly interacting bosons [1, 2, 3, 4], or in photonic
systems subjected to parametric driving, generating squeezing of light [5, 6].
They also describe magnonic [7, 8, 9], phononic [10], and mechanical systems
[11]. Different from their fermionic (BCS) counterparts, BBdG systems are
diagonalized in a symplectic manner via a paraunitary transformation [2, 12].
Their topological features [13, 14] have first been studied in the context of
magnonic crystals [8, 7] by utilizing a generalized symplectic Chern number
based on a symplectic Berry curvature for Bogoliubov bands. A generalization
of the conventional bulk-boundary correspondence [15, 16] to the symplectic
case can even lead to novel chiral edge states, whose properties are
fundamentally distinct from their particle-number-conserving counterparts [17,
18, 19, 20, 21]. Recent works, building on the ten-fold Altland-Zirnbauer
classification of topological insulators [22, 23, 24, 25, 26], corroborate the
notions of the symplectic Berry curvature and Chern number [27, 28] and
provide a topological classification of quadratic BBdG systems [29, 30, 31,
32, 33, 34].

Although topological features of BBdG systems have recently received some
attention [35, 36, 21, 20, 37, 33, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48,
49, 50, 51, 52, 53, 54], the geometry of the underlying Bogoliubov state space
is still lacking a complete characterization. In particle-number-conserving
systems, the Berry curvature is related to the imaginary part of the so-called
quantum geometric tensor (QGT) [55, 56] and the real part of the QGT, known as
quantum metric, provides a natural distance measure in state space [57, 56].

Figure 1: (a) Illustration of the symplectic quantum metric providing a
distance measure $ds$ between infinitesimally close Bogoliubov modes
$\bm{w}(\bm{\lambda}_{0})$ and $\bm{w}(\bm{\lambda}_{0}+d\bm{\lambda})$ [cf.
Eq. 13]. The integrated excitation rate $\Gamma^{n}_{\text{int}}$ of a
Bogoliubov system in response to (weak) periodic perturbations is dictated by
the symplectic quantum geometry [cf. Eqs. 16 and 18]. (b) Illustration of the
transverse velocity a Bogoliubov Bloch wave packet experiences in response to
an external force, which is shown to be proportional to the symplectic Berry
curvature [cf. Eq. 20].

Here, we generalize these concepts to the non-number-conserving BBdG case and
propose a symplectic quantum geometric tensor (SQGT) 111We note that non-
hermitian generalizations of the quantum geometric tensor have also recently
been put forward [117, 118, 119], although not explicitly applied to the case
of bosonic BdG systems.. We show that the imaginary part of the SQGT is
related to the previously investigated symplectic Berry curvature and that its
real part defines a natural distance measure in the space of bosonic
Bogoliubov modes, which we dub symplectic quantum metric. Previous proposals
for measurable signatures of the topology of BBdG systems have solely focused
on probing bosonic chiral edge modes, predicted for non-zero symplectic Chern
numbers [17, 21]. Here, inspired by Tran2017,Ozawa2018,Refs. Ref. [59, 60], we
propose to measure the full SQGT from excitation rates to other Bogoliubov
modes in response to weak periodic driving [Fig. 1(a)]. As a further probe, we
also show that, analogous to the number-conserving case [61, 62, 63], the
symplectic Berry curvature naturally arises as a transverse velocity term,
when a Bogoliubov Bloch wave packet is subjected to an external force [Fig.
1(b)].

_Framework._ — Consider a BBdG Hamiltonian

$\displaystyle\hat{H}$
$\displaystyle=\sum_{i,j=1}^{N}K_{ij}\hat{a}_{i}^{\dagger}\hat{a}_{j}+\frac{1}{2}\left(G_{ij}\hat{a}_{i}^{\dagger}\hat{a}_{j}^{\dagger}+G_{ji}^{*}\hat{a}_{i}\hat{a}_{j}\right)$
$\displaystyle=\frac{1}{2}\hat{\Psi}^{\dagger}\mathcal{M}\hat{\Psi}-\frac{1}{2}\mathrm{Tr}\left\\{{K}\right\\},\
\mathcal{M}=\begin{pmatrix}K&G\\\ G^{*}&K^{*}\end{pmatrix},$ (1)

with $K_{ij}=K_{ji}^{*}\in\mathbb{C}$ (hermitian),
$G_{ij}=G_{ji}\in\mathbb{C}$ (symmetric) forming the hermitian $2N\times 2N$
matrix $\mathcal{M}$ 222Throughout this letter, we assume a
(thermodynamically) stable system, i.e., the coefficient matrix $\mathcal{M}$
is positive definite such that a diagonalization of the BdG Hamiltonian (1)
satisfying the bosonic commutation relations with strictly positive
eigenvalues is always possible [65, 2, 120] as dictated by Williamson’s
theorem [121, 122] and Nambu spinor
$\hat{\Psi}=(\hat{a}_{1}\,\cdots\,\hat{a}_{N}\,\hat{a}_{1}^{\dagger}\,\cdots\,\hat{a}_{N}^{\dagger})^{T}$
of bosonic annihilation (creation) operators $\hat{a}^{(\dagger)}_{i}$ [2, 12,
65]. The Hamiltonian can be diagonalized [2, 12] (see also supplemental
material (SM) [66]),

$\displaystyle\hat{H}=\frac{1}{2}\hat{\Phi}^{\dagger}W^{\dagger}\mathcal{M}W\hat{\Phi}-\frac{1}{2}\mathrm{Tr}\left\\{{K}\right\\}=\sum_{n=1}^{N}\omega_{n}\hat{b}_{n}^{\dagger}\hat{b}_{n}+C,$
(2)

by a Bogoliubov transformation $\hat{\Psi}=W\hat{\Phi}$, where
$\hat{\Phi}=(\hat{b}_{1}\,\cdots\,\hat{b}_{N}\,\hat{b}_{1}^{\dagger}\,\cdots\,\hat{b}_{N}^{\dagger})^{T}$
and

$\displaystyle W=\begin{pmatrix}U&V^{*}\\\ V&U^{*}\end{pmatrix},\text{ with
}\left\\{\begin{array}[]{l}W^{\dagger}\tau_{z}W=W\tau_{z}W^{\dagger}=\tau_{z},\\\
W^{\dagger}\mathcal{M}W=\Omega.\end{array}\right.$ (5)

Here $C\equiv\sum_{n}\omega_{n}/2-\mathrm{Tr}\left\\{{K}\right\\}/2$; $U$ and
$V$ are complex $N\times N$ matrices, so that
$\hat{a}_{i}=\sum_{n}[U_{in}\hat{b}_{n}+(V^{*})_{in}\hat{b}^{\dagger}_{n}]$;
and
$\Omega=\text{diag}\begin{pmatrix}\omega_{1},\dots,\omega_{N},\omega_{1},\dots,\omega_{N}\end{pmatrix}$
with real strictly positive excitation energies, $\omega_{n}>0$, assumed in
the following [64]. The transformation $W$ is paraunitary, as indicated by the
relation involving $\tau_{z}=\sigma_{z}\otimes\mathbbm{1}_{N}$, with Pauli
matrix $\sigma_{z}$ and $N\times N$ identity matrix $\mathbbm{1}_{N}$. This
ensures that the new operators $\hat{b}^{(\dagger)}_{n}$ fulfill bosonic
commutation relations. For a different ordering of the bosonic operators in
the Nambu spinors $\hat{\Psi}$ and $\hat{\Phi}$, $W$ becomes an element of the
real symplectic group $\mathrm{Sp}(2N,\mathbb{\mathbb{R}})$ [67, 68]. Using
$\tau_{z}^{2}=\mathbbm{1}_{2N}$, we identify
$W^{-1}=\tau_{z}W^{\dagger}\tau_{z}$, such that the inverse transformation
$\hat{\Phi}=W^{-1}\hat{\Psi}$ reads
$\hat{b}_{n}=\sum_{m}[(U^{\dagger})_{ni}\hat{a}_{i}-(V^{\dagger})_{ni}\hat{a}^{\dagger}_{i}]$.

_Symplectic quantum geometric tensor._ — The Bogoliubov ground state
$|\psi_{0}\rangle$ is the quasiparticle vacuum,
$\hat{b}_{n}{\lvert}\psi_{0}{\rangle}=0\,\forall n$. It is generally a multi-
mode squeezed state with a fluctuating non-zero number of $\hat{a}_{i}$ bosons
[45, 68, 17]. Although $|\psi_{0}\rangle$ may display geometrical features
itself, here we will focus on the geometrical properties of the quasiparticle
excitations on top of it [17].

The Bogoliubov modes are associated with the vectors $\bm{w}_{n}$ given by the
columns of $W=(\bm{w}^{1},\cdots,\bm{w}^{2N})$. For $n\leq N$, they describe
quasiparticles,
$\hat{b}^{\dagger}_{n}=\hat{\Psi}^{\dagger}\tau_{z}\bm{w}^{n}$, and for $N>n$
to quasiholes, $\hat{b}_{n}=-\hat{\Psi}^{\dagger}\tau_{z}\bm{w}^{n}$ [18, 45,
66]. They play a role akin to single-particle wavefunctions [17] and solve the
eigenvalue equation [66],

$\displaystyle D\bm{w}^{n}=\tilde{\Omega}_{n}\bm{w}^{n},\,\text{ with
}\,D\equiv\tau_{z}\mathcal{M}\,\text{ and
}\,\tilde{\Omega}_{n}=(\tau_{z}\Omega)_{nn},$ (6)

where $D$ is called dynamical matrix. The excitation energies read
$\tilde{\Omega}_{n}=(\tau_{z})_{nn}\omega_{n\bmod N}=s_{n}\omega_{n}$, which
are positive ($s_{n}=+1$) for quasiparticles ($n\leq N$) and negative
($s_{n}=-1$) for quasiholes ($n>N$). They also obey the generalized (Krein)
$\tau_{z}$ inner product structure
$(\bm{w}^{m})^{\dagger}\tau_{z}\bm{w}^{n}=(\tau_{z})_{mn}=\delta_{mn}s_{n}$
[69] (see SM [66]). The projector $P^{n}$ onto the $n$th Bogoliubov mode and
its orthogonal complement $Q^{n}$ are naturally defined as [7]

$\displaystyle
P^{n}=W\Gamma^{n}W^{-1}=W\Gamma^{n}\tau_{z}W^{\dagger}\tau_{z},\;Q^{n}=\mathbbm{1}_{2N}-P^{n},$
(7)

where $(\Gamma^{n})_{lm}=\delta_{nl}\delta_{lm}$. Then
$P^{n}P^{m}=\delta_{nm}P^{n}$ and $\sum_{n=1}^{2N}P^{n}=\mathbbm{1}_{2N}$ [7].
We stress that $P^{n}$ and $Q^{n}$ are not hermitian, but
$\tau_{z}X^{n}\tau_{z}=(X^{n})^{\dagger}$ for $X^{n}\in\\{P^{n},Q^{n}\\}$, a
property which is called Krein- or pseudo-hermiticity [69, 28]. Below, we
assume $\tilde{\Omega}_{n}\neq\tilde{\Omega}_{m}$ for $n\neq m$, but the
generalization to the degenerate case is straightforward.

Let the Hamiltonian (1) now depend on a set of dimensionless parameters,
$\bm{\lambda}=(\lambda^{1},\ldots,\lambda^{r})$, i.e.
$\hat{H}\equiv\hat{H}(\bm{\lambda})$. Then, motivated by the definition of the
conventional quantum geometric tensor (QGT) [55, 57], we define the symplectic
quantum geometric tensor (SQGT)

$\displaystyle\eta^{n}_{\mu\nu}=\mathrm{Tr}\left\\{{\partial_{\mu}P^{n}Q^{n}\partial_{\nu}P^{n}}\right\\},$
(8)

where $\partial_{\mu}\equiv\partial/\partial\lambda^{\mu}$. The SQGT is $U(1)$
gauge-invariant, since the projectors $P_{n}$ and $Q_{n}$ do not depend on the
phase of the modes $\bm{w}^{n}(\bm{\lambda})$. The Krein-hermiticity of
$P^{n}$ and $Q^{n}$ (7), implies that the SQGT is hermitian,
$(\eta^{n}_{\mu\nu})^{*}=\eta^{n}_{\nu\mu}$. By employing completeness,
$\sum_{n}P^{n}=\mathbbm{1}_{2N}$, and orthogonality,
$(\bm{w}^{m})^{\dagger}\tau_{z}\bm{w}^{n}=\delta_{mn}s_{n}$, the SQGT can be
expressed as [66]

$\displaystyle\eta^{n}_{\mu\nu}=s_{n}(\partial_{\mu}\bm{w}^{n})^{\dagger}\tau_{z}Q^{n}\partial_{\nu}\bm{w}^{n}.$
(9)

Moreover, can be obtained [66]

$\displaystyle\eta^{n}_{\mu\nu}=s_{n}\sum_{m\neq
n}s_{m}\frac{(\bm{w}^{n})^{\dagger}(\partial_{\mu}\mathcal{M})\bm{w}^{m}(\bm{w}^{m})^{\dagger}(\partial_{\nu}\mathcal{M})\bm{w}^{n}}{(\tilde{\Omega}_{n}-\tilde{\Omega}_{m})^{2}}.$
(10)

A first indication that the so-defined SQGT (8) is meaningful, is that its
imaginary part is proportional to the previously defined symplectic Berry-
curvature
$B_{\mu\nu}^{n}=is_{n}[(\partial_{\mu}\bm{w}^{n})^{\dagger}{\tau_{z}}(\partial_{\nu}\bm{w}^{n})-(\partial_{\nu}\bm{w}^{n})^{\dagger}{\tau_{z}}(\partial_{\mu}\bm{w}^{n})]$
[7, 8, 21],

$\displaystyle
B^{n}_{\mu\nu}=-2\mathrm{Im}[\eta^{n}_{\mu\nu}]=-2\mathrm{Im}\left[\mathrm{Tr}\left\\{{\partial_{\mu}P^{n}Q^{n}\partial_{\nu}P^{n}}\right\\}\right],$
(11)

which is real-valued and antisymmetric, $B^{n}_{\mu\nu}=-B^{n}_{\nu\mu}$. From
this quantity a symplectic (first) Chern number
$C^{n}\equiv\frac{1}{2\pi}\int_{S}B^{n}$ was defined for closed two-
dimensional parameter spaces $S$ [7, 8], which describes a generalized bulk-
boundary correspondence [17, 18, 19, 20, 21]. Like in the conventional case,
we show in Note3,Refs. Ref. [66] that the symplectic Berry curvature admits a
local conservation law, i.e., $\sum_{n}B^{n}_{\mu\nu}=0$. We, moreover, show
below that the symplectic Berry curvature describes the anomalous (transverse)
velocity induced by an external force in Bogoliubov Bloch bands.

We now show that also the real part of the SQGT,

$\displaystyle
g^{n}_{\mu\nu}=\mathrm{Re}[\eta^{n}_{\mu\nu}]=\mathrm{Re}\left[\mathrm{Tr}\left\\{{\partial_{\mu}P^{n}Q^{n}\partial_{\nu}P^{n}}\right\\}\right],$
(12)

which is real-valued and symmetric, $g^{n}_{\mu\nu}=g^{n}_{\nu\mu}$, carries
physical meaning. We dub it symplectic quantum metric. Unlike the symplectic
Berry curvature, which is only non-zero for systems with broken time-reversal
or inversion symmetry [61], the symplectic quantum metric can already be non-
zero without these constraints. It generalizes the conventional quantum metric
[57, 70, 71, 55], also known as the Fubini-Study metric [72, 73, 57, 56],
which is given by the real part of the conventional QGT and defines a natural
distance measure in the (projective) Hilbert space. The latter is employed to
describe quantum phase transitions [74, 75, 76, 77, 78, 79, 80] as well as
excitations rates and quantum fluctuations [56, 60, 81] (properties which have
been exploited to measure the quantum metric in various experimental setups
[82, 83, 84, 85, 86, 87, 88, 89]).

The symplectic quantum metric $g_{\mu\nu}$ naturally describes the distance
$ds^{2}$ between two infinitesimally close Bogoliubov modes
$\bm{w}(\bm{\lambda})$ and $\bm{w}(\bm{\lambda}+d\bm{\lambda})$,

$\displaystyle
ds^{2}=1-\left\lvert\bm{w}^{\dagger}(\bm{\lambda})\tau_{z}\bm{w}(\bm{\lambda}+d\bm{\lambda})\right\rvert^{2}=g_{\mu\nu}d\lambda^{\mu}d\lambda^{\nu},$
(13)

with summation over repeated indices. Here we first defined $ds^{2}$ via the
fidelity
$\left\lvert\bm{w}^{\dagger}(\bm{\lambda})\tau_{z}\bm{w}(\bm{\lambda}+d\bm{\lambda})\right\rvert^{2}$
between both modes corresponding to the inner-product structure described
above. The second identity follows from expanding
$\bm{w}(\bm{\lambda}+d\bm{\lambda})$ to second order in $d\bm{\lambda}$,
giving
$ds^{2}=\\{(\bm{w}^{\dagger}\tau_{z}\bm{w})[(\partial_{\mu}\bm{w})^{\dagger}\tau_{z}(\partial_{\nu}\bm{w})]-(\partial_{\mu}\bm{w})^{\dagger}\tau_{z}\bm{w}\bm{w}^{\dagger}\tau_{z}(\partial_{\nu}\bm{w})\\}d\lambda^{\mu}d\lambda^{\nu}=\eta_{\mu\nu}d\lambda^{\mu}d\lambda^{\nu}=\frac{1}{2}(\eta_{\mu\nu}+\eta_{\nu\mu})d\lambda^{\nu}d\lambda^{\mu}=g_{\mu\nu}d\lambda^{\mu}d\lambda^{\nu}$
[66].

_Relation to linear response and probing._ — We will now show that the SQGT
determines the transition rates of Bogoliubov excitations between different
modes in linear response to weak external forcing. This provides another
physical interpretation of this quantity and opens the door for measuring its
components similarly to the protocols that have been used to extract the
quantum geometry of free fermions [59, 60, 81, 90].

Assuming a quasiparticle in a superposition of (unperturbed) Bogoliubov modes,
$\bm{\psi}(t)=\sum_{n=1}^{N}\psi_{n}(t)\bm{w}^{n}(\bm{\lambda_{0}})\in\mathbb{C}^{2N}$,
its dynamics is described by the effective time-dependent Schrödinger equation
$-i\partial_{t}\bm{\psi}(t)=D(t)\bm{\psi}(t)$ [45] (see SM 333See
supplementary material, which includes Refs. [2, 12, 65, 123, 124, 67, 125,
69, 28, 45, 7, 12, 126, 57, 56, 97, 127, 59, 60, 20, 128, 129, 130, 131, 17,
132, 62, 63, 133, 61, 21, 92, 134, 135, 121, 122, 136, 102, 95], where we
provide details regarding: the paraunitary Bogoliubov transformation and the
associated indefinite inner product space, the derivation of alternative
representations of the SQGT, the relation between the symplectic quantum
metric and the infinitesimal distance measure, the local conservation law for
the symplectic Berry curvature, the derivation of the equation of motion for
Bogoliubov modes and its solution using time-dependent perturbation theory,
the application of the measurement protocol to BBdG lattice systems, the
multi-quasiparticle case in Fock space, adiabatic perturbation theory of the
equation of motion, the derivation of the symplectic anomalous velocity, the
Bogoliubov-Haldane model, and observable signatures for the original
particles.), with $D(t)$ being the dynamical matrix (6), which becomes time-
dependent in response to external forcing (external forces generally also
induce terms in the Hamiltonian that do not conserve the number of Bogoliubov
quasiparticles, which are not captured by $D(t)$, which will be discussed
below).

