Longitudinal magnetization dynamics in Heisenberg magnets: Spin Green functions approach
In spite of the fact that dynamical properties of magnets have been extensively studied over the past years, the longitudinal magnetization dynamics is still much less understood than transverse one even in the equilibrium state of a system. In this paper, we give a review of existing, based on quan...
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irk-123456789-1762292021-02-05T01:26:04Z Longitudinal magnetization dynamics in Heisenberg magnets: Spin Green functions approach Krivoruchko, V.N. Специальный выпуск К 80-летию со дня рождения А.И. Звягина In spite of the fact that dynamical properties of magnets have been extensively studied over the past years, the longitudinal magnetization dynamics is still much less understood than transverse one even in the equilibrium state of a system. In this paper, we give a review of existing, based on quantum-mechanical approach, theoretical descriptions of the longitudinal magnetization dynamics for ferro-, ferri- and antiferromagnetic dielectrics. The aim is to reveal specific features of this type of magnetization vibrations under description a system within the framework of one of the basic model theory of magnetism— the Heisenberg model. Related experimental investigations as well as open questions are also briefly discussed. We hope that understanding of the longitudinal magnetization dynamics distinctive features in the equilibrium state have to be a reference point for a theory uncovering the physical mechanisms that govern ultrafast spin dynamics after femtosecond laser pulse demagnetization when a system is far beyond an equilibrium state. 2018 Article Longitudinal magnetization dynamics in Heisenberg magnets: Spin Green functions approach / V.N. Krivoruchko // Физика низких температур. — 2018. — Т. 44, № 11. — С. 1565-1574. — Бібліогр.: 56 назв. — англ. 0132-6414 PACS: 75.78.–n, 76.50.+g, 75.78.Jp http://dspace.nbuv.gov.ua/handle/123456789/176229 en Физика низких температур Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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Специальный выпуск К 80-летию со дня рождения А.И. Звягина Специальный выпуск К 80-летию со дня рождения А.И. Звягина |
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In spite of the fact that dynamical properties of magnets have been extensively studied over the past years, the longitudinal magnetization dynamics is still much less understood than transverse one even in the equilibrium state of a system. In this paper, we give a review of existing, based on quantum-mechanical approach, theoretical descriptions of the longitudinal magnetization dynamics for ferro-, ferri- and antiferromagnetic dielectrics. The aim is to reveal specific features of this type of magnetization vibrations under description a system within the framework of one of the basic model theory of magnetism— the Heisenberg model. Related experimental investigations as well as open questions are also briefly discussed. We hope that understanding of the longitudinal magnetization dynamics distinctive features in the equilibrium state have to be a reference point for a theory uncovering the physical mechanisms that govern ultrafast spin dynamics after femtosecond laser pulse demagnetization when a system is far beyond an equilibrium state. |
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Krivoruchko, V.N. |
| title |
Longitudinal magnetization dynamics in Heisenberg magnets: Spin Green functions approach |
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Longitudinal magnetization dynamics in Heisenberg magnets: Spin Green functions approach |
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Longitudinal magnetization dynamics in Heisenberg magnets: Spin Green functions approach |
| title_fullStr |
Longitudinal magnetization dynamics in Heisenberg magnets: Spin Green functions approach |
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Longitudinal magnetization dynamics in Heisenberg magnets: Spin Green functions approach |
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longitudinal magnetization dynamics in heisenberg magnets: spin green functions approach |
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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Специальный выпуск К 80-летию со дня рождения А.И. Звягина |
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| citation_txt |
Longitudinal magnetization dynamics in Heisenberg magnets: Spin Green functions approach / V.N. Krivoruchko // Физика низких температур. — 2018. — Т. 44, № 11. — С. 1565-1574. — Бібліогр.: 56 назв. — англ. |
| series |
Физика низких температур |
| work_keys_str_mv |
AT krivoruchkovn longitudinalmagnetizationdynamicsinheisenbergmagnetsspingreenfunctionsapproach |
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2025-07-15T13:55:00Z |
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2025-07-15T13:55:00Z |
| _version_ |
1837721401631440896 |
| fulltext |
Low Temperature Physics/Fizika Nizkikh Temperatur, 2017, v. 43, No. 11, pp. 1565–1574
Longitudinal magnetization dynamics in Heisenberg
magnets: Spin Green functions approach
(Review Article)
V.N. Krivoruchko
Donetsk Institute for Physics and Engineering the National Academy of Sciences of Ukraine
46 Nauki Avenue, Kyiv 03028, Ukraine
E-mail: krivoruc@gmail.com
Received March 29, 2017, published online September 25, 2017
In spite of the fact that dynamical properties of magnets have been extensively studied over the past years,
the longitudinal magnetization dynamics is still much less understood than transverse one even in the equilibrium
state of a system. In this paper, we give a review of existing, based on quantum-mechanical approach, theoretical
descriptions of the longitudinal magnetization dynamics for ferro-, ferri- and antiferromagnetic dielectrics. The
aim is to reveal specific features of this type of magnetization vibrations under description a system within the
framework of one of the basic model theory of magnetism — the Heisenberg model. Related experimental inves-
tigations as well as open questions are also briefly discussed. We hope that understanding of the longitudinal
magnetization dynamics distinctive features in the equilibrium state have to be a reference point for a theory un-
covering the physical mechanisms that govern ultrafast spin dynamics after femtosecond laser pulse demagneti-
zation when a system is far beyond an equilibrium state.
