# Emergent Non-Abelian Gauge Theory in Coupled Spin-Electron Dynamics

Nicolas Lenzing  I. Institute of Theoretical Physics, Department of Physics,
University of Hamburg, Notkestraße 9-11, 22607 Hamburg, Germany  Alexander I.
Lichtenstein  I. Institute of Theoretical Physics, Department of Physics,
University of Hamburg, Notkestraße 9-11, 22607 Hamburg, Germany  The Hamburg
Centre for Ultrafast Imaging, Luruper Chaussee 149, 22761 Hamburg, Germany
Michael Potthoff  I. Institute of Theoretical Physics, Department of Physics,
University of Hamburg, Notkestraße 9-11, 22607 Hamburg, Germany  The Hamburg
Centre for Ultrafast Imaging, Luruper Chaussee 149, 22761 Hamburg, Germany

###### Abstract

A clear separation of the time scales governing the dynamics of “slow” and
“fast” degrees of freedom often serves as a prerequisite for the emergence of
an independent low-energy theory. Here, we consider (slow) classical spins
exchange coupled to a tight-binding system of (fast) conduction electrons. The
effective equations of motion are derived under the constraint that the
quantum state of the electron system at any instant of time $t$ lies in the
$n$-dimensional low-energy subspace for the corresponding spin configuration
at $t$. The effective low-energy theory unfolds itself straightforwardly and
takes the form of a non-abelian gauge theory with the gauge freedom given by
the arbitrariness of the basis spanning the instantaneous low-energy sector.
The holonomic constraint generates a gauge covariant spin-Berry curvature
tensor in the equations of motion for the classical spins. In the non-abelian
theory for $n>1$, opposed to the $n=1$ adiabatic spin dynamics theory, the
spin-Berry curvature is generically nonzero, even for time-reversal symmetric
systems. Its expectation value with the representation of the electron state
is gauge invariant and gives rise to an additional geometrical spin torque.
Besides anomalous precession, the $n\geq 2$ theory also captures the spin
nutational motion, which is usually considered as a retardation effect. This
is demonstrated by proof-of-principle numerical calculations for a minimal
model with a single classical spin. Already for $n=2$ and in parameter regimes
where the $n=1$ adiabatic theory breaks down, we find good agreement with
results obtained from the full (unconstrained) theory.

##  I Introduction

Classical spin models Nowak (2007); Bertotti  _et al._ (2009) are a highly
useful and widely employed tool to understand the non-equilibrium dynamics of
magnetic materials. At the expense of disregarding the quantum nature of the
magnetic moments and related phenomena, such as the Kondo effect Kondo (1964);
Hewson (1993), they provide a numerically tractable framework for spin
dynamics on an atomistic length scale Tatara  _et al._ (2008); Skubic  _et
al._ (2008); Fähnle and Illg (2011); Evans  _et al._ (2014). Typically,
classical spin models may comprise a short-range isotropic Heisenberg-type
exchange, various anisotropic couplings and long-range, e.g., dipole
interactions. The classical equations of motion are usually supplemented by
Gilbert-damping terms to account for dissipation effects.

Spin-only models can actually be seen as effective low-energy theories
emerging from a more fundamental level of modelling, where the local magnetic
moments (classical spins ${\boldsymbol{S}}_{i}$) at sites $i$ of a lattice are
coupled to the local spins ${\boldsymbol{s}}_{i}$ of a system of conduction
electrons via a local exchange coupling $J$. Such quantum-classical spin-
electron hybrid models are necessary to explain various phenomena, including
indirect spin exchange interactions, like the Ruderman-Kittel-Kasuya-Yoshida
(RKKY) interaction rkk , Gilbert spin damping due to coupling to electronic
degrees of freedom llg , spin inertia effects (nutation) Butikov (2006);
Wegrowe and Ciornei (2012), and other more strongly retarded effective spin-
spin interactions mediated by the conduction-electron system.

The standard formal approach Onoda and Nagaosa (2006); Umetsu  _et al._
(2012); Bhattacharjee  _et al._ (2012); Sayad and Potthoff (2015); Bajpai and
Nikolic (2019) that achieves the derivation of the effective spin-only theory
is based on the (usually realistic) assumption that the local exchange
coupling $J$ is weak as compared to the typical energy scales of the electron
system. Consider the $s$-$d$ model vz  with Hamiltonian $H=H_{\rm
el.}+J\sum_{i}{\boldsymbol{s}}_{i}{\boldsymbol{S}}_{i}$ as a prototype. The
torque on the classical spin at site $i$ is given by
$J\langle{\boldsymbol{s}}_{i}\rangle_{t}\times{\boldsymbol{S}}_{i}$, where the
expectation value of the local electron spin ${\boldsymbol{s}}_{i}$ at site
$i$ is obtained from the many-body state $|\Psi(t)\rangle$ of the electron
system (Hamiltonian $H_{\rm el.}$) at time $t$. Since the electron state
itself must be computed in the presence of the local exchange interaction term
$\propto J$ for the (time-dependent) classical spin configuration
$\\{{\boldsymbol{S}}\\}$, there is a retarded mutual effective interaction
emerging. This is uncovered, for example, by linear-response theory, i.e., by
lowest-order time-dependent perturbation theory in $J$. This leads to an
integro-differential equation of motion for ${\boldsymbol{S}}_{i}$,

$\dot{{\boldsymbol{S}}}_{i}(t)=J^{2}\sum_{i^{\prime}}\int_{0}^{t}dt^{\prime}\underline{\chi}_{ii^{\prime}}(t-t^{\prime})\boldsymbol{S}_{i^{\prime}}(t^{\prime})\times{\boldsymbol{S}}_{i}(t)$
(1)

which involves the retarded magnetic susceptibility tensor with elements
${\chi}_{ii^{\prime}}^{(\alpha\alpha^{\prime})}(t-t^{\prime})$ of the electron
ground state as the integral kernel ($\alpha,\alpha^{\prime}=x,y,z$). The
resulting spin dynamics is non-conservative, as Eq. (1) describes an open
quantum system and is known from Redfield theory Breuer and Petruccione
(2002).

Assuming that $\underline{\chi}_{ii^{\prime}}(t-t^{\prime})$ is strongly
peaked at $t^{\prime}=t$, we can replace
${\boldsymbol{S}}_{i^{\prime}}(t^{\prime})$ by the first few terms in its
Taylor expansion around $t^{\prime}=t$, i.e.,
${\boldsymbol{S}}_{i^{\prime}}(t^{\prime})\approx{\boldsymbol{S}}_{i^{\prime}}(t)+\dot{{\boldsymbol{S}}}_{i^{\prime}}(t)(t^{\prime}-t)+\ddot{{\boldsymbol{S}}}_{i^{\prime}}(t)(t^{\prime}-t)^{2}/2$.
Keeping the first term on the right-hand side only and extending the
integration over $t^{\prime}$ to infinity, one obtains an effective
Hamiltonian equation of motion for the spins ${\boldsymbol{S}}_{i}$, which
involves the instantaneous spin-spin interaction mediated by the RKKY coupling
$J_{ii^{\prime}}^{\rm(RKKY)}=J^{2}\chi_{ii^{\prime}}(\omega=0)$. Including the
second term in addition, gives rise to a (non-local) Gilbert damping tensor
$\underline{\alpha}_{ii^{\prime}}=-iJ^{2}\partial_{\omega}\underline{\chi}_{ii^{\prime}}(\omega)|_{\omega=0}$,
while the third term leads to spin-inertia effects, i.e., additional nutation
of the spins. This derivation has been put forward in Refs. Bhattacharjee  _et
al._ (2012); Sayad and Potthoff (2015) and can be employed in the context of
strongly correlated electron models Sayad  _et al._ (2016a) or, when combined
with band-structure theory, for an ab initio computation of the Gilbert
damping Antropov  _et al._ (1995); Kuneš and Kamberský (2002); Capelle and
Gyorffy (2003); Ebert  _et al._ (2011). Nutation effects, as have been
discussed in Refs. Fähnle  _et al._ (2011); Kikuchi and Tatara (2015); Sayad
_et al._ (2016b), for example, find a natural explanation in the same
framework set by Eq. (1). Furthermore, at least in principle, systematic
extensions of the resulting low-energy spin-only theory can be achieved by
taking into account terms of higher order in the expansion. One may also drop
the approximation on the $t^{\prime}$-integration range. This leads to a time-
dependent RKKY coupling $J^{(RKKY)}_{ii^{\prime}}(t)$ and a time-dependent
Gilbert damping $\underline{\alpha}_{ii^{\prime}}(t)$, as has been mentioned
in Ref. Sayad and Potthoff (2015); Bajpai and Nikolic (2019).

The above-sketched standard theory misses, however, an important effect
pointed out recently Stahl and Potthoff (2017): The slow dynamics of the
classical spins results in a non-trivial Berry curvature of the electronic
quantum system as is well known since long Berry (1984); Xiao  _et al._
(2010). Quite generally, however, this Berry curvature in turn has a feedback
on the classical spin dynamics Wen and Zee (1988); Niu and Kleinman (1998);
Bohm  _et al._ (2003); Niu  _et al._ (1999); Stahl and Potthoff (2017).
Namely, there is a geometrical spin torque which comes with the same prefactor
$J^{2}$ as the RKKY coupling and the Gilbert damping. This torque can give
rise to unconventional spin dynamics as has been demonstrated Stahl and
Potthoff (2017); Bajpai and Nikolić (2020) not only for a quantum-classical
system as is considered here as well, but also for slow classical spins
locally exchange coupled to a system of fast classical spins Elbracht  _et
al._ (2020); Michel and Potthoff (2021) and even for the dynamics of a quantum
spin in a Kondo model Stahl and Potthoff (2017).

This geometrical spin torque emerges in an effective low-energy spin-only
theory that is derived by starting from full theory of classical spins coupled
to conduction electrons by imposing the constraint that, at any instant of
time $t$, the electron system is in its ground state, i.e.,
$|\Psi(t)\rangle=|\Psi_{0}(\\{{\boldsymbol{S}}(t)\\})\rangle$, for the spin
configuration $\\{{\boldsymbol{S}}(t)\\}$ at time $t$. This is analogous to
molecular dynamics approaches Marx and Hutter (2000); Bohm  _et al._ (2003);
Zhang and Wu (2006) where the slow nuclear coordinates are treated
classically. If the exchange coupling $J$ is weak, the classical spin dynamics
is slow compared to typical energy scales of the electron systems. The
adiabatic spin dynamics (ASD) thus addresses the same parameter regime as the
standard perturbative linear-response approach discussed above.

With the present paper we explore a systematic extension of the ASD by
relaxing the adiabatic constraint. The impact of electronic low-energy
excitations from the instantaneous ground state
$|\Psi_{0}(\\{{\boldsymbol{S}}(t)\\})\rangle$ on the classical spin dynamics
can be taken into account by imposing, as a weaker constraint, that the
electron state $|\Psi(t)\rangle$ be at time $t$ in the subspace of the Fock
space spanned by the first $n>1$ eigenstates of the Hamiltonian for the spin
configuration $\\{{\boldsymbol{S}}(t)\\}$ at $t$. This beyond-adiabatic
constraint leads to a non-abelian Berry connection and curvature Xiao  _et
al._ (2010); Wilczek and Zee (1984). Here, we will work out the general
formalism of the non-abelian gauge theory that emerges as the effective low-
energy theory. The formally correct incorporation of the constraint is
achieved within conventional Lagrange formalism. A simple toy model will be
considered and solved numerically to study the effect of the geometric torque
on the classical spin dynamics in the non-abelian case. We discuss the
anomalies in the precessional spin dynamics and demonstrate that spin nutation
arises naturally in our framework. The previously developed ASD represents the
$n=1$ limit of our non-abelian spin-dynamics (NA-SD) theory. In the ASD for a
single classical spin, the presence of an anomalous precession frequency has
been found Stahl and Potthoff (2017) for an odd number of conduction electrons
only, while the full solution of the coupled equations of motion for spin and
electron dynamics yields an anomalous frequency for both, odd and even
electron numbers. In the broader framework of NA-SD we can resolve this open
issue.

The paper is organized as follows: The next section II presents the general
Hamiltonian and Lagrangian formulation of the theory. The equations of motion
of the non-abelian gauge theory in the instantaneous low-energy sector are
worked out in Sec. III, and various formal aspects of the theory are discussed
in Sec. IV. Sections V and VI are particularly devoted to a discussion of the
impact of time-reversal symmetry and of gauge transformations, respectively. A
minimal model, suitable for proof-of-principle studies, is introduced in Sec.
VII. In Sec. VIII we present and discuss the results of numerical
calculations. Conclusions are given in Sec. IX.

##  II General Theory

Geometric forces or torques originate in the adiabatic limit of hybrid systems
consisting of quantum degrees of freedom interacting with classical degrees of
freedom. Here, we consider a quantum lattice model of $N$ conduction electrons
interacting with $M$ classical “spins” ${\boldsymbol{S}}_{m}$ of unit length
$|{\boldsymbol{S}}_{m}|=1$. The system dynamics is governed by a quantum-
classical Hamiltonian of the form

$\hat{H}(\\{\boldsymbol{S}\\})=\hat{H}_{\text{qu}}+{H}_{\text{cl}}(\\{\boldsymbol{S}\\})+\hat{H}_{\rm
int}(\\{\boldsymbol{S}\\})\>.$ (2)

The quantum Hamiltonian $\hat{H}_{\text{qu}}$ is constructed in terms of
fermion creation and annihilation operators
$c^{\dagger}_{{\boldsymbol{r}}\sigma}$ and $c_{{\boldsymbol{r}}\sigma}$, where
${\boldsymbol{r}}$ refers to the sites of the lattice and
$\sigma=\uparrow,\downarrow$ is the spin projection. Additional orbital
degrees of freedom may be considered as well. The formulation of the theory is
largely independent of $\hat{H}_{\text{qu}}$ but requires a well-defined local
quantum spin ${\boldsymbol{s}}_{{\boldsymbol{r}}}$ at lattice site
${\boldsymbol{r}}$:

${\boldsymbol{s}}_{{\boldsymbol{r}}}=\frac{1}{2}\sum_{\sigma\sigma^{\prime}}c^{\dagger}_{{\boldsymbol{r}}\sigma}{\boldsymbol{\sigma}}_{\sigma\sigma^{\prime}}c_{{\boldsymbol{r}}\sigma^{\prime}}\>.$
(3)

Here, ${\boldsymbol{\sigma}}$ is the vector of $2\times 2$ Pauli matrices (and
$\hbar\equiv 1$).

The dynamics of the subsystem of $M$ classical spins
$\\{{\boldsymbol{S}}\\}\equiv\\{{\boldsymbol{S}}_{1},...,{\boldsymbol{S}}_{M}\\}$
derives from a classical Hamilton function
${H}_{\text{cl}}(\\{\boldsymbol{S}\\})$ and may comprise an external magnetic
field and isotropic or anisotropic spin exchange couplings. The third term in
Eq. (2) represents a quantum-classical interaction term. Here, we choose an
isotropic local exchange interaction

$\hat{H}_{\text{int}}(\\{\boldsymbol{S}\\})=J\sum_{m=1}^{M}\boldsymbol{S}_{m}\boldsymbol{s}_{{\boldsymbol{r}}_{m}}\>,$
(4)

between the $m$-th classical spin ${\boldsymbol{S}}_{m}$ and the local spin
${\boldsymbol{s}}_{{\boldsymbol{r}}_{m}}$ of the conduction-electron system at
the site ${\boldsymbol{r}}_{m}$. The coupling strength is $J>0$. The theory is
developed for an arbitrary number of classical spins $M$, but we will later
focus on a single-classical-spin Kondo model ($M=1$) for the sake of
simplicity.

If the classical spins $\\{{\boldsymbol{S}}\\}$ were replaced by quantum
spins, Eq. (2) would represent the Hamiltonian of the multi-impurity or
lattice Kondo model. With the classical-spin approximation we disregard
typical correlation effects, such as Kondo screening and heavy-fermion
behavior, and hence we are essentially working on a mean-field-type level. The
approximation may be justified in cases where there are well-formed spin
moments which are stable on time scales exceeding all remaining time scales of
the problem, e.g., in cases, where the Kondo effect is suppressed by magnetism
or in case of quantum spins with large spin quantum numbers. An example has
been given in Ref. Sayad  _et al._ (2016a), where anomalous quantum-classical
dynamics due to a geometrical torque has also been found in the corresponding
full quantum system. A consistent theory for a system that is entirely quantum
with a least two largely different time scales has yet to be developed. This
means that the presence of slow classical degrees of freedom is necessarily
required for the very concept of geometrical forces and torques. The classical
degrees of freedom are required to define the smooth manifold onto which the
quantum dynamics is restricted in the adiabatic limit.

