# Geometrical torque on magnetic moments coupled to a correlated
antiferromagnet

Nicolas Lenzing  University of Hamburg, Department of Physics, Notkestraße
9-11, 22607 Hamburg, Germany  David Krüger  University of Hamburg, Department
of Physics, Notkestraße 9-11, 22607 Hamburg, Germany  Michael Potthoff
University of Hamburg, Department of Physics, Notkestraße 9-11, 22607 Hamburg,
Germany  The Hamburg Centre for Ultrafast Imaging, Luruper Chaussee 149, 22761
Hamburg, Germany

###### Abstract

The geometrical spin torque mediates an indirect interaction of magnetic
moments, which are weakly exchange coupled to a system of itinerant electrons.
It originates from a finite spin-Berry curvature and leads to a non-
Hamiltonian magnetic-moment dynamics. We demonstrate that there is an
unprecedentedly strong geometrical spin torque in case of an electron system,
where correlations cause antiferromagnetic long-range order. The key
observation is that the anomalous torque is strongly boosted by low-energy
magnon modes emerging in the two-electron spin-excitation spectrum due to
spontaneous breaking of SU(2) spin-rotation symmetry. As long as single-
electron excitations are gapped out, the effect is largely universal, i.e.,
essentially independent of the details of the electronic structure, but
decisively dependent on the lattice dimension and spatial and spin
anisotropies. Analogous to the reasoning that leads to the Mermin-Wagner
theorem, there is a lower critical dimension at and below which the spin-Berry
curvature diverges.

##  Introduction.

A magnetic moment coupled to a system of itinerant electrons via a local
exchange interaction of strength $J$ experiences a spin torque which leads to
precession dynamics. For several magnetic moments ${\boldsymbol{S}}_{m}$ (with
$m=1,...,M$), usually described as classical fixed-length spins, there are
further torques caused by, e.g., indirect exchange interactions mediated by
the electron system. These Hamiltonian spin torques, well known in
micromagnetics Bertotti et al. (2009) and in the theory of coupled spin-
electron dynamics Koshibae et al. (2009); Bhattacharjee et al. (2012); Evans
et al. (2014); Sayad and Potthoff (2015); Sayad et al. (2016); Chern et al.
(2018); Bajpai and Nikolic (2019), all derive from interaction terms in the
quantum-classical Hamiltonian Elze (2012) for the spin and electron degrees of
freedom. In addition, there is a non-Hamiltonian spin torque that has a purely
geometric nature. This geometrical spin torque represents the feedback of the
Berry physics Berry (1984) on the classical magnetic-moment dynamics.

Generally, such feedback effects have been pointed out early Kuratsuji and
Iida (1985); Moody et al. (1986); Zygelman (1987) but have not been studied in
spin dynamics theory until recently Stahl and Potthoff (2017). For weak $J$
compared to the typical energy scales of the electron system, the classical
spin dynamics is slow, such that the electron system accumulates a geometrical
phase which is gauge independent in case of a cyclic motion Berry (1984);
Simon (1983); Wilczek and Zee (1984). This Berry phase is closely related to
the Berry curvature, a two-form which, when integrated in classical parameter
space over a two-dimensional surface bounded by a closed path ${\cal C}$
yields the Berry phase associated with ${\cal C}$. For example, in molecular
physics Bohm et al. (2003) and when treating the coordinates of the nuclei
classically, the feedback of the Berry physics produces an additional
geometrical force, where the Berry curvature plays the role of a magnetic
field in the nuclei equations of motion. This effect is known as “geometrical
magnetism” Berry and Robbins (1993); Campisi et al. (2012).

The geometrical spin torque resulting from the spin-Berry curvature (SBC)
Stahl and Potthoff (2017) is the analogous concept in the field of atomistic
spin dynamics Skubic et al. (2008); Evans et al. (2014). As opposed to the
closely related geometrical friction term Berry and Robbins (1993); Campisi et
al. (2012), i.e., Gilbert damping llg , it is energy conserving. But,
importantly, the SBC is non-Hamiltonian and emerges for weak $J$, i.e., in the
limit of slow classical spin dynamics. However, the effects are typically weak
Michel and Potthoff (2022) for a solid Ihm (1991), such that it appears
difficult to disentangle the effect of the geometrical spin torque from other
contributions Bajpai and Nikolić (2020).

Table 1:  Spin-Berry curvature of a spontaneously symmetry-broken antiferromagnetic state with gapped single-particle excitations. ${\boldsymbol{k}}$: wave vector. See text for discussion.  lattice | spin-Berry | distance | magnetic
---|---|---|---
dimension | curvature (SBC) | dependence | ground state
1 | divergent | — | —
2 | log. divergent | — | stable
3 | regular | $1/R$ | stable
$D\geq 4$ | $\sim\int_{0}^{\Lambda_{\sf cutoff}}dk\,k^{D-3}$ | $1/R^{D-2}$ | stable

In this Letter we study the geometrical spin torque for magnetic moments
coupled to a magnetic solid: a correlated $D$-dimensional antiferromagnetic
(AF) insulator. This is a generic situation realized, e.g., by magnetic
impurities in the bulk or by magnetic adatoms on the surface of the
antiferromagnet. We demonstrate that the magnitude of the SBC is governed by
the magnon-excitation spectrum. This has very general consequences: the SBC
must diverge for $D=1$ but is regular for $D\geq 3$, see Tab. 1. For $D=2$ the
SBC generically exhibits a logarithmic divergence as a function of any
perturbation causing a gap in the magnon dispersion, such as magnetic
anisotropies or external magnetic fields. The magnitude of the SBC and thus
the impact on the magnetic-moment dynamics is studied for the Hubbard model at
half-filling and zero temperature as a prototype of a correlation-induced
insulator.

##  Time-reversal symmetry (TRS).

Within adiabatic spin-dynamics theory Stahl and Potthoff (2017); Michel and
Potthoff (2022), the geometrical spin torque is obtained from the SBC of the
electron system, see Eq. (2) below. Importantly, a finite SBC generally
requires TRS breaking in the electron system Michel and Potthoff (2022). If
$J$ is strong, as assumed in Ref. Stahl and Potthoff, 2017, TRS is broken by
the classical spin moment itself, as this acts like a local symmetry-breaking
field. TRS breaking can be waived only at the cost of working with a non-
Abelian extension of the theory well beyond the adiabatic limit Lenzing et al.
(2022), where the dynamics is governed by the generically finite non-Abelian
spin-Berry curvature. Another approach is to replace the electron system by an
entirely classical model composed of “slow” and “fast” spin moments Elbracht
et al. (2020); Michel and Potthoff (2021). This circumvents the necessity of
TRS breaking altogether but still exhibits the feedback of holonomy effects in
purely classical systems Hannay (1985). For magnetic moments coupled to
quantum systems and in the physically relevant weak-$J$ regime, a finite SBC
can be achieved with an external magnetic field, or with a (staggered) orbital
field as considered recently Michel and Potthoff (2022) with the Haldane model
Haldane (1988) as a prototype of a TRS-breaking Chern insulator Bernevig
(2013). However, fine tuning of the parameters is required to achieve
considerable effects Michel and Potthoff (2022). Here we consider an electron
system in which correlations induce a TRS-breaking AF state. The AF order not
only enables a finite SBC but also strongly boosts its magnitude due to magnon
modes in the spin-excitation spectrum.

##  Dynamics of magnetic moments.

We are interested in the slow dynamics of $M$ magnetic moments, described as
classical spins ${\boldsymbol{S}}_{m}$ of unit length, which are coupled to a
correlated electron system with Hamiltonian $H_{\rm el}$ via a local exchange
interaction $H_{\rm
int}=J\sum_{m=1}^{M}{\boldsymbol{s}}_{i_{m}}{\boldsymbol{S}}_{m}$. Here,
$i_{m}$ is the site, at which the $m$-th moment is coupled to, and
${\boldsymbol{s}}_{i}=1/2\sum_{\sigma\sigma^{\prime}}c^{\dagger}_{i\sigma}{\boldsymbol{\tau}}_{\sigma\sigma^{\prime}}c_{i\sigma^{\prime}}$,
where ${\boldsymbol{\tau}}$ is the vector of Pauli matrices, is the local spin
moment at site $i$ of the electron system. The total Hamiltonian is
$H=H({\boldsymbol{S}})=H_{\rm el}+H_{\rm int}({\boldsymbol{S}})$ and depends
on the configuration
${\boldsymbol{S}}=({\boldsymbol{S}}_{1},...,{\boldsymbol{S}}_{m})$ of the
magnetic moments.

Assuming that the electron system at any instant of time $t$ is in its
instantaneous ground state for the spin configuration ${\boldsymbol{S}}(t)$,
i.e., $|\Psi(t)\rangle=|\Psi_{0}({\boldsymbol{S}}(t))\rangle$, the equation of
motion of adiabatic spin dynamics is given by Stahl and Potthoff (2017);
Michel and Potthoff (2022)

$\dot{{\boldsymbol{S}}}_{m}=({\boldsymbol{T}}_{m}^{(\rm
H)}+{\boldsymbol{T}}_{m}^{\rm(geo)})\times{\boldsymbol{S}}_{m}\>.$ (1)

Here ${\boldsymbol{T}}_{m}^{\rm(H)}\times{\boldsymbol{S}}_{m}$ with
${\boldsymbol{T}}_{m}^{\rm(H)}=\partial\langle
H({\boldsymbol{S}})\rangle/\partial{\boldsymbol{S}}_{m}=J\langle{\boldsymbol{s}}_{i_{m}}\rangle$
is the conventional (Hamiltonian) spin torque, where $\langle\cdots\rangle$ is
the instantaneous ground-state expectation value.