PACS: 75.78.–n Magnetization dynamics;
76.50.+g Ferromagnetic, antiferromagnetic, and ferrimagnetic resonances; spin-wave resonance;
75.78.Jp Ultrafast magnetization dynamics and switching.
Keywords: longitudinal magnetization dynamics, spin Green functions, Heisenberg magnets.
Contents
1. Introduction ....................................................................................................................................... 1565
2. Longitudinal spin dynamics. Spin Green functions approach ........................................................... 1566
3. Ferromagnet ...................................................................................................................................... 1567
4. Ferrimagnet ....................................................................................................................................... 1569
5. Antiferromagnet ................................................................................................................................ 1572
6. Concluding remarks .......................................................................................................................... 1572
References ............................................................................................................................................. 1573
1. Introduction
Ultrafast magnetization dynamics, i.e., spins dynamics
on sub-picosecond (0.01–1 ps) or picosecond (10–100 ps)
time scale (1 ps = 10–12 s) is an important research direc-
tion in modern magnetism. This new research field gained
widespread attention by the seminal observation of ultra-
fast laser-induced demagnetization of metallic Ni [1], the
discovery of all-optical helicity dependent magnetic switch-
ing [2], the demonstration of toggle switching in the metallic
ferrimagnet GdFeCo [3,4], and the recent discovery of all-
optical switching in a broad class of metallic materials [5,6].
These experiments revealed that excitation of a magnet
system by intense femtosecond laser pulses causes a col-
lapse of magnetic order much faster than any characteristic
time (energy) scale of spin-spin interactions known at the
time. Despite experimental progress, microscopic under-
standing of the ultrafast magnetization dynamics still re-
mains an open question. At the same time, uncovering the
physical mechanisms that govern ultrafast spin behavior is
crucial for designing the next generation magnetic record-
ing media and spin-based electronics. The state of the art
of the ultrafast spin dynamics and its prospects can be
found by the reader in the reviews [7–11].
© V.N. Krivoruchko, 2017
V.N. Krivoruchko
At present, it is commonly accepted that the short time
scale of a laser pulse and high temperatures following the
excitation lead to processes when the longitudinal magnet-
ization dynamics becomes pronounced [12–14]. In spite of
the progress in phenomenological modeling the magnetic
moment evolution at high temperatures during a time scale
approaching femtoseconds [15–21], a number of questions
still remain open. What is a magnetic configuration of a
system just after ultrafast laser-induced heating? What is a
local (quasi-) equilibrium state that the system acquires
after the first sub-picosecond of ultrafast magnetization
switching? Can the magnetic dynamics be treated as a qua-
si-static evolution if the system is in out-of-equilibrium
state? What are the roles of various spin-resonance modes
in these processes? Giving only these as examples. Natu-
rally, to achieve a theoretical explanation of ultrafast mag-
netization dynamics, the microscopic consideration based
on quantum-mechanical theory is indispensable. Any mod-
el claiming of correct description of ultrafast magnetization
dynamics in the limit of a long-time evolution should re-
produce longitudinal dynamics of the system in equilibri-
um state. Surprisingly, in spite of the fact that different
aspects of magnetic system properties have been studied
quite extensively over past years, rather little attention has
been paid to the longitudinal magnetization dynamics. This
type of magnetization excitations is still less well under-
stood even for an equilibrium state of a magnetic system.
