Einstein-de Haas effect: a bridge linking mechanics, magnetism, and topology

Xin Nie1 and Dao-Xin Yao1, ∗

1
Guangdong Provincial Key Laboratoryd of Magnetoelectric Physics and Devices,
State Key Laboratory of Optoelectronic Materials and Technologies,
Center for Neutron Science and Technology, School of Physics,
Sun Yat-Sen University, Guangzhou, 510275, China
The Einstein-de Haas (EdH) effect is an interesting phenomenon linking mechanics and
magnetism, in which changes in magnetization induce mechanical rotation. Despite being discovered
more than a century ago, the EdH effect still shows crucial significance in modern science,
arXiv:2409.17245v1 [cond-mat.mes-hall] 25 Sep 2024

particularly in the realms of spintronics and ultrafast magnetism. Recently, it has been predicted
that the EdH effect can be realised in topological magnon systems, which undoubtedly brings even
richer properties. In this perspective, we aim to review the recent progress of the EdH effect and
discuss its developments in three key aspects: the microscopic mechanism of the EdH effect, the
EdH effect of topological magnons, and the chiral phonon–magnon conversion. These discussions
are poised to inspire further explorations in physics and promising applications in different areas.

The underlying mechanism about how the angular momentum of electrons is converted into that of macroscopic
rotation is a central topic of the EdH effect. It has been elusive owing to the intricate interplay among three
fundamental degrees of freedom–electron spin, orbital motion, and lattice dynamics. Numerous theoretical studies
can also only focus on the coupled dynamics of some of these degrees of freedom, rather than all of them. Nevertheless,
recent advances in ultrafast demagnetization experiments have gradually unraveled this mystery [1]: Upon femtosecond
laser excitation, the incident photon energy is immediately absorbed by electrons, propelling them to higher energy
states within their atomic orbitals. In the presence of spin-orbit coupling, the spin begins to flip and its angular
momentum changes. It is noteworthy that the angular momentum brought by the photon during this process is
extremely small and is negligible. Since the electronic orbit is incapable of accommodating the majority of the
angular momentum lost by the spin, the spin redirects its angular momentum to the ions in the lattice. In this way,
a physical picture of ultrafast demagnetization is established, where the main demagnetization occurs during the
transfer of angular momentum from the spin to the lattice. This channel explains why the EdH effect can occur in
most transition metals in which the orbital angular momentum is frozen by the crystal field. Thus, the follow-up task
becomes to elucidate the dynamics of angular momentum transfer from spin to phonons or lattice.

Magnetic moment Stimulate ◆Magnetic field

◆Heating bath
◆Laser
Gyromagnetic ratio
◆ Initial spin configuration
𝑩 ◆ Local rotation
SOC ◆ Macroscopic rotation
Quantum angular momentum Magnetism
⚫ Spin contribution
⚫ Orbit contribution

⚫ EdH effect
Conversion ⚫ Barnett effect

Lattice dynamics
Mechanical angular Mechanical
momentum property
(a) (b) (c)

FIG. 1: (a) Fundamentals of the EdH effect. (b) EdH experiment [2]. (c) Spin-rotation coupling mechanism [3].

In previous research, the EdH effect, especially the Barnett effect, was often explained by a phenomenological
model that incorporates the spin–rigid-body-rotation coupling S · Ω, where S denotes the spin angular momentum
and Ω denotes the angular velocity of the rigid-body rotation. Although these theories satisfy the conservation of
angular momentum, they pose a subtle question at the microscopic level: Whether the spin angular momentum can