We assume driving in the form of a sinusoidal modulation
$\lambda^{\nu}=\lambda^{\nu}_{0}+2(A_{\nu}/\omega)\cos(\omega
t-\varphi_{\nu})$ of the system parameters around a mean value
$\bm{\lambda}_{0}$. For weak driving ($A_{\nu}/\omega\ll 1$), the resulting
time dependence of the dynamical matrix, $D(t)=D[\bm{\lambda}(t)]$, is
captured by the linear approximation $D[\bm{\lambda}(t)]\simeq
D(\bm{\lambda}_{0})+\sum_{\nu}\partial_{\nu}D(\bm{\lambda}_{0})2(A_{\nu}/\omega)\cos(\omega
t-\varphi_{\nu})$. For the sake of simplicity and the design of a probing
scheme inspired by Ozawa2018,Refs. Ref. [60], we focus on a two-parameter
drive with relative phase $\varphi$, for which

$\displaystyle D[\bm{\lambda}(t)]=$ $\displaystyle
D(\bm{\lambda}_{0})+\partial_{1}D(\bm{\lambda}_{0})2(A_{1}/\omega)\cos(\omega
t)$ $\displaystyle\pm\partial_{2}D(\bm{\lambda}_{0})2(A_{2}/\omega)\cos(\omega
t-\varphi).$ (14)

The driving amplitudes $A_{\nu}$ shall be switched on suddenly at time $t=0$,
and we assume, for simplicity, that the system is prepared in mode $n$,
$\bm{\psi}(0)=\bm{w}^{n}(\bm{\lambda}_{0})$. Applying time-dependent
perturbation theory in first order (linear response), we obtain the
probabilities

$\displaystyle p^{\pm}_{mn}(\omega,t)=$
$\displaystyle\left\lvert(\bm{w}^{m})^{\dagger}(A_{1}\partial_{1}\mathcal{M}\pm
A_{2}e^{i\varphi}\partial_{2}\mathcal{M})\bm{w}^{n}\right\rvert^{2}$
$\displaystyle\times(2\pi t/\omega^{2})\delta(\tilde{\Omega}_{mn}-\omega).$
(15)

for finding the system in state $\bm{w}^{m}$ at time $t$, where
$\tilde{\Omega}_{mn}\equiv\tilde{\Omega}_{m}-\tilde{\Omega}_{n}$. Note that
all quantities are evaluated for $\bm{\lambda}=\bm{\lambda}_{0}$, whenever the
argument $(\bm{\lambda})$ is dropped, i.e.,
$\bm{w}^{m}\equiv\bm{w}^{m}(\bm{\lambda}_{0})$ and
$\tilde{\Omega}_{m}\equiv\tilde{\Omega}_{m}(\bm{\lambda}_{0})$. The total
excitation rate $\Gamma(\omega)$ into all other Bogoliubov modes $m\neq n$ is
then given by $\Gamma^{n,\pm}(\omega)=t^{-1}\sum_{m\neq
n}s_{m}p^{\pm}_{mn}(\omega,t)$. For an initial quasiparticle state, $n\leq N$,
the sum can be restricted to $m\leq N$, since there is no quasiparticle-to-
quasihole transition for a single-quasiparticle initial state [66].

To probe the SQGT, we consider the total excitation rate integrated over all
(relevant) probing frequencies $\omega$ [59, 60],
$\Gamma^{n}_{\text{int}}=\int d\omega\,\Gamma^{n,\pm}(\omega)$. For the
modulation of a single parameter ($A_{1}\equiv A$, $A_{2}=0$), we find

$\displaystyle\Gamma^{n}_{\text{int}}=2\pi A^{2}\sum_{m\neq
n}\frac{\left\lvert(\bm{w}^{m})^{\dagger}(\partial_{1}\mathcal{M})\bm{w}^{n}\right\rvert^{2}}{s_{m}(\tilde{\Omega}_{m}-\tilde{\Omega}_{n})^{2}}=2\pi
A^{2}g^{n}_{11}.$ (16)

For the second identity, we have used the spectral representation (12) of the
SQGT, whose diagonal elements correspond to the symmetric part given by the
symplectic quantum metric, i.e. $\eta_{\nu\nu}^{n}=g_{\nu\nu}^{n}$. Thus, we
can probe these diagonal elements by measuring the frequency integrated
response to a modulation of $\lambda^{\nu}$.

Guided once more by Refs. [59, 60] for conventional systems, we now consider
the driving of both parameters $\lambda^{1}$ and $\lambda^{2}$ with two
different phase lags $\varphi$ and equal amplitudes $A_{1}=A_{2}=A$, to obtain
the frequency integrated transition rates

$\displaystyle\Gamma^{n,\pm}_{\text{int}}=2\pi A^{2}\begin{cases}g^{n}_{11}\pm
2g^{n}_{12}+g^{n}_{22},\\!\\!\\!&\,\varphi=0\\\ g^{n}_{11}\mp
B^{n}_{12}+g^{n}_{22},\\!\\!\\!&\,\varphi=\pi/2,\end{cases}$ (17)

the differences of which,

$\displaystyle\Delta\Gamma^{n}_{\text{int}}\equiv\Gamma^{n,+}_{\text{int}}-\Gamma^{n,-}_{\text{int}}=\begin{cases}\
\ 8\pi A^{2}g^{n}_{12},\\!\\!\\!&\,\varphi=0\\\ -4\pi
A^{2}B^{n}_{12},\\!\\!\\!&\,\varphi=\pi/2,\end{cases}$ (18)

are directly proportional to the real and imaginary part of the off diagonal
element $\eta^{n}_{12}$ of the SQGT, given by $g^{n}_{12}$ and
$-B_{12}^{n}/2$. In this way, all off-diagonal components
$\eta^{n}_{\mu\nu}=g^{n}_{\mu\nu}-iB^{n}_{\mu\nu}/2$ can be extracted from the
linear response of the system, allowing (in principle) for a complete
reconstruction of the SQGT from quasiparticle scattering rates.

We can generalize the above results to initial states of multiple
quasiparticles. A generic perturbation,
$V(t)=\sum_{ij}\\{A_{ij}(t)\hat{a}_{i}^{\dagger}\hat{a}_{j}+\frac{1}{2}[B_{ij}(t)\hat{a}_{i}^{\dagger}\hat{a}_{j}^{\dagger}+\text{h.c.}]\\}+\sum_{i}[C_{i}(t)\hat{a}_{i}^{\dagger}+\text{h.c.}]$,
takes the form
$V(t)=\sum_{nm}\\{A^{\prime}_{nm}(t)\hat{b}^{\dagger}_{n}\hat{b}_{m}+\frac{1}{2}[B^{\prime}_{nm}(t)\hat{b}^{\dagger}_{n}\hat{b}^{\dagger}_{m}+\text{h.c.}]\\}+\sum_{n}[C^{\prime}_{n}(t)\hat{b}_{n}^{\dagger}+\text{h.c.}]+D^{\prime}(t)$
with respect to the unperturbed Bogoliubov operators
$\hat{b}^{(\dagger)}_{n}$. Here terms $\propto A^{\prime}$ correspond to
scattering between quasiparticle (or hole) states, whereas terms $\propto
B^{\prime}$ are associated with the scattering of a quasiparticle into a
quasihole or vice versa. The terms $\propto C^{\prime}$, which correspond to
the creation of a single quasiparticle (or hole), can result from the
scattering of an $\hat{a}_{i}$ boson out of (or into) a Bose condensate, which
is described by a macroscopic wavefunction that is not in equilibrium with
respect to the perturbation. All these processes are associated with
corresponding bosonic enhancement factors. To reconstruct the full SQGT, it is
sufficient to consider the scattering of single quasiparticles. In case of
discrete energies $\omega_{n}$, this can be achieved by exploiting the fact
that different processes can be addressed selectively by appropriate driving
frequencies [91]. Starting from the ground state, one can first exploit the
$B^{\prime}$ or $C^{\prime}$ terms to populate the state $n$ and then modulate
the system resonantly with to transitions $n\to m$ of $A^{\prime}$ type to
extract information about the SQGT [Fig. 2(a)].

Figure 2: (a) Sketch of the measurement protocol: Selecting transitions by the
frequency of the drive, first the Bogoliubov mode $n$ is populated (red),
before, in a second step, a transition to another mode $m$ is induced (green).
(b) Illustration of the Bogoliubov-Haldane model. (c) Bogoliubov energy
spectrum along high-symmetry path in the first Brillouin zone (inset) for
$J_{2}=0.1J_{1}$, $\theta=\pi/2$, $\Delta=3\sqrt{3}J_{2}/2$ and different
interaction strengths $Un/J_{1}$.

_Application to Bogoliubov-Haldane model._ — As a concrete example, we study
weakly interacting bosons in a two-dimensional hexagonal tight-binding lattice
with two sublattice states, $s=A,B$, per unit cell. The single-particle terms
correspond to the Haldane model [92], as it was recently realized with
ultracold atoms in optical lattices [93, 90, 94]. They are characterized by
real (complex) tunneling matrix elements $-J_{1}$ ($-J_{2}e^{i\theta}$)
between nearest (next-nearest) neighbors, and on-site potentials $\Delta$
($-\Delta$) on $A$ ($B$) sites [Fig. 2(b)]. Due to the lattice symmetry,
quasimomentum $\bm{\mathbf{q}}=(q_{x},q_{y})$ is conserved by these processes,
which is treated as a parameter in the following. By treating a weak repulsive
on-site interaction $U$ at a filling of $n$ bosons per site in a self-
consistent Bogoliubov approximation with quantum fluctuations on top of a
Gross-Piteavskii mean-field condensate, we obtain a Bogoliubov-Haldane model,
$\hat{H}^{(B)}=\sum_{\bm{\mathbf{q}}\neq\bm{0}}\sum_{s,s^{\prime}}\hat{a}^{\dagger}_{\bm{\mathbf{q}}s}[\mathcal{H}(\bm{\mathbf{q}})+\mathcal{H}_{1}]_{ss^{\prime}}\hat{a}^{\dagger}_{\bm{\mathbf{q}}s^{\prime}}+(\hat{a}^{\dagger}_{\bm{\mathbf{q}}s}[\mathcal{H}_{2}]_{ss^{\prime}}\hat{a}^{\dagger}_{-\bm{\mathbf{q}}s^{\prime}}+\text{h.c.})$,
where $\mathcal{H}(\bm{\mathbf{q}})+\mathcal{H}_{1}$ contains the single-
particle tight-binding Hamiltonian including a mean-field interaction
contribution, and $\mathcal{H}_{2}$ stems exclusively from the interactions
[66]. The Bogoliubov spectrum is shown in Fig. 2(c).

Figure 3: Integrated and ideal symplectic quantum metric component
$g_{xx}(\bm{\mathbf{q}})$ and Berry curvature $B_{xy}(\bm{\mathbf{q}})$ for
the upper particle band of the Bogoliubov-Haldane model [66]. (a) Integrated
(left) and ideal (right) symplectic quantum metric component
$g_{xx}(\bm{\mathbf{q}})$ for $Un/J_{1}=3$ shown in 2D quasimomentum space.
(b) Same as in (a) but for the symplectic Berry curvature
$B_{xy}(\bm{\mathbf{q}})$. (c) Ideal (solid line) and integrated (crosses)
symplectic quantum metric component $g_{xx}(\bm{\mathbf{q}})$ along a high-
symmetry path (cf. inset in Fig. 2(c)) for two interactions strengths
$Un/J_{1}=0$ (non-interacting Haldane model) and $Un/J_{1}=3$. (d) Same as in
(c) but for the symplectic Berry curvature $B_{xy}(\bm{\mathbf{q}})$.
Simulation parameters: $A/J_{1}=0.01$, $J_{1}t/\hbar=10$ (final integration
time), $\hbar\omega/J_{1}\in[0.1(0.2),5.5]$ for $Un/J_{1}=3\,(0)$ with a
frequency spacing of $\delta\omega=0.05$. Other parameters are given in Fig.
2.

We test our proposal for extracting the symplectic quantum geometric tensor
via excitation rates to other Bogoliubov modes by simulating the time
evolution for initially populated Bogoliubov modes corresponding to the upper
particle band. The parameter modulation (14) is induced by periodically
shaking the Bogoliubov-Haldane model along the lattice directions $x$ and $y$.
In the reference frame co-moving with the lattice, the corresponding inertial
forces directly induce modulations of the parameters $q_{x}$ and $q_{y}$,
respectively [66]. In Figs. 3(a)-(b) we present the ideal symplectic quantum
metric $g_{xx}^{+}(\bm{\mathbf{q}})$ and Berry curvature
$B_{xy}^{+}(\bm{\mathbf{q}})$ of the upper quasiparticle for $Un/J_{1}=3$
together with the integrated rates (16),(18) obtained from the time evolution.
In Figs. 3(c)-(d) we compare the ideal and integrated values for the same
quantities along a high-symmetry path in quasimomentum space for $Un/J_{1}=0$
(corresponding to a number-conserving model) and $Un/J_{1}=3$. We find very
good agreement between the ideal and integrated symplectic quantum geometric
quantities, validating our scheme for extracting the components of the SQGT.
Results for $g^{+}_{xx}(\bm{\mathbf{q}})$ and $g^{+}_{xy}(\bm{\mathbf{q}})$
are presented in the SM [66]. Experimentally, the Bogoliubov excitation
fraction can be measured by relating the presence of Bogoliubov quasiparticles
to the quasimomentum distribution of the original particles, which can be
observed using standard time-of-flight imaging techniques [95, 96, 66].

_Symplectic anomalous velocity._ — As a final result, we now show that, like
in the number-conserving case [61, 62, 63, 97], a Bogoliubov lattice system
acquires an anomalous velocity transverse to an applied homogeneous
$\bm{\mathbf{F}}$ proportional to the symplectic Berry curvature. Consider a
quasiparticle prepared in the Bogoliubov-Bloch mode
$\bm{w}^{n}(\bm{\mathbf{q}}_{0})$. In response to the force, the quasimomentum
is translated according to
$\bm{\mathbf{q}}(t)=\bm{\mathbf{q}}_{0}+\bm{\mathbf{F}}t$, inducing an
effective modulation of the dynamical matrix $D[\bm{\mathbf{q}}(t)]$. In
first-order adiabatic perturbation theory with respect to
$\dot{\bm{\mathbf{q}}}=\bm{\mathbf{F}}$, the time evolved state is given by
[66]

$\displaystyle\bm{w}^{n}(\bm{\mathbf{q}})+i\sum_{m\neq
n}\bm{w}^{m}(\bm{\mathbf{q}})\frac{(\bm{w}^{m}(\bm{\mathbf{q}}))^{\dagger}(\nabla_{\bm{\mathbf{q}}}\mathcal{M})\bm{w}^{n}(\bm{\mathbf{q}})\cdot\bm{\mathbf{F}}}{s_{m}[\tilde{\Omega}_{n}(\bm{\mathbf{q}})-\tilde{\Omega}_{m}(\bm{\mathbf{q}})]^{2}}.$
(19)

In first order, the expectation value of the Bogoliubov velocity operator
$\bm{\mathbf{v}}(\bm{\mathbf{q}})=\nabla_{\bm{\mathbf{q}}}D(\bm{\mathbf{q}})$
then reads

$\displaystyle\langle\bm{\mathbf{v}}(\bm{\mathbf{q}})\rangle=\nabla_{\bm{\mathbf{q}}}\tilde{\Omega}_{n}(\bm{\mathbf{q}})+\bm{\mathbf{B}}_{n}(\bm{\mathbf{q}})\times\bm{\mathbf{F}},$
(20)

with
$\bm{\mathbf{B}}_{n}(\bm{\mathbf{q}})\equiv\epsilon_{\mu\nu\kappa}B^{n}_{\mu\nu}(\bm{\mathbf{q}})\bm{e}_{\kappa}$.
The first term corresponds to the slope of the occupied Bogoliubov energy band
and the second term provides a transverse velocity component proportional to
the symplectic Berry curvature, which is the symplectic generalization of the
anomalous velocity of number-conserving systems 444We note that the dynamics
of Bogoliubov Bloch wave packets have also been studied using a time-dependent
variational principle in Ref. [97], but no connection to the symplectic Berry
curvature has been made therein.. The symplectic anomalous velocity can be
measured in the semiclassical dynamics of Bogoliubov-Bloch wave packets.

_Conclusion._ — We have introduced the manifestly gauge-invariant symplectic
quantum geometric tensor (SQGT) for BBdG systems, whose imaginary part is
proportional to the previously studied symplectic Berry curvature. The real
part of the SQGT defines a symplectic quantum metric, which is shown to
provide a natural distance measure in the space of Bogoliubov modes. All
components of the symplectic quantum geometric tensor can be measured via
tracking Bogoliubov excitation rates and the symplectic Berry curvature
describes an anomalous velocity in Bogoliubov Bloch systems. State-of-the-art
experiments with bosonic ultracold atoms [99, 100, 101, 102] and nonlinear
photonic crystal arrays [103, 104, 105, 106, 107, 17, 40, 108] are well-suited
to test our proposals. For future work, it will be interesting to connect
recent topological notions for mixed quantum states [109, 110, 111] to thermal
BBdG systems and to see whether invariants as the Euler characteristic [112,
113, 114] may be generalized to the BBdG context.

###### Acknowledgements.

We thank Christof Weitenberg, Luca Asteria, Nathan Harshman and Alexander
Schnell for fruitful and insightful discussions. Part of the work has been
conducted using the QuSpin project [115, 116]. This research was funded by the
Deutsche Forschungsgemeinschaft via the Research Unit FOR 2414 under Project
No. 277974659. I. T. acknowledges support from the Studienstiftung des
deutschen Volkes.