One of the basic microscopic model of magneto-di-
electrics is the Heisenberg model [22]. In this paper, we
give a brief review of the existing microscopic descriptions
of the longitudinal magnetization dynamics in ferro-, ferri-
and antiferromagnets within the framework of this model.
The paper is organized as follows. We begin by briefly
reminding the key equations describing the magnetic sys-
tem’s linear response within the spin Green functions ap-
proach. Based on the results of earlier works of different
authors, it will be shown that the system’s longitudinal
dynamic susceptibility χzz(q,ω) reduces to a calculation of
the (sublattice) longitudinal correlation functions (the spin
Green functions, GFs) ( , )zz
ij nG iωq . A diagrammatic repre-
sentation for the spin GFs as well as related distinction
between the system’s isothermal and isolated susceptibility
tensors are briefly discussed here, too. The results obtained
for the Heisenberg ferromagnet are expounded in Sec. 3.
Investigations involving strongly renormalized spin-wave
excitations existing at high temperatures confirm the multi-
spin-wave nature of longitudinal spin dynamics in a fer-
romagnet suggested in the early studies of the model. Sec-
tions 4 and 5 are devoted to the results obtained for the
Heisenberg ferrimagnets and antiferromagnets, respective-
ly. Multisublattice nature of the system’s magnetic struc-
ture reveals in the longitudinal magnetization precession
through a few additional channels of multi-spin-wave crea-
tion/annihilation processes. The paper is concluded with
some general remarks on further progress in the field.
2. Longitudinal spin dynamics. Spin Green functions
approach
The Heisenberg model is a statistical mechanical model,
in which the spins of the magnetic systems are treated
quantum mechanically [22]. This model describes magnet-
ic dielectrics which magnetism of ions in ith sites can be
described by spin operator ( , , )yx z
i i iiS S S=S , and the inter-
action between spins is of the so-called exchange type:
ij i jJ S S . Depending on a sign of the exchange interaction
( – )ij i jJ J= r r parallel (ferromagnetic) or antiparallel (an-
tiferromagnetic or ferrimagnetic) spins states is realized
below the critical temperature.
Within the microscopic approach (the spin Green func-
tion method [23–25]) description of a system’s magnetiza-
tion dynamics implies a calculation of the dynamic suscep-
tibility tensor χ(q,ω) and investigation its properties as
function of frequency ω and momentum q. This, in turn, is
reduced to calculations of the retarded spin Green func-
tions: ( )( , ) ( , )RGχ ω → ωq q . In particular, to find the sys-
tem’s longitudinal susceptibility χzz(q,ω) one should find
the retarded longitudinal spin GF ( )
tot ( , )zz RG ωq
2
( )
tot tot , tot
0
ˆ( , ) ( ) | (0) ( , )zz Rzz z z BTM t M G
vω
µ
χ ω = 〈〈 〉〉 = − ωqq q .
(1)
Here tot
zM is a z component of a total magnetization
tot B i i ig= µ ΣM S , μB is the Bohr magneton, gi stands for
the g factor of the ith sublattice and v0 stands for the vol-
ume of a primitive magnetic cell. The symbol <<…>>q,ω
denotes the Fourier transform of the trace of ρ(…) with
( )exp ( ) / Sp exp ( )H Hρ = −β −β
; T̂ is the time-ordering
operator; β–1 = T is the temperature and H
is the Hamilto-
nian. There are theorems proving that the poles of the re-
tarded GFs correspond to the natural frequencies of mag-
netization excitations that are (i) transverse magnetization
oscillations of the spins or ordinary spin-waves, and (ii)
longitudinal spin oscillations. By turn, the retarded GFs of
the system can be restored from the temperature GFs
( , )nG iωq by analytic continuation from the Matsubara
frequencies iωn = i2πnT (n = 0, ±1, ±2, …) onto the real
frequency axis iωn → ω + iδ (δ → 0) (for more details, see,
e.g., Refs. 23–25).
For a ferromagnetic system, the total GF ( )
tot ( , )zz RG ωq
reads
( ) ( )2
tot tot
,
ˆ( , ) ( ) | .zz zz z z z z
nG i G q g T S S S S
ω
ω = = − 〈 〉 − 〈 〉f g
q
q
(2)
Here and below we use the notation q = {q,iωn}, where q
stands for the momentum and the Matsubara frequency iωn.