∗ Electronic address: yaodaox@mail.sysu.edu.cn

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be directly transferred to the macroscopic rotation without traversing phonons? To answer the question, we envision
such a scenario [4] in which a magnon is removed from the magnetic system and the reduced angular momentum is
fully converted into macroscopic rotation. Based on the law of conservation of angular momentum, the rotational
angular velocity of the magnet can be expressed as Ω = ℏS/I, with the Planck’s constant ℏ, the spin S, and the
moment of inertia I. Therefore, we can deduce the kinetic energy of lattice 21 IΩ 2 , which is much lower than the
energy needed to flip a spin, ℏw, where w is the magnon frequency. This discrepancy indicates that there should
be an intermediary that involves in the process of exchanging angular momentum and energy, which is most likely
phonons. In 2022, an ultrafast experiment utilizing electron diffraction strongly supports the view [5]. Since then,
the spin–lattice angular momentum transfer mechanism mediated by circularly polarized phonons has attracted much
attention.
Researchers also characterize the phonon-mediated mechanism by exploiting spin–rotation coupling [3], which
comprises local and macroscopic rotations, as illustrated in Fig. 1c. Their investigation demonstrates that, by
applying an external field to an elastic ferromagnet, angular momentum is first transferred from the electron spin to
circularly polarized phonons, then to the entire lattice, eventually resulting in a macroscopic EdH rotation. Besides,
it estimates the timescale for angular momentum transfer from spin to lattice and suggests the existence of phonon
angular momentum. Nevertheless, to determine phonon angular momentum within the framework of continuum
field theory remains a great challenge due to the difficulty in distinguishing the local rotation from the macroscopic
dynamics. This problem might be addressed by magneto-molecular dynamics, as long as the rigid-body rotation and
localized rotation can be decoupled. On the other hand, the transfer of angular momentum from spins to phonons
occurs on a timescale that is significantly shorter than that from phonons to the lattice, which further complicates
theoretical calculation and experimental detection of phonon angular momentum.
Experimental progress. The EdH technique has long been used to measure the g-factor of materials because of its
accuracy over electron-spin and ferromagnetic resonance means. Over the past two decades, the EdH experiments have
achieved remarkable advancements. Firstly, the experimental systems have evolved from the macroscopic scale to the
nanoscale; Secondly, the experimental materials have expanded from conventional ferromagnets to antiferromagnets
[6]. In addition, the quantum EdH experiment [2] and the ultrafast Barnett experiment [7] have been successfully
realized. These progresses extend the utilization of materials and pave the way for the development of magnetic-
rotation sensors and attitude control designs in aerospace engineering.
EdH effect of topological magnons. Magnons are quasiparticles of spin wave excitation and can carry magnetic
moment, which can exhibit topological characteristics after introducing interactions such as the Dzyaloshinskii-Moriya
interaction (DMI). It has been predicted that a non-zero Berry curvature in magnon Chern insulators can endow
magnons with orbital angular momentum, which consists of two components: an edge current and a self-rotation [9].
Recently, it has been proposed that this angular momentum, originating from the Berry curvature induced by the
DMI, can contribute to the EdH effect [8]. This phenomenon is called as the EdH effect of topological magnons and
L
its gyromagnetic ratio is defined as γm = ∆M , where L is the topological angular momentum and ∆M is magnon
moment.

DM interaction
Kitaev interaction Einstein-de Haas
Altermagnet effect
…

Magnon angular
momentum
Topological magnons

Berry curvature Temperature

Chern number Magnetic field

Thermal Hall conductivity

(a) (b)

FIG. 2: EdH of toplogical magnons [8, 9]. (a) Theoretical mechanism. (b) Experimental design.

Topological magnon formalism on the square-octagon, honeycomb and kagome lattices [8, 10, 11] further
 3

