# Spin Berry curvature of the Haldane model

Simon Michel  I. Institute of Theoretical Physics, Department of Physics,
University of Hamburg, Notkestraße 9-11, 22607 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

The feedback of the geometrical Berry phase, accumulated in an electron
system, on the slow dynamics of classical degrees of freedom is governed by
the Berry curvature. Here, we study local magnetic moments, modelled as
classical spins, which are locally exchange coupled to the (spinful) Haldane
model for a Chern insulator. In the emergent equations of motion for the slow
classical-spin dynamics there is a an additional anomalous geometrical spin
torque, which originates from the corresponding spin-Berry curvature. Due to
the explicitly broken time-reversal symmetry, this is nonzero but usually
small in a condensed-matter system. We develop the general theory and compute
the spin-Berry curvature, mainly in the limit of weak exchange coupling, in
various parameter regimes of the Haldane model, particularly close to a
topological phase transition and for spins coupled to sites at the zigzag edge
of the model in a ribbon geometry. The spatial structure of the spin-Berry
curvature tensor, its symmetry properties, the distance dependence of its
nonlocal elements and further properties are discussed in detail. For the case
of two classical spins, the effect of the geometrical spin torque leads to an
anomalous non-Hamiltonian spin dynamics. It is demonstrated that the magnitude
of the spin-Berry curvature is decisively controlled by the size of the
insulating gap, the system size and the strength of local exchange coupling.

##  I Introduction

The time evolution of a quantum system with a non-degenerate and gapped ground
state, steered by external time-dependent classical degrees of freedom, is
governed by the adiabatic theorem Messiah (1961); Jansen  _et al._ (2007): If
prepared at time $t=0$ in its ground state, the system state at $t>0$ is given
by its instantaneous ground state for the then existing configuration of
classical degrees of freedom. Roughly, the adiabatic theorem applies, if the
typical time scale $\tau$ of the classical time evolution is large compared to
the inverse of the gap $\Delta E$ between the ground and the first excited
state. Besides the dynamical phase, the system accumulates a geometrical phase
during the adiabatic time evolution, which cannot be gauged away in case of a
cyclic motion in the classical state space Berry (1984); Simon (1983); Wilczek
and Zee (1984). This Berry phase and related phenomena, such as the molecular
Aharonov-Bohm effect, have been studied extensively in molecular physics Mead
(1992); Resta (2000), assuming that the coordinates ${\boldsymbol{R}}_{m}$ of
the nuclei can be treated as classical observables.

The Berry-phase physics of the quantum system also feeds back to the dynamical
state of the classical observables as has been pointed out early Kuratsuji and
Iida (1985); Moody  _et al._ (1986); Zygelman (1987). In molecular systems,
for example, this leads to an additional geometric force term in the classical
Newtonian set of equations of motion, which has the form of a Lorentz force
despite the absence of a physical magnetic field.

The geometric force plays an important role also in other contexts as, for
example, in the semiclassical theory of electron dynamics in crystals, where
the wave vector ${\boldsymbol{k}}$ is treated as a dynamical classical
variable Resta (2000); Bohm  _et al._ (2003). Its pendant on the side of the
quantum system is the ${\boldsymbol{k}}$-space Berry phase, an important
quantity in topological band theory Hasan and Kane (2010); Qi and Zhang
(2011); Chiu  _et al._ (2016). In fact, Berry phase and geometrical force are
rather general concepts that can be applied to configuration spaces of various
kinds of classical degrees of freedom.

Hence, a similar situation arises for another type of quantum-classical hybrid
systems as well, namely for a quantum-mechanical electron system with an
exchange coupling to the local magnetic moments ${\boldsymbol{S}}_{m}$ of
magnetic atoms, which are modelled as classical spins, i.e., vectors of fixed
length ${\boldsymbol{S}}_{m}^{2}=1$. The typical (picosecond) time scale for
classical spin dynamics is much longer than that of the fast (femtosecond)
electron quantum dynamics, such that the electron system can adiabatically
follow the classical spin configuration and accumulate a finite spin-Berry
phase Tatara (2019). The feedback of the Berry physics on the classical
system, however, is different and given by a geometrical spin torque rather
than a force, and can thereby strongly affect the precessional spin dynamics
Stahl and Potthoff (2017).

This spin torque is given in terms of the gauge invariant spin-Berry curvature
Stahl and Potthoff (2017), which must be distinguished from the
${\boldsymbol{k}}$-space Berry curvature and the ${\boldsymbol{R}}_{m}$-space
or atomic-nuclei-Berry curvature of molecular dynamics. A simplified variant
of the approach presented in Ref. Stahl and Potthoff, 2017 has been used to
discuss the effect of the geometrical spin torque on the spin-wave spectrum of
magnetic systems Niu and Kleinman (1998); Niu  _et al._ (1999).

In the recent years, the spin-Berry curvature and the geometrical spin torque
have been studied more intensively. A numerical study for an exactly solvable
one-dimensional model of local magnetic moments has been carried out to
quantitatively disentangle the geometrical spin torque from other
contributions to the spin dynamics Bajpai and Nikolic (2020). This work also
emphasizes the close relation to geometric-friction effects Head-Gordon and
Tully (1995); Berry and Robbins (1993), i.e., Gilbert spin damping, which can
be derived within linear-response theory Onoda and Nagaosa (2006);
Bhattacharjee  _et al._ (2012); Sayad and Potthoff (2015); Sayad  _et al._
(2016); Bajpai and Nikolic (2019) or adiabatic-response theory Campisi  _et
al._ (2012). To go beyond the strict adiabatic limit, a generalized
geometrical spin torque originating from the non-Abelian spin-Berry curvature
has been proposed recently Lenzing  _et al._ (2022), in the spirit of Ref.
Wilczek and Zee (1984). Effects of the geometrical spin torque in purely
classical spin systems, where a single slow spin Elbracht  _et al._ (2020) or
several slow spins Michel and Potthoff (2021) interact with an exchange-
coupled system of fast spins, have been studied within a formal framework
largely analogous to the quantum-classical hybrid case. This goes back to
earlier work that drew attention to holonomy effects in purely classical
systems Hannay (1985).

The main subject of this paper is to study the spin-Berry curvature for a
model for local magnetic moments (classical spins) coupled to an insulator.
The latter is described by a two-band tight-binding model for electrons
hopping on a two-dimensional lattice. With the (spinful) Haldane model on the
honeycomb lattice Haldane (1988); Bernevig (2013) we choose a prototypical
model for a Chern insulator.

Our motivation for this choice is twofold: First, the Haldane model stands at
the origin of the development topological band theory Thouless  _et al._
(1982); Hasan and Kane (2010); Qi and Zhang (2011); Chiu  _et al._ (2016) and
represents the first model for a quantum anomalous Hall insulator. It can be
experimentally realized in a setup with ultracold fermions Jotzu  _et al._
(2014). Furthermore, there is generally a rapidly increasing interest in the
study of magnetic impurities in topological insulators and at the boundaries
of topological materials in particular Wray  _et al._ (2011); Honolka  _et
al._ (2012); Scholz  _et al._ (2012); Valla  _et al._ (2012); Goth  _et al._
(2013); Schlenk  _et al._ (2013); Eelbo  _et al._ (2014); Li  _et al._ (2014);
Jiang  _et al._ (2015); Chen  _et al._ (2015); Pieper and Fehske (2016);
Rüßmann  _et al._ (2018); Sumida  _et al._ (2019) as well as in their mutual
interaction Liu  _et al._ (2009); Gao  _et al._ (2009); Kurilovich  _et al._
(2017); Hosseini  _et al._ (2020); Yevtushenko and Yudson (2018).

Second, if the Hamiltonian of the quantum system is time-reversal symmetric,
there may be a finite Berry phase originating from a singularity in the
classical configuration space, while the Berry curvature vanishes Ihm (1991).
A finite Berry curvature (${\boldsymbol{k}}$-, ${\boldsymbol{R}}_{m}$\- or
spin-Berry curvature), however, requires an explicit breaking of time-reversal
symmetry. In case of the spin-Berry curvature in a model including an exchange
coupling to the classical spins, time-reversal symmetry is already broken
intrinsically, since the classical spins act like local symmetry-breaking
magnetic fields. This effect, however, becomes irrelevant in the (physical)
limit of weak exchange-interaction strength $J$. As is shown later, the spin-
Berry curvature generally vanishes in perturbation theory up to ${\cal
O}(J^{2})$, if the quantum system itself is time-reversal symmetric. The
Haldane model is constructed such that the symmetry is broken due to a
“magnetic field” coupling to the orbital degrees of freedom rather than to the
local electron spin (note that the original model is spinless anyway), which
averages to zero in a unit cell. The mechanism by which the Haldane model
leads to a nonzero Chern number can be realized in actual materials, where the
spin-orbit coupling acts as a source of the orbital magnetic field Ren  _et
al._ (2016); Mokrousov (2018).

The spinful Haldane model is interesting, as it allows us to study the impact
of the spin-Berry curvature on the effective interaction between local
magnetic moments and thus on the resulting effective spin dynamics. The
standard indirect magnetic RKKY exchange is of the Bloembergen-Rowland type
Bloembergen and Rowland (1955) in case of an insulator, and its strength
depends exponentially on the insulating gap $\Delta E$, opposed to RKKY theory
for metallic phases Kit ; rkk . A central question is thus if the geometric
spin torque is able to “boost” the indirect coupling between two classical
spins.

To this end, we generalize the theory of Ref. Stahl and Potthoff (2017) to an
arbitrary number of classical spins and derive the nonlocal geometrical spin
torque that emerges in the adiabatic limit, see Sec. II. This requires to
deduce explicit expressions for the spin-Berry curvature, which can be
evaluated numerically, and an analysis of time-reversal and spatial symmetry
transformations. In particular, we show that the spin-Berry curvature is
closely related to the spin susceptibility of the electron system. For two
classical spins, we explicitly demonstrate that the resulting adiabatic spin
dynamics is anomalous and cannot be derived from an effective Hamiltonian.

On this basis, we numerically analyze the dependence of the bulk spin-Berry
curvature on the positions and on the distance between two sites, at which two
classical spins are exchange coupled to the electron system, see Sec. III.
Furthermore, we study its dependence on the parameters of the Haldane model in
the weak-$J$ regime both, for topologically trivial and nontrivial phases, and
especially in the parametric vicinity of a topological phase transition.
Invoking a phase-space argument, one can show that the spin-Berry curvature is
in fact continuous across the transition. It diverges, however, for the
Haldane model in a ribbon geometry, if the classical spins couple to sites at
the zigzag edge.

##  II Theory

###  II.1 Effective low-energy theory

We consider a quantum-classical hybrid system consisting of $M$ classical
spins ${\boldsymbol{S}}\equiv({\boldsymbol{S}}_{1},...,{\boldsymbol{S}}_{M})$
of fixed length $S_{m}=|{\boldsymbol{S}}_{m}|=1$, the intrinsic dynamics of
which are governed by a classical Hamilton function
${H}_{\text{cl}}({\boldsymbol{S}})$, and $N$ electrons with corresponding
quantum tight-binding Hamiltonian $\hat{H}_{\text{qu}}$ constructed with the
help of creation and annihilation operators $c^{\dagger}_{i\sigma}$ and
$c_{i\sigma}$. Here $i$ refers to a site of a given lattice and
$\sigma=\uparrow,\downarrow$ to the electron spin projection. The orthonormal
states $|i\sigma\rangle$ span the one-particle Hilbert space. To develop the
general theory, it is not yet necessary to further specify
${H}_{\text{cl}}({\boldsymbol{S}})$ and $\hat{H}_{\text{qu}}$. Concrete
calculations will be performed for ${H}_{\text{cl}}({\boldsymbol{S}})=0$ and
for $\hat{H}_{\text{qu}}$ given by the Haldane model Haldane (1988); Bernevig
(2013), but trivially generalized for spinful electrons, see Sec. II.7. Spins
and electrons interact via a local exchange

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

between the $m$th classical spin ${\boldsymbol{S}}_{m}$ and the local spin
${\boldsymbol{s}}_{i_{m}}$ of the electron system at the site $i_{m}$ given by
${\boldsymbol{s}}_{i}=\frac{1}{2}\sum_{\sigma\sigma^{\prime}}c^{\dagger}_{i\sigma}{\boldsymbol{\tau}}_{\sigma\sigma^{\prime}}c_{i\sigma^{\prime}}$,
where ${\boldsymbol{\tau}}=(\tau_{x},\tau_{y},\tau_{z})^{T}$ is the vector of
Pauli matrices. With $J>0$ the coupling is antiferromagnetic. The total
Hamiltonian is

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

The coupled equations of motion for the spin configuration
${\boldsymbol{S}}={\boldsymbol{S}}(t)$ and the pure $N$-electron state
$|\Psi\rangle=|\Psi(t)\rangle$ are given by:

$\displaystyle i\partial_{t}|\Psi(t)\rangle$ $\displaystyle=$
$\displaystyle\hat{H}({\boldsymbol{S}})|\Psi(t)\rangle\>,$
$\displaystyle\dot{{\boldsymbol{S}}}_{m}(t)$ $\displaystyle=$
$\displaystyle\frac{\partial\langle\hat{H}({\boldsymbol{S}})\rangle}{\partial{\boldsymbol{S}}_{m}}\times{\boldsymbol{S}}_{m}(t)\>.$
(3)

Here, $\langle\cdots\rangle$ is the expectation value with the state
$|\Psi(t)\rangle$, and the dot denotes the time derivative. The equations can
be derived from the Hamiltonian $\hat{H}({\boldsymbol{S}})$ or, equivalently,
via the action principle, from the Lagrangian
$L=L({\boldsymbol{S}},\dot{{\boldsymbol{S}}},|\Psi\rangle,\dot{|\Psi\rangle},\langle\Psi|,\dot{\langle\Psi|})$:

$L=\sum_{m}{\boldsymbol{A}}({\boldsymbol{S}}_{m})\boldsymbol{\dot{S}}_{m}+\langle\Psi|i\partial_{t}|\Psi\rangle-\langle\Psi|\hat{H}({\boldsymbol{S}})|\Psi\rangle\>.$
(4)

Here,

${\boldsymbol{A}}({\boldsymbol{S}}_{m})=-\frac{1}{S_{m}^{2}}\frac{{\boldsymbol{e}}\times{\boldsymbol{S}}_{m}}{1+{\boldsymbol{e}}{\boldsymbol{S}}_{m}/S_{m}}\;,$
(5)

with a fixed unit vector ${\boldsymbol{e}}$, which is usually chosen as
${\boldsymbol{e}}={\boldsymbol{e}}_{z}$ Dirac (1931). We have

$\nabla\times{\boldsymbol{A}}({\boldsymbol{S}}_{m})=-{\boldsymbol{S}}_{m}/S_{m}^{3}\>,$
(6)

and thus ${\boldsymbol{A}}({\boldsymbol{S}}_{m})$ can be interpreted as the
vector potential of a unit magnetic monopole located at
${\boldsymbol{S}}_{m}=0$. For the derivation of Eq. (3) from Eq. (4) see the
supplemental material of Ref. Elbracht  _et al._ (2020).