In the case of a system with two antiparallel sublattices,
the total GF ( )
tot ( , )zz RG ωq can be reduced to four sublattice
longitudinal GFs ( , )zz
ij nG iωq (i, j = 1, 2) as follows:
1566 Low Temperature Physics/Fizika Nizkikh Temperatur, 2017, v. 43, No. 11
Longitudinal magnetization dynamics in Heisenberg magnets: Spin Green functions approach
tot 1 1 2 2 1 1 2 2
ˆ( ) ( ) | ( )zz z z z zG q T g S g S g S g S= δ − δ δ − δ =
2 2
1 11 1 2 12 21 2 22( ) ( ) ( ) ( ),zz zz zz zzg G q g g G q G q g G q = − + + (3)
where z z z
i i iS S Sδ ≡ − 〈 〉 and ˆ ( ) | ( ) .zz z z z z
ij i i j jG S S ST Sτ= 〈〈 − 〈 〉 − 〉 〉〉〈
Thus, for a multisublattice system, calculation of the dy-
namic susceptibility χzz(q,ω) reduces to calculation of the
sublattice longitudinal GFs ( , )zz
ij nG iωq .
To find the spin GFs it is convenient to use the dia-
grammatic technique for spin operators. General rules for
constructing particular diagrams can be found, e.g., in
Refs. 23–25. We note only here that there are two sorts of
the Sz vertices: (a) without attached Green’s lines of trans-
verse spin-waves and (b) with one incoming and one out-
going Green’s lines. There is one sort of the – x yS S iS= −
vertices with one outgoing Green’s line, while two sorts of
the x yS S iS+ = + vertices are present, with one incoming
Green’s line or, with two incoming and one outgoing Green
lines. All internal vertices are joined by lines of interaction Jij.
To calculate the longitudinal spin GFs, we use the Larkin
equation derived earlier in the framework of the diagram-
matic technique for spin operators. Without dwelling into
the details of summing up the diagram series (Refs. 23–25
contain technical details concerning the construction of the
diagram technique for the Heisenberg magnet), we present
here the final analytic results. One can show that the graph
series for the Gzz(q) functions of a ferromagnet can be pre-
sented analytically in the form
1
( ) ( ) 1 ( )zz z z z
qG q q J q
−
= Σ − Σ . (4)
In terms of the diagrammatic technique, the quantity ( )z qΣ
is called an irreducible (by the Larkin’s method of isolating
the irreducible diagrams) part. Note that the irreducibility
is understood here in the sense that ( )z qΣ is represented by
the collection of all diagrams from the series for ( )zzG q
that cannot be cut across one line of interaction Jq (Jq is
the Fourier transform of the exchange interaction).
Within this diagrammatic technique, the graphical rep-
resentation of the longitudinal susceptibility χzz(q,ω) con-
tains two external Sz vertices which could be classified into
three types of diagrams. Namely, diagrams where: (i) all
external Sz vertices are isolated, (ii) one external Sz vertex
is isolated and another is connected by incoming and out-
going transverse Green’s lines, and (iii) all external Sz ver-
tices are connected by incoming and outgoing transverse
spin-waves lines. Thus, the mathematical problem amounts
to summing up all the diagrams describing these processes.
A complexity of the problem arises from the fact that a
commutator of two spin operators is not a c number but an
operator, too. Therefore, the series of the diagrams turns
out to be rather complicated in comparison with Bose or
Fermi operators.
In accordance with the diagram technique rules, the an-
alytical expressions of the (i) and (ii) types diagrams for
( , )z
niΣ ωq will contain a singular discrete frequency part
~ δωn,0 [26] and, thus, will not depend on the thermo-
dynamic time (here δn,0 is the Kronecker delta, being 0 if
the two arguments are different and 1 if they are equal). In
fact, we get here into touch with the difference between the
isolated and isothermal susceptibilities of a system [27–29].
The isolated (the Kubo) susceptibility is defined for the
case when a system is initially in thermal equilibrium and
isolated. The isothermal susceptibility is defined for a sys-
tem assumed to be in thermal equilibrium with thermostat
in the presence of a time-independent external «force». In
accordance with the general analysis of different suscepti-
bilities [27–29] the distinction between them points to the
nonergodicity of the system. Below we will be interested
in the Kubo (or isolated) susceptibility of the system and
discuss the related analytical expressions derived from the
quantity tot ( , )zz
nG iωq by analytic continuation from the
Matsubara frequencies onto a real axis ni iω →ω+ δ
( 0) δ → .