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Supplementary Material for
“Quantum geometry of bosonic Bogoliubov quasiparticles”

###### Contents

  1. S1 Paraunitary Bogoliubov transformation from bosonic commutation relations
  2. S2 Indefinite Bogoliubov inner product space
  3. S3 Symplectic quantum geometric tensor
    1. S3A Symplectic quantum metric
    2. S3B Local conservation law for symplectic Berry curvature
  4. S4 Time-dependent Schrödinger equation for Bogoliubov modes
  5. S5 Time-dependent perturbation theory for the effective Schrödinger equation
  6. S6 Measuring the symplectic quantum geometry in lattice systems
  7. S7 Time-dependent perturbation theory for many-body Bogoliubov Fock states
    1. S7A Transitions between Bogoliubov quasiparticle excitations lead to SQGT
  8. S8 Adiabatic perturbation theory for the effective Schrödinger equation
  9. S9 Symplectic anomalous velocity
  10. S10 Bogoliubov Haldane model
    1. S10A Algorithm for constructing paraunitary Bogoliubov transformation matrix
    2. S10B SQGT from integrated Bogoliubov excitation rates
    3. S10C Signatures for the distribution of the original particles

##  S1 Paraunitary Bogoliubov transformation from bosonic commutation
relations

First, for ease of reference, we restate the bosonic BdG Hamiltonian [Eq. 1 in
the main text] here

$\displaystyle\hat{H}=\sum_{i,j=1}^{N}K_{ij}\hat{a}_{i}^{\dagger}\hat{a}_{j}+\frac{1}{2}\left(G_{ij}\hat{a}_{i}^{\dagger}\hat{a}_{j}^{\dagger}+G_{ji}^{*}\hat{a}_{i}\hat{a}_{j}\right)=\frac{1}{2}\hat{\Psi}^{\dagger}\mathcal{M}\hat{\Psi}-\frac{1}{2}\mathrm{Tr}\left\\{{K}\right\\},\
\mathcal{M}=\begin{pmatrix}K&G\\\ G^{*}&K^{*}\end{pmatrix}.$ (S1)

The hermiticity of $\hat{H}=\hat{H}^{\dagger}$ imposes $K$ to be hermitian,
i.e., $K_{ij}=K_{ji}^{*}\in\mathbb{C}$, and the bosonic commutation relations
for $\hat{a}_{i}^{(\dagger)}$ lead to $G$ being symmetric, i.e.,
$G_{ij}=G_{ji}\in\mathbb{C}$ [2, 12, 65]. In the second equality, the BdG
Hamiltonian is reformulated in terms of the bosonic Nambu spinors
$\hat{\Psi}^{(\dagger)}$,

$\displaystyle\hat{\Psi}^{\dagger}=\begin{pmatrix}\hat{a}_{1}^{\dagger}&\cdots&\hat{a}_{N}^{\dagger}&\hat{a}_{1}&\cdots&\hat{a}_{N}\end{pmatrix},\quad\hat{\Psi}=\begin{pmatrix}\hat{a}_{1}&\cdots&\hat{a}_{N}&\hat{a}_{1}^{\dagger}&\cdots&\hat{a}_{N}^{\dagger}\end{pmatrix}^{T},$
(S2)

and the $2N\times 2N$ coefficient matrix $\mathcal{M}$ consisting of $N\times
N$ matrix blocks $K$ and $G$. The bosonic operators $\hat{a}^{(\dagger)}_{i}$
in (S1) preserve the bosonic commutation relations (BCR),

$\displaystyle[\hat{a}_{i},\hat{a}_{j}]=0,\quad[\hat{a}_{i},\hat{a}^{\dagger}_{j}]=\delta_{ij},\Leftrightarrow\,[\hat{\Psi}_{i},\hat{\Psi}^{\dagger}_{j}]=(\tau_{z})_{ij},\quad\tau_{z}=\begin{pmatrix}\mathbbm{1}_{N}&0_{N}\\\
0_{N}&-\mathbbm{1}_{N}\end{pmatrix},\ $ (S3)

where in the last equation, we have reformulated the bosonic commutation
relations via the Nambu spinors (S2) with
$\tau_{z}=\sigma_{z}\otimes\mathbbm{1}_{N}$, where
$\sigma_{z}=\text{diag}\begin{pmatrix}1,-1\end{pmatrix}$ is the Pauli-z matrix
and $\mathbbm{1}_{N}$ ($0_{N}$) is the $N\times N$ identity (zero) matrix.
Now, our goal is to find a (Bogoliubov) transformation from the original set
of bosonic operators $\hat{a}^{(\dagger)}_{i}$ to a new set of bosonic
operators $\hat{b}^{(\dagger)}_{n}$ that diagonalizes the Hamiltonian (S1).

Let us first discuss the constraints on this transformation. Importantly, this
transformation must preserve the BCR (S3), i.e., the BCR must also hold for
the new bosonic operators $\hat{b}^{(\dagger)}_{n}$,

$\displaystyle[\hat{b}_{n},\hat{b}_{m}]=0,\quad[\hat{b}_{n},\hat{b}^{\dagger}_{m}]=\delta_{nm},\Leftrightarrow\
[\hat{\Phi}_{n},\hat{\Phi}^{\dagger}_{m}]=(\tau_{z})_{nm},$ (S4)

where we defined
$\hat{\Phi}=(\hat{b}_{1}\,\cdots\,\hat{b}_{N}\,\hat{b}^{\dagger}_{1}\,\cdots\,\hat{b}^{\dagger}_{N})^{T}$
and
$\hat{\Phi}^{\dagger}=(\hat{b}^{\dagger}_{1}\,\cdots\,\hat{b}^{\dagger}_{N}\,\hat{b}_{1}\,\cdots\,\hat{b}_{N})$
as the bosonic Nambu spinors (S2) for the new bosonic operators
$\hat{b}^{(\dagger)}_{n}$.

We parametrize such a general Bogoliubov transformation from
$\hat{a}^{(\dagger)}_{i}$ to $\hat{b}^{(\dagger)}_{n}$ by [68, 2, 3]

$\displaystyle\hat{a}_{i}=\sum_{n}U_{in}\hat{b}_{n}+(V^{*})_{in}\hat{b}^{\dagger}_{n},\quad\hat{a}_{i}^{\dagger}=\sum_{n}\hat{b}^{\dagger}_{n}(U^{\dagger})_{ni}+\hat{b}_{n}(V^{T})_{ni},$
(S5) $\displaystyle\hat{\Psi}=W\hat{\Phi},\
\hat{\Psi}^{\dagger}=\hat{\Phi}^{\dagger}W^{\dagger},\quad
W=\begin{pmatrix}U&V^{*}\\\ V&U^{*}\end{pmatrix},\
W^{\dagger}=\begin{pmatrix}U^{\dagger}&V^{\dagger}\\\
V^{T}&U^{T}\end{pmatrix},$ (S6)

where in the second line we have reformulated the Bogoliubov transformation
via a transformation between the original and new Nambu spinors, $\hat{\Psi}$
and $\hat{\Phi}$ respectively.

The requirement that also the $\hat{b}^{(\dagger)}_{n}$ are bosonic operators
(S4) will impose constraints on the complex $2N\times 2N$ transformation
matrix $W$, or equivalently on its constituents, the $N\times N$ complex
matrices $U$ and $V$. Using (S5) these constraints from the BCR (S3)-(S4) are
given by

$\displaystyle\delta_{ij}=$
$\displaystyle[\hat{a}_{i},\hat{a}^{\dagger}_{j}]=\sum_{n,n^{\prime}}[(U_{in}\hat{b}_{n}+(V^{*})_{in}\hat{b}^{\dagger}_{n}),(\hat{b}^{\dagger}_{n^{\prime}}(U^{\dagger})_{n^{\prime}j}+\hat{b}_{n^{\prime}}(V^{T})_{n^{\prime}j})]=\sum_{n}(U_{in}(U^{\dagger})_{nj}-(V^{*})_{in}(V^{T})_{nj}),$
(S7) $\displaystyle 0=$
$\displaystyle[\hat{a}_{i},\hat{a}_{j}]=\sum_{n,n^{\prime}}[(U_{in}\hat{b}_{n}+(V^{*})_{in}\hat{b}^{\dagger}_{n}),(U_{jn^{\prime}}\hat{b}_{n^{\prime}}+(V^{*})_{jn^{\prime}}\hat{b}^{\dagger}_{n^{\prime}})]=\sum_{n,}(U_{in}(V^{*})_{jn}-(V^{*})_{in}(U)_{jn}).$
(S8)

In matrix form, these conditions read

$\displaystyle UU^{\dagger}-(V^{*})V^{T}=\mathbbm{1}_{N},\qquad
V^{*}U^{T}-UV^{\dagger}=0_{N},$ (S9)

where $\mathbbm{1}_{N}$ ($0_{N}$) is again the $N\times N$ identity (zero)
matrix. We can also formulate the conditions for the entire transformation
matrix $W$ (S6) by using BCR in terms of the Nambu spinors (S3)-(S4)

$\displaystyle(\tau_{z})_{ij}=[\hat{\Psi}_{i},\hat{\Psi}^{\dagger}_{j}]=\sum_{m,m^{\prime}}[(W_{im}\hat{\Phi}_{m}),(\hat{\Phi}^{\dagger}_{m^{\prime}}(W^{\dagger})_{m^{\prime}j})]=\sum_{m,m^{\prime}}W_{im}(W^{\dagger})_{m^{\prime}j}[\hat{\Phi}_{m},\hat{\Phi}^{\dagger}_{m^{\prime}}]=\sum_{m,m^{\prime}}W_{im}(\tau_{z})_{mm^{\prime}}(W^{\dagger})_{m^{\prime}j}.$
(S10)

In matrix form these conditions for the full transformation matrix $W$ read

$\displaystyle\tau_{z}=W\tau_{z}W^{\dagger}=W^{\dagger}\tau_{z}W,$ (S11)

which lead to Eq. 5 in the main text. Note that we have added a second
equality in (S11) in addition to the first from (S10). To see that the second
equality holds, note that the first equality, $\tau_{z}=W\tau_{z}W^{\dagger}$,
is equivalent to $W^{\dagger}=\tau_{z}W^{-1}\tau_{z}$. Then we immediately see
that $W^{\dagger}\tau_{z}W=(\tau_{z}W^{-1}\tau_{z})\tau_{z}W=\tau_{z}$ proving
the second equality in (S11). The Bogoliubov transformation matrices $W$ which
fulfill the constraints of Eq. S11 are called _paraunitary_ matrices [2].

Conditions (S11) further allow us to obtain the useful relations for the
inverse transformation of (S6), from the new bosonic operators
$\hat{b}^{(\dagger)}_{n}$ to the original bosonic operators
$\hat{a}^{(\dagger)}_{i}$ via

$\displaystyle\hat{b}_{n}$
$\displaystyle=\sum_{i}(U^{\dagger})_{ni}\hat{a}_{i}-(V^{\dagger})_{ni}\hat{a}^{\dagger}_{i},\quad\hat{b}^{\dagger}_{n}=\sum_{i}\hat{a}^{\dagger}_{i}U_{in}-\hat{a}_{i}V_{in},$
(S12) $\displaystyle\hat{\Phi}$ $\displaystyle=W^{-1}\hat{\Psi},\
\hat{\Phi}^{\dagger}=\hat{\Psi}^{\dagger}(W^{\dagger})^{-1}=\hat{\Psi}^{\dagger}\tau_{z}W\tau_{z},$
(S13) $\displaystyle W^{-1}$
$\displaystyle=\tau_{z}W^{\dagger}\tau_{z}=\begin{pmatrix}U^{\dagger}&-V^{\dagger}\\\
-V^{T}&U^{T}\end{pmatrix},\
(W^{\dagger})^{-1}=\tau_{z}W\tau_{z}=\begin{pmatrix}U&-V^{*}\\\
-V&U^{*}\end{pmatrix}.$ (S14)

If we denote the $n$-th column of $W=(\bm{w}^{1},\cdots,\bm{w}^{2N})$ by
$\bm{w}^{n}$, we can directly see from (S12) that the new bosonic
quasiparticle operators $\hat{b}^{(\dagger)}_{n}$ can be compactly expressed
as

$\displaystyle\hat{b}_{n}$
$\displaystyle=(\bm{w}^{n})^{\dagger}\tau_{z}\hat{\Psi},\quad$
$\displaystyle\hat{b}^{\dagger}_{n}=\hat{\Psi}^{\dagger}\tau_{z}\bm{w}^{n},\quad$
$\displaystyle\text{for}\ n\leq N,\ \text{or equivalently}$ (S15)
$\displaystyle\hat{b}_{n}$
$\displaystyle=-\hat{\Psi}^{\dagger}\tau_{z}\bm{w}^{N+n},\quad$
$\displaystyle\hat{b}^{\dagger}_{n}=-(\bm{w}^{N+n})^{\dagger}\tau_{z}\hat{\Psi},\quad$
$\displaystyle\text{for}\ n>N.$ (S16)

Having discussed the constraints on the Bogoliubov transformation matrix $W$,
we turn to the diagonalization of the BdG Hamiltonian (S1). With the
Bogoliubov transformation (S6) the Hamiltonian (S1) can be written in terms of
the new bosonic quasiparticle operators $\hat{b}^{(\dagger)}_{n}$,

$\displaystyle\hat{H}=\frac{1}{2}\hat{\Phi}^{\dagger}W^{\dagger}\mathcal{M}W\hat{\Phi}-\frac{1}{2}\mathrm{Tr}\left\\{{K}\right\\}.$
(S17)

For the Hamiltonian (S1) to become diagonal in $\hat{b}^{(\dagger)}_{n}$, the
coefficient matrix $\mathcal{M}$ must become diagonal via the Bogoliubov
transformation,

$\displaystyle W^{\dagger}\mathcal{M}W=\Omega=\begin{pmatrix}\bm{\omega}&0\\\
0&\bm{\omega}\end{pmatrix},\quad\bm{\omega}=\text{diag}\begin{pmatrix}\omega_{1},\ldots,\omega_{N}\end{pmatrix},$
(S18)

where $\Omega$ is the $2N\times 2N$ diagonal matrix consisting of the $N\times
N$ diagonal matrix $\bm{\omega}$ with Bogoliubov excitation energies
$\omega_{n}$. As dictated by Williamson’s theorem [121, 122] such a
diagonalization (S18) via Bogoliubov matrices $W$ (S5) is always possible for
a positive definite coefficient matrix $\mathcal{M}$ [2, 68]. This then
eventually leads to the diagonal form of the Hamiltonian (S1)

$\displaystyle\hat{H}=\frac{1}{2}\hat{\Phi}^{\dagger}\Omega\hat{\Phi}-\frac{1}{2}\mathrm{Tr}\left\\{{K}\right\\}=\sum_{n=1}^{N}\omega_{n}\left(\hat{b}_{n}^{\dagger}\hat{b}_{n}+\frac{1}{2}\right)-\frac{1}{2}\mathrm{Tr}\left\\{{K}\right\\}=\sum_{n=1}^{N}\omega_{n}\hat{b}_{n}^{\dagger}\hat{b}_{n}+C,$
(S19)

with $C\equiv\sum_{n}\omega_{n}/2-\mathrm{Tr}\left\\{{K}\right\\}/2$. The
ground state ${\lvert}\psi_{0}{\rangle}$ of this Hamiltonian (S19) is
generally given by a multi-mode squeezed state with a non-zero number of
original $\hat{a}^{(\dagger)}_{i}$ bosons, which is also called the Bogoliubov
vacuum. This Bogoliubov vacuum is defined by the condition that it contains no
Bogoliubov quasiparticle excitations, i.e.,
$\hat{b}_{n}{\lvert}\psi_{0}{\rangle}=0$ for all $n$ [123, 124].

##  S2 Indefinite Bogoliubov inner product space

We collect some useful relations for the indefinite inner product space
spanned by the Bogoliubov modes $\bm{w}^{n}$. For a more rigorous introduction
into these concepts pertaining indefinite inner product spaces, we refer to
Bognar1974b,Refs. Ref. [125]. Given a Bogoliubov transformation matrix
$W\in\mathbb{C}^{2N\times 2N}$ which satisfies (S11)

$\displaystyle W\tau_{z}W^{\dagger}=W^{\dagger}\tau_{z}W=\tau_{z}$ (S20)

and whose $n$-th column corresponds to the Bogoliubov mode $\bm{w}^{n}$, we
can define a projector $P^{n}$ onto the $n$-th non-degenerate Bogoliubov
eigenspace (and its complement $Q^{n}$) (7)

$\displaystyle
P^{n}=W\Gamma^{n}W^{-1}=W\Gamma^{n}\tau_{z}W^{\dagger}\tau_{z}=s_{n}\bm{w}^{n}(\bm{w}^{n})^{\dagger}\tau_{z},\quad
Q^{n}=\mathbbm{1}_{2N}-P^{n},$ (S21)

with $(\Gamma^{n})_{ml}=\delta_{nm}\delta_{ml}$ being the $2N\times 2N$ matrix
whose only non-zero element is the $n$-th diagonal element being equal to $+1$
and where $s_{n}=1\,(-1)\,\text{for}\ n\leq N\,(n>N)$ for the so-called
particle (hole) states. We note that these projectors $X^{n}$ are necessarily
invariant under (local) $U(1)$ gauge transformations for the Bogoliubov modes
of the form $\bm{w}^{n}\rightarrow e^{i\phi}\bm{w}^{n}$, i.e.,
$X^{n}\rightarrow X^{n}$ for $X^{n}\in\\{P^{n},Q^{n}\\}$ (S21).

Both $P^{n}$ and $Q^{n}$ are Krein-hermitian (pseudo-hermitian)
$(X^{n})^{\dagger}=\tau_{z}X^{n}\tau_{z}$ for $X^{n}\in\\{P^{n},Q^{n}\\}$ [69,
28], which directly follows from (S20). Moreover, the projectors are
idempotent $P^{n}P^{k}=\delta_{nk}P^{n}$ and span the entire inner product
space $\sum_{n}P^{n}=\mathbbm{1}_{2N}$. This allows for a resolution of the
identity in the indefinite Bogoliubov inner product space,

$\displaystyle\mathbbm{1}_{2N}=\sum_{n=1}^{2N}P^{n}=\sum_{n=1}^{2N}s_{n}\bm{w}^{n}(\bm{w}^{n})^{\dagger}\tau_{z}=\sum_{l=1}^{N}\left(\bm{w}^{l}_{+}(\bm{w}^{l}_{+})^{\dagger}-\bm{w}^{l}_{-}(\bm{w}^{l}_{-})^{\dagger}\right)\tau_{z},$
(S22)

where in the second equality we have separated the Bogoliubov modes into so-
called particle $\bm{w}^{l}_{+}$ and hole parts $\bm{w}^{l}_{-}$ [45], which
correspond to the first $N$ and last $N$ columns of the Bogoliubov
transformation matrix $W$ respectively. Furthermore, note that the particle
and hole parts are not independent but are directly related via
$\bm{w}^{l}_{-}=\tau_{x}(\bm{w}^{l}_{+})^{*}$, with
$\tau_{x}=\sigma_{x}\otimes\mathbbm{1}_{N}$, where $\sigma_{x}$ is the Pauli-x
matrix, which can also be seen from the parametrization of $W$ (S6).