The Lagrangian formulation is convenient to implement the central constraint

$|\Psi(t)\rangle=|\Psi_{0}({\boldsymbol{S}}(t))\rangle$ (7)

of adiabatic spin dynamics (ASD) theory Stahl and Potthoff (2017); Elbracht
_et al._ (2020); Michel and Potthoff (2021). This is a holonomic constraint
without explicit time dependence. We thereby assume that the fast electron
dynamics is perfectly constrained to the manifold of instantaneous ground
states $|\Psi_{0}({\boldsymbol{S}})\rangle$ of $\hat{H}({\boldsymbol{S}})$,
and that the ground states are non-degenerate. The smooth map
${\boldsymbol{S}}\mapsto|\Psi_{0}({\boldsymbol{S}})\rangle$ is characterized
by the spin-Berry connection:

${\boldsymbol{C}}_{m}({\boldsymbol{S}})=i\langle\Psi_{0}({\boldsymbol{S}})|\frac{\partial}{\partial{\boldsymbol{S}}_{m}}|\Psi_{0}({\boldsymbol{S}})\rangle\>.$
(8)

Unter a local (${\boldsymbol{S}}$-dependent) gauge transformation

$|\Psi_{0}({\boldsymbol{S}})\rangle\mapsto|\Psi^{\prime}_{0}({\boldsymbol{S}})\rangle=e^{i\chi({\boldsymbol{S}})}|\Psi_{0}({\boldsymbol{S}})\rangle$
(9)

given by a smooth function $\chi({\boldsymbol{S}})$, the spin-Berry connection
transforms as

${\boldsymbol{C}}_{m}({\boldsymbol{S}})\mapsto{\boldsymbol{C}}^{\prime}_{m}({\boldsymbol{S}})={\boldsymbol{C}}_{m}({\boldsymbol{S}})-\frac{\partial\chi({\boldsymbol{S}})}{\partial{\boldsymbol{S}}_{m}}\>,$
(10)

while the spin-Berry curvature

$\Omega_{m\alpha,m^{\prime}\alpha^{\prime}}({\boldsymbol{S}})=\frac{\partial}{\partial
S_{m\alpha}}C_{m^{\prime}\alpha^{\prime}}({\boldsymbol{S}})-\frac{\partial}{\partial
S_{m^{\prime}\alpha^{\prime}}}C_{m\alpha}({\boldsymbol{S}})$ (11)

($\alpha,\alpha^{\prime}=x,y,z$) is gauge invariant.

Using the constraint Eq. (7), we can eliminate the electron degrees of freedom
in the Lagrangian (4) so that a spin-only low-energy theory is obtained. The
effective Lagrangian depends on the spin degrees of freedom only and is given
by:

$\displaystyle L_{\rm eff}({\boldsymbol{S}},\dot{{\boldsymbol{S}}})$
$\displaystyle=$
$\displaystyle\sum_{m}{\boldsymbol{A}}({\boldsymbol{S}}_{m})\dot{{\boldsymbol{S}}}_{m}+\langle\Psi_{0}({\boldsymbol{S}})|i\partial_{t}|\Psi_{0}({\boldsymbol{S}})\rangle$
(12) $\displaystyle-$
$\displaystyle\langle\Psi_{0}({\boldsymbol{S}})|\hat{H}({\boldsymbol{S}})|\Psi_{0}({\boldsymbol{S}})\rangle-\sum_{m}\lambda_{m}{\boldsymbol{S}}_{m}^{2}\>.$

The last term, with Lagrange multipliers $\lambda_{m}$ takes care of the
normalization condition $S_{m}=1$. The Euler-Lagrange equations are
straightforwardly derived. We find

$\displaystyle 0$ $\displaystyle=$ $\displaystyle\frac{d}{dt}\frac{\partial
L_{\rm eff}}{\partial\dot{{\boldsymbol{S}}}_{m}}-\frac{\partial L_{\rm
eff}}{\partial{\boldsymbol{S}}_{m}}=\frac{\dot{{\boldsymbol{S}}}_{m}\times{\boldsymbol{S}}_{m}}{S_{m}^{3}}+\frac{\partial\langle\hat{H}({\boldsymbol{S}})\rangle}{\partial{\boldsymbol{S}}_{m}}$
(13) $\displaystyle-$
$\displaystyle\sum_{\alpha}\sum_{m^{\prime}\alpha^{\prime}}\Omega_{m\alpha
m^{\prime}\alpha^{\prime}}\dot{S}_{m^{\prime}\alpha^{\prime}}{\boldsymbol{e}}_{\alpha}+2\lambda_{m}{\boldsymbol{S}}_{m}\>,$

where ${\boldsymbol{e}}_{\alpha}$ is the $\alpha$th canonical unit vector,
$\langle\cdots\rangle$ is the expectation value in the ground state
$|\Psi_{0}({\boldsymbol{S}})\rangle$, and where we have made use of Eqs. (6),
(8), (11) and the antisymmetry of the spin-Berry curvature

$\Omega_{m\alpha,m^{\prime}\alpha^{\prime}}({\boldsymbol{S}})=-\Omega_{m^{\prime}\alpha^{\prime},m\alpha}({\boldsymbol{S}})\>.$
(14)

Scalar multiplication of Eq. (13) with ${\boldsymbol{S}}_{m}$ yields an
expression for $\lambda_{m}$. Taking the cross product with
${\boldsymbol{S}}_{m}$ from the right and using ${\boldsymbol{S}}_{m}^{2}=1$,
we find the effective equation of motion

$\dot{{\boldsymbol{S}}}_{m}=\frac{\partial\langle\hat{H}({\boldsymbol{S}})\rangle}{\partial{\boldsymbol{S}}_{m}}\times{\boldsymbol{S}}_{m}+{\boldsymbol{T}}_{m}\times{\boldsymbol{S}}_{m}\;,$
(15)

where the last term, ${\boldsymbol{T}}_{m}\times{\boldsymbol{S}}_{m}$, with

${\boldsymbol{T}}_{m}={\boldsymbol{T}}_{m}({\boldsymbol{S}},\dot{{\boldsymbol{S}}})=\sum_{\alpha}\sum_{m^{\prime}\alpha^{\prime}}\Omega_{m^{\prime}\alpha^{\prime}m\alpha}({\boldsymbol{S}})\dot{S}_{m^{\prime}\alpha^{\prime}}{\boldsymbol{e}}_{\alpha}$
(16)

is the geometric spin torque. We note that for $M=1$ this reproduces the
result given in Ref. Stahl and Potthoff (2017).

The spin-Berry curvature represents the feedback of the geometrical Berry
physics of the quantum system on the classical spin dynamics in the deep
adiabatic limit. In addition to the first term in Eq. (15), it gives rise to
an anomalous spin torque. The strength of this geometrical spin torque is
determined by the elements of the spin-Berry curvature tensor.

###  II.2 Spin-Berry curvature

There are various useful general representations of the spin-Berry curvature.
First, starting from the definition Eq. (11) and using Eq. (8), we get

$\Omega_{m\alpha,m^{\prime}\alpha^{\prime}}({\boldsymbol{S}})=i\left(\frac{\partial\langle\Psi_{0}|}{\partial
S_{m\alpha}}\frac{\partial|\Psi_{0}\rangle}{\partial
S_{m^{\prime}\alpha^{\prime}}}-\frac{\partial\langle\Psi_{0}|}{\partial
S_{m^{\prime}\alpha^{\prime}}}\frac{\partial|\Psi_{0}\rangle}{\partial
S_{m\alpha}}\right)=-2\,\mbox{Im}\left(\frac{\partial\langle\Psi_{0}|}{\partial
S_{m\alpha}}\frac{\partial|\Psi_{0}\rangle}{\partial
S_{m^{\prime}\alpha^{\prime}}}\right)\>,$ (17)

where $|\Psi_{0}\rangle=|\Psi_{0}({\boldsymbol{S}})\rangle$ has been written
for short.

Second, let $|\Psi_{n}({\boldsymbol{S}})\rangle$ for $n>0$ be the (possibly
degenerate) $n$th excited eigenstate of $\hat{H}({\boldsymbol{S}})$, i.e.,
$\hat{H}({\boldsymbol{S}})|\Psi_{n}({\boldsymbol{S}})\rangle=E_{n}({\boldsymbol{S}})|\Psi_{n}({\boldsymbol{S}})\rangle$.
Differentiating with respect to $S_{m\alpha}$ yields the following identity,
valid for $n\neq 0$:

$\langle\Psi_{0}({\boldsymbol{S}})|\nabla_{m\alpha}|\Psi_{n}({\boldsymbol{S}})\rangle=\frac{\langle\Psi_{0}({\boldsymbol{S}})|\nabla_{m\alpha}\hat{H}({\boldsymbol{S}})|\Psi_{n}({\boldsymbol{S}})\rangle}{E_{n}-E_{0}}\;,$
(18)

where $\nabla_{m\alpha}\equiv\partial/\partial S_{m\alpha}$. Inserting a
resolution of the unity with a local orthonormal basis of eigenstates
$\\{|\Psi_{n}({\boldsymbol{S}})\rangle\\}_{n=0,1,...}$ in Eq. (17) and using
Eq. (18), we find:

$\Omega_{m\alpha,m^{\prime}\alpha^{\prime}}({\boldsymbol{S}})=-2\,\mbox{Im}\sum_{n\neq
0}\frac{\langle\Psi_{0}({\boldsymbol{S}})|\nabla_{m\alpha}\hat{H}({\boldsymbol{S}})|\Psi_{n}({\boldsymbol{S}})\rangle\,\langle\Psi_{n}({\boldsymbol{S}})|\nabla_{m^{\prime}\alpha^{\prime}}\hat{H}({\boldsymbol{S}})|\Psi_{0}({\boldsymbol{S}})\rangle}{(E_{0}({\boldsymbol{S}})-E_{n}({\boldsymbol{S}}))^{2}}\>,$
(19)

and with Eqs. (1) and (2),

$\Omega_{m\alpha,m^{\prime}\alpha^{\prime}}({\boldsymbol{S}})=-2J^{2}\,\mbox{Im}\sum_{n\neq
0}\frac{\langle\Psi_{0}({\boldsymbol{S}})|s_{i_{m}\alpha}|\Psi_{n}({\boldsymbol{S}})\rangle\,\langle\Psi_{n}({\boldsymbol{S}})|s_{i_{m^{\prime}}\alpha^{\prime}}|\Psi_{0}({\boldsymbol{S}})\rangle}{(E_{0}({\boldsymbol{S}})-E_{n}({\boldsymbol{S}}))^{2}}\>.$
(20)

Note that a possibly present classical-spin-classical-spin coupling in $H_{\rm
cl}({\boldsymbol{S}})$ does not contribute here.

Third, we note that the energy eigenstates for $J=0$ are trivially independent
of the spin configuration ${\boldsymbol{S}}$. Expanding
$|\Psi_{n}({\boldsymbol{S}})\rangle=|\Psi^{(0)}_{n}\rangle+{\cal O}(J)$ and
$E_{n}=E^{(0)}_{n}+{\cal O}(J)$ and noting the explicit $J^{2}$ prefactor in
Eq. (20), the spin-Berry curvature takes the form

$\Omega_{m\alpha,m^{\prime}\alpha^{\prime}}({\boldsymbol{S}})=\Omega_{m\alpha,m^{\prime}\alpha^{\prime}}=-2J^{2}\,\mbox{Im}\sum_{n\neq
0}\frac{\langle\Psi^{(0)}_{0}|s_{i_{m}\alpha}|\Psi^{(0)}_{n}\rangle\,\langle\Psi^{(0)}_{n}|s_{i_{m^{\prime}}\alpha^{\prime}}|\Psi^{(0)}_{0}\rangle}{\big{(}E^{(0)}_{0}-E^{(0)}_{n}\big{)}^{2}}+{\cal
O}(J^{3})$ (21)

in the weak-$J$ limit. We see that the spin-Berry curvature becomes
independent of the classical spin configuration in this case.