A pure state of the quantum-classical hybrid system at time $t$ is specified
by a Hilbert-space vector $|\Psi(t)\rangle$ and by the classical spin
configuration $\\{{\boldsymbol{S}}(t)\\}$, see Refs. Heslot (1985); Hall
(2008); Elze (2012) for a general discussion of hybrid dynamics. The
trajectory of the system state is obtained as the solution of a system of
coupled ordinary differential equations. These consist of the Schrödinger
equation, involving the quantum Hamiltonian and the interaction term, which
depends on the classical-spin configuration,

$i\partial_{t}\ket{\Psi(t)}=[\hat{H}_{\text{qu}}+\hat{H}_{\rm
int}(\\{\boldsymbol{S}(t)\\})]\ket{\Psi(t)}\>,$ (5)

and the Hamilton equations of motion for the classical-spin configuration,
involving the classical Hamilton function and the expectation value of the
interaction term in the quantum state $|\Psi(t)\rangle$:

$\dot{{\boldsymbol{S}}}_{m}(t)=\Big{\\{}{\boldsymbol{S}}_{m}(t),{H}_{\text{cl}}(\\{\boldsymbol{S}(t)\\})+\langle\hat{H}_{\rm
int}(\\{\boldsymbol{S}(t)\\})\rangle\Big{\\}}_{S}\>.$ (6)

Here, the dot denotes the time derivative, and $\\{\cdot,\cdot\\}_{S}$ is the
Poisson bracket. In case of spin systems, the latter is defined for two
arbitrary functions $A(\\{\boldsymbol{S}\\})$ and $B(\\{\boldsymbol{S}\\})$ as
Bulgac and Kusnezov (1990)

$\\{A,B\\}_{S}=\sum_{m}\frac{\partial
A}{\partial{\boldsymbol{S}}_{m}}\times\frac{\partial
B}{\partial{\boldsymbol{S}}_{m}}\cdot{\boldsymbol{S}}_{m}\>.$ (7)

The coupled equations of motion, Eq. (5) and Eq. (6), are generated as Euler-
Lagrange equations by requiring stationarity of an action functional ${\cal
S}=\int Ldt$ with the Lagrangian
$L=L(\\{\boldsymbol{S}\\},\\{\dot{\boldsymbol{S}}\\},\ket{\Psi},\dot{\ket{\Psi}},\bra{\Psi},\dot{\bra{\Psi}})$:

$L=\sum_{m}\boldsymbol{A}(\boldsymbol{S}_{m})\dot{\boldsymbol{S}}_{m}+\bra{\Psi(t)}i\partial_{t}-\hat{H}\ket{\Psi(t)}\>.$
(8)

Here, ${\boldsymbol{A}}({\boldsymbol{S}})$ is a function satisfying
$\nabla\times{\boldsymbol{A}}({\boldsymbol{S}})=-{\boldsymbol{S}}/S^{3}$,
which can thus be interpreted as the vector potential of a unit magnetic
monopole located at ${\boldsymbol{S}}=0$. We have

${\boldsymbol{A}}({\boldsymbol{S}})=-\frac{1}{S^{2}}\frac{{\boldsymbol{e}}\times{\boldsymbol{S}}}{1+{\boldsymbol{e}}{\boldsymbol{S}}/S}\;,$
(9)

with a unit vector ${\boldsymbol{e}}$. In the standard gauge Dirac (1931) this
is chosen as ${\boldsymbol{e}}={\boldsymbol{e}}_{z}$. In this gauge, another
representation is
${\boldsymbol{A}}({\boldsymbol{S}})=-(1/S)\tan(\vartheta/2){\boldsymbol{e}}_{\rm\varphi}$,
using spherical coordinates $(S,\vartheta,\varphi)$. For details of deriving
Eq. (5) and Eq. (6) from $\delta{\cal S}=0$, see Ref. Elbracht  _et al._
(2020) (supplemental material).

We will address the parameter regime of the Hamiltonian, where the system
dynamics is characterized by two strongly different time scales, a slow spin
dynamics and a fast dynamics of the electron state, which almost
instantaneously follows the motion of the spins. In the extreme adiabatic
limit, the quantum many-body state $|\Psi(t)\rangle$ of the electron system at
time $t$ is given by the ground state,
$|\Psi_{0}(\\{{\boldsymbol{S}}(t)\\})\rangle$ of
$\hat{H}_{\text{qu}}+\hat{H}_{\text{int}}(\\{\boldsymbol{S}(t)\\})$, for the
spin configuration $\\{{\boldsymbol{S}}(t)\\}$ at time $t$. When approaching
the adiabatic limit in parameter space, the fast electron dynamics will be
more and more constrained to the ground manifold
$\\{|\Psi_{0}(\\{{\boldsymbol{S}}(t)\\})\rangle\\}$. Adiabatic spin-dynamics
(ASD) theory Stahl and Potthoff (2017); Elbracht  _et al._ (2020); Michel and
Potthoff (2021) assumes that the dynamics is perfectly constrained to the
ground-state manifold and employs

$|\Psi(t)\rangle=|\Psi_{0}(\\{{\boldsymbol{S}}(t)\\})\rangle$ (10)

as a holonomic constraint to completely eliminate the electron degrees of
freedom from the Lagrangian Eq. (8). In this way, one arrives at a spin-only
effective Lagrangian $L_{\rm
eff}(\\{{\boldsymbol{S}}\\},\\{\dot{{\boldsymbol{S}}}\\})$, and the resulting
effective equations of motion include the geometrical spin torque as an
holonomy effect Stahl and Potthoff (2017). The unconventional spin dynamics
originating from the corresponding geometrical spin torque is missed by other
approaches, such as the standard linear-response approach to a spin-only
theory that has been discussed in the introduction. On the other hand,
retardation effects, e.g., nutational motion, are excluded within ASD by the
very construction.

The validity of the basic assumption, Eq. (10), strongly depends on the
specific system considered and on the considered parameter range. Even for
gapped systems, however, the strict adiabatic approximation is never perfectly
satisfied, and the true slow spin dynamics will be affected to some degree by
admixtures from (low-energy) excited electron states. As a systematic
generalization of ASD, we therefore propose to relax the constraint Eq. (10)
and to replace it by the weaker constraint

$\ket{\Psi(t)}=\sum_{i=0}^{n-1}\alpha_{i}(t)\ket{\Psi_{i}(\\{\boldsymbol{S}(t)\\})}\>.$
(11)

Here, $\ket{\Psi_{i}(\\{\boldsymbol{S}(t)\\})}$ is the $i$-th excited state of
$\hat{H}_{\text{qu}}+\hat{H}_{\text{int}}(\\{\boldsymbol{S}(t)\\})$, i.e., we
assume that at any instant of time $t$ the conduction-electron state
$\ket{\Psi(t)}$ is contained in the low-energy subspace ${\cal
E}_{n}(\\{{\boldsymbol{S}}\\})$ spanned by the instantaneous ground state and
the lowest $n-1$ instantaneous eigenstates for the spin configuration
$\\{{\boldsymbol{S}}\\}=\\{{\boldsymbol{S}}(t)\\}$ at time $t$. Choosing a
fixed orthonormal basis

$\\{\ket{\Psi_{i}(\\{\boldsymbol{S}\\})}\,|\,{i=0,...,n-1}\\}$ (12)

of ${\cal E}_{n}(\\{{\boldsymbol{S}}\\})$ for any spin configuration, the
electron state at time $t$ is fully specified by the set of expansion
coefficients $\\{\alpha(t)\\}\equiv\\{\alpha_{0}(t),...,\alpha_{n-1}(t)\\}$
via Eq. (11).

For $n=1$, we recover conventional ASD, and thus obtain a true spin-only
theory. For small $n>1$, the effective Lagrangian is obtained from Eq. (8) by
substituting $\ket{\Psi}$, $\partial_{t}|{\Psi}\rangle$, $\bra{\Psi}$,
$\partial_{t}\langle{\Psi}|$ using Eq. (11). It thereby becomes a function of
$\\{{\boldsymbol{S}}\\}$ and $\\{\dot{{\boldsymbol{S}}}\\}$ and furthermore a
function of the set of expansion coefficients $\\{\alpha\\}$, i.e., we get
$L_{\text{eff}}=L_{\text{eff}}(\\{\boldsymbol{S}\\},\\{\dot{\boldsymbol{S}}\\},\\{\alpha\\},\\{\alpha^{\ast}\\},\\{\dot{\alpha}\\}),\\{\dot{\alpha}^{\ast}\\})$.
Hence, besides the spin degrees of freedom, the resulting low-energy theory
contains a few electronic degrees of freedom as well.

We also define the eigenenergies $E_{i}=E_{i}(\\{{\boldsymbol{S}}\\})$ of
$\hat{H}_{\text{qu}}+\hat{H}_{\text{int}}(\\{\boldsymbol{S}(t)\\})$
corresponding to the basis states $|\Psi_{i}(\\{\boldsymbol{S}\\})\rangle$.
$E_{i}(\\{{\boldsymbol{S}}\\})$ is the analog of the $i$-th potential-energy
(Born-Oppenheimer) surface known from molecular-dynamics theory Marx and
Hutter (2000); Bohm  _et al._ (2003). The spin configuration
$\\{{\boldsymbol{S}}\\}$ takes the role of the configuration of atomic nuclei.
Note that the strict adiabatic approximation, Eq. (10), becomes invalid, if
the trajectory of the spin configuration $\\{{\boldsymbol{S}}(t)\\}$ passes a
configuration $\\{{\boldsymbol{S}}_{\rm cr}\\}$, at which there is a crossing
of the ground state with the first excited state, i.e.,
$E_{0}(\\{{\boldsymbol{S}}_{\rm cr}\\})=E_{1}(\\{{\boldsymbol{S}}_{\rm
cr}\\})$, since this is in conflict with the adiabatic theorem Kato (1950);
Avron and Elgart (1999); Comparat (2009).

For $n>1$, the relaxed condition (11) corresponds to a generalized adiabatic
theorem, see Ref. Kato (1950), stating that the condition is respected, if the
low-energy sector ${\cal E}_{n}(\\{{\boldsymbol{S}}\\})$ and its orthogonal
complement (the “high-energy sector”) remain gapped for all
$\\{{\boldsymbol{S}}(t)\\})$ and, of course, if the electron dynamics is
sufficiently slow. In other words, for a given $n$, NA-SD applies if there is
no crossing $E_{n-1}(\\{{\boldsymbol{S}}_{\rm
cr}\\})=E_{n}(\\{{\boldsymbol{S}}_{\rm cr}\\})$, while crossings of states
within the low-energy sector are irrelevant. One should note, however, that a
crossing of two states belonging to the low- and the high-energy sector,
respectively, is in fact unproblematic, if the expansion coefficient
$\alpha_{n-1}(t)=0$ for all $t$, since in this case the $n-1$-th excited
eigenstate would not contribute to $|\Psi(t)\rangle$ anyway. This argument can
be extended to $k<n-1$, as long as there are crossings between “unoccupied”
states with $\alpha_{i}(t)=0$ and $\alpha_{j}(t)=0$ for $k\leq i,j\leq n$
only. We conclude that the relaxed condition (11) for $n>1$ also implies a
less severe, relaxed approximation.

##  III Effective equations of motion

The effective Lagrangian that is obtained by using the constraint Eq. (11) to
eliminate $\ket{\Psi(t)}$ from the original Lagrangian Eq. (8), is given by:

$L_{\text{eff}}=L_{\text{eff}}(\\{\boldsymbol{S}\\},\\{\dot{\boldsymbol{S}}\\},\\{\alpha\\},\\{\alpha^{\ast}\\},\\{\dot{\alpha}\\},\\{\dot{\alpha}^{\ast}\\})=\sum_{m}\boldsymbol{A}_{m}(\boldsymbol{S}_{m})\dot{\boldsymbol{S}}_{m}+i\sum_{ij}\alpha_{i}^{\ast}\bra{\Psi_{i}}\partial_{t}(\alpha_{j}\ket{\Psi_{j}})-\sum_{ij}\alpha_{i}^{\ast}\alpha_{j}\bra{\Psi_{i}}\hat{H}\ket{\Psi_{j}}\>,$
(13)

where $\ket{\Psi_{i}}=\ket{\Psi_{i}(\\{\boldsymbol{S}_{m}\\})}$, and where the
$\\{\dot{{\boldsymbol{S}}}\\}$-dependence, besides the first term, is due to
$\bra{\Psi_{i}}\partial_{t}\ket{\Psi_{j}}=\sum_{m}\bra{\Psi_{i}}\partial_{{\boldsymbol{S}}_{m}}\ket{\Psi_{j}}\dot{{\boldsymbol{S}}}_{m}$.
The Euler-Lagrange equation $\partial_{t}(\partial
L_{\text{eff}}/\partial\dot{\alpha}^{\ast}_{i})-\partial
L_{\text{eff}}/\partial\alpha^{\ast}_{i}=0$ for the “wave function”
$\alpha_{i}$ is straightforwardly obtained as:

$i\partial_{t}\alpha_{i}=\sum_{j}\bra{\Psi_{i}}(\hat{H}_{\text{qu}}+\hat{H}_{\text{int}})\ket{\Psi_{j}}\alpha_{j}-i\sum_{m}\sum_{j}\alpha_{j}\bra{\Psi_{i}}\partial_{{\boldsymbol{S}}_{m}}\ket{\Psi_{j}}\dot{{\boldsymbol{S}}}_{m}\>.$
(14)

The complex conjugate of this equation is just the equation of motion that is
obtained for $\alpha_{i}^{\ast}$.

Note that the second term involves the non-abelian spin-Berry connection
$\underline{{\boldsymbol{C}}}_{m}=\underline{{\boldsymbol{C}}}_{m}(\\{{\boldsymbol{S}}\\})$.
Opposed to the (abelian) spin-Berry connection
${\boldsymbol{C}}_{m}=i\langle\Psi_{0}|\partial_{{\boldsymbol{S}}_{m}}|\Psi_{0}\rangle$
of the (abelian) ASD, this is, for each $m$, a matrix-valued vector with
elements:

${\boldsymbol{C}}^{(ij)}_{m}=i\bra{\Psi_{i}}\partial_{{\boldsymbol{S}}_{m}}\ket{\Psi_{j}}=i\sum_{\gamma}\bra{\Psi_{i}}\partial_{S_{m\gamma}}\ket{\Psi_{j}}{\boldsymbol{e}}_{\gamma}=\sum_{\gamma}C^{(ij)}_{m\gamma}{\boldsymbol{e}}_{\gamma}\>.$
(15)

The matrix dimension is given by the dimension of the low-energy subspace
$n=\dim{\cal E}_{n}(\\{{\boldsymbol{S}}\\})$. It is easy to see that this is a
real quantity. Its transformation behavior under gauge transformations will be
discussed in Sec. VI.