The Bogoliubov modes $\bm{w}^{n}$ adopt a generalized orthonormalization
condition [45, 7, 12, 2] following from (S20),

$\displaystyle(\bm{w}^{n})^{\dagger}\tau_{z}\bm{w}^{m}=s_{n}\delta_{nm},\,\Leftrightarrow(\bm{w}^{l}_{\pm})^{\dagger}{\tau_{z}}{\bm{w}^{k}_{\pm}}=\pm\delta_{lk},\
(\bm{w}^{l}_{\pm})^{\dagger}{\tau_{z}}{\bm{w}^{k}_{\mp}}=0.$ (S23)

Next, note that the trace of a matrix $A\in\mathbb{C}^{2N\times 2N}$ can be
evaluated in any basis which spans the entire $\mathbb{C}^{2N}$. The
Bogoliubov modes are such a basis which can be seen from the resolution of the
identity (S22). Thus, the trace of a matrix $A$ can be expressed in terms of
the Bogoliubov modes as

$\displaystyle\mathrm{Tr}\left\\{{A}\right\\}$
$\displaystyle=\sum_{n,m}\mathrm{Tr}\left\\{{P^{n}AP^{m}}\right\\}=\sum_{n,m,i}(\bm{e}_{i})^{\dagger}(P^{n}AP^{m})\bm{e}_{i}=\sum_{n,m,i}s_{n}s_{m}[(\bm{e}_{i})^{\dagger}\bm{w}^{n}(\bm{w}^{n})^{\dagger}\tau_{z}A\bm{w}^{m}(\bm{w}^{m})^{\dagger}\tau_{z}\bm{e}_{i}]$
$\displaystyle=\sum_{n,m}s_{n}s_{m}[(\bm{w}^{n})^{\dagger}\tau_{z}A\bm{w}^{m}(\bm{w}^{m})^{\dagger}\tau_{z}\big{(}\sum_{i}\bm{e}_{i}(\bm{e}_{i})^{\dagger}\big{)}\bm{w}^{n}]=\sum_{n}s_{n}[(\bm{w}^{n})^{\dagger}\tau_{z}A\bm{w}^{n}],$
(S24)

where $\bm{e}_{i}$ are the standard basis vectors in $\mathbb{C}^{2N}$. In the
last equality of (S24) we have further used that
$\mathbbm{1}_{2N}=\sum_{i}\bm{e}_{i}(\bm{e}_{i})^{\dagger}$ and the
orthonormalization condition of the Bogoliubov modes (S23).

Any vector $\bm{\psi}\in\mathbb{C}^{2N}$ can be expanded in the basis of the
Bogoliubov modes by making use of the resolution of identity (S22), such that

$\displaystyle\bm{\psi}=\sum_{j=1}^{2N}P^{n}\bm{\psi}=\sum_{n=1}^{2N}s_{n}\bm{w}^{n}(\bm{w}^{n})^{\dagger}\tau_{z}\bm{\psi}\equiv\sum_{n}c_{n}^{\psi}\bm{w}^{n},$
(S25)

where $c_{n}^{\psi}=s_{n}(\bm{w}^{n})^{\dagger}\tau_{z}\bm{\psi}$. Moreover,
since $\bm{\psi}$ lives in the indefinite inner product space, its norm is
only correctly defined via the generalized inner product
$(\bm{\psi})^{\dagger}\tau_{z}\bm{\psi}=\pm 1$, which is also called Krein
signature [126, 137]. Using the generalized orthonormality condition (S23)
this norm can consequently also be expressed as

$\displaystyle(\bm{\psi})^{\dagger}\tau_{z}\bm{\psi}=\sum_{nm}(c_{n}^{\psi})^{*}c_{m}^{\psi}(\bm{w}^{n})^{\dagger}\tau_{z}\bm{w}^{m}=\sum_{nm}(c_{n}^{\psi})^{*}c_{m}^{\psi}s_{m}\delta_{nm}=\sum_{n=1}^{2N}s_{n}\left\lvert
c_{n}^{\psi}\right\rvert^{2}=\pm 1,$ (S26)

which may be viewed as a generalization of Parseval’s identity for this
indefinite inner product space.

##  S3 Symplectic quantum geometric tensor

In addition to the manifestly gauge-invariant form of the symplectic quantum
geometric tensor
$\eta^{n}_{\mu\nu}=\mathrm{Tr}\left\\{{\partial_{\mu}P^{n}Q^{n}\partial_{\nu}P^{n}}\right\\}$
[Eq. 8 in the main text], we derive two equivalent representations of the
SQGT, which are more suitable for practical calculations. We again assume
implicitly that the BdG Hamiltonian (S1) depends on a set of $\bm{\lambda}$
parameters and all derivatives should be understood as derivates with respect
to these parameters, i.e.,
$\partial_{\mu}\equiv\frac{\partial}{\partial\lambda^{\mu}}$.

First, note that the diagonalization of the BdG Hamiltonian (S1) via the
paraunitary matrix $W$ (S6) leads to the diagonalization of the positive
definite coefficient matrix $\mathcal{M}$ (S18),
$W^{\dagger}\mathcal{M}W=\Omega$. This in turn corresponds to the generalized
eigenvalue equation [2, 12],

$\displaystyle\begin{split}\mathcal{M}W&=(W^{\dagger})^{-1}\Omega=\tau_{z}W\tau_{z}\Omega=\tau_{z}W\tilde{\Omega},\
\text{with}\
\tilde{\Omega}\equiv\tau_{z}\Omega=\begin{pmatrix}\bm{\omega}&0\\\
0&-\bm{\omega}\end{pmatrix},\end{split}$ (S27) $\displaystyle\Leftrightarrow\
DW$ $\displaystyle=W\tilde{\Omega},\ \text{with}\ D\equiv\tau_{z}\mathcal{M},$
(S28)

which rewritten in terms of the Bogoliubov modes $\bm{w}^{n}$ ($n$-th column
of $W$) reads

$\displaystyle D\bm{w}^{n}=\tilde{\Omega}_{n}\bm{w}^{n},\ \text{with}\
D=\tau_{z}\mathcal{M}=\begin{pmatrix}K&G\\\ -G^{*}&-K^{*}\end{pmatrix},$ (S29)

where $D\equiv\tau_{z}\mathcal{M}$ is the so-called dynamical matrix of the
system and $\bm{w}^{n}$ is the $n$-th Bogoliubov mode corresponding to the
excitation energy $\tilde{\Omega}_{n}=s_{n}\omega_{n\bmod\\!N}$ [7, 12]. Even
though $D$ is generally non-hermitian, the positive definiteness of
$\mathcal{M}$ ensures that the eigenvalues $\tilde{\Omega}_{n}$ are all real
[2, 68].

Using the definition of the trace via the Bogoliubov modes (S24), we see that
the SQGT
$\eta^{n}_{\mu\nu}=\mathrm{Tr}\left\\{{\partial_{\mu}P^{n}Q^{n}\partial_{\nu}P^{n}}\right\\}$
can equivalently be expressed as

$\displaystyle\eta^{n}_{\mu\nu}$
$\displaystyle=\sum_{m}s_{m}[(\bm{w}^{m})^{\dagger}\tau_{z}(\partial_{\mu}P^{n}Q^{n}\partial_{\nu}P^{n})\bm{w}^{m}]=\sum_{m}s_{m}[(\bm{w}^{m})^{\dagger}\tau_{z}\bm{w}^{n}\
(\partial_{\mu}\bm{w}^{n})^{\dagger}\tau_{z}Q^{n}(\partial_{\nu}\bm{w}^{n})(\bm{w}^{n})^{\dagger}\tau_{z}\bm{w}^{m}]$
$\displaystyle=s_{n}(\partial_{\mu}\bm{w}^{n})^{\dagger}{\tau_{z}Q^{n}}(\partial_{\nu}\bm{w}^{n}).$
(S30)

Here, in the second equality we have used that
$Q^{n}\bm{w}^{n}=(\bm{w}^{n})^{\dagger}\tau_{z}Q^{n}=0$ which directly follows
from (S21). In the final equality, the orthonormality conditions (S23) has
been used. Eq. S30 is the first equivalent representation of the SQGT.

To obtain the spectral representation of the SQGT, we first derive a
symplectic version of the Hellmann-Feynman theorem. To this end, we first
observe from eigenvalue equation (S29) that

$\displaystyle(\partial_{\mu}D)\bm{w}^{n}+D\partial_{\mu}\bm{w}^{n}=(\partial_{\mu}\tilde{\Omega}_{n})\bm{w}^{n}+\tilde{\Omega}_{n}\partial_{\mu}\bm{w}^{n},$
(S31)

Then we project a distinct ($m\neq n$) Bogoliubov mode
$(\bm{w}^{m})^{\dagger}\tau_{z}$ from the left onto (S31) and obtain with use
of the orthonormality conditions (S23) the symplectic version of the Hellmann-
Feynman theorem,

$\displaystyle({\bm{w}^{m}})^{\dagger}(\partial_{\mu}\mathcal{M}){\bm{w}^{n}}=(\tilde{\Omega}_{n}-\tilde{\Omega}_{m})({\bm{w}^{m}})^{\dagger}\tau_{z}(\partial_{\mu}\bm{w}^{n}).$
(S32)

Next, by inserting two resolutions of the identity (S22) into Eq. S30 and
using the idempotence property of the projectors $P^{n}$, i.e.,
$P^{n}P^{m}=\delta_{nm}P^{n}$ (S21), one obtains the spectral representation
of the SQGT,

$\displaystyle\eta^{n}_{\mu\nu}$
$\displaystyle=s_{n}(\partial_{\mu}\bm{w}^{n})^{\dagger}{\tau_{z}Q^{n}}(\partial_{\nu}\bm{w}^{n})=\sum_{m,m^{\prime}}s_{n}(\partial_{\mu}\bm{w}^{n})^{\dagger}{\tau_{z}P^{m}Q^{n}P^{m^{\prime}}}(\partial_{\nu}\bm{w}^{n})$
$\displaystyle=\sum_{m,m^{\prime}}s_{n}(\partial_{\mu}\bm{w}^{n})^{\dagger}{\tau_{z}P^{m}(\mathbbm{1}_{2N}-P^{n})P^{m^{\prime}}}(\partial_{\nu}\bm{w}^{n})=\sum_{m,m^{\prime}}s_{n}(\partial_{\mu}\bm{w}^{n})^{\dagger}{\tau_{z}(\delta_{mm^{\prime}}P^{m}-\delta_{nm^{\prime}}\delta_{nm}P^{m})}(\partial_{\nu}\bm{w}^{n})$
$\displaystyle=s_{n}\sum_{m}\left[(\partial_{\mu}\bm{w}^{n})^{\dagger}\tau_{z}(P^{m})(\partial_{\nu}\bm{w}^{n})\right]-s_{n}(\partial_{\mu}\bm{w}^{n})^{\dagger}\tau_{z}(P^{n})(\partial_{\nu}\bm{w}^{n})$
$\displaystyle=s_{n}\sum_{\begin{subarray}{c}m\neq n\\\
m=1\end{subarray}}^{2N}\left[(\partial_{\mu}\bm{w}^{n})^{\dagger}\tau_{z}(P^{m})(\partial_{\nu}\bm{w}^{n})\right]=s_{n}\sum_{\begin{subarray}{c}m\neq
n\\\
m=1\end{subarray}}^{2N}s_{m}\frac{(\bm{w}^{n})^{\dagger}(\partial_{\mu}\mathcal{M})\bm{w}^{m}\times(\bm{w}^{m})^{\dagger}(\partial_{\nu}\mathcal{M})\bm{w}^{n}}{(\tilde{\Omega}_{n}-\tilde{\Omega}_{m})^{2}},$
(S33)

where in the last step, the symplectic Hellmann-Feynman theorem (S32) has been
used. Eq. S33 is the spectral representation of the SQGT.

###  S3A Symplectic quantum metric

We show that the symplectic quantum metric naturally arises when considering
the distance $ds^{2}$ between two infinitesimally close Bogoliubov modes. In
analogy to Provost1980a,Kolodrubetz2017,Refs. Ref. [57, 56], we define the
distance $ds^{2}$ between nearby states $\bm{w}^{n}(\bm{\lambda})$ and
$\bm{w}^{n}(\bm{\lambda}+d\bm{\lambda})$ (omitting the index $n$ in the
following) via

$\displaystyle
ds^{2}=1-f^{2}=1-\left\lvert(\bm{w}(\bm{\lambda}))^{\dagger}{\tau_{z}}{\bm{w}(\bm{\lambda}+d\bm{\lambda})}\right\rvert^{2},$
(S34)

with the fidelity
$f=\left\lvert(\bm{w}(\bm{\lambda}))^{\dagger}{\tau_{z}}{\bm{w}(\bm{\lambda}+d\bm{\lambda})}\right\rvert$.
We first expand $\bm{w}(\bm{\lambda}+d\bm{\lambda})$ to second order in
$d\bm{\lambda}$,

$\displaystyle\bm{w}(\bm{\lambda}+d\bm{\lambda})=\bm{w}(\bm{\lambda})+\partial_{\mu}\bm{w}(\bm{\lambda})d\lambda^{\mu}+\frac{1}{2}\partial_{\mu}\partial_{\nu}\bm{w}(\bm{\lambda})d\lambda^{\mu}d\lambda^{\nu},$
(S35)

where we use the Einstein summation convention over Greek indices in the
following. We then insert this expression (S35) into (S34), using
orthonormalization conditions (S23). Collecting only terms up to second order
in $d\bm{\lambda}$ then yields

$\displaystyle ds^{2}=$ $\displaystyle
1-\Big{\\{}(\bm{w}^{\dagger}\tau_{z}\bm{w})^{2}+(\bm{w}^{\dagger}\tau_{z}\bm{w})[((\partial_{\mu}\bm{w})^{\dagger}\tau_{z}\bm{w})+(\bm{w}^{\dagger}\tau_{z}(\partial_{\mu}\bm{w}))]d\lambda^{\mu}+[((\partial_{\mu}\bm{w})^{\dagger}\tau_{z}\bm{w})(\bm{w}^{\dagger}\tau_{z}(\partial_{\nu}\bm{w}))]d\lambda^{\mu}d\lambda^{\nu}$
$\displaystyle\ \
+\frac{1}{2}(\bm{w}^{\dagger}\tau_{z}\bm{w})[((\partial_{\nu}\partial_{\mu}\bm{w})^{\dagger}\tau_{z}\bm{w})+(\bm{w}^{\dagger}\tau_{z}(\partial_{\nu}\partial_{\mu}\bm{w}))]d\lambda^{\mu}d\lambda^{\nu}\Big{\\}}+\mathcal{O}(\left\lvert
d\bm{\lambda}\right\rvert^{3})$ $\displaystyle=$
$\displaystyle\\{(\bm{w}^{\dagger}\tau_{z}\bm{w})[(\partial_{\mu}\bm{w})^{\dagger}\tau_{z}(\partial_{\nu}\bm{w})]-(\partial_{\mu}\bm{w})^{\dagger}\tau_{z}\bm{w}\
\bm{w}^{\dagger}\tau_{z}(\partial_{\nu}\bm{w})\\}d\lambda^{\mu}d\lambda^{\nu}+\mathcal{O}(\left\lvert
d\bm{\lambda}\right\rvert^{3})$ $\displaystyle=$
$\displaystyle\left[(\bm{w}^{\dagger}\tau_{z}\bm{w})(\partial_{\mu}\bm{w})^{\dagger}{\tau_{z}Q}(\partial_{\nu}\bm{w})\right]d\lambda^{\mu}d\lambda^{\nu}+\mathcal{O}(\left\lvert
d\bm{\lambda}\right\rvert^{3})$ (S36)

where the explicit parameter dependence of the states has been omitted for
clarity. In the second equality above, we have on the one hand used
$(\bm{w}^{\dagger}\tau_{z}\bm{w})^{2}=1$ (S23) and on the other hand that the
linear terms ($\sim d\lambda^{\mu}$) vanish, which follows from the relation
$(\bm{w})^{\dagger}{\tau_{z}}(\partial_{\mu}\bm{w})=-(\partial_{\mu}\bm{w})^{\dagger}{\tau_{z}}(\bm{w})$
(S23). From this relation, it moreover follows that
$((\partial_{\nu}\partial_{\mu}\bm{w})^{\dagger}\tau_{z}\bm{w})+(\bm{w}^{\dagger}\tau_{z}(\partial_{\nu}\partial_{\mu}\bm{w}))=-[(\partial_{\mu}\bm{w})^{\dagger}\tau_{z}(\partial_{\nu}\bm{w})+(\partial_{\nu}\bm{w})^{\dagger}\tau_{z}(\partial_{\mu}\bm{w})]$
which has also been used in the second equality above (S36). Finally,
comparing this expression to the definition of the symplectic quantum
geometric tensor in Eq. S30, we see that the symplectic quantum geometric
tensor (SQGT) naturally emerges as a distance measure in the space of the
Bogoliubov modes

$\displaystyle
ds^{2}=\eta_{\mu\nu}d\lambda^{\mu}d\lambda^{\nu}+\mathcal{O}(\left\lvert
d\bm{\lambda}\right\rvert^{3})=\frac{1}{2}\left(\eta_{\mu\nu}+\eta_{\nu\mu}\right)d\lambda^{\mu}d\lambda^{\nu}+\mathcal{O}(\left\lvert
d\bm{\lambda}\right\rvert^{3})=g_{\mu\nu}d\lambda^{\mu}d\lambda^{\nu}+\mathcal{O}(\left\lvert
d\bm{\lambda}\right\rvert^{3}).$ (S37)

More precisely, since the distance measure $ds^{2}$ (S36) is a symmetric
bilinear form, the antisymmetric part of the SQGT does not contribute to
$ds^{2}$, i.e., only the symmetric part, i.e., the symplectic quantum metric
$g_{\mu\nu}=\frac{1}{2}\left(\eta_{\mu\nu}+\eta_{\nu\mu}\right)=\mathrm{Re}[\eta_{\mu\nu}]$,
determines the distance measure between two infinitesimally close Bogoliubov
states, hence, the second equality in Eq. S37.