Fourth, in the weak-$J$ limit, there is a close relation with the retarded
magnetic spin susceptibility of the unperturbed ($J=0$) electron system, which
is defined as

$\chi_{i\alpha,i^{\prime}\alpha^{\prime}}(t)=-i\Theta(t)e^{-\eta
t}\langle[s_{i\alpha}(t),s_{i^{\prime}\alpha^{\prime}}(0)]\rangle^{(0)}\;.$
(22)

Here, $\Theta$ is the step function, $\eta$ is a positive infinitesimal, and
$\langle\cdots\rangle^{(0)}$ denotes the expectation value with the ground
state of the $J=0$ electron system. Furthermore,
$s_{i\alpha}(t)=e^{i\hat{H}_{0}t}s_{i\alpha}e^{-i\hat{H}_{0}t}$ with
$\hat{H}_{0}=\hat{H}|_{J=0}$. Expanding the commutator and inserting a
resolution of the unity with eigenstates of $\hat{H}_{0}$ yields the Lehmann
representation via Fourier transformation:

$\chi_{i\alpha,i^{\prime}\alpha^{\prime}}(\omega)=\int dt\,e^{i\omega
t}\chi_{i\alpha,i^{\prime}\alpha^{\prime}}(t)=\sum_{n\neq
0}\left(\frac{\langle\Psi^{(0)}_{0}|s_{i\alpha}|\Psi^{(0)}_{n}\rangle\,\langle\Psi^{(0)}_{n}|s_{i^{\prime}\alpha^{\prime}}|\Psi^{(0)}_{0}\rangle}{\omega+i\eta-(E^{(0)}_{n}-E^{(0)}_{0})}-\frac{\langle\Psi^{(0)}_{0}|s_{i^{\prime}\alpha^{\prime}}|\Psi^{(0)}_{n}\rangle\,\langle\Psi^{(0)}_{n}|s_{i\alpha}|\Psi^{(0)}_{0}\rangle}{\omega+i\eta-(E^{(0)}_{0}-E^{(0)}_{n})}\right)\>.$
(23)

The susceptibility at $\omega=0$ and its $\omega$-derivative at $\omega=0$ are
given by:

$\chi_{i\alpha,i^{\prime}\alpha^{\prime}}(0)=-2\,\mbox{Re}\sum_{n\neq
0}\frac{\langle\Psi^{(0)}_{0}|s_{i\alpha}|\Psi^{(0)}_{n}\rangle\,\langle\Psi^{(0)}_{n}|s_{i^{\prime}\alpha^{\prime}}|\Psi^{(0)}_{0}\rangle}{E^{(0)}_{n}-E^{(0)}_{0}}\>,\;\frac{\partial}{\partial\omega}\chi_{i\alpha,i^{\prime}\alpha^{\prime}}(0)=-2i\,\mbox{Im}\sum_{n\neq
0}\frac{\langle\Psi^{(0)}_{0}|s_{i\alpha}|\Psi^{(0)}_{n}\rangle\,\langle\Psi^{(0)}_{n}|s_{i^{\prime}\alpha^{\prime}}|\Psi^{(0)}_{0}\rangle}{\big{(}E^{(0)}_{n}-E^{(0)}_{0}\big{)}^{2}}\>.$
(24)

Here, for a gapped system, we can disregard the infinitesimal $i\eta$.
Comparing with Eq. (21), we conclude:

$\Omega_{m\alpha,m^{\prime}\alpha^{\prime}}({\boldsymbol{S}})=\Omega_{m\alpha,m^{\prime}\alpha^{\prime}}=-iJ^{2}\frac{\partial}{\partial\omega}\chi_{i_{m}\alpha,i^{\prime}_{m}\alpha^{\prime}}(\omega)\Big{|}_{\omega=0}+{\cal
O}(J^{3})\>.$ (25)

We see that, in the weak-coupling limit, the spin-Berry curvature describes
the linear magnetic response of the electron system due to a slow time-
dependent perturbation and in fact represents the first nontrivial correction
to a static perturbation.

###  II.3 Time reversal and spin rotation

Equations (15) and (16) tell us that anomalous adiabatic spin dynamics under a
geometrical spin torque requires a finite spin-Berry curvature. In this
context, it is remarkable that the spin-Berry curvature vanishes in the
weak-$J$ regime, if the underlying quantum system is time-reversal symmetric.

This is easily seen as follows: Let $\Theta$ be the usual anti-unitary
representation of time reversal in the Fock space. We have
$\Theta{\boldsymbol{s}}_{i_{m}}\Theta^{\dagger}=-{\boldsymbol{s}}_{i_{m}}$,
and thus $\Theta\hat{H}_{\rm int}\Theta^{\dagger}=-\hat{H}_{\rm int}$.
Obviously, the interaction term Eq. (1), involving classical spins that may be
seen as local magnetic fields, explicitly breaks time-reversal symmetry.
However, this is irrelevant in the weak-$J$ limit, see Eqs. (24) and (25),
since here the spin-Berry curvature is a property of the unperturbed ($J=0$)
electron system only. The unperturbed electron system is time-reversal
symmetric if $[\hat{H}_{\text{qu}},\Theta]=0$. The case of a non-degenerate
ground state, as considered here, implies that there is no Kramers degeneracy
and that the total number of spin-$1/2$ electrons must be even. Hence, the
time-reversal operator squares to unity, $\Theta^{2}=+1$, and we can choose an
orthonormal basis of time-reversal-symmetric eigenstates
$|\Psi^{(0)}_{n}\rangle$ of $\hat{H}_{\text{qu}}$, i.e., we have
$\Theta|\Psi^{(0)}_{n}\rangle=|\Psi^{(0)}_{n}\rangle$ for the ground state
($n=0$) and all excited states ($n>0$). Therewith, we have
$\langle\Psi^{(0)}_{0}|s_{i\alpha}|\Psi^{(0)}_{n}\rangle=\langle\Theta\Psi^{(0)}_{0}|s_{i\alpha}|\Theta\Psi^{(0)}_{n}\rangle$
for the matrix element in Eq. (24). Anti-linearity of $\Theta$ implies
$\langle\Theta\Psi^{(0)}_{0}|s_{i\alpha}|\Theta\Psi^{(0)}_{n}\rangle=\langle\Psi^{(0)}_{0}|\Theta^{\dagger}s_{i\alpha}\Theta|\Psi^{(0)}_{n}\rangle^{\ast}$,
and with $\Theta^{\dagger}s_{i\alpha}\Theta=-{\boldsymbol{s}}_{i\alpha}$ and
$\Theta^{\dagger}\Theta={\boldsymbol{1}}$, we eventually find
$\langle\Psi^{(0)}_{0}|s_{i\alpha}|\Psi^{(0)}_{n}\rangle=-\langle\Psi^{(0)}_{0}|s_{i\alpha}|\Psi^{(0)}_{n}\rangle^{\ast}$.
We conclude that time-reversal symmetry enforces that the matrix elements are
purely imaginary and that

$\frac{\partial}{\partial\omega}\chi_{i\alpha,i^{\prime}\alpha^{\prime}}(0)=0\>.$
(26)

The spin-Berry curvature vanishes in the weak-$J$ limit for a time-reversal
symmetric electron system. As we are particularly interested in the
geometrical spin torque, this motivates us to study systems with broken time-
reversal symmetry already at $J=0$. Beyond ${\cal O}(J^{2})$, time-reversal
symmetry is broken explicitly.

Next, we consider the usual unitary representation
$U=U({\boldsymbol{n}},\varphi)=\exp(-i{\boldsymbol{s}}_{\rm
tot}{\boldsymbol{n}}\varphi)$ of spin rotations on the Fock space, where the
unit vector ${\boldsymbol{n}}$ defines the rotation axis and $\varphi$ the
rotation angle. The generators are given by the components of the total spin
${\boldsymbol{s}}_{\rm tot}=\sum_{i}{\boldsymbol{s}}_{i}$. Invariance of
$\hat{H}_{\text{qu}}$ under spin rotations,
$[\hat{H}_{\text{qu}},{\boldsymbol{s}}_{\rm tot}]=0$, implies that its
eigenstates $\\{|\Psi_{n}\rangle\\}$ can be simultaneously chosen as
eigenstates of $U$. As ${\boldsymbol{s}}_{i_{m}}$ is a vector operator, we
have $U^{\dagger}s_{i_{m}\alpha}U=\sum_{\beta}R_{\alpha\beta}s_{i_{m}\beta}$,
where $R=R({\boldsymbol{n}},\varphi)$ is the defining SO(3) matrix
representation. With this, the left matrix element in the expression for
$\Omega_{m\alpha,m^{\prime}\alpha^{\prime}}$ [see Eqs. (24) and (25)] can be
written as
$\langle\Psi^{(0)}_{0}|UU^{\dagger}s_{i_{m}\alpha}UU^{\dagger}|\Psi^{(0)}_{n}\rangle=\sum_{\beta}R_{\alpha\beta}e^{i\phi_{0}}\langle\Psi^{(0)}_{0}|s_{i_{m}\beta}|\Psi^{(0)}_{n}\rangle
e^{-i\phi_{n}}$, where the phase factors $e^{i\phi_{n}}$ are the corresponding
eigenvalues of $U$. These cancel with those obtained from the right matrix
element and, hence, we find
$\Omega_{m\alpha,m^{\prime}\alpha^{\prime}}=\sum_{\beta\beta^{\prime}}R_{\alpha\beta}\Omega_{m\beta,m^{\prime}\beta^{\prime}}R^{T}_{\beta^{\prime}\alpha^{\prime}}$.
Irreducibility of $R$ and Schur’s lemma thus imply
$\Omega_{m\alpha,m^{\prime}\alpha^{\prime}}=\Omega_{mm^{\prime}}\delta_{\alpha\alpha^{\prime}}$.
Furthermore, the antisymmetry condition Eq. (14) requires that the matrix
$\underline{\Omega}$ with elements $\Omega_{mm^{\prime}}$ is skew-symmetric.
In case of two impurity spins ($M=2$) this means

$(\Omega_{m\alpha,m^{\prime}\alpha^{\prime}})=\underline{\Omega}\otimes{\boldsymbol{1}}=\Omega\begin{pmatrix}0&1\\\
-1&0\end{pmatrix}\otimes{\boldsymbol{1}}\>,$ (27)

i.e., the spin-Berry curvature is fully determined by a single real number
$\Omega$. Note that spin-rotation symmetry requires $\Omega=0$ in the case of
a single impurity spin ($M=1$), i.e., a finite spin-Berry curvature is
obtained beyond the weak-$J$ limit only.

###  II.4 RKKY interaction

The static ($\omega=0$) magnetic susceptibility
$\chi_{i\alpha,i^{\prime}\alpha^{\prime}}(0)$ is generally nonzero, even if
$\hat{H}_{\text{qu}}$ is time-reversal symmetric and the matrix elements are
purely imaginary. This quantity just describes the linear-in-$J$ response of
the electron system at time $t$. In the adiabatic limit, the response of the
electron system is in fact static: The local magnetic moment
$\langle{\boldsymbol{s}}_{i_{m}}\rangle$ at site $i_{m}$ in the
(instantaneous) ground state $|\Psi_{0}({\boldsymbol{S}}(t))\rangle$ at time
$t$, which is induced by the classical perturbations
$J{\boldsymbol{S}}_{m^{\prime}}(t)$ that couple to
${\boldsymbol{s}}_{i_{m^{\prime}}}$, is given by:

$\langle
s_{i_{m}\alpha}\rangle=\sum_{m^{\prime}\alpha^{\prime}}\chi_{i_{m}\alpha,i_{m^{\prime}}\alpha^{\prime}}(0)JS_{m^{\prime}\alpha^{\prime}}(t)+{\cal
O}(J^{2})\>.$ (28)

Invariance of $\hat{H}_{\text{qu}}$ under SU(2) spin rotations implies that
$\chi_{i_{m}\alpha,i_{m^{\prime}}\alpha^{\prime}}(0)=\delta_{\alpha\alpha^{\prime}}\chi_{i_{m},i_{m^{\prime}}}(0)\equiv\delta_{\alpha\alpha^{\prime}}\chi_{mm^{\prime}}(0)$,
so that we can write
$\langle{\boldsymbol{s}}_{i_{m}}\rangle=\sum_{m^{\prime}}J\chi_{m,m^{\prime}}(0){\boldsymbol{S}}_{m^{\prime}}(t)$
up to linear order in $J$. With the Hellmann-Feynman theorem we have
$\partial\langle\hat{H}({\boldsymbol{S}})\rangle/\partial{\boldsymbol{S}}_{m}=J\langle{\boldsymbol{s}}_{i_{m}}\rangle+\partial
H_{\rm cl}({\boldsymbol{S}})/\partial{\boldsymbol{S}}_{m}$. Assuming that
$H_{\rm cl}\equiv 0$, for the sake of simplicity, and using Eqs. (15) and
(28), we find

$\dot{{\boldsymbol{S}}}_{m}=\sum_{m^{\prime}}J^{2}\chi_{m,m^{\prime}}(0){\boldsymbol{S}}_{m^{\prime}}\times{\boldsymbol{S}}_{m}+{\boldsymbol{T}}_{m}({\boldsymbol{S}},\dot{{\boldsymbol{S}}})\times{\boldsymbol{S}}_{m}\;.$
(29)

For a time-reversal-symmetric Hamiltonian, the second, geometrical term
vanishes while the first is just the standard classical spin dynamics
described by the effective classical Hamiltonian $H_{\rm
RKKY}=\sum_{mm^{\prime}}J^{\rm
RKKY}_{mm^{\prime}}{\boldsymbol{S}}_{m}{\boldsymbol{S}}_{m^{\prime}}$ with
effective RKKY rkk  exchange coupling $J^{\rm
RKKY}_{mm^{\prime}}=J^{2}\chi_{mm^{\prime}}(\omega=0)$. We note, that both
spin torques in Eq. (29), the RKKY torque and the geometric torque, come with
the same prefactor $J^{2}$.