##  Geometrical spin torque.

The second term, the geometrical spin torque
${\boldsymbol{T}}_{m}^{\rm(geo)}\times{\boldsymbol{S}}_{m}$ is necessary to
enforce the constraint $|\Psi(t)\rangle=|\Psi_{0}({\boldsymbol{S}}(t))\rangle$
and has been derived within a quantum-classical Lagrange formalism in Refs.
Stahl and Potthoff, 2017; Michel and Potthoff, 2022. This assumes that the
ground state is non-degenerate (otherwise non-Abelian spin-dynamics theory
Lenzing et al. (2022) must be used) and that $J$ is sufficiently weak so that
the classical spin dynamics is much slower than typical relaxation time scales
of the quantum system $H_{\rm el}$. Alternatively, the term may be derived
within adiabatic response theory Berry and Robbins (1992, 1993); Campisi et
al. (2012) as the first nontrivial correction in a systematic expansion of the
response of a driven system with respect to the driving speed, when applied to
spin dynamics Lenzing and Potthoff (2023). It is given by

${\boldsymbol{T}}_{m}^{\rm(geo)}=\sum_{\alpha}\sum_{m^{\prime}\alpha^{\prime}}\Omega_{m^{\prime}m,\alpha^{\prime}\alpha}({\boldsymbol{S}})\dot{S}_{m^{\prime}\alpha^{\prime}}{\boldsymbol{e}}_{\alpha}\;,$
(2)

with $\alpha=x,y,z$ and the $\alpha$-th unit vector
${\boldsymbol{e}}_{\alpha}$, and where

$\Omega_{mm^{\prime},\alpha\alpha^{\prime}}({\boldsymbol{S}})=\frac{\partial}{\partial
S_{m\alpha}}A_{m^{\prime}\alpha^{\prime}}({\boldsymbol{S}})-\frac{\partial}{\partial
S_{m^{\prime}\alpha^{\prime}}}A_{m\alpha}({\boldsymbol{S}})$ (3)

is the spin-Berry curvature. At each spin configuration ${\boldsymbol{S}}$,
this is a real antisymmetric tensor
($\Omega_{m^{\prime}m,\alpha^{\prime}\alpha}=-\Omega_{mm^{\prime},\alpha\alpha^{\prime}}$),
which is invariant under local gauge transformations of the ground states
$|\Psi_{0}({\boldsymbol{S}})\rangle\mapsto
e^{i\phi({\boldsymbol{S}})}|\Psi_{0}({\boldsymbol{S}})\rangle$. It is the
exterior derivative of the spin-Berry connection
${\boldsymbol{A}}_{m}=i\langle\Psi_{0}|\frac{\partial}{\partial{\boldsymbol{S}}_{m}}|\Psi_{0}\rangle$,
which describes parallel transport of the ground state
$|\Psi_{0}({\boldsymbol{S}})\rangle$ on the manifold of spin configurations
${\cal M}$. For $M$ classical spins ${\boldsymbol{S}}_{m}\in S^{2}$, this is
given by the $M$-fold Cartesian product of 2-spheres ${\cal M}\equiv
S^{2}\times\cdots\times S^{2}$.

##  Spontaneous antiferromagnetic order.

We consider a coupling of the magnetic spin moments to the single-band Hubbard
model Gebhard (1997); Essler et al. (2005) on a $D$-dimensional hypercubic
lattice as a prototypical model for itinerant magnetic order. Its Hamiltonian
is $H_{\rm
el}=-t\sum_{ij}^{\rm{n.n.}}\sum_{\sigma=\uparrow,\downarrow}c^{\dagger}_{i\sigma}c_{j\sigma}+U\sum_{i}n_{i\uparrow}n_{i\downarrow}$,
where the nearest-neighbor hopping $t=1$ fixes the energy and (with
$\hbar\equiv 1$) the time scales. $c_{i\sigma}$ annihilates an electron at
site $i$ with spin projection $\sigma$, and
$n_{i\sigma}=c_{i\sigma}^{\dagger}c_{i\sigma}$. The sums over $i,j$ are
restricted to nearest neighbors, and $L$ is the total number of sites. It is
well known Anderson (1950); Schrieffer et al. (1989); Manousakis (1991);
Auerbach (1994); Schäfer et al. (2015) that at half-filling, repulsive
Hubbard-$U$ and for $D\geq 2$, the ground state of the system in the
thermodynamical limit $L\to\infty$ develops long-range AF correlations. SU(2)
spin-rotation symmetry and therewith TRS are spontaneously broken, and the
ordered state is characterized by a finite staggered magnetization
${\boldsymbol{m}}=m{\boldsymbol{e}}_{z}$ with $m=L^{-1}\sum_{i}z_{i}\langle
n_{i\uparrow}-n_{i\downarrow}\rangle$ and $z_{i}=\pm 1$ for $i$ in sublattice
A or B, respectively. We assume $m>0$ for sublattice A.

At weak $U$, AF order is driven by the Slater mechanism and perturbatively
accessible Schrieffer et al. (1989); Auerbach (1994). Within self-consistent
Hartree-Fock theory Moriya (1985), the one-electron excitation spectrum
displays a gap $\Delta=Um$ at wave vector $Q=(\pi,\pi,...)$ in the
conventional Brillouin zone. The two-electron spin-excitation spectrum is well
described by standard random-phase approximation (RPA) but for the symmetry-
broken AF state Rowe et al. (2012); Del Re and Toschi (2021); Luttinger and
Ward (1960); Rohringer et al. (2018).

In the strong-$U$ limit, the one-electron spectrum is dominated by a large
Hubbard gap $\Delta\sim U$ and well developed local spin moments, coupled via
Anderson’s superexchange Anderson (1950); Auerbach (1994). Here, the model
maps onto the Heisenberg spin-1/2 Hamiltonian with AF exchange $J_{\rm
H}=4t^{2}/U$ and AF long-range order, see Refs. Chakravarty et al., 1989;
Manousakis, 1991; Sandvik, 1997, for example. To compute the low-energy magnon
dispersion and states, we can apply spin-wave theory (SWT) Anderson (1952) to
the AF Heisenberg model and use the Holstein-Primakoff transformation Holstein
and Primakoff (1940) at linear order. Linear SWT is motivated by the fact that
single-magnon decay requires overlap with the two-magnon continuum, so that
the picture of a stable magnon gas is protected by kinematic restrictions at
low energies Hamer et al. (1992); Chernyshev and Maksimov (2016); McClarty et
al. (2018); McClarty (2022).

##  Spin-Berry curvature of an antiferromagnet.

To compute the geometrical spin torque, we make use of a Lehmann-type
representation of the SBC starting from Eq. (3). This is straightforwardly
derived Michel and Potthoff (2022) using a resolution of the the unity,
${\boldsymbol{1}}=\sum_{n}|\Psi_{n}({\boldsymbol{S}})\rangle\langle\Psi_{n}({\boldsymbol{S}})|$,
with an orthonormal basis of instantaneous eigenstates of $H_{\rm el}+H_{\rm
int}({\boldsymbol{S}})$:

$\Omega_{mm^{\prime},\alpha\alpha^{\prime}}=-2J^{2}\mbox{Im}\\!\sum_{n\neq
0}\frac{\langle\Psi_{0}|s_{i_{m}}^{\alpha}|\Psi_{n}\rangle\,\langle\Psi_{n}|s_{i_{m^{\prime}}}^{\alpha^{\prime}}|\Psi_{0}\rangle}{(E_{n}-E_{0})^{2}}\>.$
(4)

Note that, due to the $J^{2}$ prefactor, the ${\boldsymbol{S}}$ dependence of
the eigenenergies and eigenstates will provide corrections to Eq. (4) only at
order $J^{3}$. As we refer to the weak-$J$ limit, these will be neglected in
the following.

In the AF phase and assuming that the order parameter is aligned to the $z$
axis, $\langle{\boldsymbol{s}}_{i}\rangle=(-1)^{i}m{\boldsymbol{e}}_{z}$,
there is a remaining SO(2) symmetry of the energy eigenstates under spin
rotations around ${\boldsymbol{e}}_{z}$. This unbroken spin-rotation symmetry,
together with the spatial inversion and translation symmetries of $H_{\rm
el}$, and the antisymmetry
$\Omega_{mm^{\prime},\alpha\alpha^{\prime}}=-\Omega_{m^{\prime}m,\alpha^{\prime}\alpha}$
[see Eq. (3)] imply that the spin-Berry curvature tensor is entirely fixed by
a single real number
$\Omega\equiv\Omega_{mm^{\prime},xy}=-\Omega_{mm^{\prime},yx}$ for each fixed
pair of sites $i_{m}$, $i_{m^{\prime}}$. All other elements must vanish, as is
detailed by the symmetry analysis in Sections A and B of the Supplemental
Material (SM) SM .

In a first step, for weak $U$, we compute the SBC via

$\Omega_{mm^{\prime}}=-iJ^{2}\frac{\partial}{\partial\omega}\chi_{i_{m}i_{m^{\prime}},xy}(\omega)\Big{|}_{\omega=0}+{\cal
O}(J^{3})\>,$ (5)

where
$\chi_{ii^{\prime},\alpha\alpha^{\prime}}(\omega)=L^{-1}\sum_{{\boldsymbol{k}}}e^{i{\boldsymbol{k}}({\boldsymbol{R}}_{i}-{\boldsymbol{R}}_{i^{\prime}})}\chi_{\alpha\alpha^{\prime}}({\boldsymbol{k}},\omega)$
is the real-space retarded susceptibility, obtained by the RPA (see SM, Sec. C
SM ). The relation Eq. (5) is easily derived by comparing the representation
Eq. (4) of the SBC with the Lehmann representation of the susceptibility (SM,
Secs. A, B SM ). Therewith, the susceptibility in the symmetry-broken AF state
is seen to play a dual role for the spin dynamics: (i) via Eq. (5) and Eq. (2)
its frequency derivative at $\omega=0$ yields the geometrical spin torque
${\boldsymbol{T}}_{m}^{\rm(geo)}\times{\boldsymbol{S}}_{m}$, and (ii) the
static susceptibility yields, in the weak-$J$ regime, the conventional RKKY
spin torque ${\boldsymbol{T}}_{m}^{\rm(H)}\times{\boldsymbol{S}}_{m}$ with
${\boldsymbol{T}}_{m}^{\rm(H)}=\partial H_{\rm
RKKY}/\partial{\boldsymbol{S}}_{m}$, where $H_{\rm
RKKY}=J^{2}\sum\chi_{i_{m}i_{m^{\prime}},\alpha\alpha^{\prime}}(\omega=0)S_{m\alpha}S_{m^{\prime}\alpha^{\prime}}$
is the perturbative RKKY Hamiltonian of the AF state.