###  S3B Local conservation law for symplectic Berry curvature

We show that the symplectic Berry curvature, admits a local conservation law
over all subspaces (Bogoliubov subspaces), including both the particle and
hole subspaces [7]. For this, note that using Eq. S30, the symplectic Berry
curvature of the subspace $n$,
$B_{\mu\nu}^{n}=-2\mathrm{Im}\,[\eta^{n}_{\mu\nu}]$ can be written as

$\displaystyle
B_{\mu\nu}^{n}=is_{n}((\partial_{\mu}\bm{w}^{n})^{\dagger}{\tau_{z}}(\partial_{\nu}\bm{w}^{n})-(\partial_{\nu}\bm{w}^{n})^{\dagger}{\tau_{z}}(\partial_{\mu}\bm{w}^{n})),$
(S38)

where we omit the explicit parameter dependence for clarity. Next, inserting
the resolution of the identity (S22) into (S38), we see that we may also write
$B_{\mu\nu}^{n}$ as

$\displaystyle B_{\mu\nu}^{n}$
$\displaystyle=is_{n}\left\\{\sum_{m}s_{m}(\partial_{\mu}\bm{w}^{n})^{\dagger}{\tau_{z}}{\bm{w}^{m}}(\bm{w}^{m})^{\dagger}{\tau_{z}}(\partial_{\nu}\bm{w}^{n})-(\partial_{\nu}\bm{w}^{n})^{\dagger}{\tau_{z}}(\partial_{\mu}\bm{w}^{n})\right\\}$
$\displaystyle=is_{n}\left\\{\sum_{m}\left[s_{m}(\partial_{\nu}\bm{w}^{m})^{\dagger}{\tau_{z}}{\bm{w}^{n}}(\bm{w}^{n})^{\dagger}{\tau_{z}}(\partial_{\mu}\bm{w}^{m})\right]-(\partial_{\nu}\bm{w}^{n})^{\dagger}{\tau_{z}}(\partial_{\mu}\bm{w}^{n})\right\\},$
(S39)

where in the second equality we used
$(\bm{w}^{m})^{\dagger}{\tau_{z}}(\partial_{\nu}\bm{w}^{n})=-(\partial_{\nu}\bm{w}^{m})^{\dagger}{\tau_{z}}(\bm{w}^{n})$
twice, which itself simply follows from Eq. S23. Now, by summing Eq. S39 over
all subspaces $n$,

$\displaystyle\sum_{n}B_{\mu\nu}^{n}$
$\displaystyle=i\left\\{\sum_{m}\left[s_{m}(\partial_{\nu}\bm{w}^{m})^{\dagger}{\tau_{z}}\left(\sum_{j}s_{n}{\bm{w}^{n}}(\bm{w}^{n})^{\dagger}{\tau_{z}}\right)(\partial_{\mu}\bm{w}^{m})\right]-\sum_{j}s_{n}(\partial_{\nu}\bm{w}^{n})^{\dagger}{\tau_{z}}(\partial_{\mu}\bm{w}^{n})\right\\},$
$\displaystyle=i\left\\{\sum_{m}\left[s_{m}(\partial_{\nu}\bm{w}^{m})^{\dagger}{\tau_{z}}(\partial_{\mu}\bm{w}^{m})\right]-\sum_{j}\left[s_{n}(\partial_{\nu}\bm{w}^{n})^{\dagger}{\tau_{z}}(\partial_{\mu}\bm{w}^{n})\right]\right\\}=0,$
(S40)

we immediately see that the symplectic Berry curvature admits a local
conservation law for all parameter values $\bm{\lambda}$,

$\displaystyle\sum_{n}B^{n}_{\mu\nu}(\bm{\lambda})=0,\ \forall\bm{\lambda}.$
(S41)

##  S4 Time-dependent Schrödinger equation for Bogoliubov modes

Starting from the Heisenberg equations of motions (EOMs) for the original
bosonic operators, we derive an effective time-dependent single-particle
Schrödinger equation for the Bogoliubov modes $\bm{w}^{n}$. Using Eq. S1, we
see that the Heisenberg EOM ($\hbar\equiv 1$ throughout) for the original
bosonic operators read

$\displaystyle i\partial_{t}\hat{a}_{m}$
$\displaystyle=[\hat{a}_{m},\hat{H}]=\sum_{j}K_{mj}\hat{a}_{j}+\frac{1}{2}[\hat{a}_{j}^{\dagger}G_{jm}+\hat{a}^{\dagger}_{j}G_{jm}],$
(S42) $\displaystyle i\partial_{t}\hat{a}_{m}^{\dagger}$
$\displaystyle=[\hat{a}_{m}^{\dagger},\hat{H}]=-\sum_{j}\hat{a}_{j}^{\dagger}K_{jm}-\frac{1}{2}[G_{mj}^{*}\hat{a}_{j}+G_{mj}^{*}\hat{a}_{j}].$
(S43)

These EOMs can also be compactly written in terms of the bosonic Nambu spinor
$\hat{\Psi}=(\hat{a}_{1},\dots,\hat{a}_{N},\hat{a}_{1}^{\dagger},\dots,\hat{a}_{N}^{\dagger})^{T}$
[12, 65]

$\displaystyle i\partial_{t}\hat{\Psi}=[\hat{\Psi},\hat{H}]=D\hat{\Psi},\
\text{with}\ D=\tau_{z}\mathcal{M},$ (S44)

where $D$ is again the dynamical matrix (S29) and $\mathcal{M}$ is the
coefficient matrix of the BdG Hamiltonian consisting of matrix blocks $K$ and
$G$ (S1). Eq. S44 is easily solved by

$\displaystyle\hat{\Psi}(t)=e^{-iDt}\hat{\Psi}(0)$ (S45)

which itself leads to the following time-dependence for the Nambu spinor of
the new bosonic operators $\hat{\Phi}(t)=W^{-1}\hat{\Psi}(t)$ (S6)

$\displaystyle\hat{\Phi}(t)=e^{-i\tilde{\Omega}t}\hat{\Phi}(0)=e^{-i\tilde{\Omega}t}W^{-1}\hat{\Psi}(0),$
(S46)

where we have utilized that
$W^{-1}DW=\tau_{z}W^{\dagger}\mathcal{M}W=\tau_{z}\Omega\equiv\tilde{\Omega}$
which follows from Eq. S28. Thus, each new bosonic operator acquires a phase
factor associated to eigenenergy
$\hat{b}_{n}(t)=e^{-i\omega_{n}t}\hat{b}_{n}(0)$ [12, 65].

Next, we note that we may also absorb the time-dependence of the original
operators $\hat{\Psi}(t)$ into the (inverse) Bogoliubov matrix,
$\hat{\Phi}(t)=W^{-1}\hat{\Psi}(t)\equiv W^{-1}(t)\hat{\Psi}(0)$, i.e., in
(S46) $W^{-1}(t)=e^{-i\tilde{\Omega}t}W^{-1}(0)=W^{-1}(0)e^{-iDt}$. This leads
to the immediate question, what kind of equation of motion $W^{-1}(t)$
fulfills. In compact matrix form, the answer is given by the following set of
equations for the Bogoliubov modes,

$\displaystyle\partial_{t}W^{-1}(t)=-iW^{-1}(t)D\
\Leftrightarrow\partial_{t}W(t)=iDW(t).$ (S47)

Since we know that $W^{-1}D=\tilde{\Omega}W^{-1}$, we see that (S47) is solved
by $W^{-1}(t)=e^{-i\tilde{\Omega}t}W^{-1}(0)$, which is in agreement with
(S46). This may also be expressed in as an effective Schrödinger equation for
the individual Bogoliubov modes $\bm{w}^{n}(t)$ [65, 97]

$\displaystyle-i\partial_{t}\bm{w}^{n}(t)=D\bm{w}^{n}(t),$ (S48)

which is the sought after effective time-dependent Schrödinger equation. From
Eq. S48, for a time-independent dynamical matrix $D$, we can define the
paraunitary time evolution operator $P(t)$,

$\displaystyle P(t)=\exp\left(iDt\right),$ (S49)

such that $\bm{w}^{n}(t)=P(t)\bm{w}^{n}(0)$. For a time-dependent dynamical
matrix $D(t)$ in (S48), the paraunitary time evolution operator reads

$\displaystyle
P(t)=\mathcal{T}\exp\left(i\int_{0}^{t}dt^{\prime}D(t^{\prime})\right),$ (S50)

where $\mathcal{T}$ denotes the time-ordering operator. Note that the
paraunitarity of $P(t)$, i.e., $\tau_{z}P^{\dagger}(t)\tau_{z}=P^{-1}(t)$,
directly follows from the pseudo-hermiticity of the dynamical matrix,
$D^{\dagger}(t)=\tau_{z}D(t)\tau_{z}$. Thus, the time evolution of the
Bogoliubov modes $\bm{w}^{n}$ (S48) directly specifies the time-dependence of
the Heisenberg-picture quasiparticle operators
$\hat{b}^{\dagger}_{n}(t)=\hat{\Psi}^{\dagger}(t)\tau_{z}\bm{w}^{n}$ (S16),

$\displaystyle\hat{b}^{\dagger}_{n}(t)=\hat{\Psi}^{\dagger}(t)\tau_{z}\bm{w}^{n}=\hat{\Psi}^{\dagger}(0)e^{i(D^{\dagger})t}\tau_{z}\bm{w}^{n}=\hat{\Psi}^{\dagger}(0)\tau_{z}e^{iDt}\bm{w}^{n}=\hat{\Psi}^{\dagger}(0)\tau_{z}\bm{w}^{n}(t),$
(S51)

where we have used (S45) and the pseudo-hermiticity of the dynamical matrix
$D^{\dagger}=\tau_{z}D\tau_{z}$.

##  S5 Time-dependent perturbation theory for the effective Schrödinger
equation

Consider the effective time-dependent single-particle Schrödinger equation
(S48) for some state $\bm{\psi}(t)\in\mathbb{C}^{2N}$ in the indefinite inner
product space spanned by the Bogoliubov modes $\bm{w}^{n}(t)$ with the
addition of a small time-dependent perturbation $V(t)$ added to the dynamical
matrix, i.e.,

$\displaystyle-i\partial_{t}\bm{\psi}(t)=D(t)\bm{\psi}(t)=\left[D+V(t)\right]\bm{\psi}(t)$
(S52)

where $V(t)$ shall only be present for $t>0$. Adapting standard time-dependent
perturbation theory [127] to the space of Bogoliubov modes, we now derive the
effect of the perturbation $V(t)$ on the population of Bogoliubov modes.

At $t=0$, where $V(0)=0$, we consider the initial state $\bm{\psi}(0)$ to be
given by the Bogoliubov eigenstate $\bm{w}^{n}$, i.e.,
$D\bm{\psi}(0)=\tilde{\Omega}_{n}\bm{w}^{n}$ (S29). At this point,
$\bm{w}^{n}$ adheres to the effective time-dependent Schrödinger equation
without the perturbation (S48), which is easily solved by
$\bm{w}^{n}(t)=e^{iDt}\bm{w}^{n}(0)=e^{i\tilde{\Omega}_{n}t}\bm{w}^{n}(0)$
(S48). To first order in the perturbation $V(t)$, we now compute the resulting
excitation probability to other distinct Bogoliubov modes $\bm{w}^{m}(t)$ at
time $t$.

First, we introduce the interaction picture

$\displaystyle\bm{\psi}_{I}(t)=\tilde{P}(t)\bm{\psi}(t)=e^{-iDt}\bm{\psi}(t).$
(S53)

The transformation is explicitly given by $\tilde{P}(t)=e^{-iDt}$ where $D$ is
the static part in Eq. S52. Due to the pseudo-hermiticity of the dynamical
matrix, $D^{\dagger}=\tau_{z}D\tau_{z}$, in the equation of motion (S48), the
transformation $\tilde{P}(t)$ must be paraunitary, i.e.,
$\tilde{P}^{\dagger}(t)=\tau_{z}\tilde{P}^{-1}(t)\tau_{z}$. This directly
follows from
$\tau_{z}\tilde{P}^{\dagger}(t)\tau_{z}=\tau_{z}e^{iD^{\dagger}t}\tau_{z}=e^{iDt}=\tilde{P}^{-1}(t)$.
The effective time-dependent Schrödinger equation in the interaction picture
then reads

$\displaystyle-i\partial_{t}\bm{\psi}_{I}(t)=D_{I}(t)\bm{\psi}_{I}(t),$ (S54)

where we used Eq. S48 and introduced $D_{I}(t)$, the dynamical matrix in the
interaction picture given by

$\displaystyle
D_{I}(t)=\tilde{P}(t)D(t)\tilde{P}^{-1}(t)-i(\partial_{t}\tilde{P}(t))\tilde{P}^{-1}(t).$
(S55)

With $\tilde{P}(t)=e^{-iDt}$, Eq. S54 then becomes

$\displaystyle-i\partial_{t}\bm{\psi}_{I}(t)=V_{I}(t)\bm{\psi}_{I}(t),$ (S56)

where $V_{I}(t)\equiv\tilde{P}(t)V(t)\tilde{P}^{-1}(t)=e^{-iDt}V(t)e^{iDt}$ is
the perturbation in the interaction picture. The solution to (S56) is then
given by a Dyson-type series,

$\displaystyle\bm{\psi}_{I}(t)=\bm{\psi}_{I}(0)+i\int_{0}^{t}dt^{\prime}V_{I}(t^{\prime})\bm{\psi}_{I}(0)+i^{2}\int_{0}^{t}dt^{\prime}\int_{0}^{t^{\prime}}dt^{\prime\prime}V_{I}(t^{\prime})V_{I}(t^{\prime\prime})\bm{\psi}_{I}(0)+\dots,$
(S57)

where we are only interested in the first order contribution. Now, given an
initial eigenstate $\bm{\psi}_{I}(0)=\bm{w}^{n}(0)$, we compute the transition
amplitude $c^{\psi}_{n}(t)$ to some other arbitrary mode $\bm{w}^{m}(t)$ to
first order at time $t$ by first transforming back to the original picture
(S53)

$\displaystyle\bm{\psi}(t)=e^{iDt}\bm{\psi}_{I}(t)=e^{i\tilde{\Omega}_{n}t}\bm{w}^{n}(0)+e^{iDt}i\int_{0}^{t}dt^{\prime}V_{I}(t^{\prime})\bm{w}^{n}$
(S58)

and then projecting onto $s_{m}(\bm{w}^{m})^{\dagger}\tau_{z}$ [cf. Eq. S25],

$\displaystyle c^{\psi}_{m}(t)$
$\displaystyle=s_{m}(\bm{w}^{m})^{\dagger}\tau_{z}\bm{\psi}(t)=s_{m}s_{n}\delta_{nm}e^{i(\tilde{\Omega}_{n}-\tilde{\Omega}_{m})t}+is_{m}\int_{0}^{t}dt^{\prime}(\bm{w}^{m})^{\dagger}\tau_{z}V(t^{\prime})\bm{w}^{n}e^{i(\tilde{\Omega}_{n}-\tilde{\Omega}_{m})t^{\prime}},$
(S59)

where we have utilized that the dynamical matrix is pseudo-hermitian,
$D^{\dagger}=\tau_{z}D\tau_{z}$, which implies that
$\exp(iDt)^{\dagger}=\tau_{z}\exp(-iDt)\tau_{z}$. Eventually, the probability
for the system to transition into some distinct orthogonal state $m\neq n$ is
given by

$\displaystyle p_{mn}\equiv\left\lvert
c^{\psi}_{m}(t)\right\rvert^{2}=\left\lvert\int_{0}^{t}dt^{\prime}((\bm{w}^{m})^{\dagger}\tau_{z}V(t^{\prime})\bm{w}^{n})e^{i(\tilde{\Omega}_{n}-\tilde{\Omega}_{m})t^{\prime}}\right\rvert^{2}.$
(S60)

Next, we consider a periodic perturbation of the form $V(t)=F\cos(\omega
t-\varphi)=\frac{1}{2}F(e^{i(\omega t-\varphi)}+e^{-i(\omega t-\varphi)})$,
where $F$ is some constant real-valued amplitude. Thus, using standard
assumptions of $t\gg 1/\omega$ and
$\omega\simeq(\tilde{\Omega}_{m}-\tilde{\Omega}_{n})$ [127], Eq. S60 becomes

$\displaystyle
p_{mn}(\omega,t)=\frac{2\pi}{4}t\left\lvert((\bm{w}^{m})^{\dagger}\tau_{z}Fe^{i\varphi}\bm{w}^{n})\right\rvert^{2}\delta_{t}(\tilde{\Omega}_{m}-\tilde{\Omega}_{n}-\omega),$
(S61)

where $\delta_{t}(\alpha)=\frac{\sin^{2}(\alpha t/2)}{\pi\alpha^{2}t}$
approaches a Dirac-Delta distribution in the limit of large observation times
$t$, which then yields Eq. 15 in the main text.

##  S6 Measuring the symplectic quantum geometry in lattice systems

In this appendix, we specifically apply the general formalism above to the
case of two-dimensional lattice BBdG systems, described by a translationally
invariant dynamical matrix $D$, which is inspired by Tran2017,Ozawa2018,Refs.
Ref. [59, 60]. For such a system, the Bogoliubov eigenmodes of the dynamical
matrix $D$ have Bloch function form,
$\bm{\psi}^{n}(\bm{\mathbf{q}})=e^{i\bm{\mathbf{q}}\cdot\bm{\mathbf{r}}}\bm{w}^{n}(\bm{\mathbf{q}})$
[20, 97], such that the Bogoliubov eigenvalue equation in the quasimomentum
space representation (S29) is given by
$D(\bm{\mathbf{q}})\bm{w}^{n}(\bm{\mathbf{q}})=\tilde{\Omega}_{n}(\bm{\mathbf{q}})\bm{w}^{n}(\bm{\mathbf{q}})$.
Here, $\bm{w}^{n}(\bm{\mathbf{q}})$ is the cell-periodic part of the
Bogoliubov Bloch function for the $n$-th band. Next, in direct analogy to
Tran2017,Ozawa2018,Refs. Ref. [59, 60], we consider periodically shaking the
underlying lattice along $x$-direction of the lattice, resulting in the
following time-dependent dynamical matrix,

$\displaystyle D_{x}(t)=D+2A\cos(\omega t)\mathcal{X}=D+V(t),\
V(t)=2A\cos(\omega t)\mathcal{X},$ (S62)

where $D$ is the initial dynamical matrix in position space and
$\mathcal{X}=\sigma_{z}\otimes x$ the (pseudo-hermitian) position operator for
the direction $x$ (extended to Nambu space), and with $x$ being the usual
position operator.

We note that such periodically shaken systems described via $D_{x}(t)$ are in
direct analogy to the general discussion following Eq. 14 in the main text. To
see this, we first note that $D_{x}(t)$ becomes translationally invariant in
the co-moving frame of the lattice [59, 60], when introducing a paraunitary
rotation

$\displaystyle\tilde{P}(t)=\exp\left(-i\int
dt^{\prime}V(t^{\prime})\right)=\exp(-i(2A/\omega)\sin(\omega t)\mathcal{X}),$
(S63)

such that the dynamical matrix in the co-moving frame (S55) becomes

$\displaystyle D_{x}^{\prime}(t)$
$\displaystyle=\tilde{P}D_{x}(t)\tilde{P}^{-1}-i(\partial_{t}\tilde{P})\tilde{P}^{-1}=\tilde{P}[D+V(t)]\tilde{P}^{-1}-i(-iV(t)\tilde{P})\tilde{P}^{-1}$
$\displaystyle=\tilde{P}D\tilde{P}^{-1}=D(\bm{\mathbf{q}})+\frac{2A}{\omega}\sin(\omega
t)\partial_{q_{x}}D(\bm{\mathbf{q}})+\mathcal{O}((A/\omega)^{2}),$ (S64)

which agrees with the formalism of Eq. 14 in the main text.