###  II.5 Effective dynamics of two classical spins

Equation (29) is an implicit nonlinear system of differential equations. It
can be further simplified in the case of two classical spins. With Eqs. (15)
and (16) we get for $M=2$:

$\displaystyle\dot{{\boldsymbol{S}}}_{1}=J^{\rm
RKKY}{\boldsymbol{S}}_{2}\times{\boldsymbol{S}}_{1}-\Omega\dot{{\boldsymbol{S}}}_{2}\times{\boldsymbol{S}}_{1}\>,$
$\displaystyle\dot{{\boldsymbol{S}}}_{2}=J^{\rm
RKKY}{\boldsymbol{S}}_{1}\times{\boldsymbol{S}}_{2}+\Omega\dot{{\boldsymbol{S}}}_{1}\times{\boldsymbol{S}}_{2}\>.$
(30)

These equations hold in the weak-$J$ limit with $J^{\rm
RKKY}=J^{2}\chi_{12}(\omega=0)=J^{2}\chi_{21}(0)$, where we can explicitly
make use of Eq. (27). Note that for finite $\Omega$ they are not forminvariant
under exchange of the two spins $1\leftrightarrow 2$. This is interesting,
since any Hamilton function
${H}_{\text{cl}}({\boldsymbol{S}}_{1},{\boldsymbol{S}}_{2})$ for identical
classical spins would be symmetric,
${H}_{\text{cl}}({\boldsymbol{S}}_{1},{\boldsymbol{S}}_{2})={H}_{\text{cl}}({\boldsymbol{S}}_{2},{\boldsymbol{S}}_{1})$,
and would lead to forminvariant equations of motion. In fact, the asymmetry of
the second term $\propto\Omega$ demonstrates the anomalous, non-Hamiltonian
character of the resulting spin dynamics. A similar discussion has been given
earlier Michel and Potthoff (2021) in a different (purely classical) context.

For $m=1,2$ we introduce real and antisymmetric $3\times 3$ matrices

${\cal A}_{m}=\begin{pmatrix}0&-S_{mz}&S_{my}\\\ S_{mz}&0&-S_{mx}\\\
-S_{my}&S_{mx}&0\\\ \end{pmatrix}\>,$ (31)

such that we can replace the cross product by a matrix-vector multiplication,
${\boldsymbol{S}}_{m}\times(...)={\cal A}_{m}(...)$, and write

$\displaystyle\dot{{\boldsymbol{S}}}_{1}$ $\displaystyle=$ $\displaystyle
J^{\rm RKKY}{\boldsymbol{S}}_{2}\times{\boldsymbol{S}}_{1}+\Omega{\cal
A}_{1}\dot{{\boldsymbol{S}}}_{2}\>,$ $\displaystyle\dot{{\boldsymbol{S}}}_{2}$
$\displaystyle=$ $\displaystyle J^{\rm
RKKY}{\boldsymbol{S}}_{1}\times{\boldsymbol{S}}_{2}-\Omega{\cal
A}_{2}\dot{{\boldsymbol{S}}}_{1}\>.$ (32)

Using this notation, we can formally solve for $\dot{{\boldsymbol{S}}}_{1}$
and $\dot{{\boldsymbol{S}}}_{2}$,

$\begin{pmatrix}\dot{{\boldsymbol{S}}}_{1}\\\
\dot{{\boldsymbol{S}}}_{2}\end{pmatrix}=J^{\rm RKKY}{\cal
M}^{-1}\begin{pmatrix}{\cal A}_{2}&0\\\ 0&{\cal
A}_{1}\end{pmatrix}\begin{pmatrix}{\boldsymbol{S}}_{1}\\\
{\boldsymbol{S}}_{2}\end{pmatrix}\;,$ (33)

where the $6\times 6$ matrix ${\cal M}$ is defined as

${\cal M}\equiv{\boldsymbol{1}}-\Omega\begin{pmatrix}0&{\cal A}_{1}\\\ -{\cal
A}_{2}&0\end{pmatrix}=\begin{pmatrix}{\boldsymbol{1}}&-\Omega{\cal A}_{1}\\\
\Omega{\cal A}_{2}&{\boldsymbol{1}}\end{pmatrix}\>.$ (34)

Therewith we have rewritten the equations of motion as explicit differential
equations. We note that the determinant of the block matrix is

$\det{\cal M}=\det({\boldsymbol{1}}+\Omega^{2}{\cal A}_{2}{\cal A}_{1})\>.$
(35)

A straightforward calculation yields the simple result

$\det{\cal
M}=\Omega^{4}({\boldsymbol{S}}_{1}{\boldsymbol{S}}_{2})^{2}-2\Omega^{2}{\boldsymbol{S}}_{1}{\boldsymbol{S}}_{2}+1\>,$
(36)

i.e., matrix inversion is possible unless $\det{\cal M}=0$ or

${\boldsymbol{S}}_{1}{\boldsymbol{S}}_{2}\Omega^{2}=1\>.$ (37)

The inversion can be done analytically:

$\displaystyle{\cal
M}^{-1}=\frac{1}{1-\Omega^{2}{\boldsymbol{S}}_{1}{\boldsymbol{S}}_{2}}\begin{pmatrix}{\boldsymbol{1}}-\Omega^{2}{\boldsymbol{S}}_{2}\\!\otimes\\!{\boldsymbol{S}}_{1}&\Omega{\cal
A}_{1}\\\ -\Omega{\cal
A}_{2}&{\boldsymbol{1}}-\Omega^{2}{\boldsymbol{S}}_{1}\\!\otimes\\!{\boldsymbol{S}}_{2}\end{pmatrix},$

where ${\boldsymbol{S}}_{1}\otimes{\boldsymbol{S}}_{2}$ is the outer (dyadic)
product of ${\boldsymbol{S}}_{1}$ with ${\boldsymbol{S}}_{2}$. Using the
identity
${\boldsymbol{S}}_{1}\otimes{\boldsymbol{S}}_{2}\cdot{\boldsymbol{S}}_{1}\times{\boldsymbol{S}}_{2}=0$
and combining the results, we find

$\displaystyle\dot{{\boldsymbol{S}}}_{1}=\frac{J^{\rm
RKKY}}{1-\Omega^{2}{\boldsymbol{S}}_{1}{\boldsymbol{S}}_{2}}\left[{\boldsymbol{S}}_{2}\times{\boldsymbol{S}}_{1}+\Omega{\boldsymbol{S}}_{1}\times({\boldsymbol{S}}_{1}\times{\boldsymbol{S}}_{2})\right]\>,$
$\displaystyle\dot{{\boldsymbol{S}}}_{2}=\frac{J^{\rm
RKKY}}{1-\Omega^{2}{\boldsymbol{S}}_{1}{\boldsymbol{S}}_{2}}\left[{\boldsymbol{S}}_{1}\times{\boldsymbol{S}}_{2}-\Omega{\boldsymbol{S}}_{2}\times({\boldsymbol{S}}_{2}\times{\boldsymbol{S}}_{1})\right]\>.$
(39)

We immediately see that $|{\boldsymbol{S}}_{1}|$, $|{\boldsymbol{S}}_{2}|$,
and the scalar product ${\boldsymbol{S}}_{1}{\boldsymbol{S}}_{2}$ are
conserved. The total impurity spin ${\boldsymbol{S}}_{1}+{\boldsymbol{S}}_{2}$
is not conserved for $\Omega\neq 0$. This is compensated by a respective
dynamics of the total quantum spin, see Ref. Michel and Potthoff (2021) for a
discussion in the purely classical case.

For $\Omega=0$ we recover the standard RKKY dynamics governed by the RKKY
coupling constant $J^{\rm RKKY}_{12}=J^{\rm
RKKY}_{21}=J^{2}\chi_{12}(\omega=0)$. Here, ${\boldsymbol{S}}_{1}$ and
${\boldsymbol{S}}_{2}$ precess around
${\boldsymbol{S}}_{1}+{\boldsymbol{S}}_{2}=\mbox{const}$ with frequency
$\omega_{\rm L}=J_{\rm RKKY}|{\boldsymbol{S}}_{1}+{\boldsymbol{S}}_{2}|$. The
spin dynamics in that case is a Hamiltonian dynamics and derives from the
effective RKKY Hamiltonian $H_{\rm RKKY}=J_{\rm
RKKY}{\boldsymbol{S}}_{1}{\boldsymbol{S}}_{2}$.

For finite $\Omega$, we can distinguish between two additional effects: First,
there is an overall renormalization of the RKKY coupling $J^{\rm
RKKY}_{12}\mapsto J^{\rm
RKKY}_{12}/(1-\Omega^{2}{\boldsymbol{S}}_{1}{\boldsymbol{S}}_{2})$, which
depends on the initial spin configuration. Second, there is an additional
(non-Hamiltonian) coupling between the spins, the relative strength of which
is given by $\Omega$. We note that for $\Omega\to\infty$ the additional
coupling cannot outweigh the renormalization effect, see Eq. (39), and there
is no spin dynamics at all in this limit.

Equation (37) shows that the theory must break down for model parameters,
where $\Omega=\Omega_{\rm c}\equiv
1/({\boldsymbol{S}}_{1}{\boldsymbol{S}}_{2})$ (recall that
${\boldsymbol{S}}_{1}{\boldsymbol{S}}_{2}$ is conserved). For a spin-Berry
curvature right at the critical value $\Omega_{\rm c}$, the dynamics of the
two spins gets arbitrarily fast, such that the adiabatic theorem does not
apply. On the other hand, Eq. (37) or Eqs. (39) show that the strongest
renormalization effect due to the geometrical spin torque occurs for model
parameters, where $\Omega$ is close to $\Omega_{\rm c}$, while the exchange
coupling $J$ at the same time must satisfy a condition

$\frac{1}{\tau}\equiv
J^{2}\frac{\chi_{12}(0)}{1-{\boldsymbol{S}}_{1}{\boldsymbol{S}}_{2}\Omega^{2}}\ll\Delta
E$ (40)

to ensure the applicability of the adiabatic theorem. As a rule of thumb, for
$M=2$ spins and for a generic value of
${\boldsymbol{S}}_{1}{\boldsymbol{S}}_{2}$ in the initial state, the value of
the (with $\hbar\equiv 1$) dimensionless quantity $\Omega$ should be
$\Omega={\cal O}(1)$ to get a strong effect of the geometric torque.

Introducing the dimensionless time scale $t^{\prime}=t/\tau$, the effective
equations of motion can be rewritten in a form that is independent of all
model parameters, except for the spin-Berry curvature:

$\displaystyle\frac{d{{\boldsymbol{S}}}_{1}}{dt^{\prime}}={\boldsymbol{S}}_{2}\times{\boldsymbol{S}}_{1}+\Omega{\boldsymbol{S}}_{1}\times({\boldsymbol{S}}_{1}\times{\boldsymbol{S}}_{2})\>,$
$\displaystyle\frac{d{{\boldsymbol{S}}}_{2}}{dt^{\prime}}={\boldsymbol{S}}_{1}\times{\boldsymbol{S}}_{2}-\Omega{\boldsymbol{S}}_{2}\times({\boldsymbol{S}}_{2}\times{\boldsymbol{S}}_{1})\>.$
(41)

Let us emphasize once more that these equations cannot be derived from a
Hamilton function $H=H({\boldsymbol{S}}_{1},{\boldsymbol{S}}_{2})$.

The nonlinear system of equations (41) is easily solved analytically. We
define

${\boldsymbol{\Sigma}}={\boldsymbol{S}}_{1}+{\boldsymbol{S}}_{2}+\Omega{\boldsymbol{S}}_{2}\times{\boldsymbol{S}}_{1}\>.$
(42)

Using Eq. (41), $d{\boldsymbol{\Sigma}}/dt=0$ is obtained straightforwardly.
Furthermore, since $d({\boldsymbol{S}}_{1}{\boldsymbol{S}}_{2})/dt=0$, the
enclosed angle $\varphi$ is conserved. Both, ${\boldsymbol{S}}_{1}$ and
${\boldsymbol{S}}_{2}$ undergo a precession around ${\boldsymbol{\Sigma}}$,
i.e., we have
$\dot{{\boldsymbol{S}}}_{1}={\boldsymbol{\Sigma}}\times{\boldsymbol{S}}_{1}$
and
$\dot{{\boldsymbol{S}}}_{2}={\boldsymbol{\Sigma}}\times{\boldsymbol{S}}_{2}$
with frequency $\omega_{\rm prec}=\sqrt{{\boldsymbol{\Sigma}}^{2}}$. With
$|{\boldsymbol{S}}_{1}|=|{\boldsymbol{S}}_{2}|=1$, we find

$\omega_{\rm prec}=\sqrt{4\cos^{2}(\varphi/2)+\Omega^{2}\sin^{2}\varphi}\>.$
(43)

For $\Omega=0$ the conventional RKKY dynamics is recovered.