Figure 1:  Left: Transversal retarded ground-state spin susceptibility
$\mbox{Im}\,\chi_{+-}({\boldsymbol{k}},\omega)$ for $U=2$ and $U=4$ along
high-symmetry directions in the conventional $D=2$ Brillouin zone, as obtained
by RPA. Right: Frequency derivative $\mbox{Im}\,\partial_{\omega}\chi_{\rm
xy}({\boldsymbol{k}},\omega)$ (absolute values) in the mBz, related to the SBC
at $\omega=0$. White dotted lines: Slater gap $\Delta=Um$ (onset of the
continuum). Lorentzian broadening $\omega\to\omega+i\eta$ with $\eta=0.045$.
Energy scale: $t=1$.

For the Hubbard model on the $D=2$ square lattice the spin-excitation spectrum
$\chi_{+-}({\boldsymbol{k}},\omega)$, see Fig. 1 (left) for $U=2$ and $U=4$,
consists of a continuum at high frequencies $\omega>\Delta=Um$ ($\Delta\approx
0.75$ for $U=2$, $\Delta\approx 2.76$ for $U=4$) and, furthermore, within the
gap an undamped transversal and doubly degenerate magnon mode. This mode takes
most of the spectral weight. The magnon contribution to the derivative
$\partial_{\omega}\chi_{xy}({\boldsymbol{k}},\omega)$ on sublattice A (Fig. 1,
right) is even more pronounced, especially for $\omega=0$, where it is related
to the SBC by Eq. (5).

##  Goldstone theorem, implications.

In our second step, we exploit the fact that the spin-excitation spectrum of
an AF insulator has a universal structure at low frequencies. This is due to
Goldstone’s theorem which enforces the presence of gapless magnon modes
Goldstone (1961); Nambu and Jona-Lasinio (1961); Brauner (2010). In the
collinear AF state and corresponding to the two broken generators of the spin
SU(2) symmetry, there are two degenerate modes with a linear and isotropic
dispersion in the vicinity of the $\Gamma$ point in the magnetic Brillouin
zone (mBz). Linear SWT applied to the Heisenberg model that emerges in the
strong $U$ limit captures this physics, i.e., the dispersion close to $\Gamma$
is given by $\frac{1}{2}J_{\rm H}\omega({\boldsymbol{k}})=c_{\rm s}k+{\cal
O}(k^{2})$, where $c_{\rm s}$ is the spin-wave velocity. Using the magnon
energies and eigenstates, we can compute the SBC in this limit from Eq. (4)
directly (SM, Secs. D, E SM ), ending up with

$\Omega_{mm^{\prime}}=\mp\frac{2J^{2}}{J_{\rm
H}^{2}}\frac{1}{(2\pi)^{D}}\int_{\rm
mBz}\\!\\!d^{D}k\frac{\cos(\boldsymbol{k}({\boldsymbol{R}}_{i_{m}}\\!-\\!{\boldsymbol{R}}_{i_{m^{\prime}}}))}{\omega({\boldsymbol{k}})^{2}}\>,$
(6)

if both, $i_{m},i_{m^{\prime}}$ belong to sublattice A ($-$ sign) or B ($+$
sign), and $\Omega_{mm^{\prime}}=0$ else.

For $D=2$, the linear dispersion close to $\Gamma$ then implies a $1/k^{2}$
singularity of the integrand and thus a logarithmic infrared divergence. For
$D\geq 3$, the local ($m=m^{\prime}$) SBC is finite. We note that the same
arguments as invoked for the Mermin-Wagner theorem Anderson (1952); Mermin and
Wagner (1966), i.e., a divergence due to the low-energy spin excitations, here
lead to a lower critical dimension ($D_{\rm c}=3$) that is shifted by one, see
Tab. 1. The numerical value for the $D=3$ local SBC is $\Omega_{\rm
loc}\approx-0.084\,J^{2}/J_{\rm H}^{2}=-0.084\,J^{2}U^{2}/16t^{4}$. When
scaling the hopping as $t=t^{\ast}/\sqrt{D}$ with $t^{\ast}=\mbox{const}$
Metzner and Vollhardt (1989); Müller-Hartmann (1989), the modulus of the SBC
decreases monotonically with $D$, and the SBC approaches a finite mean-field
value $|\Omega_{\rm loc}|\to J^{2}U^{2}/32t^{\ast}{}^{4}$ for $D\to\infty$
(SM, Sec. F SM ).

Figure 2:  Local SBC as function of $U$ for $D=3$, as obtained from RPA. Left:
$\Omega_{\rm loc}/J^{2}$. Right: $\Omega_{\rm loc}/J^{2}U^{2}$. Diamonds:
$t_{\perp}/t=0.1$, see also Fig. 3 (left). Red arrow: $D=3$ SWT ($U\to\infty$)
result. $\eta=0.035$.

##  Magnitude of the SBC.

SWT predicts a $U^{2}$ dependence of the SBC in the Heisenberg limit for
strong $U$. For $U=0$, on the other hand, TRS of the resulting paramagnetic
state implies that it must vanish. For $U\to 0$, there is an intricate
competition between the exponential suppression of the order parameter
$m\propto e^{-1/U}$, i.e., of the “strength” of TRS breaking and thus of the
SBC and, on the other hand, the exponential closure of the single-electron
Slater gap $\Delta=Um$ and thus of the onset of the continuum in the spin-
excitation spectrum resulting in continuum contributions that favor a large
SBC. Our numerical results for the local SBC in $D=3$, as obtained from weak-
coupling RPA and strong-coupling SWT are displayed in Fig. 2. With increasing
$U$ we find a smooth crossover from the Slater to the Heisenberg limit with a
monotonically increasing $|\Omega_{\rm loc}|$.

The nonlocal SBC at large distances
$R\equiv\|{\boldsymbol{R}}_{i_{m}}-{\boldsymbol{R}}_{i_{m^{\prime}}}\|$ is
again governed by the linear dispersion at low frequencies. Carrying out the
integration in Eq. (6) for $R\to\infty$ we find $\Omega(R)\propto 1/R^{D-2}$
(see Tab. 1 and SM, Sec. F SM ). For $D=3$ this implies that the geometrical
spin torque mediates a long-range coupling in the spin dynamics.

Compared to previous studies Stahl and Potthoff (2017); Lenzing et al. (2022);
Michel and Potthoff (2022, 2021); Bajpai and Nikolić (2020); Elbracht et al.
(2020) the $D=3$ value of the local SBC $|\Omega_{\rm loc}|\approx
0.084\,J^{2}/J_{\rm H}^{2}$ is several orders of magnitude larger for
realistic parameters $J,J_{\rm H}\ll t,U$. Renomalization of $c_{s}\to
c^{\prime}_{s}\approx 1.1c_{s}$ due to magnon interaction Oguchi (1960) leads
to a slightly smaller SBC, $|\Omega_{\rm loc}|\to(c_{\rm s}/c_{\rm
s}^{\prime})^{2}|\Omega_{\rm loc}|$.

There are at least two routes that lead to an even larger $|\Omega_{\rm
loc}|$: Namely, we can take advantage of the formally infinite SBC in $D=2$
and regularize the theory (i) by dimensional crossover to $D=3$ Liu (1990);
Majlis et al. (1992, 1993), i.e., by switching on a small hopping $t_{\perp}$
in the third dimension (Fig. 2) implying $J_{\rm H}^{\perp}\ll J_{\rm
H,x}=J_{\rm H,y}=J_{\rm H}$, see Fig. 3 (left), or (ii) by switching on a
magnetic anisotropy to open a small gap in the magnon spectrum (Fig. 3,
right), i.e., by adding an Ising term $\delta\,J_{\rm H}S_{iz}S_{jz}$ to the
standard Heisenberg coupling $J_{\rm
H}{\boldsymbol{S}}_{i}{\boldsymbol{S}}_{j}$. A moderate $J_{\rm
H}^{\perp}/J_{\rm H}=0.1$ yields a SBC $|\Omega_{\rm loc}|\approx
0.22J^{2}/J_{\rm H}^{2}$. About the same enhancement is obtained for an
anisotropy parameter $\delta\sim 10^{-2}$.

Figure 3:  SWT results (dots) for anisotropic systems. Left, dimensional
crossover: local SBC for $D=3$ but with a spatially anisotropic nearest-
neighbor Heisenberg exchange $J_{\rm H}^{\perp}\leq J_{\rm H}$. Right, spin
anisotropy: SBC as function of the coupling anisotropy parameter $\delta$.

##  Geometrical spin dynamics.

For the AF ordered phase, Eq. (1) tells us that the dominating effect in the
magnetic-moment dynamics is a precession around the staggered magnetization
${\boldsymbol{m}}$ on a time scale $1/J$. This effect dominates the weaker
(and slower) anisotropic RKKY-type exchange on the scale $J^{2}$. Importantly,
the SBC $\Omega\sim J^{2}$ enters the equations of motion as a renormalization
factor (for $M>1$ classical spins as a matrix factor) rather than a summand
and thus does not compete with the stronger direct exchange of order $J$ (SM,
Sec. G SM ). For $M=1$ this factor amounts to $1/(1-\Omega_{\rm loc}S_{z})$,
such that the most pronounced effects are found for a SBC of intermediate
strength, $\Omega_{\rm loc}={\cal O}(1)$. This holds true for $M=2$ as well,
as is detailed in the SM, Sec. G SM . Note that a singular renormalization
indicates a breakdown of the theory as this is the point where the condition
for nearly adiabatic spin dynamics is invalidated. Note further that the
precession comes with an inverted orientation beyond the singular point.

##  Conclusions.

A hitherto unknown but generic interplay of electron correlations, spontaneous
symmetry breaking, gapless Goldstone bosons, and a holonomy on the
configuration space of classical spin degrees of freedom leads to non-
Hamiltonian effects, such as renormalization of precession frequencies,
inverted orientation of the precessional motion, or long-range interactions,
in the spin dynamics. This is due to a geometrical spin torque which is finite
for correlated AF ground states in lattice models with dimension $D\geq 3$ and
diverges for $D\leq 2$, caused by the same mechanism that leads to the Mermin-
Wagner theorem, however, shifted by one dimension. With a SBC $\Omega_{\rm
loc}={\cal O}(1)$ for typical parameters, the effect is unexpectedly large. It
is boosted by electron correlations and further enhanced by spatial and by
spin anisotropies. We expect a strong overall impact on the phenomenology of
atomistic spin dynamics.