Note here, that typically the irreducible parts ( )z
ij qΣ
could be calculated by summing up the ladder diagrams
with antiparallel lines. Such summing up corresponds to
the so-called generalized random-phase approximation (RPA)
and rather well describes the ground state and dynamics
of a magnetic system (see, for example, Ref. 25). Note also
here, that as it is known, the RPA results hold valid beyond
the hydrodynamic regime, i.e., when the wave vector
is larger than the inverse correlation length ξ:
( )1/21/ ~ 1 – / Cq T T> ξ , and, of course, beyond the criti-
cal region (here TC is the Curie temperature; for more de-
tails, see, e.g., Ref. 30).
3. Ferromagnet
An isotropic Heisenberg ferromagnet is described by
the Hamiltonian
( )
, ,
1 1 .
2 2
z z
fg f g fg f g g f f g
f g f g
H J J S S S S S S+ − + − = − = − + + ∑ ∑S S
(5)
Here the circular spin operators gS± are defined as
( ) / 2x y
g g gS S iS± = ± , and fgJ (> 0) stands for the ex-
change integral between localized spins in f th and gth
sites.
In the pioneering paper [31] Vaks, Larkin and Pikin
proposed original diagram technic for spin operators and
studied magnetization dynamics in the Heisenberg ferro-
magnet. Using the standard perturbation approach (an ex-
pansion in the terms of 1/z, where z is the first coordination
lattice number) they showed that the transverse coherent
precession of magnetization (elementary spin-wave excita-
tions or magnons) survive in the long wavelength limit at
all temperatures below the critical one TC. For the longitu-
dinal correlations, they obtained an expression of the form
Low Temperature Physics/Fizika Nizkikh Temperatur, 2017, v. 43, No. 11 1567
V.N. Krivoruchko
(0)
,0( , )
1
zz
n n
bG i
J b
′
ω = δ
′−β q
q (6)
in the zeroth order of 1/z, and
–
( ) ( )1( , )zz
n
n n
G i
N i
−ε − ε
ω =
ω− ε + ε + δ∑ k k q
k k qk
q (7)
in the first order of the terms of 1/z. Here b′ is the first
derivative of the function ( ) ( )Sb z SB z= , and ( )SB z stands
for the Brillouin function, ,0,0 nn ωδ = δ , εk is the spin-wave
energy and [ ]–1( ) exp ( ) –/ 1k kn Tε = ε is the Bose distribu-
tion function. Equation (6) represents the isothermal sus-
ceptibility of the system. Note that within this approxima-
tion its characterizes static fluctuations of the longitudinal
spin components, which are characterized by the Brillouin
functions derivatives. The isolated (the Kubo) susceptibil-
ity is described by expression (7).
Thus, in accordance with Eq. (7), the uniform suscepti-
bility , ) 0(zzχ ω =0 for ω ≠ 0. This is a consequence of the
total spin conservation law. Generally, in the noninteract-
ing spin-wave approximation, the longitudinal spin excita-
tions eventuate as virtual processes of coherent creation
and annihilation of transverse spin-waves. For a ferromag-
net, the longitudinal excitation frequency — a pole of the
denominator in Eq. (7) — is ( ) ±ω = ε − εk k qq (here the
wave vector k is a variable). Also, the excitation processes
are controlled by the occupation factor, determined
through the Bose distribution function ( )n εk , which makes
the spin-waves with ~ 0k to be dominant ones.
Note that this peculiarity of the longitudinal spin dy-
namics is a general one and is determined by specific
quantum properties of the spin operators. Namely, the
transverse operators S+ and S– are operators of one spin-
wave creation (annihilation), and the commutator of these
operators is the operator Sz. As a result, the longitudinal
vibrations of magnetization by external magnetic field,
~ z
zh S , may be realized only as virtual processes of coher-
ent creation and annihilation of two transverse spin-waves:
–( )z
z zh S h S S+→ . For multisublattice magnets, the mag-
netization longitudinal dynamics can also be realized
through virtual processes of two different transverse modes
creation and annihilation (see below).