###  II.6 Computation of the spin-Berry curvature

The actual calculations of the spin Berry curvature are performed for a gapped
tight-binding model of non-interacting electrons with Hamiltonian

$\displaystyle\hat{H}_{\rm qu}$ $\displaystyle=$
$\displaystyle\sum_{ii^{\prime},\sigma}t_{ii^{\prime}}c_{i\sigma}^{\dagger}c_{i^{\prime}\sigma}=\sum_{{\boldsymbol{I}}r,{\boldsymbol{I}}^{\prime}r^{\prime},\sigma}t_{{\boldsymbol{I}}r,{\boldsymbol{I}}^{\prime}r^{\prime}}c_{{\boldsymbol{I}}r\sigma}^{\dagger}c_{{\boldsymbol{I}}^{\prime}r^{\prime}\sigma}$
$\displaystyle=$
$\displaystyle\sum_{{\boldsymbol{k}}}t_{rr^{\prime}}({\boldsymbol{k}})c^{\dagger}_{{\boldsymbol{k}}r\sigma}c_{{\boldsymbol{k}}r^{\prime}\sigma}=\sum_{{\boldsymbol{k}}\nu}\varepsilon_{\nu}({\boldsymbol{k}})c^{\dagger}_{{\boldsymbol{k}}\nu\sigma}c_{{\boldsymbol{k}}\nu\sigma}\>.$

Here, $t_{ii^{\prime}}$ is the (spin-independent) hopping matrix, and we write
$i=({\boldsymbol{I}},r)$, where ${\boldsymbol{I}}$ are the lattice translation
vectors and $r$ labels the sites within a unit cell. For the honeycomb lattice
considered below, $r=A,B$, corresponding to A- or B-sublattice sites. The
hopping matrix is diagonalized with repect to
${\boldsymbol{I}},{\boldsymbol{I}}^{\prime}$ by a unitary transformation of
the form
$c^{\dagger}_{{\boldsymbol{k}}r\sigma}=\sum_{{\boldsymbol{I}}}U^{(r)}_{{\boldsymbol{I}}{\boldsymbol{k}}}c^{\dagger}_{{\boldsymbol{I}}r\sigma}$,
such that
$t_{{\boldsymbol{I}}r,{\boldsymbol{I}}^{\prime}r^{\prime}}=\sum_{{\boldsymbol{k}}}U^{(r)}_{{\boldsymbol{I}}{\boldsymbol{k}}}t_{rr^{\prime}}({\boldsymbol{k}})U^{(r^{\prime})\dagger}_{{\boldsymbol{k}}{\boldsymbol{I}}^{\prime}}$.
In case of a translationally symmetric lattice with periodic boundary
conditions, ${\boldsymbol{k}}$ runs over the discrete set of $L$ wave vectors
in the first Brillouin zone, where $L$ is the number of unit cells, and
$U_{{\boldsymbol{I}}{\boldsymbol{k}}}^{(r)}=U_{{\boldsymbol{I}}{\boldsymbol{k}}}=L^{-1/2}e^{i{\boldsymbol{k}}\cdot{\boldsymbol{I}}}$.
In a second step, the ${\boldsymbol{k}}$-dependent diagonalization of
$t_{rr^{\prime}}({\boldsymbol{k}})=\sum_{\nu}\overline{U}_{r\nu}({\boldsymbol{k}})\varepsilon_{\nu}({\boldsymbol{k}})\overline{U}^{\dagger}_{\nu
r^{\prime}}({\boldsymbol{k}})$ is achieved by the unitary transformation
$c^{\dagger}_{{\boldsymbol{k}}\nu\sigma}=\sum_{r}\overline{U}_{r\nu}({\boldsymbol{k}})c^{\dagger}_{{\boldsymbol{k}}r\sigma}$
for each ${\boldsymbol{k}}$. Here, $\nu$ is the band index, if $\hat{H}$ is
translationally symmetric. We have
$t_{{\boldsymbol{I}}r,{\boldsymbol{I}}^{\prime}r^{\prime}}=\sum_{{\boldsymbol{k}}}U^{(r)}_{{\boldsymbol{I}}{\boldsymbol{k}}}\overline{U}_{r\nu}({\boldsymbol{k}})\varepsilon_{\nu}({\boldsymbol{k}})\overline{U}^{\dagger}_{\nu
r^{\prime}}({\boldsymbol{k}})U^{(r^{\prime})\dagger}_{{\boldsymbol{k}}{\boldsymbol{I}}^{\prime}}$.
With the combined transformation

$S_{{\boldsymbol{I}}r,{\boldsymbol{k}}\nu}=U^{(r)}_{{\boldsymbol{I}}{\boldsymbol{k}}}\overline{U}_{r\nu}({\boldsymbol{k}})\>,$
(45)

this reads as
$t_{{\boldsymbol{I}}r,{\boldsymbol{I}}^{\prime}r^{\prime}}=\sum_{{\boldsymbol{k}}\nu}S_{{\boldsymbol{I}}r,{\boldsymbol{k}}\nu}\varepsilon_{\nu}({\boldsymbol{k}})S^{\dagger}_{{\boldsymbol{k}}\nu,{\boldsymbol{I}}^{\prime}r^{\prime}}$.

Evaluating the matrix elements in Eq. (20) for $N$-electron Slater
determinants
$|\Psi^{(0)}_{n}\rangle=\prod_{k\nu\sigma}c_{k\nu\sigma}^{\dagger}|\mbox{vac}\rangle$
is straightforward. We find

$\Omega_{m\alpha,m^{\prime}\alpha^{\prime}}=-J^{2}\delta_{\alpha\alpha^{\prime}}\,\mbox{Im}\sum_{{\boldsymbol{k}}\nu}^{\rm
occ.}\sum_{{\boldsymbol{k}}^{\prime}\nu^{\prime}}^{\rm
unocc.}\sum_{rr^{\prime}}\frac{S^{\dagger}_{{\boldsymbol{k}}\nu,{\boldsymbol{I}}r}S_{{\boldsymbol{I}}r,{\boldsymbol{k}}^{\prime}\nu^{\prime}}S^{\dagger}_{{\boldsymbol{k}}^{\prime}\nu^{\prime},{\boldsymbol{I}}^{\prime}r^{\prime}}S_{{\boldsymbol{I}}^{\prime}r^{\prime},{\boldsymbol{k}}\nu}}{(\varepsilon_{\nu^{\prime}}({\boldsymbol{k}}^{\prime})-\varepsilon_{\nu}({\boldsymbol{k}}))^{2}}\>,$
(46)

where $({\boldsymbol{I}},r)=i_{m}$ and
$({\boldsymbol{I}}^{\prime},r^{\prime})=i_{m^{\prime}}$. This representation
applies to the translationally symmetric model $\hat{H}=\hat{H}_{\rm qu}$ for
$J=0$ and can be used for the case, where $J$ is treated perturbatively.
Beyond the weak-$J$ regime, it applies for each spin configuration
${\boldsymbol{S}}$ in $\hat{H}=\hat{H}_{\rm qu}+\hat{H}_{\rm
int}({\boldsymbol{S}})$. But as translational symmetry is broken in this case,
$S_{{\boldsymbol{I}}r,{\boldsymbol{k}}\nu}$ does no longer contain the Fourier
factor $L^{-1/2}e^{i{\boldsymbol{k}}{\boldsymbol{I}}}$ and must be computed
numerically and explicitly include the $\alpha$ index.

###  II.7 Spinful Haldane model

For generic parameters the half-filled Haldane model Haldane (1988) is a
prototypical Chern insulator with broken time-reversal symmetry. Numerical
calculations have been performed for the spinful Haldane model at half-
filling, which consists of two identical copies of the Haldane model for
electrons with spin projection $\sigma=\uparrow$ and $\sigma=\downarrow$,
respectively. The spinful Haldane model is trivially invariant under SU(2)
spin rotations. Rotation symmetry is broken in the full model Eq. (2), where
the two copies are coupled by the interaction term Eq. (1).

The corresponding tight-binding quantum Hamiltonian in Eq. (2) is given by

$\displaystyle\hat{H}_{\rm qu}$ $\displaystyle=$ $\displaystyle
M\sum_{i\sigma}z_{i}c_{i\sigma}^{\dagger}c_{i\sigma}-\tau_{1}\\!\\!\sum_{\langle
ii^{\prime}\rangle,\sigma}c_{i\sigma}^{\dagger}c_{i^{\prime}\sigma}$ (47)
$\displaystyle-$ $\displaystyle\tau_{2}\\!\\!\sum_{\langle\langle
ii^{\prime}\rangle\rangle,\sigma}e^{i\xi_{ii^{\prime}}}c_{i\sigma}^{\dagger}c_{i^{\prime}\sigma}\>.$

Here, $i$, $i^{\prime}$ run over the sites of the two-dimensional bipartite
honeycomb lattice, see Fig. 1. $M$ is the strength of a staggered on-site
potential, where the sign factor $z_{i}=+1$ for a site $i$ in the A sublattice
($r=A$) and $z_{i}=-1$ for $i$ in the B sublattice ($r=B$). For $M\neq 0$ the
potential term breaks particle-hole symmetry. The amplitude for hopping
between nearest-neighbor sites on the lattice, $\langle ii^{\prime}\rangle$,
is denoted by $\tau_{1}$. We set $\tau_{1}=1$ to fix the energy scale (and
with $\hbar=1$) also the time scale. The next-nearest-neighbor hopping
amplitude $e^{i\xi_{ii^{\prime}}}\tau_{2}$ with real $\tau_{2}$ includes a
phase factor with $\xi_{ii^{\prime}}=-\xi$ for hopping from $i^{\prime}$ to
$i$ in clockwise direction (see purple lines in Fig. 1), and with
$\xi_{ii^{\prime}}=\xi$ in counterclockwise direction (light blue). This
ensures that the total flux of the corresponding orbital magnetic field
through a unit cell vanishes. Time-reversal symmetry is broken for
$\tau_{2}\neq 0$ and $\xi\neq 0,\pm\pi$.

Figure 1:  Haldane model on the honeycomb lattice. A sites indicated by blue
dots with on-site potential $+M$. B sites: red dots, potential $-M$. Nearest-
neighbor hopping with amplitude $\tau_{1}$: dashed lines. Next-nearest-
neighbor hopping $\tau_{2}$ with additional Peierls factor $e^{-i\xi}$
($e^{i\xi}$) for hopping in clockwise (counterclockwise) direction: purple
(light-blue) arrows.  Figure 2:  Topological phase diagram of the Haldane
model in the $\xi$-$M/\tau_{2}$ plane. $C$ denotes the first Chern number. The
nearest-neighbor hopping $\tau_{1}=1$ sets the energy scale.

The band structure of the spinful Haldane model consists of two bands with
trivial twofold spin degeneracy each. The band structure is gapped for all $M$
and all $\xi$, if $|\tau_{2}/\tau_{1}|<1/3$, except for parameter values
satisfying the condition Haldane (1988) $|M/\tau_{2}|=3\sqrt{3}|\sin\xi|$, or,
in terms of the band gap given by

$\Delta E=|M-3\sqrt{3}\tau_{2}\sin\xi|\stackrel{{\scriptstyle!}}{{=}}0\>,$
(48)

for $M,\tau_{2}>0$ and $0<\xi<\pi$. In the $M/\tau_{2}$ vs. $\xi$ phase
diagram, see Fig. 2, this condition defines lines of topological phase
transitions (see red and blue lines in the figure). At a phase transition,
there is a band closure in the first Brillouin zone, either at
$K=2\pi(1/3\sqrt{3},1/3)$, when $M=-3\sqrt{3}\,\tau_{2}\sin(\xi)$, or
$K^{\prime}=2\pi(2/3\sqrt{3},0)$, when $M=3\sqrt{3}\,\tau_{2}\sin(\xi)$
Bernevig (2013).

In the Altland-Zirnbauer classification Altland and Zirnbauer (1997), the
model belongs to symmetry class A. For parameters inside the boundary in Fig.
2, i.e., in the orange or green regions for $|M/\tau_{2}|<3\sqrt{3}|\sin\xi|$,
the model represents a Chern insulator with finite Chern number $C=-1$
($\xi<0$) and $C=+1$ ($\xi>0$). Outside the boundary, we have $C=0$, and the
system is a topologically trivial band insulator.

The Haldane model $\hat{H}_{\text{qu}}$ is invariant under SU(2) spin
rotations, i.e., we can make use of Eq. (27). Furthermore, it is invariant
under the discrete $C_{3}$ rotations of the lattice around a fixed site, and
thus $\Omega$ is $C_{3}$ invariant. Time-reversal symmetry, on the other hand,
is broken explicitly. Under time reversal, we have
$\langle\Psi^{(0)}_{0}|s_{i\alpha}|\Psi^{(0)}_{n}\rangle\mapsto-\langle\Psi^{(0)}_{0}|s_{i\alpha}|\Psi^{(0)}_{n}\rangle^{\ast}$.
With Eq. (21), this implies
$\Omega_{m\alpha,m^{\prime}\alpha^{\prime}}\mapsto\Omega_{m^{\prime}\alpha^{\prime},m\alpha}$,
and using Eq. (27) we find $\Omega\mapsto-\Omega$. Under a reflection at a
mirror symmetry axis of the hexagonal lattice, the Hamiltonian transforms in
the same way as under time reversal, $H\mapsto\Theta H\Theta^{\dagger}$. The
first two terms on the right-hand side of Eq. (47) are invariant. Reflection
of the hopping term for two next-nearest-neighbor sites $i,j$ is represented
by complex conjugation, since $\xi_{ij}=-\xi_{ji}$. Hence, we have

$\Omega\mapsto-\Omega\quad\mbox{under time reversal or reflection}\>.$ (49)

The Hamiltonian and thus $\Omega$ is invariant under the combined
transformation.

###  II.8 Spin-Chern number

Let us emphasize that the different topological phases of the Haldane model
are characterized by the first Chern number $C$, which is obtained by
integrating the conventional ${\boldsymbol{k}}$-space Berry curvature
$F({\boldsymbol{k}})$ over the entire BZ, i.e., the torus $T^{2}$. The spin-
Berry curvature is a different concept, motivated by the geometrical spin
torque contribution to the adiabatic impurity-spin dynamics.