##  Acknowledgments.

This work was supported by the Deutsche Forschungsgemeinschaft (DFG, German
Research Foundation) through the research unit QUAST, FOR 5249 (project P8),
project ID 449872909, and through the Cluster of Excellence “Advanced Imaging
of Matter” - EXC 2056 - project ID 390715994.

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Geometrical torque on magnetic moments coupled to a correlated antiferromagnet

— Supplemental Material —

Nicolas Lenzing,1 David Krüger,1 and Michael Potthoff1,2

1 University of Hamburg, Department of Physics, Notkestraße 9-11, 22607
Hamburg, Germany

2The Hamburg Centre for Ultrafast Imaging, Luruper Chaussee 149, 22761
Hamburg, Germany

##  Section A: SO(2) symmetry analysis.

The retarded spin susceptibility is defined as

$\chi_{ii^{\prime},\alpha\alpha^{\prime}}(t)=-i\Theta(t)\langle[s_{i\alpha}(t),s_{i^{\prime}\alpha^{\prime}}(0)]\rangle\>,$
(7)

where $\Theta$ is the step function, $s_{i\alpha}(t)=e^{iH_{\rm
el}t}s_{i\alpha}e^{-iH_{\rm el}t}$, and $\langle\cdots\rangle$ is the ground-
state expectation value. Fourier transformation to frequency space yields the
Lehmann representation in terms of an energy eigenbasis
$\\{|\Psi_{n}\rangle\\}$:

$\displaystyle\chi_{ii^{\prime},\alpha\alpha^{\prime}}(\omega)$
$\displaystyle=$
$\displaystyle\sum_{n}\Bigg{(}\frac{\langle\Psi_{0}|s_{i\alpha}|\Psi_{n}\rangle\langle\Psi_{n}|s_{i^{\prime}\alpha^{\prime}}|\Psi_{0}\rangle}{\omega+i\eta-(E_{n}-E_{0})}$
(8) $\displaystyle-$
$\displaystyle\frac{\langle\Psi_{0}|s_{i^{\prime}\alpha^{\prime}}|\Psi_{n}\rangle\langle\Psi_{n}|s_{i\alpha}|\Psi_{0}\rangle}{\omega+i\eta-(E_{0}-E_{n})}\Bigg{)}\>.$

We have the relation
$\chi_{ii^{\prime},\alpha\alpha^{\prime}}(\omega)^{\ast}=\chi_{ii^{\prime},\alpha\alpha^{\prime}}(-\omega)$.
The spectral density
$-(1/\pi)\mbox{Im}\chi_{ii^{\prime},\alpha\alpha^{\prime}}(\omega)=\chi_{ii^{\prime},\alpha\alpha^{\prime}}(\omega)-\chi_{ii^{\prime},\alpha\alpha^{\prime}}(-\omega)/2i$
is an antisymmetric function of $\omega$.

In the AF phase with order parameter ${\boldsymbol{m}}=m{\boldsymbol{e}}_{z}$,
there is a remaining SO(2) symmetry of the nondegenerate energy eigenstates
under spin rotations around ${\boldsymbol{e}}_{z}$, which is unitarily
represented by $U_{\rm R}=e^{-is_{\rm tot,z}\varphi}$ on the Fock space with
the $z$-component of the total spin ${\boldsymbol{s}}_{\rm
tot}=\sum_{i}{\boldsymbol{s}}_{i}$ as the unbroken generator and the rotation
angle $\varphi$. We have $U_{\rm
R}^{\dagger}|\Psi_{n}\rangle=e^{i\phi_{n}}|\Psi_{n}\rangle$ with phases
$\phi_{n}$. Since ${\boldsymbol{s}}_{i}$ is a vector operator, we have $U_{\rm
R}^{\dagger}s_{i\alpha}U_{\rm R}=\sum_{\beta}R_{\alpha\beta}s_{i\beta}$, where
$\underline{R}=\underline{R}(\varphi)$ is the standard real $3\times 3$ matrix
representation of SO(2) rotations around ${\boldsymbol{e}}_{z}$. Hence, the
first matrix element in Eq. (4) can be written as $\langle\Psi_{0}|U_{\rm
R}U_{\rm R}^{\dagger}s_{i\alpha}U_{\rm R}U_{\rm
R}^{\dagger}|\Psi_{n}\rangle=\sum_{\beta}R_{\alpha\beta}(\varphi)e^{i\phi_{0}}\langle\Psi_{0}|s_{i\beta}|\Psi_{n}\rangle
e^{-i\phi_{n}}$. The phase factors cancel with those from the second matrix
element, and we thus find:
$\chi_{ii^{\prime},\alpha\alpha^{\prime}}(\omega)=\sum_{\beta\beta^{\prime}}R_{\alpha\beta}(\varphi)\chi_{ii^{\prime},\beta\beta^{\prime}}(\omega)R^{T}_{\beta^{\prime}\alpha^{\prime}}(\varphi)$,
i.e., $[\underline{\chi}_{ii^{\prime}}(\omega),\underline{R}(\varphi)]=0$ for
all $i,i^{\prime}$ and $\varphi$. It is easily verified directly that this
implies

$\underline{\chi}_{ii^{\prime}}(\omega)=\begin{pmatrix}\chi_{ii^{\prime},xx}(\omega)&\chi_{ii^{\prime},xy}(\omega)&0\\\
-\chi_{ii^{\prime},xy}(\omega)&\chi_{ii^{\prime},xx}(\omega)&0\\\
0&0&\chi_{ii^{\prime},zz}(\omega)\\\ \end{pmatrix}\>,$ (9)

i.e., there are only 3 independent entries for each pair $i,i^{\prime}$ (note
that $R$ is reducible, and furthermore Schur’s lemma does not apply to
representations over $\mathbb{R}$).

##  Section B: Spatial symmetries.

With Eq. (8), we immediately see that the spin-Berry curvature is related to
the spin susceptibility via

$\Omega_{mm^{\prime},\alpha\alpha^{\prime}}=-iJ^{2}\frac{\partial}{\partial\omega}\chi_{i_{m}i_{m^{\prime}},\alpha\alpha^{\prime}}(\omega)\Big{|}_{\omega=0}$
(10)

up to correction terms of order $J^{3}$, see Eq. (5). Therefore, the same
reasoning as above can be applied to the spin-Berry curvature tensor and
yields the same result for its structure:

$\underline{\Omega}_{mm^{\prime}}=\begin{pmatrix}\Omega_{mm^{\prime},xx}&\Omega_{mm^{\prime},xy}&0\\\
-\Omega_{mm^{\prime},xy}&\Omega_{mm^{\prime},xx}&0\\\
0&0&\Omega_{mm^{\prime},zz}\\\ \end{pmatrix}\>.$ (11)

In addition, for a given pair of sites $i_{m}$ and $i_{m^{\prime}}$, we may
consider a combined transformation $T\circ I$, composed of the space inversion
${\boldsymbol{R}}_{i}\mapsto{\boldsymbol{R}}_{i_{m}}-{\boldsymbol{R}}_{i}$
with respect to $i_{m}$ followed by the translation
${\boldsymbol{R}}_{i}\mapsto{\boldsymbol{R}}_{i}+({\boldsymbol{R}}_{i_{m}}-{\boldsymbol{R}}_{i_{m^{\prime}}})$
with the translation vector
${\boldsymbol{R}}_{i_{m}}-{\boldsymbol{R}}_{i_{m^{\prime}}}$. $T\circ I$ is a
discrete symmetry of the hypercubic lattice and interchanges $i_{m}$ with
$i_{m^{\prime}}$. This implies that the Hamiltonian commutes with the standard
unitary (and also Hermitian) representation $U_{\rm TI}$ of $T\circ I$.

For $i_{m}$ and $i_{m^{\prime}}$ in the same sublattice, the symmetry-broken
ground state is an eigenstate of $U_{\rm TI}$ as well. Analogously to the
SO(2) spin-rotation symmetry discussed above, we can thus immediately see from
the analysis of the matrix elements in Eq. (4) that
$\Omega_{mm^{\prime},\alpha\alpha^{\prime}}=\Omega_{m^{\prime}m,\alpha\alpha^{\prime}}$.
For the spin-Berry curvature we have the additional antisymmetry,
$\Omega_{mm^{\prime},\alpha\alpha^{\prime}}=-\Omega_{m^{\prime}m,\alpha^{\prime}\alpha}$,
which follows from Eq. (3). With this we get
$\Omega_{mm^{\prime},\alpha\alpha^{\prime}}=-\Omega_{mm^{\prime},\alpha^{\prime}\alpha}$,
i.e., for each pair $m,m^{\prime}$, the spin-Berry curvature tensor is
antisymmetric in the indices $\alpha,\alpha^{\prime}$ separately. Hence, with
Eq. (11), we see that only the elements
$\Omega_{mm^{\prime},xy}=-\Omega_{mm^{\prime},yx}$ can be nonzero.
Analogously, for the spin susceptibility, the $T\circ I$ symmetry of the
Hamiltonian implies
$\chi_{ii^{\prime},\alpha\alpha^{\prime}}(\omega)=\chi_{i^{\prime}i,\alpha\alpha^{\prime}}(\omega)$.
With the additional symmetry,
$\chi_{ii^{\prime},\alpha\alpha^{\prime}}(0)=\chi_{i^{\prime}i,\alpha^{\prime}\alpha}(0)$,
which follows from Eq. (8), this implies that the susceptibility matrix Eq.
(9) is diagonal for $\omega=0$.