We proceed by deriving the second set of equations of motion from the
effective Lagrangian $\partial_{t}(\partial L_{\rm
eff}/\partial\dot{{\boldsymbol{S}}}_{m})-\partial L_{\rm
eff}/\partial{\boldsymbol{S}}_{m}=0$. With Eq. (13) we straightforwardly find:

$\frac{\partial
L_{\text{eff}}}{\partial\boldsymbol{S}_{m}}=\frac{\partial}{\partial{\boldsymbol{S}}_{m}}({\boldsymbol{A}}_{m}\dot{{\boldsymbol{S}}}_{m})+i\sum_{k}\sum_{ij}\alpha^{\ast}_{i}\alpha_{j}\frac{\partial}{\partial{\boldsymbol{S}}_{m}}(\bra{\Psi_{i}}\partial_{{\boldsymbol{S}}_{k}}\ket{\Psi_{j}}\dot{{\boldsymbol{S}}}_{k})-\sum_{ij}\alpha^{\ast}_{i}\alpha_{j}\frac{\partial}{\partial{\boldsymbol{S}}_{m}}(\bra{\Psi_{i}}\hat{H}\ket{\Psi_{j}})\>,$
(16)

and with $\partial
L_{\text{eff}}/\partial\dot{\boldsymbol{S}}_{m}={\boldsymbol{A}}_{m}+i\sum_{ij}\alpha^{\ast}_{i}\alpha_{j}\bra{\Psi_{i}}\partial_{{\boldsymbol{S}}_{m}}\ket{\Psi_{j}}$,

$\frac{\text{d}}{\text{d}t}\frac{\partial
L_{\text{eff}}}{\partial\dot{\boldsymbol{S}}_{m}}=\sum_{\gamma}\frac{\partial{\boldsymbol{A}}_{m}}{\partial
S_{m\gamma}}\dot{S}_{m\gamma}+i\sum_{ij}[(\partial_{t}\alpha^{\ast}_{i})\alpha_{j}+\alpha^{\ast}_{i}(\partial_{t}\alpha_{j})]\bra{\Psi_{i}}\partial_{{\boldsymbol{S}}_{m}}\ket{\Psi_{j}}+i\sum_{k}\sum_{ij}\alpha^{\ast}_{i}\alpha_{j}(\dot{{\boldsymbol{S}}}_{k}\partial_{{\boldsymbol{S}}_{k}})(\bra{\Psi_{i}}\partial_{{\boldsymbol{S}}_{m}}\ket{\Psi_{j}})\>.$
(17)

Both, Eq. (16) and Eq. (17) involve the spin-Berry connection. The third term
in Eq. (16) can be rewritten using the identity

$\frac{\partial}{\partial{\boldsymbol{S}}_{m}}(\bra{\Psi_{i}}\hat{H}\ket{\Psi_{j}})=\bra{\Psi_{i}}(\partial_{{\boldsymbol{S}}_{m}}\hat{H})\ket{\Psi_{j}}-(E_{j}-E_{i})\bra{\Psi_{i}}\partial_{{\boldsymbol{S}}_{m}}\ket{\Psi_{j}}\>,$
(18)

and for the second term in Eq. (17) it is convenient to get rid of the time
derivatives by using

$i[(\partial_{t}\alpha^{\ast}_{i})\alpha_{j}+\alpha^{\ast}_{i}(\partial_{t}\alpha_{j})]=i\sum_{k}\sum_{l}\left[\alpha^{\ast}_{l}\alpha_{j}\bra{\Psi_{l}}\partial_{{\boldsymbol{S}}_{k}}\ket{\Psi_{i}}-\alpha^{\ast}_{i}\alpha_{l}\bra{\Psi_{j}}\partial_{{\boldsymbol{S}}_{k}}\ket{\Psi_{l}}\right]\dot{{\boldsymbol{S}}}_{k}+\alpha^{\ast}_{i}\alpha_{j}(E_{j}-E_{i})\>,$
(19)

which directly follows from the equation of motion for the wave functions Eq.
(14). Therewith, we arrive at

$\displaystyle 0$ $\displaystyle=\frac{\text{d}}{\text{d}t}\frac{\partial
L_{\text{eff}}}{\partial\dot{\boldsymbol{S}}_{m}}-\frac{\partial
L_{\text{eff}}}{\partial\boldsymbol{S}_{m}}$
$\displaystyle=\sum_{\beta\gamma}\left(\frac{\partial A_{m\beta}}{\partial
S_{m\gamma}}-\frac{\partial A_{m\gamma}}{\partial
S_{m\beta}}\right)\dot{S}_{m\gamma}\hat{e}_{\beta}+\sum_{ij}\alpha^{\ast}_{i}\alpha_{j}\bra{\Psi_{i}}(\partial_{{\boldsymbol{S}}_{m}}\hat{H})\ket{\Psi_{j}}$
$\displaystyle+i\sum_{k}\sum_{ij}\sum_{\gamma}\alpha^{\ast}_{i}\alpha_{j}\left[\partial_{S_{k\gamma}}(\bra{\Psi_{i}}\partial_{{\boldsymbol{S}}_{m}}\ket{\Psi_{j}})-\partial_{{\boldsymbol{S}}_{m}}(\bra{\Psi_{i}}\partial_{S_{k\gamma}}\ket{\Psi_{j}})\right]\dot{S}_{k\gamma}$
$\displaystyle+i\sum_{k}\sum_{ijl}\sum_{\gamma}\left[\alpha^{\ast}_{l}\alpha_{j}\bra{\Psi_{l}}\partial_{S_{k\gamma}}\ket{\Psi_{i}}-\alpha^{\ast}_{i}\alpha_{l}\bra{\Psi_{j}}\partial_{S_{k\gamma}}\ket{\Psi_{l}}\right]\bra{\Psi_{i}}\partial_{{\boldsymbol{S}}_{m}}\ket{\Psi_{j}}\dot{S}_{k\gamma}\>.$
(20)

The first on the right-hand side is a twofold cross product,
$-\dot{\boldsymbol{S}}_{m}\times(\nabla_{\boldsymbol{S}_{m}}\times\boldsymbol{A}_{m})$,
and with Eq. (9) and with the normalization $|{\boldsymbol{S}}_{m}|=1$, the
curl can be written as
$\nabla\times{\boldsymbol{A}}({\boldsymbol{S}})=-{\boldsymbol{S}}$. The second
term is an expectation value $\langle\partial_{{\boldsymbol{S}}_{m}}H\rangle$
of the “effective field” $\partial_{{\boldsymbol{S}}_{m}}H$ in the state of
the electron system $|\Psi\rangle$, see Eq. (11). With Eq. (15), the third
term reads
$\sum_{k}\sum_{ij}\sum_{\gamma}\alpha^{\ast}_{i}\alpha_{j}\left[\partial_{S_{k\gamma}}{\boldsymbol{C}}^{(ij)}_{m}-\partial_{{\boldsymbol{S}}_{m}}C^{(ij)}_{k\gamma}\right]\dot{S}_{k\gamma}$.
Its $\beta$-th component involves the “curl”

$\underline{\Omega}^{\rm(A)}_{k\gamma,m\beta}=\partial_{S_{k\gamma}}\underline{C}_{m\beta}-\partial_{S_{m\beta}}\underline{C}_{k\gamma}$
(21)

of the spin-Berry connection. Here, the underlines indicate that the spin-
Berry connection and its curl are matrices in the indices $i,j$ labelling the
basis of the low-energy subspace for given spin configuration.
$\underline{\Omega}^{\rm(A)}$ has the form of the spin-Berry curvature in the
abelian ($n=1$) theory. We refer to this as the “abelian spin-Berry
curvature”. Again with Eq. (15), the $\beta$-th component of the fourth term
in Eq. (20) reads
$-i\sum_{k}\sum_{ijl}\sum_{\gamma}\left[\alpha^{\ast}_{l}\alpha_{j}C_{k\gamma}^{(li)}-\alpha^{\ast}_{i}\alpha_{l}C_{k\gamma}^{(jl)}\right]C_{m\beta}^{(ij)}\dot{S}_{k\gamma}$.
This involves the commutator
$[\underline{C}_{k\gamma},\underline{C}_{m\beta}]$ of the spin-Berry
connection.

We define the (non-abelian) spin-Berry curvature

$\underline{\Omega}_{k\gamma,m\beta}=\partial_{S_{k\gamma}}\underline{C}_{m\beta}-\partial_{S_{m\beta}}\underline{C}_{k\gamma}-i[\underline{C}_{k\gamma},\underline{C}_{m\beta}]=\underline{\Omega}^{\rm(A)}_{k\gamma,m\beta}-i[\underline{C}_{k\gamma},\underline{C}_{m\beta}]\>,$
(22)

which differs from the abelian one by the additional commutator. Furthermore,
we define the “expectation value” of the spin-Berry curvature in the state
given by the wave function $\\{\alpha\\}$ as:

$\langle{\Omega}\rangle_{k\gamma,m\beta}=\sum_{ij}\alpha_{i}^{\ast}\Omega^{(ij)}_{k\gamma,m\beta}\alpha_{j}\>.$
(23)

With this, the effective equation of motion (20) for the classical-spin
configuration can be written in the compact form

$0=\frac{\text{d}}{\text{d}t}\frac{\partial
L_{\text{eff}}}{\partial\dot{\boldsymbol{S}}_{m}}-\frac{\partial
L_{\text{eff}}}{\partial\boldsymbol{S}_{m}}=\dot{\boldsymbol{S}}_{m}\times\boldsymbol{S}_{m}+\langle\partial_{{\boldsymbol{S}}_{m}}\hat{H}\rangle+\sum_{k}\sum_{\beta\gamma}\dot{S}_{k\gamma}\langle{\Omega}\rangle_{k\gamma,m\beta}{\boldsymbol{e}}_{\beta}\>,$
(24)

or, exploiting the structure of the quantum-classical Hamiltonian Eq. (2) and
the normalization of the wave functions, $\sum_{i}|\alpha_{i}|^{2}=1$,

$0=\dot{\boldsymbol{S}}_{m}\times\boldsymbol{S}_{m}+\langle\partial_{{\boldsymbol{S}}_{m}}\hat{H}_{\rm
int}\rangle+\partial_{{\boldsymbol{S}}_{m}}H_{\rm
cl}+\sum_{k}\sum_{\beta\gamma}\dot{S}_{k\gamma}\langle{\Omega}\rangle_{k\gamma,m\beta}{\boldsymbol{e}}_{\beta}\>.$
(25)

This equation is an implicit equation for $\dot{\boldsymbol{S}}_{m}$. An
explicit form is derived in Appendix A. Finally, we rewrite Eq. (14) using the
definition of the spin-Berry connection Eq. (15):

$i\partial_{t}\alpha_{i}=\sum_{j}\bra{\Psi_{i}}(\hat{H}_{\text{qu}}+\hat{H}_{\text{int}})\ket{\Psi_{j}}\alpha_{j}-\sum_{m}\sum_{j}\dot{{\boldsymbol{S}}}_{m}{\boldsymbol{C}}_{m}^{(ij)}\alpha_{j}\>.$
(26)

Eqs. (25) and (26) represent a closed coupled set of non-linear first-order
differential equations for the effective many-body wave function
$\\{\alpha\\}$ and for the classical spin configuration
$\\{{\boldsymbol{S}}\\}$.

##  IV Discussion

The respective last terms in the equations of motion (25) and (26) originate
from the strict treatment of the holonomic constraint (11). Although the first
time derivative of the local spins is reminiscent of a dissipative Gilbert-
like damping, the resulting dynamics is strictly conserving, i.e., the total
energy given by the expectation value of the total Hamiltonian (2) with the
quantum state of the conduction-electron system is a constant of motion.
Unlike the standard approach discussed in the introduction, the equations of
motion thus describe the dynamics of a closed quantum system (at low
energies).

For the derivation of the equations of motion, we have treated all components
of the spins and of the wave function as independent and have thereby
disregarded the normalization conditions for the length of the classical spin
and for the norm of the wave function

$\absolutevalue{\boldsymbol{S}_{m}(t)}=1\;,\quad\sum_{i}\absolutevalue{\alpha_{i}(t)}^{2}=1\;,$
(27)

which must hold at any instant of time $t$. One can easily check directly,
however, that these are respected. The normalization condition for the wave
function can also be derived by noting that the effective Lagrangian is
invariant under global $U(1)$ phase transformations. Noether’s theorem yields
$Q=\sum_{i}\absolutevalue{\alpha_{i}(t)}^{2}$ as a conserved charge.
Alternatively, the conditions can be treated as additional constraints via
appropriate Lagrange multipliers. As is shown in Appendix B, the resulting
Euler-Lagrange equations are in fact unchanged.

Adiabatic spin dynamics (ASD) theory Stahl and Potthoff (2017) is recovered
for $n=1$, where the conduction-electron dynamics is constrained to the
ground-state manifold $\ket{\Psi(t)}=\ket{\Psi_{0}(\\{\boldsymbol{S}(t)\\})}$
and where the wave function is $\alpha_{0}\equiv 1$ trivially, see Eq. (11).
In this case, the spin-Berry connection
${\boldsymbol{C}}^{(ij)}_{m}=i\bra{\Psi_{i}}\partial_{{\boldsymbol{S}}_{m}}\ket{\Psi_{j}}$
with $i,j=0,...,n-1$, reduces to a vector with scalar entries only,
${\boldsymbol{C}}_{m}=i\bra{\Psi_{0}}\partial_{{\boldsymbol{S}}_{m}}\ket{\Psi_{0}}$.
Hence, the commutator in Eq. (22) vanishes, and the spin-Berry curvature
$\underline{\Omega}_{k\gamma,m\beta}$ reduces to the corresponding expression
$\underline{\Omega}^{\rm(A)}_{k\gamma,m\beta}$, Eq. (21), of (abelian) ASD
theory.

In the opposite extreme case, i.e., when $n$ is chosen as the dimension of the
full many-electron Fock space $\cal H$, Eq. (11) is actually no longer a
constraint but rather represents the expansion of the electron state
$|\Psi(t)\rangle$ with respect to a complete orthonormal system of time-
dependent basis states $\\{|\Psi_{i}(t)\rangle\\}$ with
$|\Psi_{i}(t)\rangle=|\Psi_{i}(\\{{\boldsymbol{S}}(t)\\}\rangle$. In this
case, it is straightforward to see that Eq. (26) is just Schrödingers’s
equation $i\partial_{t}|\Psi(t)\rangle=\hat{H}|\Psi(t)\rangle$, i.e., Eq. (5),
but formulated for the coefficients $\alpha_{i}(t)$ of $|\Psi(t)\rangle$ in
that basis. The spin-Berry connection merely takes care of the fact that the
basis changes smoothly with the parameters $\\{{\boldsymbol{S}}\\}$. Eq. (25)
trivializes as well in this case: We can rewrite the (non-abelian) spin-Berry
curvature in the form (see Appendix C):

$\Omega^{(ij)}_{k\gamma,m\beta}=i\left[\bra{\partial_{S_{k\gamma}}\Psi_{i}}\mathcal{Q}_{n}\ket{\partial_{S_{m\beta}}\Psi_{j}}-(k\gamma\leftrightarrow
m\beta)\right]\>,$ (28)

where
$\mathcal{Q}_{n}:=\mathbb{1}-\sum_{i=0}^{n-1}\ket{\Psi_{i}}\bra{\Psi_{i}}$
projects onto the orthogonal complement of the low-energy space ${\cal
E}_{n}(\\{{\boldsymbol{S}}\\})$. If $n=\dim\cal H$, the complement is zero,
and the spin-Berry curvature vanishes identically, so that

$0=\dot{\boldsymbol{S}}_{m}\times\boldsymbol{S}_{m}+\langle\partial_{{\boldsymbol{S}}_{m}}\hat{H}_{\rm
int}\rangle+\partial_{{\boldsymbol{S}}_{m}}H_{\rm cl}\>.$ (29)

Taking the cross product with ${\boldsymbol{S}}_{m}$ from the right on both
sides of Eq. (25) and exploiting the normalization condition for the spin
length, we get:

$\dot{\boldsymbol{S}}_{m}=\frac{\partial\hat{H}(\\{\boldsymbol{S}\\})}{\partial{\boldsymbol{S}}_{m}}\times{\boldsymbol{S}}_{m}\>.$
(30)

This is just the explicit form of Eq. (6).

Some general properties of the spin-Berry curvature can be derived from Eq.
(28). One immediately notes the antisymmetry

$\Omega^{(ij)}_{k\gamma,m\beta}=-\Omega^{(ij)}_{m\beta,k\gamma}$ (31)

for fixed $i,j$. Furthermore, complex conjugation yields

$\Omega^{(ij)^{*}}_{k\gamma,m\beta}=-\Omega^{(ji)}_{m\beta,k\gamma}\>.$ (32)

With these properties, one can immediately conclude that

$\langle{\Omega}\rangle_{k\gamma,m\beta}=\sum_{ij}\alpha_{i}^{\ast}\Omega^{(ij)}_{k\gamma,m\beta}\alpha_{j}=\langle{\Omega}\rangle_{k\gamma,m\beta}^{\ast}\>,$
(33)

i.e., the expectation value, which enters the effective equation of motion Eq.
(25), is real.

Quite generally, the (abelian) Berry connection and Berry curvature arise in
the adiabatic problem, where a quantum Hamiltonian
$\hat{H}=\hat{H}({\boldsymbol{\lambda}})$ depends on a family of slowly
varying parameters ${\boldsymbol{\lambda}}$ and has a non-degenerate ground
state for all ${\boldsymbol{\lambda}}$. This gives rise to the famous Berry
phase Berry (1984), which the ground state picks up during a closed loop in
parameter space and which can be computed, e.g., as an integral of the Berry
curvature over the surface bounded by the loop. Mathematically, the phase is a
holonomy, i.e., it results from a twist of the line bundle
$\\{({\boldsymbol{\lambda}},|\Psi_{0}\rangle)\,|\,\hat{H}({\boldsymbol{\lambda}})|\Psi_{0}\rangle=E_{0}(\\{{\boldsymbol{\lambda}}\\})|\Psi_{0}\rangle\\}$
Simon (1983). The Berry phase is gauge invariant and thus observable and
depends on the geometry of the closed loop only. Similarly, non-abelian gauge
fields arise in the adiabatic time evolution of an $n>1$-fold degenerate
ground state of a quantum system Wilczek and Zee (1984) and produce a non-
trivial phase after completing a loop in parameter space.