Still it can be employed to define a topological invariant: The Hamiltonian of
the quantum system smoothly depends on the parameters
${\boldsymbol{S}}=({\boldsymbol{S}}_{1},...,{\boldsymbol{S}}_{M})\in{\cal S}$,
where ${\cal S}$ is the Cartesian product of two-spheres (recall
$|{\boldsymbol{S}}_{r}|=1$). The parameter manifold ${\cal S}$ is closed,
i.e., has no boundary, and is $2M$-dimensional. Hence, according to the
general theory (see Ref. Nakahara (1998), for example), the $M$th spin-Chern
number $C^{(S)}_{M}$ is given by

$C^{(S)}_{M}=\frac{i^{M}}{(2\pi)^{M}}\frac{1}{M!}\oint_{S}\mbox{tr}\,\omega^{M}\>,$
(50)

where $\omega=dA+A^{2}$ is the spin-Berry-curvature two-form derived from the
one-form $A$, the spin-Berry connection. If the ground state of the
Hamiltonian is nondegenerate on the entire manifold ${\cal S}$, the spin-Chern
number $C^{(S)}_{M}$ is well-defined and quantized.

Figure 3:  Nonzero element $\Omega$ of the spin-Berry curvature
$\Omega_{m\alpha,m^{\prime}\alpha^{\prime}}$ [see Eq. 27] for various
positions $i_{m}$ and $i_{m^{\prime}}$ of the two impurity sites, to which the
classical spins are coupled. The first spin resides at the central site
$i_{1}$ (see green dot), and the second one is varied over the remaining
$2L-1$ sites of the hexagonal lattice. Calculations for a lattice with
$L=39\times 39$ unit cells and periodic boundary conditions. Model parameters:
$\tau_{2}=0.1$, $\xi=\pi/4$. $\tau_{1}=1$ sets the energy scale. Left:
$M=0.8M_{\rm crit}$, such that the system is in a topologically nontrivial
phase. Right: $M=1.2M_{\rm crit}$, topologically trivial phase. Results for
the weak-$J$ regime. The color code indicates the value of $\Omega/J^{2}$.

The computation of $C^{(S)}_{M}$ is a highly nontrivial task. Here we note
that $C^{(S)}_{M}=0$ in the weak-$J$ limit, on which we concentrate in the
present study. This is easily verified, since for $J=0$ the spin-Berry
curvature is independent of ${\boldsymbol{S}}$ (see Eq. (21)) and hence
$C^{(S)}_{M}(J=0)=0$. Quantization of the spin-Chern number and continuity
with respect $J$ then imply $C^{(S)}_{M}(J)=0+{\cal O}(J^{2})=0$.

##  III Numerical Results

The spin-Berry curvature of the Haldane model can be computed numerically
using the representation Eq. (46), if the exchange coupling $J$ is weak. As
mentioned above in the discussion of Eq. (46), a slightly generalized formula
applies to the general, non-perturbative case. As translational symmetry is
broken in this case, however, the accessible system sizes are considerably
smaller. We will, therefore, start with a discussion of results, obtained for
a bulk system in the weak-$J$ limit.

###  III.1 Spatial structure of the spin-Berry curvature

The $m,m^{\prime}$ element of the spin-Berry curvature tensor
$\Omega_{m\alpha,m^{\prime}\alpha^{\prime}}=\Omega_{mm^{\prime}}\delta_{\alpha\alpha^{\prime}}$
describes the strength of the mutual geometric spin torque of two classical
spins ${\boldsymbol{S}}_{m}$ and ${\boldsymbol{S}}_{m^{\prime}}$, locally
exchange coupled to local spins of the electron system at two sites $i_{m}$
and $i_{m^{\prime}}$ of the lattice. Since $\Omega_{mm^{\prime}}$ is
antisymmetric, it is sufficient to specify a single real number $\Omega$ for
each pair of lattice sites, see Eq. (27). We consider a translationally
invariant system with periodic boundary conditions, fix the position of one
site, and compute the spin-Berry curvature $\Omega$ as a function of the
position of the second site, using Eq. (46).

In Fig. 3, the fixed site in the center of the plot is marked with green
color, and the $J$-independent quantity $\Omega/J^{2}$ is given by the color
code on each of the remaining lattice sites. Sites of the A sublattice are
indicated by circles, those of the B sublattice sites by squares. A large
lattice is chosen with $L=39\times 39$ unit cells, each consisting of two
sites, i.e., with a total number of $2L=3042$ sites. Calculations have been
done for parameters $\tau_{2}=0.1$ and $\xi=\pi/4$, where the system is in the
topologically nontrivial and the trivial phase, with $M=0.8M_{\rm crit}$ (left
plot) and $M=1.2M_{\rm crit}$ (right plot), respectively. For this parameter
set we have $M_{\rm crit}\approx 0.37$. Note that the parameters are chosen
such that the gap size $\Delta E$ is the same for both, the topologically
nontrivial (left) and the trivial phase (right).

Figure 4:  The same as Fig. 3 but for $\tau_{2}=0.001$. Left: topologically
nontrivial phase at $M=0.8M_{\rm crit}$. Right: trivial phase, $M=1.2M_{\rm
crit}$.

One can nicely see the invariance of $\Omega$ under discrete $2\pi/3$
rotations of the lattice around the fixed site. This set $C_{3}$ of spatial
rotations forms in fact a symmetry group of the Haldane model. Furthermore,
consistent with Eq. (27), $\Omega$ changes sign if $i_{m}$ and
$i_{m^{\prime}}$, i.e., $m$ and $m^{\prime}$ are exchanged. In the figure,
where site $i_{m}$ is kept fixed, this is seen be comparing $\Omega$ for
A-sublattice sites $i_{m^{\prime}}$ at opposite positions
${\boldsymbol{I}}_{m}$ and $-{\boldsymbol{I}}_{m^{\prime}}$. Finally,
consistent with Eq. (49), $\Omega$ changes its sign under a reflection at the
horizontal axis through the central site $i_{m}$ and at the axes rotated by
$2\pi/3$ and $-2\pi/3$ against the horizontal. This also implies that directly
on these mirror axes $\Omega$ vanishes.

Concerning the distance dependence, we can distinguish between two different
ranges. For small distances, up to about 3 unit cells, $\Omega$ has an
oscillatory distance dependence. In this close range, the spatial structure is
rather complicated generally. On the other hand, in the far range $\Omega$
does not change sign and monotonically decreases with increasing distance
between $i_{m}$ and $i_{m^{\prime}}$ along any spatial direction.

We recall that, in the weak-$J$ limit, the spin-Berry curvature $\Omega$ is
purely a property of the spinful Haldane model. However, it is not directly
related to its band topology, as the weight factors in Eq. (46) are
constructed from the same Bloch states but in a different way as compared to
the ${\boldsymbol{k}}$-space Berry curvature. Nevertheless, the sign structure
of $\Omega$ in the far range is quite universal, and it is different for the
topologically nontrivial and the trivial phase, as can be seen in Fig. 3,
although the gap $\Delta E$ is the same. This implies that, to some degree,
the spin dynamics is sensitive to the respective topological phase. In this
sense, the spin-Berry curvature can be seen as a marker for the topological
properties of the model.

Differences between the two phases become more pronounced in the case of a
smaller gap $\Delta E$. Fig. 4 displays results for a next-nearest-neighbor
hopping amplitude $\tau_{2}$, which is smaller by two orders of magnitude as
compared to Fig. 3. Choosing $M=0.8M_{\rm crit}$ and $M=1.2M_{\rm crit}$,
respectively, as in Fig. 3, a small $\tau_{2}$ leads an overall small gap
$\Delta E$ in the entire phase diagram, as can be inferred from Eq. (48).
Again, the gap $\Delta E$ is the same for both, the topologically nontrivial
and the trivial phase.

But now, for small $\tau_{2}$, the spatial structure of $\Omega$ is largely
different for both phases. In the nontrivial phase (left plot), the close
range now spreads over the entire (finite) system, and there is hardly any
visible decrease of $\Omega$ with increasing distance. In the trivial phase
(right), on the contrary, the close range is limited to distances of a few
unit cells only, while $\Omega$ is clearly reduced in size in the far range.
For still larger distances from the central site, $\Omega$ increases again.
This is due to finite-size effects, which are more pronounced for small
$\Delta E$. Since, as compared to Fig. 3, the gap is smaller by two orders of
magnitude, this is not unexpected.

###  III.2 Finite systems in the small $\tau_{2}$ regime

Interestingly, the absolute value of $\Omega$ in Fig. 4 is much larger in the
nontrival case, where the far range extends over the whole lattice. Quite
generally, finite systems (with periodic boundary conditions) in the regime of
small next-nearest-neighbor hopping $\tau_{2}$ are interesting as here the
spin-Berry curvature can be extremely large.

Figure 5:  $\Omega$ for next-nearest-neighbor sites as function of $\tau_{2}$
for $\xi=\pi/4$ and $M=0$. Note the log-log scale. Calculations for finite
systems with periodic boundary conditions of different size $L=l\times l$ as
indicated.

This is seen in Fig. 5, where $\Omega$ is plotted as a function of $\tau_{2}$.
Here, we have set $M=0$, such that the model is topologically nontrivial for
all values of $\tau_{2}$. Exactly at $\tau_{2}=0$, however, the model is a
time-reversal symmetric semimetal, such that the Chern number is zero. Hence,
in the thermodynamical limit $L\to\infty$, there is a phase transition at
$\tau_{2}=0$.

From the log-log plot in Fig. 5 we can infer that $\Omega\propto
1/\tau_{2}^{2}$ for $\tau_{2}\to 0$. This has interesting consequences, as has
already been discussed above in Sec. II.5: Namely, for $\tau_{2}\to 0$, the
spin dynamics is entirely dominated by the anomalous contribution from the
geometrical spin torque. The adiabatic spin dynamics slows down, and the
system in this limiting case ultimately shows no dynamics at all. At
intermediate values for $\tau_{2}$, however, depending on the value for $J$,
the value of $\Omega$ can be of order one. This is exactly the range, where
dynamic effects of the geometrical spin torque are most pronounced, as argued
in Sec. II.5.

Fig. 5 also shows that the $1/\tau_{2}^{2}$ behavior is realized for strictly
finite systems only. Comparing the results for different $L$ with linear
extension $l$ up to $l=210$ (88200 sites), we see that rather a linear
dependence $\Omega(\tau_{2})\propto\tau_{2}$ is obtained in the
thermodynamical limit. Note that this is just the $\tau_{2}$ dependence that
must be expected, when expanding $\Omega(\tau_{2})$ around $\tau_{2}=0$, where
the model is time-reversal symmetric and thus $\Omega=0$, up to linear order
in $\tau_{2}$. We conclude that in the thermodynamical limit $\Omega$ is
continuous at the phase transition, when this is steered via $\tau_{2}$, while
$\Omega$ diverges as $1/\tau_{2}^{2}$ for any finite system.

The underlying mechanism is the following: For small systems, where the
${\boldsymbol{k}}$-space is strongly discretized, the relative contribution to
$\Omega$ from regions in the Brillouin zone close to $K$ or $K^{\prime}$ is
comparatively large, so that this becomes the dominating contribution to
$\Omega$, for model parameters close to the transition. For larger systems,
however, the relative contribution diminishes, as can be seen in Fig. 5. If,
for a finite system, $\tau_{2}$ is sufficiently small, only the contributions
from $K$ or from $K^{\prime}$ are relevant in the double sum over
${\boldsymbol{k}}$ and ${\boldsymbol{k}}^{\prime}$ in Eq. (46). In this case,
the $\tau_{2}$ dependence of the spin-Berry curvature is essentially given via
the energy denominator in Eq. (46) only, and since $\Delta E\propto\tau_{2}$,
see Eq. (48), we have the scaling $\Omega\propto 1/\tau_{2}^{2}$.

The different scalings $\Omega\propto\tau_{2}$ and $\Omega\propto
1/\tau_{2}^{2}$, distinguish between asymptotic behavior in the thermodynamic
limit and for a finite-size system. Fig. 4 (left) for the nontrivial phase is
in fact representative for a system, where finite-size effects dominate, while
in the case of Fig. 3 the system size is sufficiently large to reflect the
spin-Berry curvature in the thermodynamic limit (for not too large distances).

###  III.3 Distance dependence

For an analysis of the distance dependence of $\Omega$, we revert to the same
parameters $\tau_{2}=0.1$ and $\xi=\pi/4$, as underlying Fig. 3, but choose a
larger system with $L=150\times 150$ unit cells. Fig. 6 shows the dependence
of the spin-Berry curvature $\Omega$ on the distance $d$ between the two
impurity sites. $d$ is defined as the Euclidean distance between the sites
$i_{m}$ and $i_{m^{\prime}}$ on the hexagonal lattice in units of the nearest-
neighbor distance. The distance of a site to all of its six next-nearest
neighbors, for example, is given by $d=\sqrt{3}$. For distances $40\lesssim
d\lesssim 100$, we find a nearly linear dependence of $\ln\Omega$ on $d$,
i.e., $\Omega\propto\exp(-d/\lambda)$ with $\lambda>0$, while for too large
distances $d\gtrsim 100$, the linear trend is disturbed by finite-size
effects. Note that for the larger distances only the sign of $\Omega$ depends
on $M$ but not its absolute value, if the gap is the same (as is the case for
$M=1.2M_{\rm crit}$ and $M=0.8M_{\rm crit}$).

Figure 6:  $\Omega/J^{2}$ as a function of the distance $d$ between the two
impurity sites for the topologically nontrivial ($M=0.8M_{\rm crit}$, blue
diamonds) and the trivial phase ($M=1.2M_{\rm crit}$, orange squares) along a
path including A-sublattice sites, starting from the central (green) site in
Fig. 3 in vertical direction. Filled symbols: positive sign of $\Omega/J^{2}$.
Hollow symbols: negative sign. Further parameters as in Fig. 3, but for a
larger lattice with $L=150\times 150$ unit cells (periodic boundary
conditions).  Figure 7:  $1/\lambda$ as a function of the gap $\Delta E$, as
obtained from calculations for systems with $L=l\times l$ with $l=150$ unit
cells for different $M$. Calculations for the nontrivial and the trivial phase
with the same gap size. See text for discussion.  Figure 8:  The same as Fig.
6 but for $J_{\rm RKKY}/J^{2}$ as a function of the distance $d$. Filled
symbols: positive sign of $J_{\rm RKKY}/J^{2}$. Hollow symbols: negative sign.