For $i_{m}$ and $i_{m^{\prime}}$ in different sublattices, we concatenate the
transformation $T\circ I$ with a flip $F$ of the $z$-component of all spins.
This is unitarily represented by $U_{\rm F}$, which is defined via
$U^{\dagger}_{\rm F}c_{i\uparrow}U_{\rm F}=c_{i\downarrow}$ and
$U^{\dagger}_{\rm F}c_{i\downarrow}U_{\rm F}=c_{i\uparrow}$. We have $[U_{\rm
F},U_{\rm TI}]=0$ and $[U_{\rm F},H_{\rm el}]=0$. The symmetry-broken ground
state and the corresponding excited states are eigenstates of $U\equiv U_{\rm
TI}U_{\rm F}$. Hence, we find for the $\alpha=\alpha^{\prime}$ matrix elements

$\displaystyle\langle\Psi_{0}|s_{i_{m}\alpha}|\Psi_{n}\rangle\langle\Psi_{n}|s_{i_{m^{\prime}}\alpha}|\Psi_{0}\rangle$
(12) $\displaystyle=$
$\displaystyle\langle\Psi_{0}|U^{\dagger}s_{i_{m}\alpha}U|\Psi_{n}\rangle\langle\Psi_{n}|U^{\dagger}s_{i_{m^{\prime}}\alpha}U|\Psi_{0}\rangle$
$\displaystyle=$
$\displaystyle\langle\Psi_{0}|s_{i_{m^{\prime}}\alpha}|\Psi_{n}\rangle\langle\Psi_{n}|s_{i_{m}\alpha}|\Psi_{0}\rangle\>,$

and thus
$\Omega_{mm^{\prime},\alpha\alpha}=\Omega_{m^{\prime}m,\alpha\alpha}$. With
the antisymmetry of the full tensor,
$\Omega_{mm^{\prime},\alpha\alpha^{\prime}}=-\Omega_{m^{\prime}m,\alpha^{\prime}\alpha}$,
we thus find $\Omega_{mm^{\prime},\alpha\alpha}=0$. On the other hand,

$\displaystyle\langle\Psi_{0}|s_{i_{m}x}|\Psi_{n}\rangle\langle\Psi_{n}|s_{i_{m^{\prime}}y}|\Psi_{0}\rangle$
(13) $\displaystyle=$
$\displaystyle\langle\Psi_{0}|U^{\dagger}s_{i_{m}x}U|\Psi_{n}\rangle\langle\Psi_{n}|U^{\dagger}s_{i_{m^{\prime}}y}U|\Psi_{0}\rangle$
$\displaystyle=$
$\displaystyle-\langle\Psi_{0}|s_{i_{m^{\prime}}x}|\Psi_{n}\rangle\langle\Psi_{n}|s_{i_{m}y}|\Psi_{0}\rangle\>,$

since $U_{\rm F}^{\dagger}s_{ix}U_{\rm F}=s_{ix}$ but $U_{\rm
F}^{\dagger}s_{iy}U_{\rm F}=-s_{iy}$. This implies
$\Omega_{mm^{\prime},xy}=-\Omega_{m^{\prime}m,xy}$, and with the antisymmetry
of the full tensor we find $\Omega_{mm^{\prime},xy}=\Omega_{mm^{\prime},yx}$.
Together with Eq. (11), we see that $\Omega_{mm^{\prime},xy}=0$, and hence the
matrix $\underline{\Omega}_{mm^{\prime}}=0$ in Eq. (11).

Summing up, for arbitrary sites $i_{m}$ and $i_{m^{\prime}}$ we have

$\underline{\Omega}_{mm^{\prime}}=\begin{pmatrix}0&\Omega&0\\\ -\Omega&0&0\\\
0&0&0\\\ \end{pmatrix}\>,$ (14)

and hence the spin-Berry curvature is fixed by a single real number
$\Omega\equiv\Omega_{mm^{\prime},xy}=\Omega_{m^{\prime}m,xy}$. Furthermore,
$\Omega=0$ if $i_{m},i_{m^{\prime}}$ belong to different sublattices.

##  Section C: Random phase approximation.

The random phase approximation (RPA) represents a standard weak-coupling
approach to the magnetic susceptibility, see, e.g., Refs. Rowe et al., 2012;
Del Re and Toschi, 2021. It can be motivated in various ways, for example, via
a partial diagrammatic summation. In general, the RPA Luttinger-Ward
functional $\Phi[{\boldsymbol{G}}]$ Luttinger and Ward (1960); Rohringer et
al. (2018) is given as the sum of the two closed and self-consistently
renormalized first-order diagrams, i.e., by the Hartree and the Fock diagram.
For the Hubbard model the Fock diagram vanishes such that we are left with

$\Phi[{\boldsymbol{G}}]=U\sum_{i}\frac{1}{\beta^{2}}\sum_{n,n^{\prime}}G_{ii,\uparrow}(i\omega_{n})G_{ii,\downarrow}(i\omega_{n^{\prime}})\>.$
(15)

Here, $i$ runs over the sites of the hypercubic lattice,
$\sigma=\uparrow,\downarrow$ refers to the spin projection relative to the $z$
axis, $n$ labels the fermionic Matsubara frequencies $i\omega_{n}$, and
$\beta$ is the inverse temperature. Computations are done in the zero-
temperature limit $1/\beta\to 0$, which is taken at the end. Furthermore,
$G_{ii,\sigma}$ denotes the local one-particle Green’s function at site $i$ in
the symmetry-broken AF state, as obtained within the self-consistent Hartree-
Fock approximation. The Hartree-Fock self-energy is generated by the
Luttinger-Ward functional:
$\Sigma_{ii,\sigma}(i\omega_{n})=\beta\delta\Phi/\delta
G_{ii\sigma}(i\omega_{n})=U\beta^{-1}\sum_{n}G_{ii,-\sigma}(i\omega_{n})=U\langle
c^{\dagger}_{i-\sigma}c_{i-\sigma}\rangle$.

On the two-particle level, the RPA yields a local and frequency-independent
irreducible vertex

$\Gamma^{\rm(loc)}_{\sigma,-\sigma}(i\omega_{n},i\omega_{n^{\prime}})=\beta^{2}\frac{\delta^{2}\Phi}{\delta
G_{ii,-\sigma}(i\omega_{n})\delta G_{ii,\sigma}(i\omega_{n^{\prime}})}=U\>.$
(16)

This means that there is no feedback of two-particle correlations on the
single-particle Green’s function. The structureless vertex allows us to easily
get the transversal magnetic susceptibility
$\chi_{rs,+-}({\boldsymbol{k}},\omega)=\langle\langle
s_{r,{\boldsymbol{k}}}^{+};s_{s,{\boldsymbol{k}}}^{-}\rangle\rangle_{\omega}$
as the solution of a strongly simplified Bethe-Salpether equation in the
particle-hole channel:

$\underline{\chi}_{+-}({\boldsymbol{k}},i\nu_{n})=\underline{\chi}^{(0)}_{+-}({\boldsymbol{k}},i\nu_{n})+\underline{\chi}^{(0)}_{+-}({\boldsymbol{k}},i\nu_{n})U\underline{\chi}_{+-}({\boldsymbol{k}},i\nu_{n})\>.$
(17)

Here, $\underline{\chi}_{+-}$ is a $2\times 2$ matrix in the sublattice
degrees of freedom, and $\underline{\chi}^{(0)}_{+-}$ the bare susceptibility
matrix, which is computed with the Hartree-Fock one-particle propagators. The
equation is diagonal in the wave vectors ${\boldsymbol{k}}$ of the first
magnetic Brillouin zone and in the bosonic Matsubara frequencies $i\nu_{n}$.
The transversal susceptibility $\chi_{+-}$ is related to the susceptibility
tensor $\chi_{\alpha\alpha^{\prime}}$ introduced in Eqs. (7) and (8), via
$\chi_{+-}=2(\chi_{xx}-i\chi_{xy})$. From the renormalized zeroth-order
diagram, we get the Hartree-Fock susceptibility in Eq. (17) as

$\displaystyle{\chi}^{(0)}_{rs,+-}({\boldsymbol{k}},i\nu_{n})$
$\displaystyle=$
$\displaystyle\frac{-1}{L}\frac{1}{\beta}\sum_{{\boldsymbol{q}},n^{\prime}}G_{sr,\uparrow}({\boldsymbol{q}},i\omega_{n^{\prime}})$
(18) $\displaystyle\times$ $\displaystyle
G_{rs,\downarrow}({\boldsymbol{q}}+{\boldsymbol{k}},i\nu_{n}+i\omega_{n^{\prime}})\>.$

After performing the summation over the fermionic frequencies $i\omega_{n}$
analytically, we can replace $i\nu_{n}\mapsto\nu+i\eta$ to find the retarded
susceptibility on the real-frequency axis. The frequency derivative in Eq.
(10) is done numerically. In the thermodynamical limit, the ${\boldsymbol{q}}$
sum over the magnetic Brillouin zone in Eq. (18) can be converted into a
${\boldsymbol{q}}$-space integration. The latter is computed in two or three
dimensions via a standard adaptive ${\boldsymbol{q}}$-space integration
technique for arbitrary $\nu\in\mathbb{R}$ and for each allowed wave vector
${\boldsymbol{k}}$ in the magnetic Brillouin zone of a finite lattice with $L$
sites and periodic boundary conditions. Practical computations are performed
at a finite Lorentzian broadening parameter $\eta>0$ replacing the
infinitesimal $\eta$, and convergence with respect to $L$ is controlled by
runs for different system sizes $L$. The main numerical error is due to
extrapolation of the data for $\eta\to 0$.

##  Section D: Magnon spectrum of an antiferromagnet.

In the strong-$U$ limit of the Hubbard model, the low-energy physics is
captured by the $s=1/2$ antiferromagnetic Heisenberg model

$H=J_{\rm H}\sum_{\langle
ij\rangle}\left(\frac{1}{2}(s^{+}_{i}s^{-}_{j}+s^{-}_{i}s^{+}_{j})+\Delta
s^{z}_{i}s^{z}_{j}\right)$ (19)

with $J_{\rm H}=4t^{2}/U$ and $\Delta=1$. An anisotropy parameter $\Delta>1$
can be used to discuss the effect of opening a gap in the dispersion. The sum
runs over all nearest-neighbor pairs $\langle ij\rangle$.