As in the general discussion in the main text, given an initially populated
Bogoliubov mode
$\bm{\psi}^{n}=e^{i\bm{\mathbf{q}}_{0}\cdot\bm{\mathbf{r}}}\bm{w}^{n}(\bm{\mathbf{q}}_{0})$,
we compute the total excitation rate into all other available Bogoliubov modes
$\bm{\psi}^{m}$,

$\displaystyle\Gamma_{x}^{n}(\omega)=\frac{1}{t}\sum_{m\neq
n}s_{m}p_{mn}(\omega,t),$ (S65)

with $p_{mn}(\omega,t)$ being given by (S61). We note that the factor $s_{m}$
comes from the indefinite inner product structure of the Bogoliubov modes [cf.
Eq. S26]. The total excitation rate can then be expressed as

$\displaystyle\begin{split}\Gamma^{n}_{x}(\omega)=&2\pi A^{2}\sum_{m\neq
n}s_{m}\left\lvert(\bm{w}^{m}(\bm{\mathbf{q}}))^{\dagger}e^{-i\bm{\mathbf{q}}\cdot\bm{\mathbf{r}}}\tau_{z}\mathcal{X}e^{i\bm{\mathbf{q}}_{0}\cdot\bm{\mathbf{r}}}\bm{w}^{n}(\bm{\mathbf{q}}_{0})\right\rvert^{2}\times\delta\left(\tilde{\Omega}_{m}(\bm{\mathbf{q}})-\tilde{\Omega}_{n}(\bm{\mathbf{q}}_{0})-\omega\right)\\\
=&2\pi A^{2}\sum_{m\neq
n}s_{m}\left\lvert(\bm{w}^{m}(\bm{\mathbf{q}}_{0}))^{\dagger}\tau_{z}i\partial_{q_{x}}\bm{w}^{n}(\bm{\mathbf{q}}_{0})\right\rvert^{2}\times\delta\left(\tilde{\Omega}_{m}(\bm{\mathbf{q}}_{0})-\tilde{\Omega}_{n}(\bm{\mathbf{q}}_{0})-\omega\right).\end{split}$
(S66)

Here, we have used that the position operator acts as a derivate in the
quasimomentum space representation, i.e.,
$(\bm{w}^{m}(\bm{\mathbf{q}}))^{\dagger}e^{-i\bm{\mathbf{q}}\cdot\bm{\mathbf{r}}}\tau_{z}\mathcal{X}e^{i\bm{\mathbf{q}}_{0}\cdot\bm{\mathbf{r}}}\bm{w}^{n}(\bm{\mathbf{q}}_{0})=i\delta_{\bm{\mathbf{q}},\bm{\mathbf{q}}_{0}}(\bm{w}^{m}(\bm{\mathbf{q}}))^{\dagger}{\tau_{z}\partial_{q_{x}}}\bm{w}^{n}(\bm{\mathbf{q}})$
[128, 60]. By integrating the excitation rate (S66) over $\omega$, we arrive
at

$\displaystyle\Gamma^{n}_{x,\text{int}}=$ $\displaystyle\int
d\omega\,\Gamma^{n}_{x}(\omega)=2\pi A^{2}\sum_{m\neq
n}s_{m}\left\lvert(\bm{w}^{m}(\bm{\mathbf{q}}_{0}))^{\dagger}\tau_{z}\partial_{q_{x}}\bm{w}^{n}(\bm{\mathbf{q}}_{0})\right\rvert^{2}$
$\displaystyle=$ $\displaystyle\sum_{\begin{subarray}{c}m\neq
n\end{subarray}}\left[(\partial_{q_{x}}\bm{w}^{n}(\bm{\mathbf{q}}_{0}))^{\dagger}\tau_{z}(P^{m})(\partial_{q_{x}}\bm{w}^{n}(\bm{\mathbf{q}}_{0}))\right]=2\pi
A^{2}g^{n}_{xx}(\bm{\mathbf{q}}_{0}),$ (S67)

where in the last step we have used (S33) to identity the symplectic quantum
metric. Thus, the integrated excitation rate after shaking the lattice along
the $x$ direction is directly proportional to the diagonal component
$g^{n}_{xx}(\bm{\mathbf{q}}_{0})$ of the symplectic quantum metric tensor. By
similarly shaking the lattice along the $y$ direction, the diagonal component
$g^{n}_{yy}(\bm{\mathbf{q}}_{0})$ is obtained.

To obtain the remaining off-diagonal components of the symplectic quantum
geometric tensor, i.e., the symplectic quantum metric
$g^{n}_{xy}(\bm{\mathbf{q}}_{0})t$ and the symplectic Berry curvature
$B^{n}_{xy}(\bm{\mathbf{q}}_{0})$, we now imagine shaking the lattice along
both the $x$ and $y$ directions [59, 60], which is described by

$\displaystyle D_{x\pm y}(t)=D+2A(\cos(\omega t)\mathcal{X}\pm\cos(\omega
t-\varphi)\mathcal{Y})=D+V(t),\ V(t)=2A(\cos(\omega
t)\mathcal{X}\pm\cos(\omega t-\varphi)\mathcal{Y}),$ (S68)

where here $\mathcal{Y}=\sigma_{z}\otimes y$ is the (pseudo-hermitian)
$y$-position operator (extended to Nambu space) with $y$ being the usual
position operator.

For completeness, we also provide the paraunitarity rotation to the co-moving
frame for this case (S68)

$\displaystyle P(t)=\exp\left(-i\int
dt^{\prime}V(t^{\prime})\right)=\exp\\{-i(A/\omega)(\sin(\omega
t)\mathcal{X}\pm\sin(\omega t-\varphi)\mathcal{Y})\\},$ (S69)

which then leads to the following dynamical matrix in the co-moving frame
(S55)

$\displaystyle D_{x\pm
y}^{\prime}(t)=D(\bm{\mathbf{q}})+\frac{2A}{\omega}\left(\sin(\omega
t)\partial_{q_{x}}D(\bm{\mathbf{q}})\pm\sin(\omega
t-\varphi)\partial_{q_{y}}D(\bm{\mathbf{q}})\right)+\mathcal{O}((A/\omega)^{2}).$
(S70)

This allows us again to see the connection to the general discussion following
Eq. 14 in the main text. We emphasize that Eq. S70 is in direct analogy to the
results for the conventional case in Tran2017,Refs. Ref. [59].

Then, in the same fashion as before (S65)-(S67), computing the integrated
excitation rate leads to the emergence of the off-diagonal components of the
SQGT depending on the phase offset $\varphi$,

$\displaystyle\begin{split}\Gamma^{n,\pm}_{\text{int}}=2\pi
A^{2}\times\begin{cases}(g^{n}_{xx}(\bm{\mathbf{q}}_{0})\pm
2g^{n}_{xy}(\bm{\mathbf{q}}_{0})+g^{n}_{yy}(\bm{\mathbf{q}}_{0})),\\!\\!\\!&\text{for}\,\varphi=0,\\\
(g^{n}_{xx}(\bm{\mathbf{q}}_{0})\mp
B^{n}_{xy}(\bm{\mathbf{q}}_{0})+g^{n}_{yy}(\bm{\mathbf{q}}_{0})),\\!\\!\\!&\text{for}\,\varphi=\pi/2.\end{cases}\end{split}$
(S71)

By taking the differential integrated rate [59]
$\Delta\Gamma^{n}_{\text{int}}=\Gamma^{n,+}_{\text{int}}-\Gamma^{n,-}_{\text{int}}$
of the two protocols with the opposite sign in $D_{x\pm y}(t)$ (S68), we
obtain

$\displaystyle\Delta\Gamma^{n}_{\text{int}}=\begin{cases}\ \ 8\pi
A^{2}g^{n}_{xy}(\bm{\mathbf{q}}_{0}),\\!\\!\\!&\text{for}\,\varphi=0,\\\ -4\pi
A^{2}B^{n}_{xy}(\bm{\mathbf{q}}_{0}),\\!\\!\\!&\text{for}\,\varphi=\pi/2.\end{cases}$
(S72)

Depending on the phase offset $\varphi$ of the periodic modulation in (S68),
we see that the differential integrated rate is directly proportional to the
off-diagonal symplectic quantum metric $g^{n}_{xy}(\bm{\mathbf{q}}_{0})$ or
the symplectic Berry curvature $B^{n}_{xy}(\bm{\mathbf{q}}_{0})$.

All the prior formalism, which assumed a single populated Bogoliubov mode at
some quasimomentum $\bm{\mathbf{q}}_{0}$, may also be generalized to an
initial Bogoliubov Bloch wave packet, e.g., a weighted average of multiple
Bogoliubov (particle) modes within the same band $n$, i.e.,
$\bm{\phi}^{n}=\sum_{\bm{\mathbf{q}}}c(\bm{\mathbf{q}})e^{i\bm{\mathbf{q}}\cdot\bm{\mathbf{r}}}\bm{w}^{n}(\bm{\mathbf{q}})$,
where the wave packet $\bm{\phi}^{n}$ is normalized with respect to the
Bogoliubov norm
$(\bm{\phi}^{n})^{\dagger}{\tau_{z}}\bm{\phi}^{n}=\sum_{\bm{\mathbf{q}}}\left\lvert
c(\bm{\mathbf{q}})\right\rvert^{2}=1$ [97]. One may further assume that the
wave packet is localized around some central quasimomentum mode
$\bm{\mathbf{q}}_{0}=\sum_{\bm{\mathbf{q}}}\left\lvert
c(\bm{\mathbf{q}})\right\rvert^{2}\bm{\mathbf{q}}$, as well as around some
central position $\bm{\mathbf{r}}_{c}$, which is given by
$\bm{\mathbf{r}}_{c}=(\bm{\phi}^{n})^{\dagger}{\tau_{z}}\bm{\mathbf{r}}\bm{\phi}^{n}$
[97]. This then leads simply leads to the integrated rates being proportional
to the weighted average of the symplectic quantum geometric quantities, e.g.,
for the linear shaking protocol of Eq. S62, the integrated rate becomes

$\displaystyle\Gamma^{n}_{x,\text{int}}=2\pi
A^{2}\sum_{\bm{\mathbf{q}}}\left\lvert
c(\bm{\mathbf{q}})\right\rvert^{2}g^{n}_{xx}(\bm{\mathbf{q}}),$ (S73)

which is in direct analogy to the proposals for probing the quantum geometry
in particle-number-conserving systems [60].

##  S7 Time-dependent perturbation theory for many-body Bogoliubov Fock states

In the main text we have shown that the symplectic quantum geometric tensor
can be extracted from integrated excitation rates to other Bogoliubov modes in
response to weak periodic modulations of bosonic BdG systems Eq. 14. For this,
we have used the results of first-order perturbation theory for the effective
Schrödinger equation (S61). In this appendix, we now show that this derivation
is equivalent to applying standard first-order time-dependent perturbation
theory (Fermi’s Golden rule) to treat transitions between many-body Bogoliubov
Fock states with quasiparticle excitations present.

Let us first consider the general case, where, in addition to the already
solved (unperturbed) problem of (S1), we apply a generic time-dependent
particle-number-conserving perturbation of the form
$V(t)=\sum_{i,j}\hat{a}_{i}^{\dagger}A_{ij}(t)\hat{a}_{j}$ to the system (S1).
To understand how this perturbation affects the population of Bogoliubov
quasiparticle excitations, we express $V(t)$ via the diagonal (quasiparticle)
basis of the unperturbed problem using (S5)

$\displaystyle V(t)=\sum_{i,j}$
$\displaystyle\hat{a}_{i}^{\dagger}A_{ij}(t)\hat{a}_{j}=\sum_{i,j}A_{ij}(t)\sum_{m,m^{\prime}}\left(\hat{b}^{\dagger}_{m}(U^{\dagger})_{mi}+\hat{b}_{m}(V^{T})_{mi}\right)A_{ij}(t)\left(U_{jm^{\prime}}\hat{b}_{m^{\prime}}+(V^{*})_{jm^{\prime}}\hat{b}^{\dagger}_{m^{\prime}}\right)$
$\displaystyle=\sum_{m,m^{\prime}}$
$\displaystyle[\hat{b}^{\dagger}_{m}\hat{b}_{m^{\prime}}\sum_{ij}(U^{\dagger})_{mi}A_{ij}(t)U_{jm^{\prime}}+\hat{b}_{m}\hat{b}^{\dagger}_{m^{\prime}}\sum_{ij}(V^{T})_{mi}A_{ij}(t)(V^{*})_{jm^{\prime}}$
$\displaystyle+$
$\displaystyle\hat{b}^{\dagger}_{m}\hat{b}^{\dagger}_{m^{\prime}}\sum_{ij}(U^{\dagger})_{mi}A_{ij}(t)(V^{*})_{jm^{\prime}}+\hat{b}_{m}\hat{b}_{m^{\prime}}\sum_{ij}(V^{T})_{mi}A_{ij}(t)U_{jm^{\prime}}].$
(S74)

Note that the ground state of the original BdG Hamiltonian (S19) contains zero
Bogoliubov quasiparticle excitations. In the main text, we have argued that
such a generic perturbation (S74), can be used to controllably create pairs of
quasiparticle excitations in desired Bogoliubov energy bands via terms
$\propto\hat{b}^{\dagger}_{m}\hat{b}^{\dagger}_{m^{\prime}}$ or
$\propto\hat{b}^{\dagger}_{m}$ [cf. Fig. 2(a) in the main text]. The presence
of such Bogoliubov quasiparticle excitations is the starting point for the
protocol to extract the symplectic quantum geometric tensor from excitation
rates between Bogoliubov modes, as will show in the following [cf. Eq. 15 in
the main text].

###  S7A Transitions between Bogoliubov quasiparticle excitations lead to SQGT

Now, as a first simple case, we consider the transition of single Bogoliubov
quasiparticle excitation from some occupied mode $\beta$ to some unoccupied
mode $\alpha$ via the perturbation $V(t)$ (S74). As an initial state
${\lvert}\phi_{\beta}{\rangle}$, we then choose the Bogoliubov Fock state with
a single quasiparticle excitation in some mode $\beta$ and, i.e.,
${\lvert}\phi_{\beta}{\rangle}={\lvert}\ldots,(0)_{\alpha},(1)_{\beta},\ldots{\rangle}$
and as a final state the state with an excitation in some mode $\alpha$, i.e.,
${\lvert}\psi_{\alpha}{\rangle}={\lvert}\ldots,(1)_{\alpha},(0)_{\beta},\ldots{\rangle}$.
Standard first-order time-dependent perturbation theory [127], tells us that
we have to look at matrix elements
${\langle}\psi_{\alpha}{\,|}V(t){|\,}\phi_{\beta}{\rangle}$ to analyze the
transitions between the initial and final states. So the only matrix elements
of the perturbation
${\langle}\psi_{\alpha}{\,|}V(t){|\,}\phi_{\beta}{\rangle}$ that contribute to
the transition are only the ones which are number-conserving in the Bogoliubov
basis, such that using (S74) we obtain

$\displaystyle{\langle}\psi_{\alpha}{\,|}V(t){|\,}\phi_{\beta}{\rangle}={\langle}\psi_{\alpha}{\rvert}\sum_{m,m^{\prime}}\left[\delta_{\alpha,m}\delta_{\beta,m^{\prime}}\hat{b}^{\dagger}_{m}\hat{b}_{m^{\prime}}\sum_{ij}(U^{\dagger})_{mi}A_{ij}(t)U_{jm^{\prime}}+\delta_{\alpha,m^{\prime}}\delta_{\beta,m}\hat{b}_{m}\hat{b}^{\dagger}_{m^{\prime}}\sum_{ij}(V^{T})_{mi}A_{ij}(t)(V^{*})_{jm^{\prime}}\right]{\lvert}\phi_{\beta}{\rangle}$
$\displaystyle=\sum_{ij}(U^{\dagger})_{\alpha
i}A_{ij}(t)U_{j\beta}{\langle}\psi_{\alpha}{\,|}\hat{b}^{\dagger}_{\alpha}\hat{b}_{\beta}{|\,}\phi_{\beta}{\rangle}+(V^{T})_{\beta
i}A_{ij}(t)(V^{*})_{j\alpha}{\langle}\psi_{\alpha}{\,|}\hat{b}_{\beta}\hat{b}^{\dagger}_{\alpha}{|\,}\phi_{\beta}{\rangle}$
$\displaystyle=\sum_{ij}(U^{\dagger})_{\alpha
i}A_{ij}(t)U_{j\beta}+(V^{\dagger})_{\alpha
j}(A^{*})_{ji}(t)V_{i\beta}=\sum_{ij}\begin{pmatrix}U^{\dagger}&V^{\dagger}\\\
V^{T}&U^{T}\end{pmatrix}_{\alpha i}\ \begin{pmatrix}A(t)&0\\\
0&A^{*}(t)\end{pmatrix}_{ij}\ \begin{pmatrix}U&V^{*}\\\
V&U^{*}\end{pmatrix}_{j\beta}$ $\displaystyle=\sum_{ij}(W^{\dagger})_{\alpha
i}\tilde{\mathcal{M}}_{ij}(t)W_{j\beta}=\bm{w}_{\alpha}^{\dagger}\tilde{\mathcal{M}}(t)\bm{w}_{\beta},\
\text{with}\ \tilde{\mathcal{M}}(t)=\begin{pmatrix}A(t)&0\\\
0&A^{*}(t)\end{pmatrix},$ (S75)

where in the second to last step we have introduced the $2N\times 2N$ matrix
$\tilde{\mathcal{M}}(t)$ and reformulated the Bogoliubov transformation in
terms of the paraunitary matrix $W$ (S6), where both the $\alpha$ and $\beta$
indices are restricted to the particle sector, i.e., $1\leq\alpha,\beta\leq N$
for the equality to hold. In the last step of (S75), we have introduced the
column vector $\bm{w}_{\beta}$ given by the $\beta$-th column of $W$ and the
row vector $\bm{w}_{\alpha}^{\dagger}$ given by the $\alpha$-th row of
$W^{\dagger}$, where again $1\leq\alpha,\beta\leq N$ (restriction to the
particle modes).