Here, we consider a quantum system coupled to dynamical classical degrees of
freedom (classical spins). In case of a clear time-scale separation between
the slow classical and the fast quantum dynamics, the classical spins induce a
spin-Berry curvature in the quantum conduction-electron system. Generically,
it is highly unlikely, however, that the classical state evolves along a
closed path. The essential observation, however, is that there is an
additional feedback of the Berry curvature on the classical spin dynamics,
seen in the last term in Eq. (25) for $\Omega=\Omega^{\rm(A)}$. Already in the
abelian case $n=1$, this leads to an anomalous geometrical spin torque Stahl
and Potthoff (2017). This geometric feedback on slow classical dynamics has
been pointed out Wen and Zee (1988); Niu and Kleinman (1998); Bohm  _et al._
(2003); Niu  _et al._ (1999); Stahl and Potthoff (2017); Elbracht  _et al._
(2020); Bajpai and Nikolić (2020); Michel and Potthoff (2021) but has not yet
been studied for spin dynamics in the non-abelian case $1<n=\dim{\cal
E}_{n}\\{{\boldsymbol{S}}\\}\ll\dim{\cal H}$.

##  V Time reversal

Time-reversal symmetry plays an important role for the presence of a finite
spin-Berry curvature in the adiabatic case ($n=1$) Stahl and Potthoff (2017).
For $n>1$, however, this is entirely different:

We assume that the electron system is time-reversal symmetric, i.e., that the
Hamiltonian $\hat{H}_{\text{qu}}$ commutes with the anti-unitary operator for
time reversal $\Theta$. The interaction term, Eq. (4), on the other hand, is
odd under time reversal, $\Theta\hat{H}_{\rm
int}\Theta^{\dagger}=-\hat{H}_{\rm int}$, since
$\Theta{\boldsymbol{s}}_{{\boldsymbol{r}}_{m}}\Theta^{\dagger}=-{\boldsymbol{s}}_{{\boldsymbol{r}}_{m}}$.
The local spins ${\boldsymbol{S}}_{m}$ are classical degrees of freedom, which
act as local magnetic fields and explicitly break time-reversal symmetry of
the quantum system.

This effect, however, can be disregarded in the weak-$J$ regime, where the
spin-Berry curvature, in the spirit of linear-response theory, is a physical
property of the electron system $\hat{H}_{\text{qu}}$ only. Namely, expanding
$E_{i}=E_{i0}+{\cal O}(J)$ and $|\Psi_{l}\rangle=|\Psi_{l}^{0}\rangle+{\cal
O}(J)$ and using the identity

$\langle\Psi_{l}|\partial_{{\boldsymbol{S}}_{m}}\Psi_{j}\rangle=\frac{\langle\Psi_{l}|\partial_{{\boldsymbol{S}}_{m}}\hat{H}(\\{\boldsymbol{S}\\})|\Psi_{j}\rangle}{E_{j}-E_{l}}\>,$
(34)

which holds for $E_{j}\neq E_{l}$, Eq. (28) can be rewritten as

$\displaystyle\Omega^{(ij)}_{k\gamma,m\beta}$ $\displaystyle=$ $\displaystyle
i\sum_{l\geq
n}\Bigg{[}\frac{\langle\Psi^{0}_{i}|\partial_{S_{k\gamma}}\hat{H}|\Psi_{l}^{0}\rangle}{E_{i0}-E_{l0}}\frac{\langle\Psi^{0}_{l}|\partial_{S_{m\beta}}\hat{H}|\Psi_{j}^{0}\rangle}{E_{j0}-E_{l0}}$
(35) $\displaystyle-$ $\displaystyle(k\gamma\leftrightarrow
m\beta)\Bigg{]}+{\cal O}(J^{3})\>,$

since $\partial_{S_{k\gamma}}\hat{H}_{\rm int}=Js_{\rm i_{k}\gamma}={\cal
O}(J)$, so that the spin-Berry curvature is of order $J^{2}$ for weak $J$ and
expressed in terms of the eigenstates and eigenenergies of
$\hat{H}_{\text{qu}}$ only. Note that $0\leq i,j\leq n-1$ in Eq. (35).

For a system with an even number of spin-$1/2$ electrons, the time-reversal
operator squares to unity, $\Theta^{2}=+1$. In this case, we can choose an
orthonormal basis of time-reversal-symmetric energy eigenstates
$|\Psi^{0}_{i}\rangle=\Theta|\Psi^{0}_{i}\rangle$. This implies that the
matrix elements,

$\displaystyle\langle\Psi^{0}_{i}|\partial_{S_{k\gamma}}\hat{H}|\Psi^{0}_{l}\rangle$
$\displaystyle=$
$\displaystyle-\langle\Psi^{0}_{i}|\Theta^{\dagger}\partial_{S_{k\gamma}}\hat{H}\Theta|\Psi^{0}_{l}\rangle$
(36) $\displaystyle=$
$\displaystyle-(\langle\Theta\Psi^{0}_{i}|\partial_{S_{k\gamma}}\hat{H}|\Theta\Psi^{0}_{l}\rangle)^{\ast}$
$\displaystyle=$
$\displaystyle-(\langle\Psi^{0}_{i}|\partial_{S_{k\gamma}}\hat{H}|\Psi^{0}_{l}\rangle)^{\ast}\;,$

are purely imaginary. Note that only the (odd) interaction term $\hat{H}_{\rm
int}(\\{{\boldsymbol{S}}\\})$ contributes. Using this in Eq. (35) shows that
$\Omega^{(ij)}_{k\gamma,m\beta}$ is purely imaginary. With Eq. (32) we can
conclude that

$\Omega^{(ij)}_{k\gamma,m\beta}=\Omega^{(ji)}_{m\beta,k\gamma}\>.$ (37)

In particular, Eq. (31) and Eq. (37) imply that the $i=j$ elements of the
spin-Berry curvature must vanish in the weak-$J$ limit for $\Theta^{2}=+1$.
This is important for the abelian case $n=1$. For $i=j=0$ we have
$\Omega^{(00)}_{k\gamma,m\beta}=0$ and, hence, there is no geometrical spin
torque in the weak-$J$ limit for a time-reversal-symmetric system with
$\Theta^{2}=+1$. In the general non-abelian case, on the other hand, we find
with Eq. (33) that

$\langle{\Omega}\rangle_{k\gamma,m\beta}=-\sum_{ij}\mbox{Im}(\alpha_{i}^{\ast}\alpha_{j})\mbox{Im}\Omega^{(ij)}_{k\gamma,m\beta}\>,$
(38)

since $\Omega^{(ij)}_{k\gamma,m\beta}$ is imaginary. Generically, the
coefficients $\alpha_{i}=\alpha_{i}(t)$ in the expansion Eq. (11) will be
complex and oscillatory functions of time. The expression above thus shows
that even in the weak-$J$ limit and for a time-reversal symmetric system, the
geometrical spin torque in the equation of motion (25) is generally finite.

Let us briefly discuss the case of an odd electron number with
$\Theta^{2}=-1$. Here, the basis states can be grouped in orthogonal and
energy-degenerate Kramers pairs
$\\{|\Psi^{0}_{i}\rangle,|\overline{\Psi}_{i}^{0}\rangle\\}$ with
$|\overline{\Psi}^{0}_{i}\rangle\equiv\Theta|\Psi_{i}^{0}\rangle$ for
$i=0,...,(n/2)-1$. An even number of states must be included in formulating
the constraint (11). For the matrix elements, we have

$\displaystyle\langle\Psi^{0}_{i}|\partial_{S_{k\gamma}}\hat{H}|\Psi^{0}_{l}\rangle$
$\displaystyle=$
$\displaystyle-\langle\Psi^{0}_{i}|\Theta^{\dagger}\partial_{S_{k\gamma}}\hat{H}\Theta|\Psi^{0}_{l}\rangle$
(39) $\displaystyle=$
$\displaystyle-(\langle\Theta\Psi^{0}_{i}|\partial_{S_{k\gamma}}\hat{H}|\Theta\Psi^{0}_{l}\rangle)^{\ast}$
$\displaystyle=$
$\displaystyle-(\langle\overline{\Psi}^{0}_{i}|\partial_{S_{k\gamma}}\hat{H}|\overline{\Psi}^{0}_{l}\rangle)^{\ast}\;.$

This can be used in Eq. (35) since in the $l$-sum with each term also the
Kramers partner is included. We find

$\Omega^{(ij)}_{k\gamma,m\beta}=\Omega^{(\overline{j}\,\overline{i})}_{m\beta,k\gamma}\>,$
(40)

where the index $\overline{i}$ refers to the Kramers partner of
$|\Psi^{0}_{i}\rangle$, and, furthermore,
$(\Omega^{(ij)}_{k\gamma,m\beta})^{\ast}=-\Omega^{(\overline{i}\,\overline{j})}_{k\gamma,m\beta}$.
As for the case $\Theta^{2}=+1$, time-reversal symmetry does not lead to a
vanishing spin-Berry curvature or a vanishing expectation value
$\langle{\Omega}\rangle_{k\gamma,m\beta}$. Note that for $\Theta^{2}=-1$ the
adiabatic theory is not applicable anyway (for the weak-coupling limit), since
the ground state is at least twofold Kramers degenerate.

##  VI Gauge transformations

The effective Lagrangian Eq. (13) can be written in a compact form as

$\displaystyle L_{\text{eff}}$ $\displaystyle=$
$\displaystyle\sum_{m}\boldsymbol{A}_{m}({\boldsymbol{S}}_{m})\dot{\boldsymbol{S}}_{m}+i\boldsymbol{\alpha}^{\dagger}\partial_{t}\boldsymbol{\alpha}$
(41) $\displaystyle+$
$\displaystyle\sum_{m}\boldsymbol{\alpha}^{\dagger}[\underline{{\boldsymbol{C}}}_{m}(\\{{\boldsymbol{S}}\\})\dot{\boldsymbol{S}}_{m}]\boldsymbol{\alpha}-\boldsymbol{\alpha}^{\dagger}\underline{H}(\\{{\boldsymbol{S}}\\})\boldsymbol{\alpha}\;,$

where ${\boldsymbol{\alpha}}=(\alpha_{0},...,\alpha_{n-1})^{T}$ and where
$\underline{H}$ is the Hamilton matrix with elements
$H_{ij}=\bra{\Psi_{i}}\hat{H}\ket{\Psi_{j}}$ and the local basis states
$|\Psi_{j}\rangle=|\Psi_{j}(\\{{\boldsymbol{S}}\\})\rangle$. We consider a
gauge transformation

$\displaystyle|\Psi_{j}(\\{{\boldsymbol{S}}\\})\rangle$ $\displaystyle\mapsto$
$\displaystyle|\Psi^{\prime}_{j}(\\{{\boldsymbol{S}}\\})\rangle=\sum_{i}U_{ij}^{\dagger}|\Psi_{i}(\\{{\boldsymbol{S}}\\})\rangle$
$\displaystyle{\boldsymbol{\alpha}}$ $\displaystyle\mapsto$
$\displaystyle{\boldsymbol{\alpha}}^{\prime}=\underline{U}{\boldsymbol{\alpha}}\>,$
(42)

where $\underline{U}$ (with elements $U_{ij}$) is the defining matrix
representation of SU(n) on the local low-energy subspace ${\cal
E}_{n}(\\{{\boldsymbol{S}}\\})$ for given spin configuration
$\\{{\boldsymbol{S}}\\}$. This transformation must leave observables
invariant, since Eq. (42) just means a rotation of the basis in ${\cal
E}_{n}(\\{{\boldsymbol{S}}\\})$, which leaves the quantum state
$|\Psi\rangle=\sum_{j=0}^{n-1}\alpha_{j}|\Psi_{j}(\\{{\boldsymbol{S}}\\})\rangle$,
and thus the constraint Eq. (11) invariant when rotating the expansion
coefficients (the wave function) accordingly. We distinguish between global
SU(n) and local SU(n) transformations. For the latter, the transformation
matrix $\underline{U}=\underline{U}(\\{{\boldsymbol{S}}\\})$ is an arbitrary
but smooth function of the spin configuration $\\{{\boldsymbol{S}}\\}$. The
effective Lagrangian is invariant under both, global and local gauge
transformations.

Note that the Hamilton matrix transforms in a covariant way,

$\underline{H}\mapsto\underline{H}^{\prime}=\underline{U}\,\underline{H}\,\underline{U}^{\dagger}\>,$
(43)

while the Berry connection transforms covariantly under a global gauge
transformation only. For a local gauge transformation we rather have:

$\underline{{\boldsymbol{C}}}_{m}\mapsto\underline{{\boldsymbol{C}}}_{m}^{\prime}=\underline{U}\,\underline{{\boldsymbol{C}}}_{m}\,\underline{U}^{\dagger}+i\underline{U}\partial_{{\boldsymbol{S}}_{m}}\underline{U}^{\dagger}\>.$
(44)

The non-abelian Berry curvature, opposed to its abelian part (21), transforms
covariantly:

$\underline{\Omega}_{k\gamma,m\beta}\mapsto\underline{\Omega}^{\prime}_{k\gamma,m\beta}=\underline{U}\,\underline{\Omega}_{k\gamma,m\beta}\,\underline{U}^{\dagger}\,\>,$
(45)

so that its expectation value in the state given by the wave function
$\alpha_{i}$ is invariant:
$\langle{\Omega^{\prime}}\rangle^{\prime}_{k\gamma,m\beta}=\langle{\Omega}\rangle_{k\gamma,m\beta}$.
Hence, Eq. (25) is invariant under local gauge transformations. The
Schrödinger-type equation Eq. (26), on the other hand, is form-invariant under
local transformations, i.e.,

$i\partial_{t}\alpha^{\prime}_{i}=\sum_{j}\bra{\Psi^{\prime}_{i}}(\hat{H}_{\text{qu}}+\hat{H}_{\text{int}})\ket{\Psi^{\prime}_{j}}\alpha^{\prime}_{j}-\sum_{mj}\dot{{\boldsymbol{S}}}_{m}{{\boldsymbol{C}}_{m}^{(ij)}}^{\prime}\alpha^{\prime}_{j}\>,$
(46)

and the spin-Berry connection term on the right-hand side is necessary to
compensate the extra term appearing on the left-hand side in case of an
$\\{{\boldsymbol{S}}\\}$-dependent transformation.

Concluding, the effective Lagrangian emerging in the low-energy sector of
hybrid spin-electron dynamics represents a non-abelian SU(n) gauge theory.
This is reminiscent of standard quantum field theories Peskin and Schroeder
(1996), where the Lagrangian is invariant under simultaneous transformations
of coupled matter and gauge fields, and where these gauge transformations
involve a gauge group, like SU(n), and are local in space-time. There are a
couple of differences though: Within non-abelian spin-dynamics theory, space-
time is not only replaced by a compact parameter manifold, namely the
Cartesian product of classical Bloch spheres representing the space of the
spin configurations, but furthermore the spin configurations have their own
dynamics. The theory is thus much more related to gauge theories that have
been devised for molecular physics Bohm  _et al._ (2003), where the state
space of the nuclei, when treated classically, define a dynamical parameter
manifold, and where the role of the gauge field is played by the non-abelian
Berry connection.

Finally, it is worth mentioning that there is a second, less important class
of gauge freedom. This concerns the vector potential
${\boldsymbol{A}}({\boldsymbol{S}}_{m})$, see the first term of $L$ in Eq.
(8), i.e., already in the full Lagrangian. Any transformation of the unit
vector ${\boldsymbol{e}}\mapsto{\boldsymbol{e}}^{\prime}$ leads to a
transformed potential
${\boldsymbol{A}}({\boldsymbol{S}}_{m})\mapsto{\boldsymbol{A}}^{\prime}({\boldsymbol{S}}_{m})$
but leaves its curl invariant. This even includes “local”, $m$-dependent
transformations
${\boldsymbol{A}}({\boldsymbol{S}}_{m})\mapsto{\boldsymbol{A}}^{\prime}_{m}({\boldsymbol{S}}_{m})$
resulting from ${\boldsymbol{e}}\mapsto{\boldsymbol{e}}^{\prime}_{m}$.
However, since only the curl
$\nabla_{{\boldsymbol{S}}}\times{\boldsymbol{A}}({\boldsymbol{S}}_{m})$ enters
the equations of motion resulting from the full or from the effective
Lagrangian, see Eq. (20) for instance, these are invariant.