The results [31] were then confirmed and generalized in
more elaborated studies based on a modern diagram tech-
nique for spin operators by Izymov et al. [32]. Authors
summed up infinite series of diagrams for the irreducible
part ( )z qΣ of the longitudinal GF (4) involving all distinct
loops built from spin-wave propagators and describing the
spin-wave interaction. In Fig. 1 one- and two-loop order
diagrams for the irreducible part ( )z qΣ of the longitudinal
spin GF (4) are shown. The diagrams representing the Kubo
susceptibility are only shown explicitly. In the figure, two
external vertices of the GFs are represented by vertices
which are a hollow point with an incoming and outgoing
Green lines. The hatched square here represents graphically
the effective four-point scattering vertex. The corresponding
analytical expression is
1( ) ( ) ( ) ( )z
k
q N G k G k q−Σ = β − +∑
2
1 2 1 2 2 2 2 2
,
( ) ( ) ( ) ( , | , ) ( ) ( ),
k p
N G k G k q k k q k k q G k q G k−+ β − Γ − + −∑
(8)
where { }, nk i= ωk . Within this perturbation approach,
the next step is a calculation of the four-point vertex
1 2 2 2( , | , )k k q k k qΓ − + — the spin-wave scattering ampli-
tude — presented graphically in Fig. 2 within the general
RPA. Here the wavy line is a graphical representation of
the interaction Jq. The graphical equation in Fig. 2 is ex-
pressed analytically as follows:
2 11 2 1 2( , | , )k k k q k q J J+ −Γ − + = + +k q k q
3
3
1
3 3 3 2 3 2( ) ( ) ( ) ( , | , )
k
N J G k G k q k q k k k q−
++ β + Γ + + +∑ k q
1
3
1
3 3 3 2 3 2( ) ( ) ( ) ( , | , )
k
J N G k q G k k q k k k q−
−+ β + Γ + +∑k q .
(9)
This integral equation can be transformed into an algebraic
one (for details, see Ref. 32).
After a simple but cumbersome algebra, in accordance
with the results of [31], for a ferromagnet the longitudinal
Fig. 1. The one- and two-loop order diagrams for irreducible part
of the longitudinal spin GF. The diagrams describing the Kubo
susceptibility are only shown explicitly. Here and in Fig. 3 exter-
nal hollow vertices correspond to external Sz spin operators. Here
and in Figs. 2–4, the solid lines represent the transverse spin-
wave GFs; the hatched squares represent graphically the effective
four-point vertices.
Fig. 2. Graphical representation the equation for the effective
four-point vertex 1 2 1 2( |, , )k k k q k qΓ − + . Here and in Fig. 4 a
wavy line corresponds to the interaction Jq.
1568 Low Temperature Physics/Fizika Nizkikh Temperatur, 2017, v. 43, No. 11
Longitudinal magnetization dynamics in Heisenberg magnets: Spin Green functions approach
excitations have been found at frequencies ω(q) which are
typical for processes of creation and annihilation of spin-
waves with energies εk and εk±q. Also, it was confirmed
that the excitation processes are controlled by the occupa-
tion factor, determined through the Bose distribution func-
tion ( )n εk . However, beyond the formal similarity, there is
a notable difference among two results due to a much more
common character of the approximation used in [32]. The
complete expression for χzz(q,ω) obtained in [32] describes
a strong renormalization of the energy longitudinal spin
fluctuations. Within the quadratic dispersion law for spin-
wave 2 ( )zS J aqε = 〈 〉q (a is the lattice spacing), the dis-
persion law of renormalized longitudinal wave excitations
was found to be of linear wave vector dependence
( ) 0.43 ( )zq S J aqω ≈ 〈 〉 . I.e., in a ferromagnet the frequency
of collective vibrations of the spin longitudinal component
lie energetically above the transverse spin-waves frequency
at the same temperature and wave vector. It was also shown
that the dynamic structure factor 1 Im ( , )zz−ω χ ωq exhibits
generally a three-peak structure including, first, two wide
maxima at frequencies ~ ( )zS J aqω ±〈 〉 corresponding to
damped longitudinal wave modes and, second, a sufficient-
ly narrow and less intensive central peak. When approach-
ing the Curie temperature, TC, the intensity of the latter
grows, and all three peaks form a broad distribution with
the linearly q-dependent width.