We can generalize this result to the case of an initial eigenstate of (S19)
with $n_{\beta}$ quasiparticle excitations in mode $\beta$ and $n_{\alpha}$
excitations in mode $\alpha$, i.e.,
${\lvert}\phi_{\beta}{\rangle}={\lvert}\ldots,(n_{\alpha})_{\alpha},(n_{\beta})_{\beta},\ldots{\rangle}$,
and a final state with $n_{\beta}-1$ quasiparticle excitations in mode
$\beta$, and $n_{\alpha}+1$ excitation in mode $\alpha$, i.e.,
${\lvert}\psi_{\alpha}{\rangle}={\lvert}\ldots,(n_{\alpha}+1)_{\alpha},(n_{\beta}-1)_{\beta},\ldots{\rangle}$.
This will yield a similar result as in (S75) but with an additional bosonic
enhancement factor of due to
$\hat{b}^{\dagger}_{\alpha}\hat{b}_{\beta}{\lvert}\phi_{\beta}{\rangle}=\sqrt{n_{\beta}(n_{\alpha}+1)}{\lvert}\phi_{\alpha}{\rangle}$,

$\displaystyle{\langle}\psi_{\alpha}{\,|}\mathcal{A}(t){|\,}\phi_{\beta}{\rangle}=\sqrt{n_{\beta}(n_{\alpha}+1)}\bm{w}_{\alpha}^{\dagger}\tilde{\mathcal{M}}(t)\bm{w}_{\beta}.$
(S76)

Now we explicitly specify the explicit form of the perturbation. Inspired by
Ozawa2018,Refs. Ref. [60], we assume an original parameter-dependent
coefficient matrix $\mathcal{M}(\bm{\lambda}(t))$, where
$\bm{\lambda}(t)=(\lambda^{1}(t),\ldots,\lambda^{r}(t))$ is a set of time-
dependent parameters. We first restrict ourselves only to have parameter
dependence in the diagonal (particle-number-conserving) blocks of the
coefficient matrix $\mathcal{M}(\bm{\lambda}(t))$, i.e., $K(\bm{\lambda}(t))$
and $G=\text{const.}$ (cf. Eq. S1). The system shall initially be prepared at
some parameter values $\bm{\lambda}=\bm{\lambda}_{0}$. Let us then, for
simplicity, consider a single parameter modulation $\lambda^{1}(t)$ of the
form $\lambda^{1}(t)=\lambda^{1}_{0}+2(A/\omega)\cos(\omega t)$ with
$A/\omega\ll 1$. The rest of the parameters are assumed to be constant, i.e.,
$\lambda^{\nu}(t)=\lambda^{\nu}_{0}$ for $\nu\neq 1$. We then Taylor-expand
the relevant particle-number-conserving perturbation $K(\bm{\lambda}(t))$ to
first order

$\displaystyle
K(\bm{\lambda}(t))=K(\bm{\lambda}_{0})+\partial_{1}K(\bm{\lambda}_{0})\times(2A/\omega)\cos(\omega
t)=K(\bm{\lambda}_{0})+\tilde{K}(t),$ (S77) $\displaystyle\text{with}\
\tilde{K}(t)=f(t)\tilde{K},\ f(t)\equiv(2A/\omega)\cos(\omega t),\
\tilde{K}\equiv\partial_{1}K(\bm{\lambda}_{0}).$

Here, the matrix $(\tilde{K})_{ij}(t)$ plays the role of the matrix
$A_{ij}(t)$ in the general perturbation considered in Eq. S74. We note that
this explicit form of the parameter modulations leading to (S77) is not
arbitrary but readily appears in the co-moving frame of periodically shaken
systems (S62) [59, 60].

On the level of the Hamiltonian (S1) the perturbation provides an additional
term of the form

$\displaystyle
V(t)=\sum_{ij}\hat{a}_{i}^{\dagger}\tilde{K}_{ij}(t)\hat{a}_{j}=\left(\sum_{ij}\hat{a}_{i}^{\dagger}\tilde{K}_{ij}\hat{a}_{j}\right)f(t)\equiv\tilde{V}f(t),$
(S78)

Now, initially, we assume the system to be in some eigenstate
${\lvert}\phi_{\beta}{\rangle}$ of (S19). As before, we let us first discuss
the case of a single excitation and generalize to multiple excitations later.
Thus, we start with a state with a single Bogoliubov quasiparticle excitation
in some mode $\beta$, i.e.,
${\lvert}\phi_{\beta}{\rangle}={\lvert}\ldots,(0)_{\alpha},(1)_{\beta},\ldots{\rangle}$
with the corresponding energy eigenvalue equation
$\hat{H}{\lvert}\phi_{\beta}{\rangle}=\omega_{\beta}{\lvert}\phi_{\beta}{\rangle}$
(S19). Our goal is to compute the transition probability to some final
eigenstate ${\lvert}\psi_{\alpha}{\rangle}$, which for now shall consist of a
single Bogoliubov quasiparticle excitation in some other mode $\alpha$, i.e.,
${\lvert}\psi_{\alpha}{\rangle}={\lvert}\ldots,(1)_{\alpha},(0)_{\beta},\ldots{\rangle}$
with corresponding energy eigenvalue equation
$\hat{H}{\lvert}\psi_{\alpha}{\rangle}=\omega_{\alpha}{\lvert}\psi_{\alpha}{\rangle}$.

Then, given the periodic perturbation of type (S77)-(S78), we can use Fermi’s
Golden rule (first-order time-dependent perturbation theory) [127] to compute
this transition probability at time $t$ with via

$\displaystyle n_{\alpha\beta}(\omega,t)$ $\displaystyle=2\pi
t\left\lvert{\langle}\psi_{\alpha}{\,|}\tilde{V}{|\,}\phi_{\beta}{\rangle}\right\rvert^{2}\delta(\omega_{\alpha}-\omega_{\beta}-\omega)=2\pi
t\left\lvert{\langle}\psi_{\alpha}{\,|}\sum_{ij}\hat{a}_{i}^{\dagger}\tilde{K}_{ij}\hat{a}_{j}{|\,}\phi_{\beta}{\rangle}\right\rvert^{2}\delta(\omega_{\alpha}-\omega_{\beta}-\omega),$
(S79)

where $\hbar\equiv 1$ in the following. Into Eq. S79 we can insert the
explicit expansion of the perturbation $\tilde{V}$ (S78) in terms of the
Bogoliubov quasiparticle operators (S74)-(S75) leading to

$\displaystyle n_{\alpha\beta}(\omega,t)$ $\displaystyle=2\pi
t\left\lvert\sum_{ij}[(U^{\dagger})_{\alpha
i}\tilde{K}_{ij}U_{j\beta}{\langle}\psi_{\alpha}{\,|}\hat{b}^{\dagger}_{\alpha}\hat{b}_{\beta}{|\,}\phi_{\beta}{\rangle}+(V^{T})_{\beta
i}\tilde{K}_{ij}(V^{*})_{j\alpha}{\langle}\psi_{\alpha}{\,|}\hat{b}_{\beta}\hat{b}^{\dagger}_{\alpha}{|\,}\phi_{\beta}{\rangle}]\right\rvert^{2}\delta(\omega_{\alpha}-\omega_{\beta}-\omega)$
$\displaystyle=2\pi
t\frac{A^{2}}{\omega^{2}}\left\lvert\bm{w}_{\alpha}^{\dagger}(\partial_{1}{\mathcal{M}})\bm{w}_{\beta}\right\rvert^{2}\delta(\omega_{\alpha}-\omega_{\beta}-\omega).$
(S80)

Next, we compute the total excitation rate by summing Eq. S80 over all
possible final states ${\lvert}\psi_{\alpha}{\rangle}$ with a single
excitation in some mode $\alpha$,

$\displaystyle\Gamma^{\alpha}(\omega)=\frac{1}{t}\sum_{\alpha\neq\beta}n_{\alpha\beta}(\omega,t)=\sum_{\alpha\neq\beta}2\pi\frac{A^{2}}{\omega^{2}}\left\lvert\bm{w}_{\alpha}^{\dagger}(\partial_{1}{\mathcal{M}})\bm{w}_{\beta}\right\rvert^{2}\delta(\omega_{\alpha}-\omega_{\beta}-\omega)$
(S81)

Then, inspired by Tran2017,Refs. Ref. [59], we introduce the integrated
excitation rate $\Gamma^{\alpha}_{\text{int}}$

$\displaystyle\Gamma^{\alpha}_{\text{int}}=\int
d\omega\,\Gamma^{\alpha}(\omega)=2\pi A^{2}\sum_{\alpha\neq\beta,\
\alpha=1}^{N}\frac{\left\lvert\bm{w}_{\alpha}^{\dagger}(\partial_{1}\mathcal{M})\bm{w}_{\beta}\right\rvert^{2}}{(\omega_{\alpha}-\omega_{\beta})^{2}}.$
(S82)

The last expression already resembles the spectral representation of the
symplectic quantum metric (S33), although not exactly, since the sum over
$\alpha$ in (S82) is only restricted to the particle Bogoliubov modes
$(\bm{w}_{\alpha}^{\dagger})_{i}=(W^{\dagger})_{\alpha i}$ with
$1\leq\alpha\leq N$. However, since the spectral representation of the
symplectic quantum metric is defined via the sum over all Bogoliubov modes,
including the hole modes with $N+1\leq\alpha\leq 2N$, we need to extend the
sum in (S82). This is allowed by the fact that the operator
$\partial_{1}\mathcal{M}$ (with only particle-number-conserving parts) does
not couple the particle and hole Bogoliubov modes. By this, we arrive at

$\displaystyle\Gamma_{\text{int}}=\int
d\omega\sum_{\alpha}\Gamma^{\alpha}(\omega)=2\pi A^{2}\sum_{\alpha\neq\beta,\
\alpha=1}^{2N}s_{\alpha}\frac{\left\lvert\bm{w}_{\alpha}^{\dagger}(\partial_{1}\mathcal{M})\bm{w}_{\beta}\right\rvert^{2}}{(\tilde{\omega}_{\alpha}-\omega_{\beta})^{2}}=2\pi
A^{2}g_{11}^{\beta}.$ (S83)

We observe that the integrated excitation rates between states
${\lvert}\phi_{\beta}{\rangle}$ and ${\lvert}\psi_{\alpha}{\rangle}$, with
single Bogoliubov quasiparticle excitations, give rise to the symplectic
quantum metric (S33). This agrees with the result derived from first-order
perturbation theory for the effective Schrödinger equation (S61) in Eq. 16 in
the main text.

As before this result can be generalized, to the case of an initial Bogoliubov
Fock state with $n_{\beta}$ quasiparticle excitations in mode $\beta$ and
$n_{\alpha}$ excitations in mode $\alpha$, i.e.,
${\lvert}\phi_{\beta}{\rangle}={\lvert}\ldots,(n_{\alpha})_{\alpha},(n_{\beta})_{\beta},\ldots{\rangle}$,
and a final state with $n_{\beta}-1$ quasiparticle excitations in mode
$\beta$, and $n_{\beta}$ excitation in mode $\alpha$, i.e.,
${\lvert}\psi_{\alpha}{\rangle}={\lvert}\ldots,(n_{\alpha}+1)_{\alpha},(n_{\beta}-1)_{\beta},\ldots{\rangle}$.
This will lead to an additional bosonic enhancement factor of
$\sqrt{n_{\beta}}$ in the transition matrix elements as in (S76). The energy
difference between the initial and final state is still given by
$[(n_{\alpha}+1)\omega_{\alpha}+\omega_{\beta}(n_{\beta}-1)]-[\omega_{\beta}n_{\beta}-\omega_{\alpha}n_{\alpha}]=\omega_{\alpha}-\omega_{\beta}$
such that we can directly generalize the Fermi’s Golden rule result of Eq. S83
to

$\displaystyle\Gamma_{\text{int}}=\int
d\omega\sum_{\alpha}\Gamma^{\alpha}(\omega)=n_{\beta}(n_{\alpha}+1)\times 2\pi
A^{2}g_{11}^{\beta}.$ (S84)

##  S8 Adiabatic perturbation theory for the effective Schrödinger equation

In this appendix, we derive the first order correction to the adiabatic
theorem for the time evolution of Bogoliubov modes (S48). This is a
prerequisite for deriving the symplectic anomalous velocity term in the
following Section S9. To this end, we adapt the standard theory of adiabatic
perturbation theory for conventional systems [129, 130] to the symplectic
Bogoliubov structure here.

We consider a time-dependent dynamical matrix $D(t)\equiv D(\bm{\lambda}(t))$,
where the time dependence is encoded via some dimensionless parameters
$\bm{\lambda}(t)=(\lambda^{1}(t),\dots,\lambda^{r}(t))$. At each point in time
$t$, we assume that the Bogoliubov system is thermodynamically stable (the
coefficient matrix $\mathcal{M}(\bm{\lambda}(t))$ is positive definite
$\forall t$ [65]), such that an instantaneous eigenvalue equation of the
following form holds (S29)

$\displaystyle D(t)\bm{w}^{n}(t)=\tilde{\Omega}_{n}(t)\bm{w}^{n}(t),$ (S85)

where the time-dependence of the Bogoliubov modes $\bm{w}^{n}(t)$ is again
specified through the parameters $\bm{\lambda}(t)$. In what follows, we assume
a non-degenerate spectrum $\tilde{\Omega}_{n}\neq\tilde{\Omega}_{m}$ for
$n\neq m$ for all times $t$. This assumption can be relaxed both regarding
states that are not coupled to $\bm{w}^{n}$ as a result of symmetry and when
considering the adiabatic theorem with respect to a degenerate eigenspace,
which is however beyond the scope of this work. We are interested in the time-
evolution an arbitrary state $\bm{\psi}(t)$ in the space spanned by the
Bogoliubov modes (S25),

$\displaystyle\bm{\psi}(t)=\sum_{n}c_{n}(t)\bm{w}^{n}(t),$ (S86)

which is governed by the effective time-dependent Schrödinger equation (S48),

$\displaystyle-i\partial_{t}\bm{\psi}(t)=D(t)\bm{\psi}(t).$ (S87)

Now, under the assumption of slow parameter variations of $\bm{\lambda}(t)$
(adiabatic limit), we would like to solve Eq. S87 perturbatively. To this end,
we first insert the instantaneous expansion of $\bm{\psi}(t)$ (S86) into the
effective Schrödinger equation (S87), then project onto a specific mode via
multiplying with $s_{m}(\bm{w}^{m})^{\dagger}\tau_{z}$ from the left, by which
we arrive

$\displaystyle\dot{c}_{m}+\sum_{n}c_{n}s_{m}(\bm{w}^{m})^{\dagger}\tau_{z}\partial_{t}\bm{w}^{n}=i\tilde{\Omega}_{m}c_{m}.$
(S88)

The explicit time-dependence will be omitted for brevity in the following.
Next, we perform a gauge transformation $c_{m}(t)\rightarrow
c_{m}(t)e^{i\xi_{m}(t)}$, where
$\xi_{m}(t)=\int_{0}^{t}dt^{\prime}\tilde{\Omega}_{m}(t^{\prime})$. This
removes the dynamical phase factor from the equations, which will, however,
obviously be reintroduced at the end. This leads to the following differential
equation for the time-dependent coefficients $c_{m}(t)$

$\displaystyle\dot{c}_{m}$
$\displaystyle=-\sum_{n}c_{n}s_{m}(\bm{w}^{m})^{\dagger}\tau_{z}\partial_{t}\bm{w}^{n}e^{i\xi_{nm}(t)}$
(S89)

with $\xi_{nm}(t)\equiv\xi_{n}(t)-\xi_{m}(t)$. We formally integrate (S89) and
obtain

$\displaystyle
c_{m}(t)=c_{m}(0)-\int_{0}^{t}dt^{\prime}\sum_{n}c_{n}(t^{\prime})s_{m}(\bm{w}^{m})^{\dagger}\tau_{z}\partial_{\mu}\bm{w}^{n}\dot{\lambda}^{\mu}e^{i\xi_{nm}(t^{\prime})}$
(S90)

where we have used that
$\partial_{t}\bm{w}^{n}(\bm{\lambda}(t))=\sum_{\mu}\partial_{\mu}\bm{w}^{n}\dot{\lambda}^{\mu}\equiv\partial_{\mu}\bm{w}^{n}\dot{\lambda}^{\mu}$
with $\partial_{\mu}\equiv\frac{\partial}{\partial\lambda^{\mu}}$.

Let us now consider the situation, where the system is initially occupying a
single Bogoliubov mode, i.e., $\bm{\psi}(0)=\bm{w}^{m}$ in (S86), such that
$c_{m}(0)=1$ and $c_{n}(0)=0$ for $n\neq m$. In the limit of
$\dot{\lambda}^{\mu}\to 0$ all transition amplitudes to other modes are highly
suppressed, and the only leading order contribution in (S90) stems from the
diagonal $n=m$ term in the sum [130]. Thus, to zeroth order, the solution for
$c_{m}(t)$ is given by

$\displaystyle c_{m}(t)=c_{m}(0)e^{i\gamma^{m}(t)}.$ (S91)

Here, $\gamma^{m}$ denotes the symplectic version of the Berry phase [131]

$\displaystyle\gamma^{m}(t)=i\int_{0}^{t}dt^{\prime}s_{m}(\bm{w}^{m})^{\dagger}\tau_{z}\partial_{\mu}\bm{w}^{m}\dot{\lambda}^{\mu}=\int_{\bm{\lambda}(0)}^{\bm{\lambda}(t)}A_{\mu}^{m}(\bm{\lambda})d\lambda^{\mu}=\int
A^{m},$ (S92)

which may also written as an integral over the symplectic Berry connection
$A^{m}=A^{m}_{\mu}d\lambda^{\mu}=[is_{m}(\bm{w}^{m})^{\dagger}\tau_{z}\partial_{\mu}\bm{w}^{m}]d\lambda^{\mu}$
[97, 7, 45, 17]. The result (S91) represents the conventional adiabatic
theorem (adiabatic limit) adapted to the time evolution Bogoliubov modes,
i.e., under slow parameter variations the system will remain in the initially
occupied mode while acquiring an additional symplectic Berry phase
$\gamma^{m}(t)$. We stress that this represents the $0$-th order contribution,
since Berry phase is independent of the rate of change in the parameters
$\dot{\lambda}^{\mu}$ [cf. Eq. S92].

Next, we aim to solve (S90) perturbatively, to self-consistently obtain
corrections to first order in $\dot{\lambda}^{\mu}$. To this end, we insert
the $0$-th order solution $c_{m}(t)$ (S91) into the right-hand side of the Eq.
S90, by which we obtain

$\displaystyle c_{m}(t)$ $\displaystyle\approx
c_{m}(0)-\int_{0}^{t}dt^{\prime}\sum_{n}c_{n}(0)e^{i\gamma^{n}(t^{\prime})}s_{m}(\bm{w}^{m})^{\dagger}\tau_{z}\partial_{\mu}\bm{w}^{n}\dot{\lambda}^{\mu}e^{i\xi_{nm}(t^{\prime})}$
(S93)

Now for all other coefficients $c_{k}(t)$ with $k\neq m$, we only need to
consider the $n=m$ term within the sum in (S93), where $m$ is again the index
of the initially occupied mode. To first order, this leads to the following
equation for $c_{k\neq m}(t)$

$\displaystyle c_{k}(t)$
$\displaystyle=-\int_{0}^{t}dt^{\prime}c_{m}(0)e^{i\gamma^{m}(t^{\prime})}s_{k}(\bm{w}^{k})^{\dagger}\tau_{z}\partial_{\mu}\bm{w}^{m}\dot{\lambda}^{\mu}e^{i\xi_{km}(t^{\prime})}$
$\displaystyle=i\int_{0}^{t}dt^{\prime}e^{i\gamma^{m}(t^{\prime})}s_{k}\frac{(\bm{w}^{k})^{\dagger}\tau_{z}\partial_{\mu}\bm{w}^{m}}{\tilde{\Omega}_{km}}\dot{\lambda}^{\mu}\frac{d}{dt^{\prime}}e^{i\xi_{km}(t^{\prime})},$
(S94)

with $\tilde{\Omega}_{km}\equiv\tilde{\Omega}_{k}-\tilde{\Omega}_{m}$ and
where we have used that
$\frac{d}{dt^{\prime}}e^{i\xi_{km}(t^{\prime})}=\tilde{\Omega}_{km}e^{i\xi_{km}(t^{\prime})}$.
Due to the fast oscillating exponential factor in Eq. S94, we can integrate
this expression by parts [130] and obtain the following solution for the off-
diagonal coefficients

$\displaystyle
c_{k}(t)=is_{k}e^{i\gamma^{m}(t)}\frac{(\bm{w}^{k})^{\dagger}\tau_{z}\partial_{\mu}\bm{w}^{m}}{\tilde{\Omega}_{km}}\dot{\lambda}^{\mu}e^{i\xi_{km}(t)}.$
(S95)

Thus, to first order in $\dot{\lambda}^{\mu}$, the state $\bm{\psi}(t)$ (S86)
is finally given by

$\displaystyle\bm{\psi}(t)=$ $\displaystyle
e^{-i(\xi_{m}(t)-\gamma^{m}(t))}\left[\bm{w}^{m}+i\sum_{k\neq
m}\bm{w}^{k}s_{k}\frac{(\bm{w}^{k})^{\dagger}\tau_{z}\partial_{\mu}\bm{w}^{m}}{(\tilde{\Omega}_{km})}\dot{\lambda}^{\mu}\right]$
$\displaystyle=$ $\displaystyle
e^{-i(\xi_{m}(t)-\gamma^{m}(t))}\left[\bm{w}^{m}+i\sum_{k\neq
m}\bm{w}^{k}s_{k}\frac{(\bm{w}^{k})^{\dagger}\partial_{\mu}\mathcal{M}\bm{w}^{m}}{(\tilde{\Omega}_{km})^{2}}\dot{\lambda}^{\mu}\right],$
(S96)

where we have reintroduced the dynamical phase $c_{n}(t)\to
c_{n}(t)e^{-i\xi_{n}(t)}$ and utilized the symplectic version of the Hellmann-
Feynman theorem (S32) in the second equality. Eq. S96 constitutes the first
order correction (in $\dot{\lambda}^{\mu}$) to the adiabatic theorem for the
time evolution of Bogoliubov modes.