##  VII Minimal model

Figure 1:  Sketch of the minimal model studied numerically. A classical spin
${\boldsymbol{S}}$ of length $|{\boldsymbol{S}}|=1$ is antiferromagnetically
exchange coupled with coupling strength $J>0$ to the local spin moment
${\boldsymbol{s}}_{i_{0}}$ at the first site $i_{0}=1$ of a system of
conduction electrons on a one-dimensional chain with open boundaries. $T$ is
the nearest-neighbor hopping. Real-time dynamics is initiated by a sudden
change of the direction of a local magnetic field ${\boldsymbol{B}}$ coupled
to ${\boldsymbol{S}}$.

For a further discussion of non-abelian spin-dynamics theory, we will present
numerical results for a minimal model, which includes a few degrees of freedom
only but is sufficient to illustrate several key aspects. Our intention is to
show by example that and how our theoretical approach can be evaluated in
practice, how the numerical results compare with the full solution of the
equations of motion, and what improvements the theory offers over the purely
adiabatic (abelian) version. This may also be seen as a preparation for future
applications to more realistic but also more complicated physical systems,
where various secondary issues become important.

The Hamiltonian our our toy model is given by

$\hat{H}=-T\sum_{\langle
i,j\rangle,\sigma}c_{i\sigma}^{\dagger}c_{j\sigma}+J\boldsymbol{s}_{i_{0}}\boldsymbol{S}-\boldsymbol{B}\boldsymbol{S}\>.$
(47)

It describes a single classical spin ($M=1$) locally exchange coupled
(coupling constant $J>0$) to a non-interacting tight-binding model in an open
chain geometry with a small number of sites $L$ hosting $N=L$ electrons, i.e.,
a half-filled conduction-electron system. The spin is coupled to the first
site of the chain $i_{0}=1$. This is the $s$-$d$ model vz  discussed in the
introduction and the same model as in Ref. Stahl and Potthoff (2017). Energy
and time units are fixed by setting the nearest-neighbor hopping amplitude to
$T=1$. In addition, the Hamiltonian includes a local magnetic field of
strength $B$ coupling to the classical spin ${\boldsymbol{S}}$. The model is
visualized in Fig. 1.

The field term is employed to initiate the real-time dynamics: At time $t=0$
the system is prepared in the ground state of $\hat{H}$ with the field in $x$
direction, i.e., the spin ${\boldsymbol{S}}=S{\boldsymbol{e}}_{x}$ is aligned
to ${\boldsymbol{B}}=B{\boldsymbol{e}}_{x}$, and the conduction-electron state
is the ground state $|\Psi(t=0)\rangle=|\Psi_{0}({\boldsymbol{S}})\rangle$.
Time propagation for $t>0$ is driven by the same Hamiltonian but with the
field pointing in $z$ direction. Dynamics is thus initiated by a sudden change
of the field direction from $x$\- to $z$ direction.

For $t>0$ one expects that the spin starts precessing around the $z$ axis. In
the adiabatic approximation with $n=1$, the electron system will follow the
respective spin direction instantaneously, and its state at time $t$ would be
the instantaneous ground state $|\Psi_{0}({\boldsymbol{S}}(t))\rangle$. The
time scale on which the precession takes place is given by the inverse of the
Larmor frequency $\omega_{\rm L}=B$. Depending on the field strength, this
time scale $\tau_{L}=1/\omega_{\rm L}=1/B$ can be much shorter than the
inverse of the finite-size gap $\Delta={\cal O}(T/L)$. With $T=1$ we thus
expect that the adiabatic approximation breaks down for $B\gg T/L$ and that
excited states $|\Psi_{j}({\boldsymbol{S}})\rangle$ with $0<j<n-1$ will be
populated. The number of states $n$ included in the
${\boldsymbol{S}}$-dependent basis controls the accuracy of the non-abelian
spin-dynamics approach.

For the single-classical-spin model the effective equations of motion Eq. (25)
and Eq. (26) are somewhat simplified. For $M=1$ we can skip the $m$-index and
take the cross product with ${\boldsymbol{S}}$ on both sides of Eq. (25).
Furthermore, we have $\langle\partial_{{\boldsymbol{S}}}\hat{H}_{\rm
int}\rangle=J\langle{\boldsymbol{s}}_{i_{0}}\rangle-{\boldsymbol{B}}$ and
$\partial_{{\boldsymbol{S}}}H_{\rm cl}=0$. Therewith we get

$\dot{\boldsymbol{S}}=\frac{J\langle\boldsymbol{s}_{i_{0}}\rangle\times\boldsymbol{S}-\boldsymbol{B}\times\boldsymbol{S}}{1-\boldsymbol{S}\langle\boldsymbol{\Omega}\rangle}\>,$
(48)

where
$\langle\boldsymbol{\Omega}\rangle=\sum_{ij}\alpha_{i}^{\ast}{\boldsymbol{\Omega}}^{(ij)}\alpha_{j}$
is the expectation value of the pseudovector ${\boldsymbol{\Omega}}^{(ij)}$
with components
$\Omega^{(ij)}_{\alpha}=\frac{1}{2}\sum_{\beta\gamma}\varepsilon_{\alpha\beta\gamma}\Omega^{(ij)}_{\beta\gamma}$
that can be constructed for $M=1$ due to the antisymmetry of the Berry
curvature tensor under $\beta\leftrightarrow\gamma$ for each pair $(ij)$, see
Eq. (31). Furthermore,
$\langle\boldsymbol{s}_{i_{0}}\rangle=\langle\boldsymbol{s}_{i_{0}}\rangle_{t}=\sum_{ij}\alpha_{i}^{\ast}(t)\langle\Psi_{i}({\boldsymbol{S}}(t))|{\boldsymbol{s}}_{i_{0}}|\Psi_{j}({\boldsymbol{S}}(t))\rangle\alpha_{j}(t)$.

Remarkably, there is a renormalization of the precession frequency resulting
from the geometrical spin torque, which has already been studied for the
adiabatic case Stahl and Potthoff (2017); Bajpai and Nikolić (2020); Elbracht
_et al._ (2020); Michel and Potthoff (2021). This manifests itself as an
additional factor $1/(1-\boldsymbol{S}\langle\boldsymbol{\Omega}\rangle)$ in
Eq. (48). In the adiabatic case $n=1$, the expectation value
$\langle\boldsymbol{\Omega}\rangle$ is strictly parallel or antiparallel to
classical-spin orientation due to symmetry reasons Stahl and Potthoff (2017).
For $\boldsymbol{S}\uparrow\uparrow\langle\boldsymbol{\Omega}\rangle$ this
results in a faster precessional dynamics, and its orientation is even
reversed if $\boldsymbol{S}\langle\boldsymbol{\Omega}\rangle>1$, while for
$\boldsymbol{S}\uparrow\downarrow\langle\boldsymbol{\Omega}\rangle$ the
precession is slowed down. Exactly at
$\boldsymbol{S}\langle\boldsymbol{\Omega}\rangle=1$ the right-hand side of Eq.
(48) becomes singular. This is linked to a divergence of the precession
frequency which, however, becomes relevant in an extreme case only: For the
adiabatic case and $L=1$, it was found in Ref. Stahl and Potthoff (2017) that
singular dynamics can in principle be approached, if the length of the
classical spin $|{\boldsymbol{S}}|\to\frac{1}{2}$. At the same time, however,
to stay in the adiabatic regime of the model, it was necessary to consider an
ever-increasing coupling strength, i.e., $J\to\infty$.

Here, we see that the same type of singularity is in principle also present in
the non-adiabatic case (for $M=1$). Generally, however, we find
$0<\boldsymbol{S}\langle\boldsymbol{\Omega}\rangle<1$ (for antiferromagnetic
exchange coupling $J>0$): A possible singularity is regularized for $n>1$ due
to contributions from excited states and partly also to due the fact that
$\langle{\boldsymbol{\Omega}}\rangle$ and ${\boldsymbol{S}}$ are no longer
necessarily collinear.

The following NA-SD studies of the minimal model are based on a numerical
solution of the coupled effective equations of motion Eq. (48) for the
classical spin ${\boldsymbol{S}}$ and Eq. (26) for the wave function
$\\{\alpha\\}$. For the computation of the expectation value of the spin-Berry
curvature $\langle{\boldsymbol{\Omega}}\rangle$ we profit from
simplifications, which hold in case of a non-interacting conduction-electron
system. These are detailed in the Appendix D.

We also compare the results of the NA-SD theory with the full solution of the
fundamental equations of motion (5) and (6), which is obtained independently.
More explicitly, Eq. (5) for the minimal model reads:

$\dot{\boldsymbol{S}}=J\langle\boldsymbol{s}_{i_{0}}\rangle_{t}\times\boldsymbol{S}-\boldsymbol{B}\times\boldsymbol{S}\>.$
(49)

Furthermore, in case of a non-interacting electron system, Eq. (6) can be
replaced by the equation of motion

$i\frac{d}{dt}\underline{\rho}=\commutator{\underline{T}^{(\text{eff})}}{\underline{\rho}}$
(50)

for the one-particle reduced density matrix $\underline{\rho}$ with elements
$\rho_{ii^{\prime}\sigma\sigma^{\prime}}(t)\ =\langle
c^{\dagger}_{i^{\prime}\sigma^{\prime}}c_{i\sigma}\rangle$, and where the
elements of the effective hopping matrix $\underline{T}^{(\text{eff})}$ are
given by:

$T^{(\text{eff})}_{ii^{\prime}\sigma\sigma^{\prime}}=T\delta_{\langle
ii^{\prime}\rangle}\delta_{\sigma\sigma^{\prime}}+\frac{J}{2}{\boldsymbol{\sigma}}_{\sigma\sigma^{\prime}}{\boldsymbol{S}}\,\delta_{ii_{0}}\delta_{i^{\prime}i_{0}}\;.$
(51)

##  VIII Numerical results

###  VIII.1 Full theory

The precession around the $z$ axis defined by the local magnetic field is
expected to be the dominant effect in the classical spin dynamics. In fact,
this is the main phenomenon found by solving the full set of equations of
motion (49) and (50). Fig. 2 displays numerical results obtained with the full
theory for a system with $L=10$ sites at half-filling $N=L$, and for generic
parameter values $J=1$ and $B=0.1$. The $x$ component of the classical spin
undergoes a quite regular oscillation with a period close to $2\pi/\omega_{\rm
L}=2\pi/B\approx 62.8$. The $y$ component exhibits the same but phase shifted
dynamics. We note that, for the selected parameter set, the geometrical spin
torque is too small to produce a sizeable renormalization of the precession
frequency.

Figure 2:  Time evolution of the $x$ and the $z$ component of the classical
spin as obtained from the full theory for a system with $L=10$ sites at half-
filling $N=L$. Parameters: $J=1$, $B=0.1$. The energy and time units are set
by fixing the nearest-neighbor hopping at $T=1$.

Damping of the spin dynamics and eventual alignment of the classical spin with
the field ${\boldsymbol{B}}=B{\boldsymbol{e}}_{z}$ is typically a weaker
effect, which takes place on a much longer time scale, see e.g. the discussion
in Refs. llg ; Bhattacharjee  _et al._ (2012); Sayad and Potthoff (2015). For
a closed, finite and with $L=10$ small system, as considered here, relaxation
will be imperfect anyway, and even in the long-time limit, the system cannot
fully approach its ground state locally, in the vicinity of $i_{0}$.
Uncovering this type of relaxation dynamics requires much larger systems, as
discussed in Refs. Elbracht and Potthoff (2020, 2021), for example.

Fig. 2 also displays the $z$ component of the spin. In case of a perfect
precessional motion, one would expect a constant $S_{z}$. As is seen in the
figure, however, an almost oscillatory motion of $S_{z}$ with some additional
irregularities is found instead. This nutation of the spin is reminiscent of
gyroscope theory Butikov (2006); Wegrowe and Ciornei (2012), but is not
understood easily. An explanation in terms of linear-response theory (see Eq.
(1)), i.e., Redfield theory for open quantum systems, involves the second-
order term in the Taylor expansion of the memory kernel Bhattacharjee  _et
al._ (2012); Sayad  _et al._ (2016b). For the parameters considered here, the
nutation effect is at least an order of magnitude smaller as compared to the
precessional dynamics (see Fig. 2). There are cases, however, where
precessional and nutational oscillations can be of the same order of
magnitude. The additional “irregularities” on top of the nutation are even
more subtle. At this level of resolution at the latest, the complexity of the
dynamics caused by the nonlinearity of the quantum-classical equations of
motion appears to prohibit a simple explanation.

###  VIII.2 Anomalous precession

Figure 3:  Time evolution of the angle enclosed by the classical spin
$\boldsymbol{S}$ and the expectation value of the local spin of the electron
system at the impurity site $\langle\boldsymbol{s}_{i_{0}}\rangle$. Results as
obtained by the full theory for $J=1$ (blue) and $J=15$ (orange). Other
parameters as in Fig. 2.

In the case of strong exchange coupling $J\gg T$, the classical spin
${\boldsymbol{S}}$ and the local magnetic moment
$\langle{\boldsymbol{s}}_{i_{0}}\rangle$ at $i_{0}$ are tightly bound
together. In this regime one would thus expect that
$\langle{\boldsymbol{s}}_{i_{0}}\rangle$ follows the classical-spin direction
almost instantaneously such that $\langle{\boldsymbol{s}}_{i_{0}}\rangle$ is
almost perfectly aligned antiferromagnetically to ${\boldsymbol{S}}$. The time
evolution of the angle enclosed by ${\boldsymbol{S}}$ and
$\langle{\boldsymbol{s}}_{i_{0}}\rangle$ is shown in Fig. 3. For $J=1$ the
mean deviation of the angle from $180^{\circ}$ is in fact about $2^{\circ}$
only, and it shrinks with increasing $J$, see the result for $J=15$. On the
other hand, the absolute value of the local moment
$\langle{\boldsymbol{s}}_{i_{0}}\rangle$ of the conduction-electron systems
that is induced by ${\boldsymbol{S}}$, increases from
$|\langle{\boldsymbol{s}}_{i_{0}}\rangle|\approx 0.18$ at $J=1$ to
$|\langle{\boldsymbol{s}}_{i_{0}}\rangle|\approx 0.49$ at $J=15$. The net
effect, however, is that the spin torque on ${\boldsymbol{S}}$ originating
from the exchange term,
$J\langle{\boldsymbol{s}}_{i_{0}}\rangle\times{\boldsymbol{S}}$ is weak
compared to the torque due to the field
$-{\boldsymbol{B}}\times{\boldsymbol{S}}$. Following naive adiabatic theory
one would therefore expect a precessional motion of ${\boldsymbol{S}}$ in the
$x$-$y$ plane with a frequency $\omega_{\rm p}$ close to the Larmor frequency
$\omega_{\rm L}=B$. However, this naive picture in principle disregards the
effect due to the geometrical spin torque, which can be sizeable. It is thus
instructive to compare the naive expectation as well as adiabatic spin
dynamics theory (ASD) with the full solution of the fundamental equations of
motion.

Numerical results for a strong coupling $J=15$ are displayed in Fig. 4. The
full theory (see red curve) does predict an oscillatory motion of $S_{x}$ as
expected for precessional dynamics. However, the precession is not perfect:
Note, e.g., that $S_{x}$ does not reach its minimum value $S_{x}=-1$, while
$S_{x}\approx+1$ after a full period. In fact, the precession does not take
place in the $x$-$y$ plane but within a plane that is a somewhat tilted and,
furthermore, the plane normal
$\boldsymbol{n}\propto{\boldsymbol{S}}\times\dot{{\boldsymbol{S}}}$ is
slightly time dependent.

The most important effect seen in Fig. 4, however, is the strongly enhanced
precession frequency $\omega_{\rm p}\approx 0.19$, which is close to twice the
Larmor freqency $\omega_{\rm L}=B=0.1$. This anomalous precession frequency
$\omega_{\rm p}$ is clearly at variance with the naive expectation and must
therefore result from the renormalization factor
$1/(1-\boldsymbol{S}\langle\boldsymbol{\Omega}\rangle)$ in Eq. (48). In fact,
the full theory (red) almost perfectly agrees with the prediction of the non-
abelian spin-dynamics (NA-SD) theory (blue), when spanning the low-energy
subspace ${\cal E}_{n}(\\{{\boldsymbol{S}}\\})$ by the instantaneous ground
and first excited state, i.e., for $n=2$.

Figure 4:  Time dependence of the $x$-component of the classical spin for
$L=10$, $J=15$, $T=1$, $B=0.1$. Results as obtained from ASD ($n=1$, orange),
NA-SD with $n=2$ (blue), and the full theory (red).