We apply the standard Holstein-Primakoff transfomation for the model on the
bipartite hypercubic lattice with dimension $D$. For sites $i$ in sublattice
A, the spin operators are expressed in terms of bosonic annihilators and
creators, i.e.,

$\displaystyle s^{z}_{i}$ $\displaystyle=$ $\displaystyle
s-a^{\dagger}_{i}a_{i}\>,$ $\displaystyle s^{+}_{i}$ $\displaystyle=$
$\displaystyle\sqrt{2s}\sqrt{1-\frac{\hat{n}_{i}}{2s}}a_{i}\>,$ $\displaystyle
s^{-}_{i}$ $\displaystyle=$
$\displaystyle\sqrt{2s}a^{\dagger}_{i}\sqrt{1-\frac{\hat{n}_{i}}{2s}}\>,$ (20)

while for sites $j\in B$

$\displaystyle s^{z}_{j}$ $\displaystyle=$
$\displaystyle-s+b^{\dagger}_{j}b_{j}\>,$ $\displaystyle s^{+}_{j}$
$\displaystyle=$
$\displaystyle\sqrt{2s}b^{\dagger}_{j}\sqrt{1-\frac{\hat{n}_{j}}{2s}}\>,$
$\displaystyle s^{-}_{j}$ $\displaystyle=$
$\displaystyle\sqrt{2s}\sqrt{1-\frac{\hat{n}_{j}}{2s}}b_{j}\>.$ (21)

The transformed Hamiltonian reads

$H=J_{\rm H}s\sum_{\langle
ij\rangle}[(a_{i}b_{j}+a^{\dagger}_{i}b^{\dagger}_{j})+\Delta(a^{\dagger}_{i}a_{i}+b^{\dagger}_{j}b_{j})]-\frac{L}{2}zJ_{\rm
H}\Delta s^{2}\;,$ (22)

where $z=2D$ is the coordination number, and $L$ is the number of lattice
sites. Quartic and higher-order magnon interaction terms resulting for the
expansion of the square root have been disregarded.

We drop the additive energy constant and block-diagonalize $H$ via Fourier
transformation:

$\displaystyle
a_{i}=\frac{1}{\sqrt{L/2}}\sum_{\boldsymbol{k}}e^{-i\boldsymbol{k}\boldsymbol{R}_{i}}a_{\boldsymbol{k}}\>,\quad
b_{j}=\frac{1}{\sqrt{L/2}}\sum_{\boldsymbol{k}}e^{i\boldsymbol{k}\boldsymbol{R}_{j}}b_{\boldsymbol{k}}\>.$

Here, ${\boldsymbol{R}}_{i}$ are the translation vectors of the magnetic A
sublattice (the same for ${\boldsymbol{R}}_{j}$ and the B sublattice),
consisting of $L/2$ unit cells, and ${\boldsymbol{k}}$ is an allowed wave
vector of the first magnetic Brillouin zone (mBz). Defining
$\gamma_{{\boldsymbol{k}}}\equiv\sum_{{\boldsymbol{\delta}}}\cos({\boldsymbol{k}}{\boldsymbol{\delta}})$
with nearest-neighbor vectors ${\boldsymbol{\delta}}$, the Fourier-transformed
model reads as

$H=J_{\rm
H}s\sum_{\boldsymbol{k}}[\gamma_{\boldsymbol{k}}(a_{\boldsymbol{k}}b_{\boldsymbol{k}}+b^{\dagger}_{\boldsymbol{k}}a^{\dagger}_{\boldsymbol{k}})+z\Delta(a_{\boldsymbol{k}}a^{\dagger}_{\boldsymbol{k}}+b^{\dagger}_{\boldsymbol{k}}b_{\boldsymbol{k}})]$
(24)

and can be diagonalized by Bogoliubov transformation

$a_{\boldsymbol{k}}=u_{\boldsymbol{k}}\alpha_{\boldsymbol{k}}+v_{\boldsymbol{k}}\beta^{\dagger}_{\boldsymbol{k}}\>,\quad
b_{\boldsymbol{k}}=u_{\boldsymbol{k}}\beta_{\boldsymbol{k}}+v_{\boldsymbol{k}}\alpha^{\dagger}_{\boldsymbol{k}}$
(25)

with real coefficients $u_{{\boldsymbol{k}}}$ and $v_{{\boldsymbol{k}}}$. We
require

$u^{2}_{\boldsymbol{k}}-v^{2}_{\boldsymbol{k}}=1\>,$ (26)

to ensure that $\alpha_{\boldsymbol{k}}$ and $\beta_{\boldsymbol{k}}$ satisfy
bosonic commutation relations, as well as

$2z\Delta
u_{\boldsymbol{k}}v_{\boldsymbol{k}}+\gamma_{\boldsymbol{k}}(u^{2}_{\boldsymbol{k}}+v^{2}_{\boldsymbol{k}})\stackrel{{\scriptstyle!}}{{=}}0\>,$
(27)

as usual, to get the Hamiltonian to the form

$H=J_{\rm
H}s\sum_{\boldsymbol{k}}\omega(\boldsymbol{k})(\alpha^{\dagger}_{\boldsymbol{k}}\alpha_{\boldsymbol{k}}+\beta^{\dagger}_{\boldsymbol{k}}\beta_{\boldsymbol{k}})\>,$
(28)

where we again dropped an unimportant constant energy term. The magnon
spectrum consists of two degenerate branches with dispersion $J_{\rm
H}s\,\omega({\boldsymbol{k}})$ given by

$\omega(\boldsymbol{k})=\sqrt{(z\Delta)^{2}-\gamma^{2}_{{\boldsymbol{k}}}}=\sqrt{z^{2}\Delta^{2}-\left(\sum_{{\boldsymbol{\delta}}}\cos({\boldsymbol{k}}{\boldsymbol{\delta}})\right)^{2}}\>.$
(29)

Close to ${\boldsymbol{k}}=0$ and in the isotropic case $\Delta=1$, the
dispersion $J_{\rm H}s\,\omega({\boldsymbol{k}})$ is linear,

$\omega({\boldsymbol{k}})=2\sqrt{D}k+{\cal O}(k^{3})\>,$ (30)

while for $\Delta>1$ the spectrum is gapped, and
$\omega({\boldsymbol{k}})=2S\sqrt{\Delta^{2}-1}+{\cal O}(k^{2})$.

From the conditions Eq. (26) and Eq. (27), we can deduce the well-known
results

$\displaystyle u^{2}_{\boldsymbol{k}}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(\frac{z\Delta}{\sqrt{(z\Delta)^{2}-\gamma^{2}_{\boldsymbol{k}}}}+1\right)=\frac{1}{2}\left(\frac{z\Delta}{\omega(\boldsymbol{k})}+1\right)\>,$
$\displaystyle v^{2}_{\boldsymbol{k}}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(\frac{z\Delta}{\sqrt{(z\Delta)^{2}-\gamma^{2}_{\boldsymbol{k}}}}-1\right)=\frac{1}{2}\left(\frac{z\Delta}{\omega(\boldsymbol{k})}-1\right)$
(31)

and

$u_{\boldsymbol{k}}v_{\boldsymbol{k}}=-\frac{\gamma_{\boldsymbol{k}}}{2\sqrt{(z\Delta)^{2}-\gamma^{2}_{\boldsymbol{k}}}}=-\frac{\gamma_{\boldsymbol{k}}}{2\omega(\boldsymbol{k})}\>,$
(32)

see Refs. Anderson, 1952; Holstein and Primakoff, 1940.

##  Section E: Computing the spin-Berry curvature from the magnon Hamiltonian.

The contribution of the magnon excitations to the spin-Berry curvature is
obtained from

$\Omega_{mm^{\prime},\alpha\alpha^{\prime}}=-2J^{2}\mbox{Im}\sum_{\boldsymbol{k},\eta=1,2}\frac{\langle
0|s^{\alpha}_{i_{m}}|\boldsymbol{k},\eta\rangle\langle\boldsymbol{k},\eta|s^{\alpha^{\prime}}_{i_{m^{\prime}}}|0\rangle}{(E_{0}-E_{\boldsymbol{k}})^{2}}\>,$
(33)

where
$|\boldsymbol{k},1\rangle\equiv\alpha^{\dagger}_{\boldsymbol{k}}|0\rangle$ and
$|\boldsymbol{k},2\rangle\equiv\beta^{\dagger}_{\boldsymbol{k}}|0\rangle$ are
the single-magnon states. Following Eq. (14), it is sufficient to compute
$\Omega_{mm^{\prime}}=\Omega_{mm^{\prime},xy}=-\Omega_{mm^{\prime},yx}$.
Furthermore, $\Omega_{mm^{\prime}}\neq 0$ only for $i_{m}$ and
$i_{m^{\prime}}$ in the same sublattice, as also argued in section B.
Expressing the spin components in terms of the Bogoliubov operators,

$\displaystyle s^{+}_{i}$ $\displaystyle=$
$\displaystyle\sqrt{2s}\frac{1}{\sqrt{L/2}}\sum_{\boldsymbol{k}}e^{-i\boldsymbol{k}\boldsymbol{R}_{i}}(u_{\boldsymbol{k}}\alpha_{\boldsymbol{k}}+v_{\boldsymbol{k}}\beta^{\dagger}_{\boldsymbol{k}})$
$\displaystyle s^{-}_{i}$ $\displaystyle=$
$\displaystyle\sqrt{2s}\frac{1}{\sqrt{L/2}}\sum_{\boldsymbol{k}}e^{i\boldsymbol{k}\boldsymbol{R}_{i}}(u_{\boldsymbol{k}}\alpha^{\dagger}_{\boldsymbol{k}}+v_{\boldsymbol{k}}\beta_{\boldsymbol{k}})$
$\displaystyle s^{+}_{j}$ $\displaystyle=$
$\displaystyle\sqrt{2s}\frac{1}{\sqrt{L/2}}\sum_{\boldsymbol{k}}e^{-i\boldsymbol{k}\boldsymbol{R}_{j}}(u_{\boldsymbol{k}}\beta^{\dagger}_{\boldsymbol{k}}+v_{\boldsymbol{k}}\alpha_{\boldsymbol{k}})$
$\displaystyle s^{-}_{j}$ $\displaystyle=$
$\displaystyle\sqrt{2s}\frac{1}{\sqrt{L/2}}\sum_{\boldsymbol{k}}e^{i\boldsymbol{k}\boldsymbol{R}_{j}}(u_{\boldsymbol{k}}\beta_{\boldsymbol{k}}+v_{\boldsymbol{k}}\alpha^{\dagger}_{\boldsymbol{k}})\>,$
(34)

we find

$\displaystyle\langle
0|s^{x}_{i}|\boldsymbol{k},1\rangle\langle\boldsymbol{k},1|s^{y}_{i^{\prime}}|0\rangle$
$\displaystyle=$ $\displaystyle
i\frac{s}{L}u^{2}_{\boldsymbol{k}}e^{-i\boldsymbol{k}(\boldsymbol{R}_{i}-\boldsymbol{R}_{i^{\prime}})}\>,$
$\displaystyle\langle
0|s^{x}_{i}|\boldsymbol{k},2\rangle\langle\boldsymbol{k},2|s^{y}_{i^{\prime}}|0\rangle$
$\displaystyle=$
$\displaystyle-i\frac{s}{L}v^{2}_{\boldsymbol{k}}e^{i\boldsymbol{k}(\boldsymbol{R}_{i}-\boldsymbol{R}_{i^{\prime}})}$
(35)