If required, one may also normalize the state $\bm{\psi}(t)$ (S96) up to
second order [132], i.e., up to
$(\dot{\lambda}^{\mu})^{2}\equiv(v^{\mu})^{2}$, which leads to

$\displaystyle\bm{\psi}(t)=e^{-i(\xi_{m}(t)-\gamma^{m}(t))}\left[(1+\beta_{\mu}^{m}(v^{\mu})^{2})\bm{w}^{m}+iv^{\mu}\sum_{k\neq
m}\alpha^{km}_{\mu}\bm{w}^{k}\right],$ (S97)

where have defined $\alpha^{km}_{\mu}\equiv
s_{k}\frac{(\bm{w}^{k})^{\dagger}\partial_{\mu}\mathcal{M}\bm{w}^{m}}{(\tilde{\Omega}_{km})^{2}}$
with $\beta_{\mu}^{m}\equiv-\frac{1}{2}s_{m}\sum_{k\neq
m}\left\lvert\alpha^{km}_{\mu}\right\rvert^{2}$.

##  S9 Symplectic anomalous velocity

In this appendix, we provide a more detailed derivation of the symplectic
anomalous velocity term, which is proportional to the symplectic Berry
curvature. Consider a translationally invariant BdG system such that the
Bogoliubov eigenmodes of the dynamical matrix $D$ admit Bloch function type
solution, i.e.,
$D\bm{\psi}^{n}(\bm{\mathbf{q}})=De^{i\bm{\mathbf{q}}\cdot\bm{\mathbf{r}}}\bm{w}^{n}(\bm{\mathbf{q}})=\tilde{\Omega}_{n}(\bm{\mathbf{q}})e^{i\bm{\mathbf{q}}\cdot\bm{\mathbf{r}}}\bm{w}^{n}(\bm{\mathbf{q}})$.
Here, $\bm{w}^{n}(\bm{\mathbf{q}})$ is the cell-periodic part of the
Bogoliubov Bloch function for the $n$-th band, which fulfill the Bogoliubov
eigenvalue equation for the quasimomentum transformed dynamical matrix
$D(\bm{\mathbf{q}})\bm{w}^{n}(\bm{\mathbf{q}})=\tilde{\Omega}_{n}(\bm{\mathbf{q}})\bm{w}^{n}(\bm{\mathbf{q}})$.
To make connections to the conventional anomalous velocity, we consider a two-
dimensional time-dependent quasimomentum parameter space
$\bm{\mathbf{q}}(t)=(q_{x}(t),q_{y}(t))$. We imagine a uniform force
$\bm{\mathbf{F}}$ to be applied to the Bogoliubov system of interest, such
that quasimomenta satisfy $\dot{\bm{\mathbf{q}}}=\bm{\mathbf{F}}$, leading to
$\bm{\mathbf{q}}(t)=\bm{\mathbf{q}}(0)+\bm{\mathbf{F}}t$.

Next, we consider an initially populated Bogoliubov particle mode
$\bm{\psi}(0)=\bm{w}^{n}(\bm{\mathbf{q}})$ and let the applied field point in
the $x$-direction, i.e., $\bm{\mathbf{F}}=F_{x}\bm{e}_{x}$. Using the results
of Eq. S96, we see that the solution to the time-dependent effective
Schrödinger equation for the Bogoliubov modes is, to first order in $F_{x}$,
given by

$\displaystyle\bm{\psi}(\bm{\mathbf{q}})=\bm{w}^{n}(\bm{\mathbf{q}})+iF_{x}\sum_{m\neq
n}\bm{w}^{m}(\bm{\mathbf{q}})s_{n}\frac{(\bm{w}^{m}(\bm{\mathbf{q}}))^{\dagger}\partial_{q_{x}}\mathcal{M}\bm{w}^{n}(\bm{\mathbf{q}})}{(\tilde{\Omega}_{n}(\bm{\mathbf{q}})-\tilde{\Omega}_{m}(\bm{\mathbf{q}}))^{2}}+\mathcal{O}(F_{x}^{2}),$
(S98)

where the irrelevant phase factors have been omitted for clarity. To obtain
the velocity in the $y$-direction to first order in $F_{x}$, we take the
expectation value of the quasiparticle velocity operator
$v_{y}(\bm{\mathbf{q}})=\partial_{q_{y}}D(\bm{\mathbf{q}})$ with respect to
the state of Eq. S98, where $D(\bm{\mathbf{q}})$ is again the dynamical matrix
in the quasimomentum space representation. Note that
$v_{y}(\bm{\mathbf{q}})=\partial_{q_{y}}D(\bm{\mathbf{q}})$ just follows from
the Heisenberg equations of motion for the position operator $\bm{\mathbf{r}}$
transformed into the quasimomentum space [61] and then extended to Nambu
space. The velocity then equates to

$\displaystyle\begin{split}\langle
v_{y}(\bm{\mathbf{q}})\rangle&=(\bm{\psi}(\bm{\mathbf{q}}))^{\dagger}\tau_{z}\partial_{q_{y}}D(\bm{\mathbf{q}})\bm{\psi}(\bm{\mathbf{q}})\\\
&=(\bm{w}^{n})^{\dagger}\partial_{q_{y}}\mathcal{M}(\bm{\mathbf{q}})\bm{w}^{n}+\left(iF_{x}\sum_{m\neq
n}s_{n}\frac{\bm{w}^{n}\partial_{q_{y}}\mathcal{M}\bm{w}^{m}\times(\bm{w}^{m})^{\dagger}\partial_{q_{x}}\mathcal{M}\bm{w}^{n}}{(\tilde{\Omega}_{n}-\tilde{\Omega}_{m})^{2}}+\text{c.c.}\right)+\mathcal{O}(F_{x}^{2}).\end{split}$
(S99)

By making use of the eigenvalue equation for the Bogoliubov modes,
$D(\bm{\mathbf{q}})\bm{w}^{n}(\bm{\mathbf{q}})=\tilde{\Omega}_{n}(\bm{\mathbf{q}})\bm{w}^{n}(\bm{\mathbf{q}})$
(S29), as well as the spectral representation of the symplectic Berry
curvature $B^{n}_{\mu\nu}=-2\,\mathrm{Im}[\eta^{n}_{\mu\nu}]$ (S33), we see
that the velocity $v_{y}$ of the Bogoliubov state in $n$-th band is given by

$\displaystyle\langle
v_{y}(\bm{\mathbf{q}})\rangle=\frac{\partial\tilde{\Omega}_{n}(\bm{\mathbf{q}})}{\partial
q_{y}}+F_{x}B^{n}_{xy}(\bm{\mathbf{q}}).$ (S100)

By symmetry, we can immediately generalize this result (S100) to general
directions of the applied force $\bm{\mathbf{F}}$

$\displaystyle\langle\bm{\mathbf{v}}(\bm{\mathbf{q}})\rangle=\nabla_{\bm{\mathbf{q}}}\tilde{\Omega}_{n}(\bm{\mathbf{q}})+\bm{\mathbf{B}}_{n}(\bm{\mathbf{q}})\times\bm{\mathbf{F}},$
(S101)

with
$\bm{\mathbf{B}}_{n}(\bm{\mathbf{q}})\equiv\epsilon_{\mu\nu\kappa}B^{n}_{\mu\nu}(\bm{\mathbf{q}})\bm{e}_{\kappa}$.
Here, the first term corresponds to the slope of the occupied Bogoliubov
energy band $\tilde{\Omega}_{n}(\bm{\mathbf{q}})$ and the second term is
proportional to the symplectic Berry curvature
$\bm{\mathbf{B}}_{n}(\bm{\mathbf{q}})$. The second term describes a transverse
velocity component and may be viewed as a symplectic generalization of the
conventional anomalous velocity term [61, 62, 63].

The results above may easily be generalized to the case of an initial
Bogoliubov wave packet consisting of a superposition (particle) Bogoliubov
modes
$\bm{\psi}^{n}(\bm{\mathbf{q}})=e^{i\bm{\mathbf{q}}\cdot\bm{\mathbf{r}}}\bm{w}^{n}(\bm{\mathbf{q}})$
of a given band $n$,

$\displaystyle\bm{\phi}^{n}=\sum_{\bm{\mathbf{q}}}c(\bm{\mathbf{q}})e^{i\bm{\mathbf{q}}\cdot\bm{\mathbf{r}}}\bm{w}^{n}(\bm{\mathbf{q}}),$
(S102)

which is normalized with respect to the Bogoliubov norm, i.e.,
${\langle}\phi^{n}{\,|}\tau_{z}{|\,}\phi^{n}{\rangle}=\sum_{\bm{\mathbf{q}}}\left\lvert
c(\bm{\mathbf{q}})\right\rvert^{2}=1$ [97]. The velocity of such a Bogoliubov
wave packet may then be related to a bulk observable, such as the total
current density $j_{y}$ in real space

$\displaystyle
j_{y}=\frac{1}{L^{2}}\sum_{\bm{\mathbf{q}}}\rho(\bm{\mathbf{q}})\langle
v_{y}(\bm{\mathbf{q}})\rangle=\int_{1.\,\text{BZ}}\frac{d^{2}\bm{\mathbf{q}}}{(2\pi)^{2}}\rho(\bm{\mathbf{q}})\langle
v_{y}(\bm{\mathbf{q}})\rangle,$ (S103)

where $L^{2}$ is the area of the system and $\rho(\bm{\mathbf{q}})=\left\lvert
c(\bm{\mathbf{q}})\right\rvert^{2}$ the distribution function of Bogoliubov
quasiparticles in the respective band. Although $j_{y}$ is not quantized by
the (symplectic) Chern number of the respective Bogoliubov band (for the
typical case of a non-uniformly populated Bogoliubov band), the current
density still gives rise to a finite transverse velocity contribution, which
is directly related to the anomalous term governed by the underlying
symplectic Berry curvature of the Bogoliubov band. We note in passing that
special care may be required when switching to the continuum of quasimomentum
states in Eq. S103, which is however beyond the scope of this work [133, 61].

##  S10 Bogoliubov Haldane model

In this appendix, we provide a brief derivation of the Bogoliubov-Haldane
model considered as an example in the main text, which mainly follows
Furukawa2015,Refs. Ref. [21]. Adding to this, we also review the numerical
algorithm described in Colpa1978,Shindou2013,Refs. Ref. [2, 7] for obtaining
the Bogoliubov matrices (S6) with the correct paraunitarity conditions (S20).

The starting point is a Bose-Hubbard model on a two-dimensional lattice with
two sublattice sites,

$\displaystyle\hat{H}=\sum_{\ell\ell^{\prime},ss^{\prime}}h_{\ell
s,\ell^{\prime}s^{\prime}}\hat{a}^{\dagger}_{\ell
s}\hat{a}_{\ell^{\prime}s^{\prime}}+\frac{U}{2}\sum_{\ell,s}\hat{n}_{\ell
s}(\hat{n}_{\ell s}-1)$ (S104)

where the elementary unit cells are labeled by $\ell$, the sublattice sites by
$s\in\\{A,B\\}$. The non-interacting (quadratic) part is governed by the
matrix elements $h_{\ell s,\ell^{\prime}s^{\prime}}$ in position space and the
interacting part by the on-site interaction strength $U$.

The first summand is the non-interacting part, which may diagonalized by first
moving into quasimomentum space via

$\displaystyle\mathcal{H}_{ss^{\prime}}(\bm{\mathbf{q}})$
$\displaystyle\equiv\sum_{\ell\ell^{\prime}}h_{\ell
s,\ell^{\prime}s^{\prime}}e^{-i(\bm{\mathbf{r}}_{\ell
s}-\bm{\mathbf{r}}_{\ell^{\prime}s^{\prime}})\cdot\bm{\mathbf{q}}}=\left(h_{0}(\bm{\mathbf{q}})\mathbbm{1}+\bm{\mathbf{h}}(\bm{\mathbf{q}})\cdot\bm{\hat{\sigma}}\right)_{ss^{\prime}},$
(S105)

where $\bm{\hat{\sigma}}=(\sigma_{x},\sigma_{y},\sigma_{z})^{T}$ is the vector
of Pauli matrices. At each quasimomentum point, $\bm{\mathbf{q}}$, the
Hamiltonian $\mathcal{H}_{ss^{\prime}}(\bm{\mathbf{q}})$ describes an operator
acting on a two-dimensional Hilbert space, whose elements can be represented
on the Bloch sphere. The north and south pole are denoted by
${\lvert}\bm{\mathbf{q}}A{\rangle}$ and ${\lvert}\bm{\mathbf{q}}B{\rangle}$
respectively. Due to the non-interacting nature of the system, the model can
be further diagonalized by rotating into the eigenbasis
${\lvert}\bm{\mathbf{q}}\pm{\rangle}$,

$\displaystyle\begin{split}{\lvert}\bm{\mathbf{q}}-{\rangle}&=\sin(\theta_{\bm{\mathbf{q}}}/2){\lvert}\bm{\mathbf{q}}A{\rangle}-\cos(\theta_{\bm{\mathbf{q}}}/2)e^{i\varphi_{\bm{\mathbf{q}}}}{\lvert}\bm{\mathbf{q}}B{\rangle},\quad{\lvert}\bm{\mathbf{q}}+{\rangle}=\cos(\theta_{\bm{\mathbf{q}}}/2){\lvert}\bm{\mathbf{q}}A{\rangle}+\sin(\theta_{\bm{\mathbf{q}}}/2)e^{i\varphi_{\bm{\mathbf{q}}}}{\lvert}\bm{\mathbf{q}}B{\rangle}.\end{split}$
(S106)

which reside at $\pm\hat{\bm{\mathbf{h}}}(\bm{\mathbf{q}})$ on the Bloch
sphere, with
$\hat{\bm{\mathbf{h}}}(\bm{\mathbf{q}})=\bm{\mathbf{h}}(\bm{\mathbf{q}})/|\bm{\mathbf{h}}(\bm{\mathbf{q}})|=(\cos(\varphi_{\bm{\mathbf{q}}})\sin(\theta_{\bm{\mathbf{q}}}),\sin(\varphi_{\bm{\mathbf{q}}})\sin(\theta_{\bm{\mathbf{q}}}),\cos(\theta_{\bm{\mathbf{q}}}))^{T}$.
The corresponding eigenenergies $\epsilon_{\pm}(\bm{\mathbf{q}})$ of
${\lvert}\bm{\mathbf{q}}\pm{\rangle}$ are given by
$\epsilon_{\pm}(\bm{\mathbf{q}})=h_{0}(\bm{\mathbf{q}})\pm\left\lvert\bm{\mathbf{h}}(\bm{\mathbf{q}})\right\rvert$
which define the band structure of the lattice. Now, an ideal Bose gas at zero
temperature [with $U=0$ in (S104)] forms a perfect Bose-Einstein condensate
(BEC) characterized by all bosons condensing into the lowest energy eigenstate
${\lvert}\bm{\mathbf{q}}_{0},-{\rangle}$ at some quasimomentum
$\bm{\mathbf{q}}_{0}$ in the lower band $\epsilon_{-}(\bm{\mathbf{q}})$, with
the ground state given by
${\lvert}\psi_{B}{\rangle}=(\sqrt{N!})^{-1}(\hat{a}^{\dagger}_{\bm{\mathbf{q}}_{0}-})^{N}{\lvert}\text{vac}{\rangle}$.
Here, $N$ shall denote the total number of bosons in the system and
$\hat{a}^{\dagger}_{\bm{\mathbf{q}},\pm}$ the creation operator for a boson at
quasimomentum $\bm{\mathbf{q}}$ in the lower or upper band, respectively.

In this work, we consider the Haldane model [92] on a hexagonal lattice for
which the non-interacting part (S105) reads,

$\displaystyle\begin{split}h_{x}(\bm{\mathbf{q}})&=J_{1}\sum_{i}\cos(\bm{\mathbf{q}}\cdot\bm{\mathbf{d}}_{i}),\
h_{y}(\bm{\mathbf{q}})=J_{1}\sum_{i}\sin(\bm{\mathbf{q}}\cdot\bm{\mathbf{d}}_{i}),\\\
h_{z}(\bm{\mathbf{q}})&=\Delta+2J_{2}\sin(\theta)\sum_{i}\sin(\bm{\mathbf{q}}\cdot\bm{\mathbf{a}}_{i}),\
h_{0}(\bm{\mathbf{q}})=2J_{2}\cos(\theta)\sum_{i}\cos(\bm{\mathbf{q}}\cdot\bm{\mathbf{a}}_{i}),\
\end{split}$ (S107)

where $J_{1}(J_{2}e^{i\theta})$ is the real nearest neighbor (complex next-
nearest neighbor) hopping amplitude and $\Delta$ the mass term (sublattice
offset). The complex next-nearest neighbor hopping amplitude breaks the time
reversal symmetry of the system, and the mass term $\Delta$ breaks the
inversion symmetry of the lattice. On the hexagonal lattice, translation
vectors connecting the nearest neighbors are denoted by $\bm{\mathbf{d}}_{i}$,
while the vectors $\bm{\mathbf{a}}_{i}$ connect the next-nearest neighbors
[Fig. S1],