Fig. 5 presents the same results of the NA-SD (blue curve) and the full theory
(red) in a classical-Bloch-sphere representation. At $t=0$, the motion of
${\boldsymbol{S}}$ starts at ${\boldsymbol{S}}=(1,0,0)$ (see blue dot) and
completes about three full periods up to the maximum propagation time $t=100$.
The dynamics is close to a planar precession but the instantaneous plane
normal ${\boldsymbol{n}}$ (green curve) exhibits a weak time dependence and
precesses itself around an axis that is somewhat tilted against the $z$ axis.
The full theory exhibits some additional wiggles which can also be seen in
Fig. 4 already and which are absent in the NA-SD. A low-energy subspace with
more than $n=2$ dimensions would be necessary to capture this effect. Apart
from that, however, there is an almost perfect agreement of the NA-SD results
with the results of the full theory.

Figure 5:  The same as in Fig. 4 for the NA-SD (blue curve) and the full
theory (red) but displayed on a classical Bloch sphere. The blue dot marks the
spin position at time $t=0$. Green curve: unit vector ${\boldsymbol{n}}$
normal to the instantaneous precession plane. The trajectories are shown for
$0\leq t\leq 100$.

While this is very satisfying and underpins the construction of the NA-SD,
there is an interesting problem remaining: Comparing with the $n=1$ theory,
i.e., with ASD, there is strong discrepancy. ASD (see orange curve in Fig. 4)
does in fact yield the same result as the naive adiabatic picture for the
present setup since the ($n=1$) spin-Berry curvature vanishes identically:
${\boldsymbol{\Omega}}=0$. This has been noted in Ref. Stahl and Potthoff
(2017) already, and the anomalous precession frequency has been explained by
referring to an effective two-spin model $H_{\rm two-
spin}=J{\boldsymbol{s}}_{i_{0}}{\boldsymbol{S}}-{\boldsymbol{B}}{\boldsymbol{S}}$
which disregards the presence of the sites $i\neq i_{0}$, which can be argued
to the justified in the strong-$J$ regime. The two-spin model indeed predicts
${\boldsymbol{\Omega}}=\frac{1}{2}{\boldsymbol{S}}$, so that the
renormalization factor $1/(1-{\boldsymbol{\Omega}}{\boldsymbol{S}})=2$, which
is in reasonable agreement with the results of the full theory.

The remaining problem is to clarify why, for the full model (47), the $n=1$
spin-Berry curvature vanishes. One should note that there is actually an odd-
even effect. For an odd number of sites $L$, the spin-Berry curvature is in
fact finite, and the agreement with the full theory is satisfying already at
the $n=1$ level, while extending the effective theory to $n=2$ yields smaller
corrections only.

The odd-even effect can in fact be explained by a combination of time-reversal
symmetry and the fact that a local spin-dependent perturbation applied to a
non-magnetic ground state cannot induced a finite spin polarization in one
dimension. For $J=0$, the ground state $|\Psi_{0}\rangle$ of an spin-
SU(2)-symmetric tight-binding model is a total-spin singlet. For $J>0$, we
have $|\Psi_{0}\rangle=|\Psi_{0}({\boldsymbol{S}})\rangle$, where the
${\boldsymbol{S}}$ dependence is induced by the local perturbation
$J{\boldsymbol{S}}{\boldsymbol{s}}_{i_{0}}$. Assuming, without loss of
generality, that ${\boldsymbol{S}}=S{\boldsymbol{e}}_{z}$, it is given by a
Slater determinant of the form
$|\Psi_{0}({\boldsymbol{S}})\rangle=\prod_{k}\prod_{k^{\prime}}c^{\dagger}_{k\uparrow}c^{\dagger}_{k^{\prime}\downarrow}|\mbox{vac.}\rangle$,
where $k,k^{\prime}$ refer to the occupied spin-$\uparrow$ and
spin-$\downarrow$ eigenstates of the full Hamiltonian, including the
perturbation, with eigenenergies $\varepsilon_{\uparrow}(k)$ and
$\varepsilon_{\downarrow}(k)$, respectively.

For a one-dimensional particle-hole symmetric tight-binding model at half-
filling, a local spin-dependent but spin-diagonal perturbation
$JS_{z}s_{i_{0}z}$ does not change the number of $\uparrow$ and of
$\downarrow$ eigenstates with eigenenergies
$\varepsilon_{\uparrow}(k),\varepsilon_{\downarrow}(k)<0$, for arbitrary
coupling strength $J$ Kulkarni  _et al._ (1999). This implies that for even
$L$ and at half-filling $N=L$, we must have $N_{\uparrow}=N_{\downarrow}=N/2$.
Consequently, the number of factors in the Slater determinant, labelled by $k$
and $k^{\prime}$, is the same, and thus $|\Psi_{0}({\boldsymbol{S}})\rangle$
is still a total-spin singlet (constructed from ${\boldsymbol{S}}$-dependent
one-particle states), irrespective of the strength of the perturbation $J$.
This argument holds for any direction of ${\boldsymbol{S}}$ and thus implies
that
$\Theta|\Psi_{0}({\boldsymbol{S}})\rangle=|\Psi_{0}({\boldsymbol{S}})\rangle$,
i.e., the ground state is invariant under time reversal $\Theta$ for all
${\boldsymbol{S}}$. Hence, the same holds for its ${\boldsymbol{S}}$
derivative:
$\Theta|\partial_{{\boldsymbol{S}}}\Psi_{0}({\boldsymbol{S}})\rangle=|\partial_{{\boldsymbol{S}}}\Psi_{0}({\boldsymbol{S}})\rangle$.
Some details on the invariance under time reversal are given in Appendix E.

Specializing Eq. (28) to the adiabatic case $n=1$ we thus have
${\boldsymbol{\Omega}}=\frac{1}{2}\sum_{\alpha\beta\gamma}\varepsilon_{\alpha\beta\gamma}{\boldsymbol{e}}_{\alpha}\Omega_{\beta\gamma}$
with

$\displaystyle\Omega_{\beta\gamma}$ $\displaystyle=$ $\displaystyle
i\left[\bra{\partial_{S_{\beta}}\Psi_{0}}\ket{\partial_{S_{\gamma}}\Psi_{0}}-(\beta\leftrightarrow\gamma)\right]$
(52) $\displaystyle=$
$\displaystyle-2\,\mbox{Im}\bra{\partial_{S_{\beta}}\Psi_{0}}\ket{\partial_{S_{\gamma}}\Psi_{0}}$
$\displaystyle=$
$\displaystyle-2\,\mbox{Im}\bra{\partial_{S_{\beta}}\Psi_{0}}\Theta^{\dagger}\Theta\ket{\partial_{S_{\gamma}}\Psi_{0}}^{\ast}$
$\displaystyle=$
$\displaystyle-2\,\mbox{Im}\bra{\partial_{S_{\beta}}\Psi_{0}}\ket{\partial_{S_{\gamma}}\Psi_{0}}^{\ast}=0\>,$

where we have exploited the anti-unitarity of $\Theta$. In an extension of the
discussion of Sec. V for the weak-$J$ case, we can thus infer that the abelian
spin-Berry curvature must vanish for even $L$ and arbitrary $J$ in one
dimension. Let us emphasize that the argument cannot be transferred to the
non-abelian case. For $n>1$, we have $\langle{\boldsymbol{\Omega}}\rangle\neq
0$ in general.

###  VIII.3 Nutation

Apart from the precessional motion, the classical-spin dynamics also exhibits
nutational oscillations with a frequency that is in general different from the
precession frequency. The nutation is most easily seen in an oscillatory
behavior of the $z$ component of the classical spin: The field points into the
$z$ direction, ${\boldsymbol{B}}=B{\boldsymbol{e}}_{z}$, such that the $z$
component of the torque on ${\boldsymbol{S}}$ due to the field must vanish
$({\boldsymbol{B}}\times{\boldsymbol{S}})_{z}=0$. A nonzero time derivative
$\dot{S}_{z}\neq 0$ is, therefore, solely due to the exchange coupling and
directly proportional to
$J(\langle{\boldsymbol{s}}_{i_{0}}\rangle\times{\boldsymbol{S}})_{z}$.

As such, a nutational motion cannot be captured by $n=1$ adiabatic spin-
dynamics (ASD) theory: The adiabatic constraint and a simple symmetry argument
immediately imply that the ground-state local moment
$\langle{\boldsymbol{s}}_{i_{0}}\rangle=\bra{\Psi_{0}({\boldsymbol{S}})}{\boldsymbol{s}}_{i_{0}}\ket{\Psi_{0}({\boldsymbol{S}})}$
must be strictly antiparallel (for $J>0$) to ${\boldsymbol{S}}$, which in turn
implies that $S_{z}$ is a constant of motion. The adiabatic spin dynamics is
thus perfectly precessional albeit, opposed to naive adiabatic theory, with a
renormalized precession frequency, as already discussed above.

Numerical results for $L=11$ as obtained from non-abelian spin-dynamics theory
with $n=2$, see Fig. 6 (blue curve), show that there can be a considerable
variation of the amplitude of the $z$ component of ${\boldsymbol{S}}$. The
nutational oscillation is perfectly harmonic and $S_{z}$ stays non-negative,
when starting with $S_{z}=0$ at $t=0$. As compared with the $S_{z}$ dynamics
predicted by the full theory (red curve), the step from $n=1$ (ASD) to $n=2$
(most simple variant of NA-SD) is in fact the essential one, and the results
for $n=2$ are already close to those of the full theory. The latter, however,
predicts a slight deviation from perfectly harmonic nutational motion, which
is not reproduced with $n=2$ but can be captured with an improved ($n=4$)
approximation within the NA-SD (dashed blue curve). A further increase of $n$
becomes technically more and more involved and has also been found to improve
the results in a non-monotonic way only. It is thus very fortunate that the
main improvement of the $n=1$ ASD is already achieved with $n=2$ NA-SD. For
the rest of the discussion, we will therefore stick to the $n=2$ case.

Figure 6:  Time dependence of the $z$-component of the classical spin as
obtained from $n=2$ (solid blue curve) and from $n=4$ (dashed blue curve) NA-
SD, compared to the result (red curve) of the full theory. $L=11$, $J=1$,
$B=0.1$.

The physical cause of the nutation can be traced back to the time-dependent
admixture of the first excited state $|\Psi_{1}({\boldsymbol{S}}(t))\rangle$
to the instantaneous ground state $|\Psi_{0}({\boldsymbol{S}}(t))\rangle$.
Fig. 7 for $J=1$ (blue curve) displays the absolute square of the ground-state
coefficient $|\alpha_{0}|^{2}$ as function of propagation time corresponding
to the $n=2$ NA-SD result for $S_{z}$ in Fig. 6. Note that we have
$|\alpha_{1}|^{2}=1-|\alpha_{0}|^{2}$ for $n=2$. As for $t=0$ the conduction-
electron system is prepared as the ground state of $\hat{H}$, the ground-state
weight $|\alpha_{0}|^{2}=1$ initially. In the course of time, there is a
weight transfer to the first excited state, which results in a significant
reduction of the ground-state weight down to a minimal value of
$|\alpha_{0}|^{2}\approx 0.72$. Within the $n=2$ NA-SD, the time-dependent
weight transfer is perfectly harmonic, and its frequency is exactly the same
as the nutation frequency of $S_{z}$ (see Fig. 6).

Figure 7:  Time dependence of the ground-state weight $|\alpha_{0}|^{2}$ as
obtained from NA-SD with $n=2$ for a system with $L=11$, for $B=0.1$, and for
various coupling strengths $J=1$ (blue), $J=2$ (orange), and $J=10$ (green).
Figure 8:  The same as Fig. 7 but for the time dependence of $S_{z}$. Coupling
strengths: $J=1$ (blue), $J=2$ (orange), and $J=10$ (green).  Figure 9:  The
same as Fig. 7 but for $L=10$. Note that the same color coding is used.
Coupling strengths: $J=1$ (blue), $J=2$ (orange), and $J=10$ (green).  Figure
10:  The same as Fig. 9 but for the time dependence of $S_{z}$. Coupling
strengths: $J=1$ (blue), $J=2$ (orange), and $J=10$ (green).

Increasing the coupling strength $J$, results in a weaker admixture of the
first excited state, as can be seen by the results for $J=2$ (orange) and
$J=10$ (green) in Fig. 7. This is accompanied by an increasing frequency of
the time-dependent weight transfer. Again, this frequency is precisely the
nutation frequency that is observed in the time dependence of $S_{z}$, which
is displayed in Fig. 8 for the different coupling strengths. We also note that
this is unrelated with the precession frequency which is much less $J$
dependent. Furthermore, also the $J$ dependence of the minimal (maximal)
amplitude shows the same trend for both, the ground-state weight and for
$S_{z}$, respectively.

Compared to the standard perturbative linear-response approach discussed in
the introduction, our approach thus provides an alternative explanation of
nutational spin dynamics. As in the standard theory, nutation is the first
phenomenon that is found in a systematic expansion starting around the
adiabatic limit, namely Taylor expansion in the retardation time on the one
hand and expansion in the dimension of the instantaneous low-energy subspace
on the other. Another important difference is that the NA-SD is formulated for
a closed system while the standard theory relies on a formalism for open
quantum systems. This also explains that the standard approach necessarily
predicts non-conserving Gilbert damping accompanying the nutation motion.

Figure 11:  $J$ dependence of the single-particle eigenenergies for $L=10$
(right) and $L=11$ (left). $B=0.1$.  Figure 12:  The finite-size energy gap
$\Delta E$ between the ground state and the first excited state as function of
$J$ for $L=10$ (blue) and $L=11$ (orange). $B=0.1$.

The time dependence of the weight $|\alpha_{0}|^{2}$, as shown in Figs. 7 and
9, is reminiscent of the Rabi oscillations of the ground-state occupation in a
simple two-level system driven by an oscillatory time-dependent external
field. In our case the driving is due to the classical spin which is
precessing around the axis of the magnetic field. However, the case is more
complicated. Opposed to the standard Rabi setup Bellac (2006), the “two-level
system” emerging in the ($n=2$) NA-SD is itself time-dependent, has a feedback
on the classical spin induced by the spin-Berry curvature via the geometrical
torque, and ${\boldsymbol{S}}$ couples locally rather than globally to a time-
dependent and in general only partially polarized local magnetic moment.

Let us return to the results for the ground-state weight for $L=11$ shown in
Fig. 7. It is tempting to interpret the decrease of the amplitude of the
oscillations of $|\alpha_{0}|^{2}$ with increasing $J$ as a consequence of
approaching the adiabatic limit, where $|\alpha_{0}|^{2}=1$. In fact, this
trend is consistent with the time-averaged angle enclosed by
${\boldsymbol{S}}$ and $\langle{\boldsymbol{s}}_{i_{0}}\rangle$ approaching
$180^{\circ}$ with increasing $J$ (see Fig. 3). However, for a tight-binding
chain with an even number sites ($L=10$), see the data in Fig. 9, we find that
the oscillation amplitude of $|\alpha_{0}|^{2}$ grows with increasing $J$. We
conclude that there is an odd-even effect not only with respect to the
precessional but also to the nutational dynamics.

For an explanation of the effect, we consider the $2L$ single-particle
eigenenergies $\varepsilon_{k}$ of the minimal model Eq. (47). Their $J$
dependence is shown in Fig. 11 for $L=10$ (right, blue lines) and $L=11$
(left, orange lines). Only at $J=0$ are the eigenenergies spin-degenerate, any
finite $J>0$ immediately lifts this degeneracy. Consistent with analytical
results available for tridiagonal pseudo-Toeplitz matrices Kulkarni  _et al._
(1999), we find that the $\varepsilon_{k}(J)$ curves do not intersect and that
a finite “critical” coupling $J\approx 2$ is necessary to split off a pair of
bound states, localized in the vicinity of $i_{0}$, from the “continuum” of
delocalized states. Importantly, however, we note that the finite-size gap
$\Delta E$ between the highest occupied and the lowest unoccupied eigenenergy,
right below and right above $\varepsilon=0$, respectively, shows opposite
trends for $L=10$ and $L=11$.

The $J$ dependence of the gap is displayed in Fig. 12. We note that $\Delta E$
monotonically shrinks with $J$ for $L=10$ (blue lines) and grows with $J$ for
$L=11$. This is also characteristic in general, for systems with an even and
odd number of sites, respectively. According to the adiabatic theorem Bellac
(2006), the real-time dynamics is close to adiabatic if the gap size is large
compared to the inverse $\tau^{-1}$ of the typical time scale $\tau$. Here,
this can be estimated as given by $\tau^{-1}\sim B=0.1$. For the case $L=11$,
this indeed implies that the adiabatic limit is approached with increasing
$J$, while for $L=10$ a decreasing $J$ favours adiabatic dynamics. This also
explains the different $J$ dependence of the amplitudes of the nutational
oscillations of $S_{z}$ shown in Figs. 8 and 10, respectively.