for $i$, $i^{\prime}$ in sublattice A, and

$\displaystyle\langle
0|s^{x}_{j}|\boldsymbol{k},1\rangle\langle\boldsymbol{k},1|s^{y}_{j^{\prime}}|0\rangle$
$\displaystyle=$ $\displaystyle
i\frac{s}{L}v^{2}_{\boldsymbol{k}}e^{-i\boldsymbol{k}(\boldsymbol{R}_{j}-\boldsymbol{R}_{j^{\prime}})}\>,$
$\displaystyle\langle
0|s^{x}_{j}|\boldsymbol{k},2\rangle\langle\boldsymbol{k},2|s^{y}_{j^{\prime}}|0\rangle$
$\displaystyle=$
$\displaystyle-i\frac{s}{L}u^{2}_{\boldsymbol{k}}e^{i\boldsymbol{k}(\boldsymbol{R}_{j}-\boldsymbol{R}_{j^{\prime}})}$
(36)

for $j$, $j^{\prime}$ in sublattice B. This implies

$\displaystyle\mbox{Im}{\sum_{\eta=1,2}\langle
0|s^{x}_{i}|\boldsymbol{k},\eta\rangle\langle\boldsymbol{k},\eta|s^{y}_{i^{\prime}}|0\rangle}$
$\displaystyle=$
$\displaystyle\frac{s}{L}\cos(\boldsymbol{k}(\boldsymbol{R}_{i}-\boldsymbol{R}_{i^{\prime}}))$
$\displaystyle\mbox{Im}{\sum_{\eta=1,2}\langle
0|s^{x}_{j}|\boldsymbol{k},\eta\rangle\langle\boldsymbol{k},\eta|s^{y}_{j^{\prime}}|0\rangle}$
$\displaystyle=$
$\displaystyle-\frac{s}{L}\cos(\boldsymbol{k}(\boldsymbol{R}_{j}-\boldsymbol{R}_{j^{\prime}}))\>,$

and finally we get

$\Omega_{mm^{\prime}}=\mp\frac{1}{s}\frac{J^{2}}{J_{\rm
H}^{2}}\frac{2}{L}\sum^{\rm
mBz}_{\boldsymbol{k}}\frac{\cos(\boldsymbol{k}(\boldsymbol{R}_{i_{m}}-\boldsymbol{R}_{i_{m^{\prime}}}))}{\omega(\boldsymbol{k})^{2}}\>,$
(38)

where the upper sign refers to $i_{m},i_{m^{\prime}}$ in sublattice A and the
lower for $i_{m},i_{m^{\prime}}$ in sublattice B. Recall that
$\Omega_{mm^{\prime}}=0$ if $i_{m},i_{m^{\prime}}$ belong to different
sublattices.

Eq. (38) can be evaluated numerically. For $D=3$, for example, we find

$\Omega_{\rm loc}\approx-0.084\frac{J^{2}}{J_{\rm H}^{2}}\>$ (39)

for the local element of the SBC with $i_{m}=i_{m^{\prime}}$ in sublattice A.

##  Section F: Different dimensions and distance dependence.

In the thermodynamic limit $L\to\infty$ (and in the isotropic case
$\Delta=1$), the convergence of the resulting integral in Eq. (38) over the
magnetic Brillouin zone decisively depends on the lattice dimension $D$. We
consider the critical contribution of the long-wave-length magnons by
integrating over a $D$-dimensional ball around ${\boldsymbol{k}}=0$ with small
cutoff radius $k_{\rm c}$, such that we can make use of Eq. (30), i.e., of the
linearity and isotropy of the magnon dispersion for $k\to 0$:

$\Omega_{mm^{\prime}}\sim\lim_{\kappa\to 0}\int_{\kappa}^{k_{\rm
c}}dk\,k^{D-1}\frac{1}{\omega(\boldsymbol{k})^{2}}\propto\lim_{\kappa\to
0}\int_{\kappa}^{k_{\rm c}}dk\,k^{D-1}\frac{1}{k^{2}}\>.$ (40)

This yields

$\Omega_{mm^{\prime}}\sim\begin{cases}\kappa^{D-2}\quad\text{ for }D\geq 3\\\
\ln\kappa\quad\text{ for }D=2\\\ 1/\kappa\quad\text{ for }D=1\end{cases}\>.$
(41)

For $\kappa\to 0$, the spin-Berry curvature diverges for $D=1$ and $D=2$. We
conclude that a meaningful theory is obtained in dimensions $D\geq 3$ only.

The magnitude of the spin-Berry curvature decreases with increasing distance
${\boldsymbol{R}}\equiv{\boldsymbol{R}}_{i_{m}}-{\boldsymbol{R}}_{i_{m^{\prime}}}$.
For $D\geq 3$ its dependence in the large-$R$ limit is governed by long-wave-
length magnon excitations, and we have:

$\Omega(R)\propto\int_{0}^{k_{c}}dk\,k^{D-1}\int
d\Omega\frac{\cos(kR\cos\theta)}{k^{2}}\>,$ (42)

where $\int d\Omega$ denotes the surface integral over the $(D-1)$-dimensional
unit sphere, and $\theta$ the angle between ${\boldsymbol{k}}$ and
${\boldsymbol{R}}$. Furthermore, we made use of Eq. (30) for $k$ smaller than
the cutoff $k_{\rm c}$. We note that the distance dependence at large $R$ is
isotropic. Substituting $kR\to k$ in the one-dimensional $k$ integral
immediately yields

$\Omega(R)\propto\frac{1}{R^{D-2}}\>.$ (43)

For $D=3$, we have $\Omega(R)\propto 1/R$. In the infinite-$D$ limit, we
expect a local spin-Berry curvature

To compute the local element $m=m^{\prime}$ of the spin-Berry curvature Eq.
(38) in this limit, we start from the representation

$\Omega_{\rm loc}=-\frac{1}{s}\frac{J^{2}}{J_{\rm
H}^{2}}\int_{-\infty}^{\infty}dx\,\rho_{D}(x)\frac{1}{z^{2}\Delta^{2}-Dx^{2}}\>,$
(44)

where, for dimension $D$, we have defined the density function

$\rho_{D}(x)=\frac{2}{L}\sum_{{\boldsymbol{k}}}^{\rm
mBz}\delta(x-\gamma_{{\boldsymbol{k}}}/\sqrt{D})\>,$ (45)

and where we have used Eq. (29). We have $z=2D$ for the $D$-dimensional
hypercubic lattice and, in the Heisenberg limit of the Hubbard model, $J_{\rm
H}=4t^{2}/U=4t^{\ast}{}^{2}/DU$, when using the scaling $t=t^{\ast}/\sqrt{D}$
with $t^{\ast}=\mbox{const}$. In the limit $D\to\infty$, this scaling of the
hopping ensures that the kinetic energy of the Hubbard model remains
nontrivial and balances the interaction term Müller-Hartmann (1989). Moreover,
the density function converges to a Gaussian Müller-Hartmann (1989):

$\rho_{D}(x)\to\rho_{\infty}(x)=\frac{1}{2\sqrt{\pi}}\exp\left(-\frac{x^{2}}{4}\right)\>.$
(46)

In the Heisenberg limit and with the scaled hopping, we thus have

$\Omega_{\rm
loc}(D)=-\frac{1}{s}\frac{J^{2}U^{2}}{16t^{\ast}{}^{4}}\int_{-\infty}^{\infty}\frac{dx\rho_{D}(x)}{4\Delta^{2}-x^{2}/D}\>,$
(47)

which for $D\to\infty$, and assuming $s=1/2$ and $\Delta=1$ converges to

$\Omega_{\rm loc}(\infty)=-\frac{1}{32t^{\ast}{}^{4}}J^{2}U^{2}\>.$ (48)

This represents the mean-field value of the (lcoal) spin-Berry curvature in
the antiferromagnetic state at large $U$.

To compare with the result obtained for $D=3$, we must use the same scaling of
the hopping. This yields

$\Omega_{\rm
loc}(3)\approx-0.084\frac{J^{2}U^{2}}{16{t^{\ast}}^{4}}D^{2}\Bigg{|}_{D=3}\approx
1.51\cdot\Omega(\infty).$ (49)

For lattice dimensions $D>3$ we find: $\Omega_{\rm loc}(4)\approx
1.22\,\Omega(\infty)$, $\Omega_{\rm loc}(5)\approx 1.16\,\Omega(\infty)$,
$\Omega_{\rm loc}(6)\approx 1.12\,\Omega(\infty)$. Hence, given the standard
scaling of the hopping with $D$, the absolute value of $\Omega_{\rm loc}(D)$
increases with decreasing $D$ and finally, for $D=2$ diverges.

Finally, when addressing the dimensional crossover Liu (1990); Majlis et al.
(1992, 1993), we consider the Heisenberg model given by Eq. (19) again, but
with spatially anisotropic nearest-neighbor exchange couplings $J_{\rm
H}\equiv J_{\rm H,x}=J_{\rm H,y}\geq J_{\rm H,z}$. Proceeding analogously to
Sec. C, one ends up with a modified magnon dispersion only:

$\omega(\boldsymbol{k})=\sqrt{(z_{\text{eff}}\Delta)^{2}-\gamma^{\prime
2}_{\boldsymbol{k}}}\>.$ (50)

Here, we have defined an effective coordination number
$z_{\text{eff}}=2(J_{\rm H,x}+J_{\rm H,y}+J_{\rm H,z})/J_{\rm H,x}$.
Furthermore, $\gamma^{\prime}_{\boldsymbol{k}}:=2(J_{\rm H,x}\cos k_{x}+J_{\rm
H,y}\cos k_{y}+J_{\rm H,z}\cos k_{z})/J_{\rm H,x}$.