These results give a natural explanation for seemingly
conflicting experimental observations of two-peaked and
single-peaked behavior, found in numerous inelastic neutron
scattering studies of ferromagnetic materials. At the same
time, the contradictory results of neutron studies of longi-
tudinal fluctuations in ferromagnets motivated researchers
to explore the longitudinal susceptibility of a ferromagnet
beyond the model (1). Authors [33,34] investigated theo-
retically an isotropic ferromagnet with dipolar-dipolar in-
teraction in different temperature regions. It was shown that
dipole-dipole forces cannot be considered as a perturbation
not only in the close vicinity of the Curie point but also at
any temperature for low enough frequencies. The longitu-
dinal susceptibility reveals the infrared discontinuity in the
zero-frequency limit. In zero internal magnetic field, Hint,
the susceptibility demonstrated the phenomenon of infra-
red divergence. At finite Hint there exists a threshold for
Im χzz(ω) at the minimal energy of two magnons creation.
Therefore, one could expect an anomalous low-frequency
behavior of the longitudinal spin susceptibility in weak ex-
ternal magnetic fields at any temperature below TC. Authors
[34] presented the experimental evidence that in the ideal
isotropic ferromagnet dipolar forces would lead to the insta-
bility of the magnetically ordered state. They attributed the
anomalous behavior of the dynamical longitudinal suscep-
tibility observed in the ordered state of a nearly isotropic
ferromagnet CdCr2Se4 to the influence of the dipole-dipole
interaction and discuss the problem related to this feature.
Rudoy [35] studied the longitudinal dynamic suscepti-
bility of easy axis anisotropic Heisenberg ferromagnet in
the presence of a weak external magnetic field (supposed
time dependent and spatially non-uniform) at low tempera-
tures T << TC. Note that, at low temperatures, the spin sys-
tem like a Heisenberg ferromagnet is very similar, but not
identical, to a non-ideal gas of Bose particles. In condensed
Bose or Fermi systems some collective motions like zero
sound or plasma oscillation can arise under certain condi-
tions. If this type of excitations could exist in a Heisenberg
ferromagnet (the so-called «zero-magnon») it would ap-
pear as a singularity in the longitudinal dynamic suscepti-
bility. In [35], linearized quantum equations of motion for
the longitudinal components of the spin operators are em-
ployed to construct the dynamic RPA for the magnon col-
lective GF. The one-parameter class of the bosonic represen-
tations of spin operators was used. The dispersion equation,
describing the poles of the longitudinal dynamic suscepti-
bility, is studied and it is shown that in the easy-axis aniso-
tropic Heisenberg ferromagnet there are no long-wavelength
excitations of the «zero-magnon» type.
4. Ferrimagnet
Dynamical properties of ferrimagnetic compounds sub-
stantially differ from those of ferromagnetic ones because
of their multisublattice magnetic structure. Let us consider
an isotropic model of a two-sublattice ferrimagnet when
interaction inside the sublattices is small and can be ne-
glected, and g factors of the sublattices are equal, g1 = g2.
In absence of any external influences, the atomistic spin
model is described purely by the exchange interaction,
given by the Heisenberg Hamiltonian:
( )–
1 2 2 1 1 2
,
1
2
z zH J S S S S S S+ + − = + +
∑ fg f g g f f g
f g
. (10)
Pikin [36] studied the transverse magnetization dynamics
in the model (10) and showed that in a such system, as in a
ferromagnet, in the long wavelength limit the elementary
spin-wave excitations survive at all temperatures below the
critical one.
As was already shown in Sec. 2, for a two-sublattice
system the calculation of the susceptibility χzz(q,ω) is re-
duced to the calculation of four sublattice longitudinal GFs
( , )zz
ij nG iωq of the form (3). As was shown in [36], within
the zeroth order of a large interaction radius (the zeroth order
of the parameter 1/z) longitudinal sublattice correlators are
(0) 1
,011 2
1 2
( , )
1 ( )
zz
n n
bG i
J b b
′
ω = δ
′ ′− β q
q ,
(0) 2
,022 2
1 2
( , )
1 ( )
zz
n n
bG i
J b b
′
ω = δ
′ ′− β q
q ,
1 2(0) (0)
,012 21 2
1 2
( , ) ( , )
1 ( )
z
zz zz
n n n
J b b
G i G i
J b b
′ ′β
ω = ω = δ
′ ′− β
q
q
q q .
Low Temperature Physics/Fizika Nizkikh Temperatur, 2017, v. 43, No. 11 1569
V.N. Krivoruchko
As for a ferromagnet, within this approximation we deal
with the isothermal susceptibility which describes static
fluctuations of the longitudinal spin components.