##  IX Concluding discussion

Systems of a single or a few quantum spins coupled to an extended lattice
fermion model pose notoriously difficult quantum many-body problems. Here, by
treating the impurity spins as classical objects with a dynamics that is slow
as compared to the typical electronic time scales, we have concentrated on a
simplified case with the ambition to exactly trace out the high-energy scales
and to arrive at an effective low-energy theory that, apart from the classical
spins, includes a minimal number of electronic degrees of freedom. Our
approach in fact represents a systematic extension of the previously proposed
adiabatic spin dynamics (ASD) theory Stahl and Potthoff (2017), where
unconventional spin dynamics was observed to result from a geometrical spin
torque.

For systems where the typical spin-dynamics time scale is much slower than the
time scale of the electron dynamics, the adiabatic theorem, in case of gapped
systems, tells us that the electron state at an instant of time $t$ is the
ground state of the electronic Hamiltonian for the given spin configuration at
$t$. Alternatively and more general, one may argue that adiabatic dynamics is
due to fast electronic relaxation processes dissipating the excess energy to
the bulk of the system or to external baths. These standard arguments and more
explicit criteria, which typically motivate a purely adiabatic theory, are
rarely controllable and hardly ever fully met in applications to realistic
systems. In most practical cases, it is a priori extremely difficult to decide
whether or not the dynamics is adiabatic. Our approach therefore aims at a
straightforward way to improve the adiabatic spin-dynamics theory in an, at
least in principle, systematic manner.

As the central and sole approximation we assume that the electronic state at
any instant of time $t$ lies in the $n$-dimensional low-energy sector spanned
by the instantaneous ground state, realized for the classical-spin
configuration at time $t$, and the corresponding lowest $n-1$ instantaneous
excited states of the electron system. The approximation is implemented as a
holonomic constraint within a Lagrange formalism. We have seen that the
effective low-energy theory unfolds itself straightforwardly and naturally
takes the form of a non-abelian gauge theory, where the non-abelian spin-Berry
connection and spin-Berry curvature enter the resulting effective equations of
motions for the electronic state and for the spins. The gauge freedom is given
by the arbitrary choice of an orthonormal basis in the instantaneous low-
energy subspace of the electron system. SU(n) gauge transformations leave
observables invariant. The number $n$ of states considered in the non-abelian
spin dynamics (NA-SD) theory can be seen as a control parameter, so that
comparing results for different $n$ allows us to check the validity of the
approach, at least in principle.

The physically interesting point of the emergent low-energy theory is that the
spin dynamics is crucially affected by the gauge-invariant expectation value
of the (gauge-covariant) spin-Berry curvature, i.e., by an additional
geometrical spin torque. In the ASD ($n=1$) a non-zero spin-Berry curvature is
obtained for systems with broken time-reversal symmetry only. Opposed to ASD
($n=1$), however, the non-abelian spin dynamics (NA-SD) theory incorporates a
spin-Berry curvature tensor, the elements of which are generically non-zero
even in the more common time-reversal-symmetric case and both, for the anti-
unitary time-reversal operator squaring to $+1$ and to $-1$. The NA-SD
formalism also provides an elegant and straightforward explanation for the
odd-even effect observed as function of the system size in the simpler ASD
Stahl and Potthoff (2017).

Applications of the NA-SD theory are promising in cases, where (i) the
classical-spin approximation is reasonable, e.g., for magnetic atoms with high
spin quantum numbers or, more generally, with well-developed local magnetic
moments, which are stable on time scales exceeding all other relevant time
scales of the full system. This excludes, e.g., Kondo systems with a fast
screening of the local moment. Strong magnetic anisotropies at surfaces or
interfaces, on the other hand, can favor extremely stable magnetic moments
with respect to both, longitudinal and transversal spin fluctuations
Wiesendanger (2009).

(ii) As regards the electron system, the amount of energy pumped in with the
initial excitation must be small compared to the lowest electron excitation
energies, such that a low-dimensional instantaneous low-energy subspace can
fully capture the essential dynamics. Such situations could be realized in
case of magnetic atoms coupled to tight-binding systems with essentially a
finite number of orbitals, e.g., to metallic nanoislands supported by an
insulating substrate Wiesendanger (2009) or in nanowires Bajpai and Nikolić
(2020), for example. Correlated molecular magnetic systems are interesting as
well, particularly in cases with a degenerate ground-state manifold (see Ref.
R. Rausch and Karrasch (2022) for an instructive example), which naturally
defines the low-energy subspace. In case of formally infinite, e.g.,
condensed-matter systems, NA-SD may be applicable whenever there is a low-
energy sector with a finite gap to excited states at higher energies, such as
insulating systems with a symmetry-induced degenerate ground state.
Topological insulators with gapless edge modes, e.g., Chern or Z2 insulators,
represent another class of systems which are worth to be considered, and the
study of the relation between different Berry curvatures, the spin-Berry
curvature considered here and the conventional Berry curvature of topological
band theory is expected to be particularly instructive. The real-time dynamics
of classical spins coupled to the edge of a one-dimensional spinful Su-
Schrieffer-Heger model Elbracht and Potthoff (2021) and to a two-dimensional
spinful Kane-Mele model Quade and Potthoff (2022) have been discussed
recently. In the former case, the low-energy subspace (at one edge) is spanned
by two quantum states only. For the Z2 Kane-Mele nanoribbon, the helical edge
modes form a continuum but with an extremely small phase space for spin
excitations, which suggests that considering a finite number of basis states
for the low-energy sector could be a reasonably good approximation.

For classical spins coupled to gapless metallic bulk systems, any low-energy
sector is formally infinite-dimensional. While the adiabatic theorem does not
apply to this case, one still expects that a low-energy subspace defined by a
certain maximum excitation energy $\Delta E$ above the many-electron ground
state could reliably capture the electron dynamics, depending on the initial
excitation energy pumped into the system. If the electron system may be
treated in the independent-electron approximation, the application of NA-SD is
well conceivable, since it merely involves diagonalization of the single-
electron hopping matrix and computation of matrix elements of two-electron
operators with two-electron and two-hole excited states above the Fermi sea
(see Appendix D). By varying $\Delta E$, the reliability of the approximation
can be tested.

Here, as a proof of principle, we performed numerical calculations for a
minimal but non-trivial model consisting of a single impurity spin coupled to
the first site of a one-dimensional non-interacting tight-binding model with a
small number of $L$ sites. The real-time dynamics is initiated by a sudden
change of the direction of a local magnetic field coupled to the impurity spin
only. Results obtained from ASD ($n=1$) and NA-SD (for $n=2$ and $n=4$) have
been checked against results obtained from the numerical solution of the full,
unconstrained set of equations of motion for the coupled spin-electron system.
We find that the NA-SD reproduces the anomalous precession frequency that is
already predicted by ASD for systems with an odd number of sites $L$. For even
$L$, NA-SD correctly predicts anomalous precession, which is absent in the
purely adiabatic approach. This deficiency of the ASD can be explained by a
symmetry analysis. Depending on the coupling strength $J$, the dynamics of the
impurity spin can exhibit a considerable nutational motion. As judged by
comparison with the full theory, this more subtle effect is almost
quantitatively covered with NA-SA for $n=2$. NA-SD calculations for $n=4$ show
an even closer agreement with the full theory.

###### Acknowledgements.

We acknowledge support from the Deutsche Forschungsgemeinschaft (DFG, German
Science Foundation) through FOR 5249-449872909 (Project P8) and the European
Research Council via Synergy Grant 854843-FASTCORR.

##  Appendix A Explicit form of the equation of motion for the classical spins

The equations of motion Eq. (25) for the classical spins derived in Sec. III
are implicit differential equations. An explicit form, however, is more
convenient for the numerical evaluation. Here, we briefly discuss a
corresponding reformulation. We start with Eq. (25) and apply
$\times\boldsymbol{S}_{m}$ from the right. This yields

$\dot{\boldsymbol{S}}_{m}=\langle\partial_{{\boldsymbol{S}}_{m}}\hat{H}_{\rm
int}\rangle\times\boldsymbol{S}_{m}+\partial_{{\boldsymbol{S}}_{m}}H_{\rm
cl}\times\boldsymbol{S}_{m}+\sum_{\delta}\sum_{k\gamma}\left(\sum_{\alpha\beta}\varepsilon_{\alpha\beta\delta}S_{m\beta}\langle{\Omega}\rangle_{k\gamma,m\alpha}\right)\dot{S}_{k\gamma}\boldsymbol{e}_{\delta}\>.$
(53)

Next, we combine the components of all $M$ spins in a single $3M$-dimensional
column,

$\boldsymbol{\mathcal{S}}:=(\boldsymbol{S}_{1},\boldsymbol{S}_{2},\dots)^{T}=\sum_{m=1}^{M}\boldsymbol{e}^{M}_{m}\otimes\boldsymbol{S}_{m}\>.$
(54)

Here $\boldsymbol{e}^{M}_{m}$ is the $m$-th canonical $M$-dimensional unit
vector and $\otimes$ denotes the Kronecker product. Writing
$\chi_{m\delta,k\gamma}=\sum_{\alpha\beta}\varepsilon_{\alpha\beta\delta}S_{m\beta}\langle{\Omega}\rangle_{k\gamma,m\alpha}$
for short, the last term on the right-hand side of Eq. (53) can be written as
$\sum_{\delta}(\underline{\chi}\dot{\boldsymbol{\mathcal{S}}})_{m\delta}\boldsymbol{e}_{\delta}$,
and we find the explicit form of the $3M$-dimensional system of differential
equations of motion:

$\dot{\boldsymbol{\mathcal{S}}}=\Big{(}\mathbb{1}-\underline{\chi}\Big{)}^{-1}\cdot\left(\sum_{m}\boldsymbol{e}^{M}_{m}\otimes\left(\langle\partial_{{\boldsymbol{S}}_{m}}\hat{H}_{\rm
int}\rangle\times\boldsymbol{S}_{m}+\partial_{{\boldsymbol{S}}_{m}}H_{\rm
cl}\times\boldsymbol{S}_{m}\right)\right)\>.$ (55)

This involves an inversion of the $3M$-dimensional matrix
$\mathbb{1}-\underline{\chi}$.

##  Appendix B Normalisation conditions

The equations of motion Eq. (25) and Eq. (26) respect the normalisation
conditions Eq. (27). We start with the wave-function normalization. Eq. (26)
implies

$i\sum_{i}\alpha^{\ast}_{i}(\partial_{t}\alpha_{i})=\sum_{ij}\alpha^{\ast}_{i}\alpha_{j}\bra{\Psi_{i}}\hat{H}\ket{\Psi_{j}}-i\sum_{ij}\alpha^{\ast}_{i}\alpha_{j}\bra{\Psi_{i}}\partial_{t}\ket{\Psi_{j}}=-i\sum_{i}(\partial_{t}\alpha^{\ast}_{i})\alpha_{i}\>.$
(56)

This yields $\partial_{t}\sum_{i}|\alpha_{i}|^{2}=0$ as required. Conservation
of the length of the classical spins can be verified directly from their
equations of motion, Eq. (25), or, more conveniently by taking the scalar
product of both sides of Eq. (53) with ${\boldsymbol{S}}_{m}$. This yields
${\boldsymbol{S}}_{m}\dot{{\boldsymbol{S}}}_{m}=0$ as required. However,
conservation of the spin length has been exploited already in deriving Eq.
(25), directly after Eq. (20).

Alternatively, we may thus explicitly take care of the normalization
conditions $\boldsymbol{S}^{2}_{m}=1$ by treating them as additional
constraints when deriving the equations of motion from the Lagrangian (13).
This is done with $M$ Lagrange multipliers $\lambda_{m}$, i.e., we replace the
Lagrangian by

$L^{\prime}_{\text{eff}}(\\{\boldsymbol{S}\\},\\{\dot{\boldsymbol{S}}\\},\\{\alpha\\},\\{\alpha^{\ast}\\},\\{\dot{\alpha}\\},\\{\dot{\alpha}^{\ast}\\},\\{\lambda\\})=L_{\text{eff}}(\\{\boldsymbol{S}\\},\\{\dot{\boldsymbol{S}}\\},\\{\alpha\\},\\{\alpha^{\ast}\\},\\{\dot{\alpha}\\},\\{\dot{\alpha}^{\ast}\\})-\sum_{m}\lambda_{m}(\boldsymbol{S}^{2}_{m}-1)\>,$
(57)

such that the Euler-Lagrange equation for $\lambda_{m}$ reads
$\boldsymbol{S}^{2}_{m}=1$. Further, the equation of motion for a classical
spin $\boldsymbol{S}_{m}$ is modified as

$0=\frac{1}{\absolutevalue{\boldsymbol{S}_{m}}^{3}}\dot{\boldsymbol{S}}_{m}\times\boldsymbol{S}_{m}+\langle\partial_{{\boldsymbol{S}}_{m}}\hat{H}_{\rm
int}\rangle+\partial_{{\boldsymbol{S}}_{m}}H_{\rm
cl}+\sum_{k}\sum_{\beta\gamma}\dot{S}_{k\gamma}\langle{\Omega}\rangle_{k\gamma,m\beta}{\boldsymbol{e}}_{\beta}+2\lambda_{m}\boldsymbol{S}_{m}\>.$
(58)

Acting on both sides of the equation with $\times\boldsymbol{S}_{m}$ and with
$\cdot\boldsymbol{S}_{m}$, respectively, gives a system of two equations,
which is equivalent with Eq. (58):

$\displaystyle 0$ $\displaystyle=$
$\displaystyle(\dot{\boldsymbol{S}}_{m}\times\boldsymbol{S}_{m})\times\frac{\boldsymbol{S}_{m}}{\absolutevalue{\boldsymbol{S}_{m}}^{3}}+\langle\partial_{{\boldsymbol{S}}_{m}}\hat{H}_{\rm
int}\rangle\times\boldsymbol{S}_{m}+\partial_{{\boldsymbol{S}}_{m}}H_{\rm
cl}\times\boldsymbol{S}_{m}+\sum_{k}\sum_{\beta\gamma}\dot{S}_{k\gamma}\langle{\Omega}\rangle_{k\gamma,m\beta}{\boldsymbol{e}}_{\beta}\times\boldsymbol{S}_{m}\>,$
$\displaystyle 0$ $\displaystyle=$
$\displaystyle\langle\partial_{{\boldsymbol{S}}_{m}}\hat{H}_{\rm
int}\rangle\cdot\boldsymbol{S}_{m}+\partial_{{\boldsymbol{S}}_{m}}H_{\rm
cl}\cdot\boldsymbol{S}_{m}+\sum_{k}\sum_{\beta\gamma}\dot{S}_{k\gamma}\langle{\Omega}\rangle_{k\gamma,m\beta}{\boldsymbol{e}}_{\beta}\cdot\boldsymbol{S}_{m}+2\lambda_{m}\boldsymbol{S}^{2}_{m}\>.$
(59)

Exploiting $\boldsymbol{S}^{2}_{m}=1$ in the second equation fixes the
Lagrange multipliers as

$\lambda_{m}=-\frac{1}{2}\left(\langle\partial_{{\boldsymbol{S}}_{m}}\hat{H}_{\rm
int}\rangle\cdot\boldsymbol{S}_{m}+\partial_{{\boldsymbol{S}}_{m}}H_{\rm
cl}\cdot\boldsymbol{S}_{m}+\sum_{k}\sum_{\beta\gamma}\dot{S}_{k\gamma}\langle{\Omega}\rangle_{k\gamma,m\beta}{\boldsymbol{e}}_{\beta}\cdot\boldsymbol{S}_{m}\right)\>,$
(60)

while using it in the first equation reproduces the familiar equation of
motion (25).