##  Section G: Spin dynamics.

The equations of motion Eq. (1) for the classical spins comprise the
conventional (Hamiltonian) and the geometrical spin torque, see Eq. (2). In
the weak-$J$ limit, the former results from the local direct exchange $J$ as
well as from the indirect RKKY-type exchange. We have:

$\dot{\boldsymbol{S}}_{m}=J\langle\boldsymbol{s}_{i_{m}}\rangle^{(0)}\times\boldsymbol{S}_{m}+J^{2}\sum_{m^{\prime}}\underline{\chi}_{i_{m}i_{m^{\prime}}}(0)\boldsymbol{S}_{m^{\prime}}\times\boldsymbol{S}_{m}+\sum_{\alpha}\sum_{m^{\prime}\alpha^{\prime}}\Omega_{m^{\prime}m,\alpha^{\prime}\alpha}({\boldsymbol{S}})\dot{S}_{m^{\prime}\alpha^{\prime}}{\boldsymbol{e}}_{\alpha}\times\boldsymbol{S}_{m}\;,$
(51)

where $\langle\dots\rangle^{(0)}$ denotes the expectation value at $J=0$. For
the non-vanishing components of the spin susceptibility and of the spin-Berry
curvature on sublattice A we have

$\chi_{ii^{\prime}}\equiv\chi_{ii^{\prime},xx}(0)=\chi_{ii^{\prime},yy}(0)=-\frac{z\Delta}{J_{\rm
H}}\frac{2}{L}\sum_{\boldsymbol{k}}^{\rm
mBz}\frac{\cos{\boldsymbol{k}(\boldsymbol{R}_{i}-\boldsymbol{R}_{i^{\prime}})}}{\omega(\boldsymbol{k})^{2}}$
(52)

and

$\Omega_{mm^{\prime}}\equiv\Omega_{mm^{\prime},xy}=-\Omega_{mm^{\prime},yx}=-\frac{1}{s}\frac{J^{2}}{J_{\rm
H}^{2}}\frac{2}{L}\sum_{\boldsymbol{k}}^{\rm
mBz}\frac{\cos(\boldsymbol{k}(\boldsymbol{R}_{i_{m}}-\boldsymbol{R}_{i_{m^{\prime}}})}{\omega(\boldsymbol{k})^{2}}\>.$
(53)

Specializing Eq. (51) for $M=1$, i.e., for a single classical spin, we get

$\dot{\boldsymbol{S}}_{1}={\boldsymbol{T}}^{\rm(H)}_{1}\times\boldsymbol{S}_{1}+\Omega_{11}({\boldsymbol{e}}_{z}\times\dot{\boldsymbol{S}}_{1})\times\boldsymbol{S}_{1}\>,$
(54)

where

${\boldsymbol{T}}^{\rm(H)}_{1}=J\langle\boldsymbol{s}_{i_{1}}\rangle^{(0)}+J^{2}\chi_{i_{1}i_{1}}({\boldsymbol{e}}_{z}\times\boldsymbol{S}_{m})\times{\boldsymbol{e}}_{z}\>.$
(55)

With

$\underline{T}^{\rm(H)}_{1}=\begin{pmatrix}0&-T^{\rm(H)}_{1,z}&T^{\rm(H)}_{1,y}\\\
T^{\rm(H)}_{1,z}&0&-T^{\rm(H)}_{1,x}\\\
-T^{\rm(H)}_{1,y}&T^{\rm(H)}_{1,x}&0\end{pmatrix}$ (56)

the cross product can be written as a matrix-vector product,
${\boldsymbol{T}}^{\rm(H)}_{1}\times\boldsymbol{S}_{1}=\underline{T}^{\rm(H)}_{1}\boldsymbol{S}_{1}$,
and the equation of motion reads:

$\dot{\boldsymbol{S}}_{1}=\frac{1}{1-\Omega_{11}S_{1z}}\underline{T}^{\rm(H)}_{1}\boldsymbol{S}_{1}\>.$
(57)

The classical spin undergoes a purely precessional dynamics around the $z$
axis, but with a renormalized precession frequency. The renormalization is due
to the local spin-Berry curvature $\Omega_{\rm loc}=\Omega_{11}$ and is the
strongest for $\Omega_{\rm loc}={\cal O}(1)$. Right at $\Omega_{\rm
loc}=1/S_{1z}$, the precession frequency diverges. This implies that the spin
dynamics is no longer adiabatic and the theory breaks down.

In case of two classical spins, $M=2$, the equations of motion (51) can be
cast into the form

$\displaystyle\dot{\boldsymbol{S}}_{1}$ $\displaystyle=$
$\displaystyle\boldsymbol{T}^{\text{(H)}}_{1}\times{\boldsymbol{S}}_{1}+\boldsymbol{T}^{\text{(geo)}}_{1}\times{\boldsymbol{S}}_{1}\>,$
$\displaystyle\dot{\boldsymbol{S}}_{2}$ $\displaystyle=$
$\displaystyle\boldsymbol{T}^{\text{(H)}}_{2}\times{\boldsymbol{S}}_{2}+\boldsymbol{T}^{\text{(geo)}}_{2}\times{\boldsymbol{S}}_{2}\>,$
(58)

where

$\displaystyle\boldsymbol{T}^{\text{(H)}}_{1}$ $\displaystyle=$ $\displaystyle
J\langle\boldsymbol{s}_{i_{1}}\rangle^{(0)}+J^{2}\chi_{i_{1}i_{1}}({\boldsymbol{e}}_{z}\times\boldsymbol{S}_{1})\times{\boldsymbol{e}}_{z}+J^{2}\chi_{i_{1}i_{2}}({\boldsymbol{e}}_{z}\times\boldsymbol{S}_{2})\times{\boldsymbol{e}}_{z}\>,$
$\displaystyle\boldsymbol{T}^{\text{(H)}}_{2}$ $\displaystyle=$ $\displaystyle
J\langle\boldsymbol{s}_{i_{2}}\rangle^{(0)}+J^{2}\chi_{i_{2}i_{2}}({\boldsymbol{e}}_{z}\times\boldsymbol{S}_{2})\times{\boldsymbol{e}}_{z}+J^{2}\chi_{i_{2}i_{1}}({\boldsymbol{e}}_{z}\times\boldsymbol{S}_{1})\times{\boldsymbol{e}}_{z}\>.$
(59)

and

$\displaystyle\boldsymbol{T}^{\rm(geo)}_{1}$ $\displaystyle=$
$\displaystyle\Omega_{11}({\boldsymbol{e}}_{z}\times\dot{\boldsymbol{S}}_{1})\times\boldsymbol{S}_{1}+\Omega_{12}({\boldsymbol{e}}_{z}\times\dot{\boldsymbol{S}}_{2})\times\boldsymbol{S}_{1}$
$\displaystyle\boldsymbol{T}^{\rm(geo)}_{2}$ $\displaystyle=$
$\displaystyle\Omega_{22}({\boldsymbol{e}}_{z}\times\dot{\boldsymbol{S}}_{2})\times\boldsymbol{S}_{2}+\Omega_{12}({\boldsymbol{e}}_{z}\times\dot{\boldsymbol{S}}_{1})\times\boldsymbol{S}_{2}\>.$
(60)

Here, we have assumed that the two spins couple to sites in the same
sublattice, as otherwise the spin-Berry curvature vanishes. The local spin-
Berry curvature term can be treated in the same way as in the $M=1$ case,
while the nonlocal term can be written as a matrix-vector product:

$\displaystyle(1-\Omega_{11}S_{1z})\dot{\boldsymbol{S}}_{1}$ $\displaystyle=$
$\displaystyle{\boldsymbol{T}}^{\rm(H)}_{1}\times\boldsymbol{S}_{1}-\Omega_{12}\underline{\mathcal{A}}^{(z)}_{1}\dot{\boldsymbol{S}}_{2}\;,$
$\displaystyle(1-\Omega_{22}S_{2z})\dot{\boldsymbol{S}}_{2}$ $\displaystyle=$
$\displaystyle{\boldsymbol{T}}^{\rm(H)}_{2}\times\boldsymbol{S}_{2}-\Omega_{12}\underline{\mathcal{A}}^{(z)}_{2}\dot{\boldsymbol{S}}_{1}\;,$
(61)

with

$\underline{\mathcal{A}}^{(z)}_{m}=\begin{pmatrix}-S_{mz}&0&0\\\
0&-S_{mz}&0\\\ S_{mx}&S_{my}&0\end{pmatrix}\>.$ (62)

This allows us to cast the equations of motion into an explicit system of
ordinary differential equations:

$\begin{pmatrix}\dot{\boldsymbol{S}}_{1}\\\
\dot{\boldsymbol{S}}_{2}\end{pmatrix}={\cal
M}^{-1}\begin{pmatrix}{\boldsymbol{T}}_{1}^{\rm(H)}\times\boldsymbol{S}_{1}\\\
{\boldsymbol{T}}_{2}^{\rm(H)}\times\boldsymbol{S}_{2}\end{pmatrix}\>.$ (63)

Here, the $6\times 6$ matrix

${\cal
M}=\begin{pmatrix}(1-\Omega_{11}S_{1z}){{\boldsymbol{1}}}&\Omega_{12}\underline{\mathcal{A}}^{(z)}_{1}\\\
\Omega_{12}\underline{\mathcal{A}}^{(z)}_{2}&(1-\Omega_{22}S_{2z}){{\boldsymbol{1}}}\end{pmatrix}$
(64)

is given in terms of the components of the spin-Berry curvature tensor. Eq.
(63) demonstrates that the effect of the geometrical spin torque is not simply
additive and hence does not directly compete with the conventional spin
torque, but enters the spin dynamics as a multiplicative (matrix) factor.

The determinant of $\mathcal{M}$ can be computed analytically:

$\det\mathcal{M}=(1-\Omega_{11}S_{1z})(1-\Omega_{11}S_{2z})\big{[}(1-\Omega_{11}S_{1z})(1-\Omega_{22}S_{2z})-\Omega_{12}^{2}S_{1z}S_{2z}\big{]}^{2}\>.$
(65)

The theory breaks down if $\det\mathcal{M}=0$. We consider $\det{\cal M}$ as a
function of the local elements $\Omega_{\rm loc}=\Omega_{11}=\Omega_{22}$ and
assume that the nonlocal elements are small, $\Omega_{\rm
nonloc}=|\Omega_{12}|\ll\Omega_{\rm loc}$. We immediately see that the zeros
of $\det{\cal M}$ are of the order of unity. This implies that anomalous spin
dynamics, which is substantially affected by the geometrical spin torque, is
expected if $\Omega_{\rm loc}={\cal O}(1)$ and thus close to, but yet
different from the zeros of ${\cal M}$.