demonstrate that the EdH might occur at suitable temperatures. Of particular interest, the square-octagon and
the square-hexagon-octagon lattices display obvious EdH effects. More recently, the EdH effect has been studied
in a collinear altermagnet [12]. To strengthen the reliability of the theory, this team computationally predicts real
materials, like CrI3 and Cu(1,3-bdc), in which the EdH effect of topological magnons may be observed. Furthermore,
they propose an experimental design to directly observe this effect by manipulating the magnetic field or varying the
temperature [8].
As the inverse of the EdH effect, the Barnett effect describes an opposite process of spin-lattice dynamics, that is,
the mechanical rotation can modulate the microscopic magnetization. In this respect, does the Barnett effect also
have a topological analog? It should be noted that the DMI associated with topology is sensitive to the mechanical
state of materials and can change with the lattice structure or position. In this sense, the topological angular
momentum arising from the DMI may change with the mechanical properties, i.e., the topological properties of the
system can be affected. However, the specific influence is unclear and deserves further exploration. We expect the
experiments can directly extract the topological angular momentum from the total angular momentum to tackle
this probelm. The triplon mode is another well-known magnetic excitation, which can have non-trivial topological
properties characterised by topological invariants. From this perspective, triplon excitations can also affect the angular
momentum transition and promote the EdH effect.
Moreover, skyrmions in two-dimensional material are also suitable candidate for implementing the EdH effect.
Magnetic skyrmion R are a∂m topologically protected domian with vortex magnetic structure, in which the topological
1
number N = 4π m · ( ∂x × ∂m ∂y )dxdy, where m = m(r) represents the magnetic moment at positionR r. When
a skyrmion is created or annihilated, or even its size changes, the total spin angular momentum M = m(r)dr3
changes. If the angular momentum is transferred to the lattice, thus the EdH effect follows. Note that due to
symmetry, the spin angular momentum of skyrmions does not necessarily change when its structure changes. With
the advantages of easy manipulation, small size, and fast driving speed, magnetic skyrmions are considered to be an
ideal information carrier that can meet the requirements of high capacity, high speed, and low power consumption
for future storage and logic computing. However, its application in practical equipments faces many challenges, and
achieving precise manipulation of it is one of the key challenges. Currently, the regulation of the skyrmions mainly
depends on external magnetic field, electric field and temperature, etc. If we can use the EdH effect to regulate
skyrmions through angular momentum exchange, it will undoubtedly provide a broader prospect for the design of
skyrmion devices.
To summarize, the EdH effect in topological magnons not only offers a new approach to probing topology through
mechanical responses, apart from existing thermal and spin-based approaches, but also connects the fields of topology,
magnetism, and mechanics, signifying the advent of an exciting new research frontier in magnonics.
Chiral phonon-magnons conversion. Phonons generally feature linear polarization with zero angular momentum.
However, a non-zero phonon angular momentum can be acquired in a system that breaks one symmetry of the time-
reversal and space-inversion. A typical example can be found in a nonmagnetic hexagonal lattice where phonons
present circular polarization and carry angular momentum at high-symmetry points [13]. These polarized phonons
are also relevant to the chirality.
Phonons with non-zero angular momentum can couple with magnons via exchanging angular momentum. This
coupling essentially originates from the spin-orbit interactions and, at a macroscopic level, is interpreted as the
magnetic anisotropy affected by atomic distances and lattice deformations [14], which includes the DMI, anisotropic
exchange interactions, and single-ion anisotropy. It has been found that strong magnon-phonon coupling can exist in
various materials and introduce interesting properties. For example, in the multiferroic Fe2 Mo3 O8 , a new quasiparticle
called the topological magnon polaro has been recently detected [15], which is the consequence of a hybridization
between magnons and phonons. In the case of FePSe3 with a zigzag-type antiferromagnetic ground state [16], the
coupling exhibits chiral selectivity, occurring exclusively when the angular momenta of magnons and phonons are
aligned.
In addition to angular momentum, phonons are also suggested to possess magnetic moments on the scale of Bohr-
magneton µB [18], which is larger than the magnetization produced by time-varying electric polarization [19, 20],
M ∝ P × ∂t P . The latter is caused by the motion of the ions and is of the order of the nuclear magneton, about
10−3 µB ∼ 10−4 µB . To date, phonon magnetic moment has been found in paramagnetic [21], ferromagnetic [22], and
non-magnetic materials [17], and various explanations have been given for this phenomenon. In paramagnetic or
magnetically ordered systems, it has been suggested that phonons can acquire magnetic moments through coupling to
magnons, however, in paramagnetic system, this is attributed to electronic excitations or electron-phonon coupling.
More recently, a microscopic model based on orbit-lattice coupling for phonon magnetic moments in paramagnetic
and magnetic materials has been developed [23]. A degenerate chiral phonon mode couples to a degenerate orbital
transition to form two hybridized branches, one with a major phonon contribution and another with a major orbital
contribution. Consequently, the phonon mode obtains a partial of the g factor from the orbital transition, which is
several orders of magnitude larger than its own [23]. This implies a unified theory about the origin is possible.
 4

The mechanism of the phonon thermal Hall effect has been inconclusive since its discovery in 2005, as phonons are
electrically neutral particles that must be modulated by magnetic fields with the help of interaction with electrons
or spins. The related conjectures can be divided into three groups: Raman-type spin-phonon interaction [26], Berry
curvature [27], and phonon scattering [28]. However, these mechanisms also fail to explain all phonon Hall effects,
especially those in some “trivial” systems, such as nonmagnetic, paramagnetic, and symmetric materials [29]. We
argue that for this effect there is a complex interplay between phonons and electrons in which the charge fluctuations
are nonnegligible. The similarity between the phonon magnetic moment generation and the mechanism of the phonon
thermal Hall effect readily suggests that the two may be related. The former may give a completely different but
more universal explanation for the latter: in the presence of a magnetic field, phonons with magnetic moments are
deflected, yielding a transverse temperature gradient.
A large phonon magnetic moment greatly challenges the general understanding that phonons cannot directly respond
to an electromagnetic field, positioning phonons as promising candidates for device design akin to electrons and
magnets. This enables researchers to explore phonon degrees of freedom from a fresh perspective. Moreover, if the
phonon magnetic moment is clearly understood, then the “magic” phonon which has both angular momentum and
magnetic moment, can itself evoke the EdH effect.

ACKNOWLEDGEMENT

This work is supported by NKRDPC2022YFA1402802, NSFC-92165204, Leading Talent Program of Guangdong

Special Projects (201626003), and Guangdong Provincial Key Laboratory of Magnetoelectric Physics and Devices
(No. 2022B1212010008).

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