##  Appendix C Spin-Berry curvature in terms of a projection operator

To prove Eq. (28) we start from the definition (22) of the non-abelian spin-
Berry curvature and insert the definition for the spin-Berry connection Eq.
(15). This gives

$\displaystyle\Omega^{(ij)}_{k\gamma,m\beta}$ $\displaystyle=$ $\displaystyle
i\left[\bra{\partial_{S_{k\gamma}}\Psi_{i}}\ket{\partial_{S_{m\beta}}\Psi_{j}}-\bra{\partial_{S_{m\beta}}\Psi_{i}}\ket{\partial_{S_{k\gamma}}\Psi_{j}}\right]$
(61) $\displaystyle+$ $\displaystyle
i\sum_{l=0}^{n-1}\left[\bra{\Psi_{i}}\partial_{S_{k\gamma}}\ket{\Psi_{l}}\bra{\Psi_{l}}\partial_{S_{m\beta}}\ket{\Psi_{j}}-\bra{\Psi_{i}}\partial_{S_{m\beta}}\ket{\Psi_{l}}\bra{\Psi_{l}}\partial_{S_{k\gamma}}\ket{\Psi_{j}}\right]\>,$

where we have exploited the commutativity of the derivatives
$\partial_{S_{k\gamma}}$ and $\partial_{S_{m\beta}}$. Using the completeness
relation and inserting a unity,

${\mathbb{1}}=\mathcal{Q}_{n}+\sum_{l=0}^{n-1}\ket{\Psi_{l}}\bra{\Psi_{l}}\>,$
(62)

where $Q_{n}=\sum_{i\geq n}\ket{\Psi_{i}}\bra{\Psi_{i}}$ is the projector onto
the orthogonal complement of the low-energy space ${\cal
E}_{n}(\\{{\boldsymbol{S}}\\})$, we find

$\displaystyle\Omega^{(ij)}_{k\gamma,m\beta}$ $\displaystyle=$ $\displaystyle
i\left[\bra{\partial_{S_{k\gamma}}\Psi_{i}}\mathcal{Q}_{n}\ket{\partial_{S_{m\beta}}\Psi_{j}}-\bra{\partial_{S_{m\beta}}\Psi_{i}}\mathcal{Q}_{n}\ket{\partial_{S_{k\gamma}}\Psi_{j}}\right]$
(63) $\displaystyle+$ $\displaystyle
i\sum_{l=0}^{n-1}\left[\bra{\partial_{S_{k\gamma}}\Psi_{i}}\ket{\Psi_{l}}\bra{\Psi_{l}}\ket{\partial_{S_{m\beta}}\Psi_{j}}-\bra{\partial_{S_{m\beta}}\Psi_{i}}\ket{\Psi_{l}}\bra{\Psi_{l}}\ket{\partial_{S_{k\gamma}}\Psi_{j}}\right]$
$\displaystyle+$ $\displaystyle
i\sum_{l=0}^{n-1}\left[\bra{\Psi_{i}}\partial_{S_{k\gamma}}\ket{\Psi_{l}}\bra{\Psi_{l}}\partial_{S_{m\beta}}\ket{\Psi_{j}}-\bra{\Psi_{i}}\partial_{S_{m\beta}}\ket{\Psi_{l}}\bra{\Psi_{l}}\partial_{S_{k\gamma}}\ket{\Psi_{j}}\right]\>.$

Noting that
$\bra{\partial_{S_{m\beta}}\Psi_{i}}\ket{\Psi_{j}}=-\bra{\Psi_{i}}\partial_{S_{m\beta}}\ket{\Psi_{j}}$,
we see that the last two terms on the right-hand side cancel, and thus

$\Omega^{(ij)}_{k\gamma,m\beta}=i\left[\bra{\partial_{S_{k\gamma}}\Psi_{i}}\mathcal{Q}_{n}\ket{\partial_{S_{m\beta}}\Psi_{j}}-\bra{\partial_{S_{m\beta}}\Psi_{i}}\mathcal{Q}_{n}\ket{\partial_{S_{k\gamma}}\Psi_{j}}\right]\>.$
(64)

##  Appendix D Numerical computation of spin-Berry curvature and connection

The equations of motion Eq. (25) and Eq. (26) form a coupled, non-linear set
of ordinary differential equations, which can be solved numerically by
standard techniques. Making use of the fact that the conduction-electron
system is non-interacting, however, is essential for an efficient computation
of the key quantities of the electron system, namely the spin-Berry curvature
and connection.

We start by specializing Eqs. (23) and (28) to the single-spin case $M=1$,

$\langle\Omega\rangle_{\beta\gamma}=i\sum_{i,j=0}^{n-1}\alpha_{i}^{\ast}\alpha_{j}\left(\bra{\partial_{\beta}\Psi_{i}}\mathcal{Q}_{n}\ket{\partial_{\gamma}\Psi_{j}}-\bra{\partial_{\gamma}\Psi_{i}}\mathcal{Q}_{n}\ket{\partial_{\beta}\Psi_{j}}\right)=2\sum_{i,j=0}^{n-1}\sum_{l\geq
n}\Im{\alpha_{i}^{\ast}\alpha_{j}\bra{\Psi_{i}}\partial_{\beta}\ket{\Psi_{l}}\bra{\Psi_{l}}\ket{\partial_{\gamma}\Psi_{j}}}\>,$
(65)

and use the identity

$\bra{\Psi_{i}}\partial_{\beta}\ket{\Psi_{l}}=\frac{\bra{\Psi_{i}}\frac{\partial\hat{H}}{\partial
S_{\beta}}\ket{\Psi_{l}}}{E_{l}-E_{i}}\qquad(E_{i}\neq E_{l})$ (66)

to express $\langle\boldsymbol{\Omega}\rangle$ in the form

$\displaystyle\langle\Omega\rangle_{\beta\gamma}$ $\displaystyle=$
$\displaystyle-2\imaginary\sum_{ij}\sum_{l}^{E_{l}\neq
E_{i},E_{j}}\alpha_{i}^{\ast}\alpha_{j}\frac{\bra{\Psi_{i}}\frac{\partial\hat{H}}{\partial
S_{\beta}}\ket{\Psi_{l}}\bra{\Psi_{l}}\frac{\partial\hat{H}}{\partial
S_{\gamma}}\ket{\Psi_{j}}}{(E_{i}-E_{l})(E_{j}-E_{l})}=-2\imaginary
J^{2}\sum_{ij}\sum_{l}^{E_{l}\neq
E_{i},E_{j}}\frac{\alpha_{i}^{\ast}\alpha_{j}\bra{\Psi_{i}}s_{i_{0}\beta}\ket{\Psi_{l}}\bra{\Psi_{l}}s_{i_{0}\gamma}\ket{\Psi_{j}}}{(E_{i}-E_{l})(E_{j}-E_{l})}\>.$

The matrix elements can be computed by plugging in the definition of the local
spin
$\boldsymbol{s}_{i}=\frac{\hbar}{2}\sum_{\sigma\sigma^{\prime}}c_{i\sigma}^{\dagger}\boldsymbol{\sigma}_{\sigma\sigma^{\prime}}c_{i\sigma^{\prime}}$
and by transforming to the eigenstates of the effective hopping matrix:

$c^{\dagger}_{i\sigma}=\sum_{k\tilde{\sigma}}U^{\dagger}_{k\tilde{\sigma},i\sigma}c^{\dagger}_{k\tilde{\sigma}}\;,\qquad
c_{i\sigma}=\sum_{k\tilde{\sigma}}U_{i\sigma,k\tilde{\sigma}}c_{k\tilde{\sigma}}\>.$
(68)

This yields

$\displaystyle\sum_{l}^{E_{l}\neq
E_{i},E_{j}}\frac{\bra{\Psi_{i}}s_{i_{0}\beta}\ket{\Psi_{l}}\bra{\Psi_{l}}s_{i_{0}\gamma}\ket{\Psi_{j}}}{(E_{i}-E_{l})(E_{j}-E_{l})}$
$\displaystyle=$
$\displaystyle\frac{1}{4}\sum_{\sigma\sigma^{\prime}\tau\tau^{\prime}}{\sum}^{\prime\prime}_{kk^{\prime}qq^{\prime}\atop\tilde{\sigma}\tilde{\sigma}^{\prime}\tilde{\tau}\tilde{\tau}^{\prime}}U^{\dagger}_{k\tilde{\sigma},i_{0}\sigma}\sigma^{(\beta)}_{\sigma\sigma^{\prime}}U_{i_{0}\sigma^{\prime},k^{\prime}\tilde{\sigma}^{\prime}}U^{\dagger}_{q\tilde{\tau},i_{0}\tau}\sigma^{(\gamma)}_{\tau\tau^{\prime}}U_{i_{0}\tau^{\prime},q^{\prime}\tilde{\tau}^{\prime}}\times$
(69) $\displaystyle\times$
$\displaystyle\frac{\bra{\Psi_{i}}c^{\dagger}_{k\tilde{\sigma}}c_{k^{\prime}\tilde{\sigma}^{\prime}}c^{\dagger}_{q\tilde{\tau}}c_{q^{\prime}\tilde{\tau}^{\prime}}\ket{\Psi_{j}}}{(E_{i}-E_{j}+\varepsilon_{q^{\prime}\tilde{\tau}^{\prime}}-\varepsilon_{q\tilde{\tau}})(\varepsilon_{q^{\prime}\tilde{\tau}^{\prime}}-\varepsilon_{q\tilde{\tau}})}\>,$

where $\sum^{\prime\prime}$ means that the indices
$k,k^{\prime},q,q^{\prime},\tilde{\sigma},\tilde{\sigma}^{\prime},\tilde{\tau},\tilde{\tau}^{\prime}$
can only take values such that
$c^{\dagger}_{k^{\prime}\tilde{\sigma}^{\prime}}c_{k\tilde{\sigma}}\ket{i}$
and $c^{\dagger}_{q\tilde{\tau}}c_{q^{\prime}\tilde{\tau}^{\prime}}\ket{j}$
are not contained in the low-energy subspace. For the summation indices it is
required that

$(k,\tilde{\sigma})\neq(k^{\prime},\tilde{\sigma}^{\prime})\quad\mbox{and}\quad(q,\tilde{\tau})\neq(q^{\prime},\tilde{\tau}^{\prime})$
(70)

since $i\neq l$ and $j\neq l$. Plugging this into (LABEL:eq:gammaim) gives an
expression that can be evaluated straightforwardly by numerical means.

We also have to compute the Berry connection, i.e., the matrix elements
$\bra{\Psi_{i}}\partial_{\beta}\ket{\Psi_{j}}$ in Eq. (26) (see also Eq.
(15)). For $i\neq j$ we can again use (66), since the single-particle energies
are generically nondegenerate for finite $J$ and since this implies that
states $\ket{\Psi_{i}}$ and $\ket{\Psi_{j}}$ with $E_{i}=E_{j}$ must differ in
more than one single-particle eigenstate. For $i=j$, one the other hand,
$\bra{\Psi_{i}}\partial_{\beta}\ket{\Psi_{i}}$ mast be computed differently.
We exploit that the many-particle state $\ket{\Psi_{i}}$ is a Slater
determinant:

$\ket{\Psi_{i}}=c^{\dagger}_{n_{1}}c^{\dagger}_{n_{2}}\cdots
c^{\dagger}_{n_{N}}\ket{\text{vac}}\>.$ (71)

Therewith, we get

$\partial_{S_{\beta}}\ket{\Psi_{i}}=\sum_{i=1}^{N}c^{\dagger}_{n_{1}}\cdots(\partial_{S_{\beta}}c^{\dagger}_{n_{i}})\cdots
c^{\dagger}_{n_{N}}\ket{\text{vac}}$ (72)

with

$\displaystyle\partial_{S_{\beta}}c^{\dagger}_{n_{i}}=\partial_{S_{\beta}}\sum_{j\sigma}U_{j\sigma,n_{i}}c^{\dagger}_{j\sigma}=\sum_{j\sigma}(\partial_{S_{\beta}}U_{j\sigma,n_{i}})c^{\dagger}_{j\sigma}=\sum_{j\sigma}\sum_{m}(\partial_{S_{\beta}}U_{j\sigma,n_{i}})U^{\dagger}_{m,j\sigma}c^{\dagger}_{m}=\sum_{m}(U^{\dagger}\partial_{S_{\beta}}U)_{mn_{i}}c^{\dagger}_{m}\>.$
(73)

Multiplying Eq. (72) with $\bra{\Psi_{i}}$ from the left yields

$\displaystyle\bra{\Psi_{i}}\partial_{\beta}\ket{\Psi_{i}}$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{N}\sum_{m}(U^{\dagger}\partial_{S_{\beta}}U)_{mn_{i}}\underbrace{\bra{\text{vac}}c_{n_{N}}\cdots
c_{n_{i}}\cdots c_{n_{1}}c^{\dagger}_{n_{1}}\cdots c^{\dagger}_{m}\cdots
c^{\dagger}_{n_{N}}\ket{\text{vac}}}_{\delta_{n_{i}m}}$ (74) $\displaystyle=$
$\displaystyle\sum_{i=1}^{N}(U^{\dagger}\partial_{S_{\beta}}U)_{n_{i}n_{i}}={\sum_{n}}^{\prime}(U^{\dagger}\partial_{S_{\beta}}U)_{nn}=\sum_{n}(U^{\dagger}\partial_{S_{\beta}}U)_{nn}\bra{\Psi_{i}}\hat{n}_{n}\ket{\Psi_{i}}\>,$

where $\sum_{n}^{\prime}$ indicates that the sum only contains those single-
particle states that are occupied in the many-particle state $\ket{\Psi_{i}}$.
The derivative of the $U$-matrix can be computed by standard numerical means.

##  Appendix E Time-reversal symmetric ground state

We consider the minimal model with Hamiltonian $H$ given by Eq. (47). For
$J=0$ the (electronic part of the) model is invariant under SU(2) spin
rotations. For a given direction of the classical spin, say
${\boldsymbol{S}}=S{\boldsymbol{e}}_{z}$, and for $J>0$ the symmetry is breaks
down to a U(1) symmetry under spin rotations around the $z$ axis. As argued in
the main text, the local spin-dependent perturbation is not strong enough to
spin-polarize the system, irrespective of the coupling strength $J$. In this
case the ground state of $H$ is invariant under time reversal, as is shown in
the following:

The antiunitary operator $\Theta$ representing time reversal in Fock space is
defined via its action on the creation and annihilation operators as

$\Theta
c^{\dagger}_{i\uparrow}\Theta^{\dagger}=c^{\dagger}_{i\downarrow}\;,\quad\Theta
c^{\dagger}_{i\downarrow}\Theta^{\dagger}=-c^{\dagger}_{i\uparrow}\>,$ (75)

where $i$ refers to lattice sites and $\sigma=\uparrow,\downarrow$ to the spin
projection with respect to the $z$ axis. Due to the remaining U(1) symmetry,
the Hamiltonian can be diagonalized in the spin-$\uparrow$ and
spin-$\downarrow$ sectors separately, i.e., the single-particle eigenstates
$c^{\dagger}_{k\sigma}|\mbox{vac}\rangle$ of $H$ are obtained via a spin-
diagonal and spin-independent unitary transformation:

$c^{\dagger}_{k\sigma}=\sum_{i}U_{ik}c^{\dagger}_{i\sigma}\>.$ (76)

For the model Eq. (47) with ${\boldsymbol{S}}=S{\boldsymbol{e}}_{z}$, the
effective hopping matrix Eq. (51) is real and symmetric, and we can thus
assume a real and orthogonal transformation matrix $U$. The creation operators
referring to the eigenbasis of $H$ in the one-particle subspace thus transform
as

$\Theta
c^{\dagger}_{k\uparrow}\Theta^{\dagger}=c^{\dagger}_{k\downarrow}\;,\quad\Theta
c^{\dagger}_{k\downarrow}\Theta^{\dagger}=-c^{\dagger}_{k\uparrow}\>$ (77)

under time reversal.

For even $N$, the ground state of $H$ is the Slater determinant

$\ket{\Psi_{0}}=\prod^{\rm occ.}_{k}c^{\dagger}_{k\uparrow}\prod^{\rm
occ.}_{k^{\prime}}c^{\dagger}_{k^{\prime}\downarrow}\ket{\text{vac}}\>,$ (78)

where $\ket{\text{vac}}$ is the time-reversal invariant vacuum,
$k=1,...,N_{\uparrow}$, and $k^{\prime}=1,...,N_{\downarrow}$ with
$N_{\uparrow}=N_{\downarrow}=N/2$, as the ground state is unpolarized.
Applying $\Theta$ yields

$\Theta\ket{\Psi_{0}}=(-1)^{N_{\downarrow}}\prod^{\rm
occ.}_{k}c^{\dagger}_{k\downarrow}\prod^{\rm
occ.}_{k^{\prime}}c^{\dagger}_{k^{\prime}\uparrow}\ket{\text{vac}}\>,$ (79)

and, after reordering,

$\Theta\ket{\Psi_{0}}=(-1)^{N_{\uparrow}N_{\downarrow}}(-1)^{N_{\downarrow}}\prod^{\rm
occ.}_{k^{\prime}}c^{\dagger}_{k^{\prime}\uparrow}\prod^{\rm
occ.}_{k}c^{\dagger}_{k\downarrow}\ket{\text{vac}}\>.$ (80)

For $N_{\uparrow}=N_{\downarrow}=N/2$, however, the total sign is $+1$, and
hence the ground state is time-reversal symmetric

$\Theta\ket{\Psi_{0}}=\ket{\Psi_{0}}\>.$ (81)

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