Performing calculations for different $M$ to vary the gap $\Delta E$, we can
extract the $\Delta E$ dependence of the slope $-1/\lambda$. This is plotted
in Fig. 7. We find a nearly linear dependence $1/\lambda\propto\Delta E$. This
means that the spin-Berry curvature has an exponential $d$ dependence for
large $d$, which is controlled by the bulk band gap $\Delta E$:
$\Omega\propto\exp(-\Delta E\,d)$.

This behavior is reminiscent of the exponential decay of the RKKY exchange
interaction with $d$ for insulating systems Bloembergen and Rowland (1955).
Fig. 8 gives an example. Here, $J_{\rm RKKY}/J^{2}$ is plotted as function of
$d$ for the same model parameters as in Fig. 6, and exponential behavior is
found for the RKKY coupling in the same range $40\lesssim d\lesssim 100$.

The range of distances with an exponential dependence of $\Omega$ or $J_{\rm
RKKY}$ exactly corresponds to the far range seen in Fig. 3, where there is a
comparatively smooth dependence of $\Omega$ on the position of the second
impurity spin. In the close range, for $d\lesssim 20$, the distance dependence
of $\Omega$ is less regular, and there are sign changes of $\Omega$ in
addition. This is somewhat reminiscent of the oscillatory distance dependence
of the RKKY exchange for a metal or semi-metal Kit ; rkk ; Liu  _et al._
(2009); Gao  _et al._ (2009); Kurilovich  _et al._ (2017); Hosseini  _et al._
(2020); Yevtushenko and Yudson (2018).

As can be seen in Figs. 6, 7 and 8, differences between topologically
nontrivial and trivial case are not very pronounced as concerns the distance
dependence. This is governed by the finite gap $\Delta E$, which has always
been chosen to be the same when comparing both phases.

Qualitatively, the exponential decay of both, the spin-Berry curvature and the
RKKY exchange, can be understood easily: In the weak-$J$ regime the coupling
of two impurity spins results from virtual second-order-in-$J$ processes
involving (de-)excitations of electron across the gap $\Delta E$. This is not
only the cause of the exponential distance dependence but also explains the
small absolute values of $\Omega/J^{2}$ and $J_{\rm RKKY}/J^{2}$, which do not
exceed values of the order of $10^{-2}$.

###  III.4 Parameter dependence of the spin-Berry curvature

The discussion of $M=2$ spin dynamics in Sec. II.5 has shown that a value of
$\Omega$ close to unity is required for a substantial impact of the
geometrical spin torque. As we have seen, this is in fact possible in case of
finite systems with periodic boundary conditions at small values for
$\tau_{2}$ (see Figs. 4 and 5). We now focus on large systems again and study
the model for small distances between $i_{m}$ and $i_{m^{\prime}}$ at
$\tau_{2}=0.1$, where finite-size effects can be neglected safely.

Figure 9:  Thick gray lines: phase diagram of the Haldane model in the
$\xi$-$M/\tau_{2}$ plane (see Fig. 2 for comparison). The color codes the
spin-Berry curvature $\Omega/J^{2}$ for next-nearest-neighbor sites $i_{m}$,
$i_{m^{\prime}}$. Calculations for a system with $27\times 27$ unit cells with
periodic boundary conditions at $\tau_{2}=10^{-1}$.  Figure 10:  The same as
Fig. 9 but for sites $i_{m}$ and $i_{m^{\prime}}$ with Euclidean distance
$d=5\sqrt{3}$ (in units of the nearest-neighbor distance).

Fig. 9 gives an overview over the dependence of the nonzero element $\Omega$
of the spin-Berry curvature on the parameters $M$ and $\xi$ for next-nearest
neighbors $i_{m}$, $i_{m^{\prime}}$. The boundaries of the topological phase
transitions are superimposed in the figure (see thick gray lines). We find a
somewhat larger absolute value of $\Omega$ within a topologically nontrivial
phase. Furthermore, we note that $\Omega\to-\Omega$ under a sign change of
$\xi$. As a sign change of phase $\xi$ has the same effect for the Hamiltonian
of the Haldane model Eq. (47) as a reflection at a mirror symmetry axis of the
hexagonal lattice, this observation is easily explained with Eq. (49).
Otherwise the parameter dependence is more or less featureless. Absolute
values for $\Omega$ do not exceed $\sim 10^{-3}$ in the entire parameter
regime.

For larger distances between the sites $i_{m}$ and $i_{m^{\prime}}$, absolute
values for $\Omega$ are smaller. But its parameter dependence can be much more
complicated. An example is given with Fig. 10, which displays results for
sites at a distance $d=\|i_{m}-i_{m^{\prime}}\|=5\sqrt{3}$. This must be
traced back to the matrix elements in Eq. (46).

In all cases we find that the parameter dependence of $\Omega$ is smooth,
including parameter ranges where the model is close to or right at a
topological phase transition. This is worth mentioning since the squared
energy denominator in Eq. (46), $(\varepsilon_{\rm
unocc.}({\boldsymbol{k}}^{\prime})-\varepsilon_{\rm
occ.}({\boldsymbol{k}}))^{2}$, suggests that the contributions of wave vectors
${\boldsymbol{k}},{\boldsymbol{k}}^{\prime}$ at (or close to) the critical
point $K$ or $K^{\prime}$ lead to a diverging or at least large spin-Berry
curvature.

###  III.5 Spin-Berry curvature close to a topological phase transition

Close to a transition, however, a careful analysis of the finite-size effects
is necessary. For systems with a finite number of units cells $L=l\times l$,
the spin-Berry curvature is in fact discontinuous at a topological phase
transition (actually, the latter is well defined for $L=\infty$ only). Fig. 11
displays results for the $M$ dependence of $\Omega$. A finite jump
$\Delta\Omega$ at the critical point $M=M_{\rm crit}=3\sqrt{3}\tau_{2}\sin\xi$
is found for various $l$. At $l=15$, the relative jump $\Delta\Omega/\Omega$
is considerable. With increasing $L$, however, it monotonically but slowly
decreases with $l$ and is about an order of magnitude smaller at $l=51$.

The data are consistent with the proposition that $\Omega$ is continuous at
$M=M_{\rm crit}$ in the limit $L\to\infty$. A numerical proof, however, is
difficult to achieve, since besides the $L\to\infty$ limit, the $\Delta
M=|M-M_{\rm crit}|\to 0$ limit must be taken simultaneously. Fig. 12 displays
the jump $\Delta\Omega$ as function of $l$ for various $\Delta M/M_{\rm
crit}$. For comparatively large $\Delta M/M_{\rm crit}=0.1$, we see that
$\Delta\Omega$ decreases but approaches a finite value for $l\to\infty$. At
smaller relative distances $\Delta M/M_{\rm crit}$ to the transition point,
however, this assumed convergence with $l$ eventually becomes invisible for
the largest feasible system sizes (see red line). Nevertheless, continuity of
$\Omega$ at a topological phase transition appears highly plausible.

Figure 11:  $\Omega$ for next-nearest-neighbor sites as function of $M/M_{\rm
crit}$ at $\tau_{2}=0.1$ and $\xi=0.1\pi$, where $M_{\rm crit}$ refers to the
critical $M$ value. Calculations for system size $L=l\times l$ with various
linear extensions $l$ as indicated.  Figure 12:  The jump
$\Delta\Omega=|\Omega_{\rm top}-\Omega_{\rm triv}|$, obtained by computing the
spin-Berry curvature at $M=M_{\rm crit}\pm\Delta M$, as function of $l$ for
various $\Delta M/M_{\rm crit}$. Calculations for next-nearest neighbors,
$\tau_{2}=0.1$, $\xi=\pi/2$.

In fact, an analytical argument can be given: Parametrically close to a
topological phase transition and in the vicinity of the band-closure points
$K$ or $K^{\prime}$ in the first Brillouin zone, the band structure of the
Haldane model takes the form of a relativistic Dirac theory Haldane (1988);
Bernevig (2013). If ${\boldsymbol{\kappa}}=(\Delta k_{x},\Delta k_{y})$ and
${\boldsymbol{\kappa}}^{\prime}=(\Delta k^{\prime}_{x},\Delta k^{\prime}_{y})$
denote the wave vectors relative to $K$ or $K^{\prime}$, i.e., if
${\boldsymbol{\kappa}},{\boldsymbol{\kappa}}^{\prime}=0$ refer to a band-
closure point, we have

$\varepsilon_{\nu}({\boldsymbol{\kappa}})=\varepsilon_{\pm}({\boldsymbol{\kappa}})\propto\pm\sqrt{\kappa_{x}^{2}+\kappa_{y}^{2}+m^{2}}\,.$
(51)

The “mass” $m$ is linearly related to the insulating gap: $m\propto\Delta$. We
can analytically check for a possible divergence of the spin-Berry curvature
in the thermodynamical limit $L\to\infty$ on approaching the phase transition
via $m\to 0$ with help of Eq. (46). To this end, we compute the contribution
$I_{\rm bulk}$ of wave vectors in a sufficiently small ball $B$ with radius
$\Lambda$ around ${\boldsymbol{\kappa}}=0$. Up to a constant factor, we find

$I_{\rm
bulk}\propto\int_{B}d^{2}\kappa\int_{B}d^{2}\kappa^{\prime}\>\frac{1}{\big{(}\sqrt{{\boldsymbol{\kappa}}^{2}+m^{2}}+\sqrt{{{{\boldsymbol{\kappa}}}^{\prime}}{}^{2}+m^{2}}\big{)}^{2}}\>,$
(52)

and the spin-Berry curvature is continuous at $m=0$, if $\lim_{m\to 0}I_{\rm
bulk}$ exists.

Note that to realize the thermodynamical limit, the wave-vector sums in Eq.
(46) have been replaced by integrations. Furthermore, $\nu=-$ (occupied) and
$\nu^{\prime}=+$ (unoccupied), see Eq. (51). The ${\boldsymbol{\kappa}}$
dependence of the matrix elements
$S_{{\boldsymbol{I}}r,{\boldsymbol{\kappa}}\nu}$ in the numerator in Eq. (46)
can be disregarded for small $\Lambda$: The first factor of each matrix
element is a Fourier factor with a smooth ${\boldsymbol{\kappa}}$ dependence,
see Eq. (45). For each $\nu$, the second factor is given by the two-component
eigenstate of the Dirac Hamiltonian Haldane (1988); Bernevig (2013)

$H_{\rm D}\propto\kappa_{y}\sigma_{x}-\kappa_{x}\sigma_{y}+m\sigma_{z}\>,$
(53)

from which the dispersion Eq. (51) is derived. Rewriting the two-dimensional
${\boldsymbol{\kappa}}$ integration (and analogously for the
${\boldsymbol{\kappa}}^{\prime}$ integration) with the help of polar
coordinates $(\kappa,\varphi)$, i.e., $\int d^{2}{\kappa}=\int
d\kappa\kappa\int d\varphi$, the ${\boldsymbol{\kappa}}$-dependent part of
this factor is of the form $e^{\pm i\varphi}$ and thus cannot lead to a
divergence.

The remaining two-dimensional integral

$I_{\rm bulk}=\int_{0}^{\Lambda}\int_{0}^{\Lambda}d\kappa
d\kappa^{\prime}\>\frac{\kappa\kappa^{\prime}}{\big{(}\sqrt{\kappa^{2}+m^{2}}+\sqrt{{\kappa^{\prime}}^{2}+m^{2}}\big{)}^{2}}\>$
(54)

can be computed analytically and turns out to stay finite in the limit $m=0$
with an additive lowest-order correction of the form $-m^{2}\ln m\to 0$ for
$m\to 0$. This implies that the spin-Berry curvature, at a topological phase
transition, is a continuous function of the model parameters.

###  III.6 Strong exchange coupling

The representation Eq. (46) for the spin-Berry curvature holds in case of weak
exchange coupling $J$. For coupling strengths beyond the perturbative regime,
we must resort to Eq. (20), where the matrix elements are defined with
eigenstates carrying a nonvanishing dependence on the spin configuration. The
numerical evaluation can be performed as described by the text below Eq. (46).

Figure 13:  Spin-Berry curvature as function of $J$ for an $L=60\times 60$
lattice at $M=0$, $\tau_{2}=0.1$, $\xi=0.25\pi$, and for a fixed classical
spin configuration ${\boldsymbol{S}}_{m}=(0,1,0)$,
${\boldsymbol{S}}_{m^{\prime}}=(-1,0,0)$, where $i_{m}$ and $i_{m^{\prime}}$
are next-nearest neighbors. Blue data (see left scale): Off-site
$m,m^{\prime}$ element $\Omega=\Omega_{m\alpha,m^{\prime}\alpha^{\prime}}$
(and $\alpha=x$, $\alpha^{\prime}=y$), divided by $J^{2}$, to be compared with
the corresponding perturbative result (orange data). Violet, green, red data
(see right scale): On-site elements $\Omega_{m\alpha,m\alpha^{\prime}}$ at the
position $i_{m}$.

Fig. 13 displays the spin-Berry curvature as a function of $J$. Comparing the
results obtained from the full theory (blue data, left scale) with those of
the perturbative-in-$J$ approach (orange), we see that perturbation theory
applies to coupling strengths $J\lesssim 0.1$ and is still a good
approximation up to $J\lesssim 1$.

For even stronger $J$ beyond the perturbative regime, the structure of the
spin-Berry-curvature tensor $\Omega_{m\alpha,m^{\prime}\alpha^{\prime}}$
changes qualitatively. In the $J\to\infty$ limit, only on-site elements
$\Omega_{m\alpha,m\alpha^{\prime}}$ are nonzero, and the blue curve (left
scale) approaches zero in this limit. These on-site elements are given by the
spin-Berry curvature of the effective two-spin model $H_{\rm
2-spin}=J{\boldsymbol{s}}_{i_{m}}{\boldsymbol{S}}_{m}$ at $i_{m}$, where
${\boldsymbol{s}}_{i_{m}}$ can be treated as an $s=1/2$ quantum spin. The two-
spin model is easily solved analytically Stahl and Potthoff (2017), and
${\boldsymbol{\Omega}}_{m}\equiv(1/2)\sum_{\alpha\alpha^{\prime}\alpha^{\prime\prime}}\varepsilon_{\alpha\alpha^{\prime}\alpha^{\prime\prime}}\Omega_{m\alpha,m\alpha^{\prime}}{\boldsymbol{e}}_{\alpha^{\prime\prime}}$
is given by the “magnetic-monopole field”

${\boldsymbol{\Omega}}_{m}=-\frac{1}{2}\frac{{\boldsymbol{S}}_{m}}{|{\boldsymbol{S}}_{m}|^{3}}=-\frac{1}{2}{\boldsymbol{S}}_{m}\>,$
(55)

since $|{\boldsymbol{S}}_{m}|=1$ and $J>0$. As ${\boldsymbol{S}}_{m}=(0,1,0)$
has been assumed to point in $y$ direction, we find that only
$\Omega_{mx,mz}=-\Omega_{m,y}=+1/2$ remains nonzero (green data, right scale)
for $J\to\infty$. On the contrary, the perturbative approach, see Eq. (27),
yields
$\Omega_{m\alpha,m^{\prime}\alpha^{\prime}}=\Omega_{mm^{\prime}}\delta_{\alpha\alpha^{\prime}}=0$
for the on-site elements $m=m^{\prime}$ due to the antisymmetry of the tensor.

In the intermediate-$J$ regime, see $J\approx 10$ in Fig. 13, there is still a
finite off-site element (blue, left scale). The on-site element
$\Omega_{mx,mz}=-\Omega_{m,y}$ (green, right scale) is still far from its
asymptotic value for $J\to\infty$, while $\Omega_{my,mz}=\Omega_{m,x}$ has a
finite negative value due to the proximity of ${\boldsymbol{S}}_{m^{\prime}}$
pointing into $-x$ direction. Finally, for the chosen classical spin
configuration, $\Omega_{mx,my}=\Omega_{m,z}=0$ in the entire $J$ range.

The data in the intermediate-coupling regime ($J\sim{\cal O}(10)$) demonstrate
that the elements of the spin-Berry curvature tensor may well assume values of
the order of one. They appear to be limited, however, by their $J\to\infty$
values $|\Omega_{m,\alpha}|\leq 1/2$. Nevertheless, one would have a strongly
anomalous spin dynamics with it.

###  III.7 Results for a ribbon geometry

We have seen that the spin-Berry curvature stays finite at a topological phase
transition, since the gap merely closes at a single critical point, $K$ or
$K^{\prime}$ in the two-dimensional Brillouin zone. This is a too weak
singularity to have a significant impact on the
${\boldsymbol{k}},{\boldsymbol{k}}^{\prime}$ sums in Eq. (46). For the model
on a one-dimensional lattice, however, this is qualitatively different.

Here, we consider the Haldane model on a one-dimensional ribbon with a large
number of unit cells $L_{x}$ and periodic boundary conditions in the $x$
direction, and with a finite number of unit cells $L_{y}$ and open boundary
conditions in the $y$ direction, so that the ribbon is bounded by zigzag edges
in the $y$ direction. Equation (46) applies accordingly for one-dimensional
wave vectors $k_{x},k_{x}^{\prime}$ and for $I,I^{\prime}=1,...,L_{y}$.

In a topologically nontrivial phase with $C=\pm 1$ and disregarding the
trivial spin degeneracy, the bulk-boundary correspondence principle requires
the existence of two gapless chiral eigenstates of $\hat{H}_{\rm qu}$
exponentially localized at the opposite edges. For finite but large $L_{y}$,
the band dispersions $\varepsilon_{\nu}(k_{x})$ for $\nu=1,...,2L_{y}$ form
two quasi-continua of bulk states in the one-dimensional Brillouin zone
separated by the bulk band gap $\Delta E$. Within the bulk band gap, the edge-
state dispersions $\varepsilon_{1}(k_{x})$ and $\varepsilon_{2}(k_{x})$ take
the form of an avoided crossing at low energies. With $\kappa\equiv\Delta
k_{x}$ we have

$\varepsilon_{1,2}(\kappa)\propto\pm\sqrt{\kappa^{2}+m^{2}}\>,$ (56)

with a gap $\propto 2m$ that is exponentially small in $L_{y}$ for large
$L_{y}$. For $L_{y}\to\infty$ the energy spectrum is gapless ($m=0$), and at
low excitation energies the edge-state dispersions are linear with positive
and negative slope, respectively:

$\varepsilon_{1}(\kappa)\propto\kappa\>,\quad\varepsilon_{2}(\kappa)\propto-\kappa\>.$
(57)

As an example, Fig. 14 shows the ribbon band structure for model parameters,
where the spectrum is particle-hole symmetric. Fixing the Fermi energy at
zero, $\varepsilon_{\rm F}=0$, we have a half-filled system.

Figure 14:  Band structure of a Haldane ribbon with zigzag edges. Calculations
for a ribbon of finite thickness with $L_{y}=50$ unit cells in $y$ direction.
Periodic boundary conditiosn are assumed along the $x$ direction. Parameters:
$\xi=\pi/2$, $\tau_{2}=0.1$ and $M=0$.

Addressing the weak-$J$ limit, we use Eq. (46) to compute the contribution
$I_{\rm rib}$ to the spin-Berry curvature, for sites $i_{m},i_{m^{\prime}}$ at
one of the edges and for $L_{x},L_{y}\to\infty$, due to relative wave vectors
$\kappa$ in a sufficiently small range around $\kappa=0$. This is given by
$\lim_{\varepsilon\to 0}I_{\rm rib}(\varepsilon)$ for positive $\varepsilon$,
where

$I_{\rm
rib}(\varepsilon)=\int_{\varepsilon}^{\Lambda}d\kappa\int_{\varepsilon}^{\Lambda}d\kappa^{\prime}\>\frac{1}{(\kappa+\kappa^{\prime})^{2}}\>.$
(58)

It is straightforward to see that the integral diverges for $\varepsilon\to 0$
as $I_{\rm rib}(\varepsilon)\sim-\ln\varepsilon$, contrary to the two-
dimensional case, see Eq. (54). We conclude that the spin-Berry curvature is
weakly (logarithmically) divergent in the topologically non-trivial phase due
to the presence of edge modes.

There are two sources for a finite gap regularizing the spin-Berry curvature.
First, for a ribbon with a large but finite $L_{x}<\infty$, the $k_{x}$-space
discretization, $\delta k_{x}=2\pi/L_{x}$, will regularize the divergence,
such that $\Omega\sim\ln L_{x}$. Second, for $L_{x}\to\infty$, but for a
finite ribbon width $L_{y}$, a gap $\propto m<\Delta E$ is produced by
overlapping edge wave functions, see Eq. (56). In this case, $\Omega\sim-\ln
m$.

Fig. 15 displays the nonzero element $\Omega$ of the spin-Berry curvature for
two next-nearest-neighbor sites at the same edge as function of $L_{x}$ and
for various finite $L_{y}$. For fixed $L_{x}$ and with increasing $L_{y}$, the
overlap between the edge states localized at different edges and thus the gap
parameter $m$ decrease exponentially fast with $L_{y}$, such that one expects
that the gap due to $k$-space discretization $\delta k_{x}=2\pi/L_{x}$ quickly
becomes the dominating factor. In fact, for any $L_{x}$, the spin-Berry
curvature $\Omega$ as a function of $L_{y}$ quickly converges to a finite
value. On the scale of the plot, there is no visible difference between the
results for $L_{y}=30$ and $L_{y}=45$ (see also the red and violet data in the
inset). For the largest $L_{y}$, one notes that $\Omega$ still increases with
$L_{x}$ at values of the order of $L_{x}={\cal O}(10^{4})$, see inset. The
dependence of $\Omega$ on $\log L_{x}$ is close to linear, as expected, but
with a small slope.

Figure 15:  $\Omega$ as function of nano-ribbon length $L_{x}$ (with periodic
boundary conditions in $x$ direction) for various ribbon widths $L_{y}$ as
indicated. Note the logarithmic scale for $L_{x}$. The classical impurity
spins are located at next-nearest-neighbor sites at the same zigzag edge.
Calculations for the particle-hole symmetric case with $M=0$ and with Fermi
energy $\varepsilon_{\rm F}=0$. Further parameters: $\tau_{2}=0.1$,
$\xi=0.1\pi$.

Importantly, the absolute values of $\Omega/J^{2}$ in the wide $L_{x}$-$L_{y}$
range considered are of the order of unity. According to Eq. (37) and if $J$
is of the order of unity as well, this is just the range, where the strongest
effects of the geometrical spin torque can be expected. We conclude that,
compared with the situation of the two-dimensional bulk system close to a
topological phase transition, the phase-space reduction in the one-dimensional
case is very favorable for a large spin-Berry curvature.

##  IV Summary and outlook

The typical time scale of the dynamics of local magnetic moments, exchange
coupled to an electron system, is controlled by the strength of the exchange
coupling $J$. Since this is usually one or even several orders of magnitudes
smaller than the characteristic energy scales of the electron system, the
local-moment dynamics is slow as compared to the characteristic electronic
time scale. In particular, if $J\ll\Delta E$, where $\Delta E$ is size of the
gap of an insulating electron system, the adiabatic theorem applies, and the
ground state of the electron system at an instant of time $t$ is well
approximated by the instantaneous ground state corresponding to the
configuration of the local moments at time $t$. Modelling the magnetic moments
as classical spins of fixed length, the spin dynamics is described by an
effective low-energy classical theory, which is much simpler than the full
coupled semiclassical electron-spin dynamics, since it involves ground-state
quantities of the electron system only.

We have developed this effective spin-only theory within a Lagrange formalism.
Besides the torque resulting from well-known indirect magnetic exchange, there
is is an additional spin torque that is given in terms of the spin-Berry
curvature. This geometrical spin torque is analogous to the geometrical force
discussed in molecular dynamics and is nonzero in case of electron systems
with broken time-reversal symmetry. As we have demonstrated explicitly for the
simple case of two classical spins, the emergent dynamics is highly
unconventional and differs from the dynamics of any spin-only Hamiltonian.
Future studies may address this non-Hamiltonian spin dynamics for systems like
the semi-classical Kondo-lattice model with a large number of spins.

For our present study, which aims at an in-principle demonstration, we have
considered the (spinful) Haldane model as a prototypical system, where time-
reversal symmetry is broken explicitly. This choice has the additional benefit
that the spin-Berry curvature can be compared with ${\boldsymbol{k}}$-space
Berry curvature, which plays a central role in topological band theory. There
is in fact a (weak) relation between both, as their Lehmann-type
representations involve exactly the same energy eigenstates, albeit different
operators must be considered in the matrix elements. This explains the
sensitivity to the topological phase that has been observed in the study of
the spatial structure of the spin-Berry curvature tensor, particularly at the
zigzag edges of the lattice. For forthcoming work, it will be interesting to
consider other condensed-matter systems with broken time reversal symmetry,
such as systems with spontaneous magnetic order.

It has been shown that the spin-Berry curvature tensor, in the weak-$J$ limit,
is closely related, namely by the frequency derivative at $\omega=0$, to the
magnetic susceptibility. This insight opens the possibility to make close
contact to well-known properties of the indirect magnetic exchange, i.e., to
RKKY theory and to its pendant for insulators and semimetals, e.g., as
concerns their distance dependence.

We have systematically studied the magnitude of the spin-Berry curvature. As
the example for two classical spins has shown, its effects, namely an overall
renormalization of the indirect magnetic exchange and an additional coupling
between the spins, are most pronounced, if this is of the order of unity.
Apart from the strong-$J$ regime, however, this is not easily achieved in case
of the Haldane model. The main reason is that virtual second-order-in-$J$
processes are exponentially damped by the finite gap size $\Delta E$, similar
to the exponential $\Delta E$ dependence of the indirect magnetic exchange.
This suggests that strong effects should be expected for systems in close
parametrical distance to a topological transition with a band closure.
However, we could argue that the spin-Berry curvature is finite and continuous
at a topological transition of the infinite bulk system. On the contrary, an
arbitrarily large curvature can be observed for finite systems (with periodic
boundaries). This underpins the view that a phase-space mechanism is at work:
A large spin-Berry curvature of a condensed-matter system in the
thermodynamical limit requires parametric vicinity to a band closure on a
$D-1$-dimensional submanifold of the $D$-dimensional Brillouin zone. In fact,
we could observe a logarithmically diverging spin-Berry curvature for the
Haldane model at the $D=1$ zigzag edge, where a ($D=0$ dimensional) gap
closure is enforced via the bulk-boundary correspondence. This insight will be
useful for forthcoming studies.

###### Acknowledgements.

This work was supported by the Deutsche Forschungsgemeinschaft (DFG, German
Research Foundation) through the SFB 925 (project B5), project ID 170620586,
and through the research unit QUAST, FOR 5249 (project P8), project ID
449872909.

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