To restore the dynamical characteristics, one needs to
find the GFs ( )zz
ijG q within a high order approximation.
These calculations have been done just recently. In the
report [37], based on the modern version diagram tech-
nique for spin operators, the analytical expression for the
longitudinal susceptibility of a two-sublattice ferrimagnet
has been obtained. Examples of the one- and two-loop or-
der diagrams for the irreducible part 11( )z qΣ of the longitu-
dinal spin Green function 11 ( )zzG q are shown in Figs. 3(a)
and (b), respectively. In the figures, the diagrams describ-
ing the Kubo susceptibility are shown only. Graphical re-
presentations for the irreducible parts 12 ( )z qΣ , 21( )z qΣ , and
22 ( )z qΣ possess a similar structure (for details, see Ref. 37).
As in the case of a ferromagnet, in Fig. 3, two external verti-
ces of the GFs are represented by vertices which are a hol-
low point with an incoming and outgoing Green lines. Other
arrangements of external vertices do not exist. Here the
hatched squares represent graphically two effective four-po-
int scattering vertexes. Graphical representation the equation
for effective four-point vertex (a) ( )11,12 1 2 1 2|, , k k k q k qΓ − +
and (b) ( )22,21 1 2 1 2, ,| k k k q k qΓ − + is shown in Fig. 4.
Within this perturbation approach the longitudinal spin
susceptibility reads as:
2
0
( , )( , ) ( / )
( , )
zz
B
Nv
D
ω
χ ω = − µ
ω
qq
q
,
where [37]
2
1 11( ) )(zN q g q= ∑ −
1 2 12 12 21(2 ) ( ) ( )z z z
qg g q J q q− ∑ − ∑ ∑ +
2
11 22 2 22( ) ( ) ( )z z z
qJ q q g q
∑+ ∑ ∑ + ,
and
2
12 21 11 22( ) 1 ( ) 1 ( ) ( ) ( )z z z zD q J q J q J q q = − Σ − Σ − Σ Σ q q q .
(11)
In the one-loop order approximation, after summing up
over the discrete Matsubara frequency analytical expres-
sions for the functions 12 ( )z qΣ and 21( )z qΣ acquire a form
12 21( ) ( )z zq qΣ = Σ =
2 2
1 1 1 11 2
1 2 1 2 1 1
( ) ( )
q -
J J n nb b
N i
− −
− −
ε − ε= +
ε + ε ε + ε ω − ε + ε
∑ p p q p p q
p p p q p q p p qp
2 2 2 2 1 1 2 2
2 2 2 1
( ) ( ) 1 ( ) ( )
q q
n n n n
i i
− −
− −
ε − ε + ε + ε
+ + −
ω + ε − ε ω + ε + ε
p q p p q p
p p q p p q
1 1 2 2
1 2
1 ( ) ( )
q
n n
i
−
−
+ ε + ε −
ω − ε − ε
p p q
p p q
. (12)
The expressions for the irreducible parts 11( )z qΣ and 22 ( )z qΣ
have the similar structure (see Ref. 37). Here within the
long-wave approximation, ( ) 1aq << ,
2
1 ( )D aqε =q , 1 2
0
1 2
2
b bD J
b b
=
−
(13)
is the energy of acoustic, and
2
2 1 2 0( ) ( )b b J D aqε = − +q , (14)
exchange spin-wave modes (in-phase and anti-phase
sublattices magnetization precession, respectively).
As it follows from the analysis of the irreducible parts
( )z
ij qΣ [see, e.g., Eq. (12)], in a ferrimagnet, dynamics of
longitudinal spin components is due to a few virtual pro-
cesses of creation and annihilation of transverse excita-
tions. Namely, the first channel — the first item in the
right-hand-site brace of Eq. (12) — represents the processes
of creation and annihilation of spin-waves with energies
1ε p and 1 ±ε p q which correspond to in-phase precession of
sublattices magnetization. This channel is controlled by the
occupation factor of acoustic magnons, 1 1( )n ε p , which
Fig. 3. The one-order (a) and two-loop orders (b) diagrams for
irreducible parts of the longitudinal spin GFs. Only the diagrams
describing the Kubo susceptibility are shown explicitly. Here and
in Fig. 4 the solid line describes the transverse spin-wave GF
G11(q); the double solid line describes the transverse spin-wave
GF G12(q) or G21(q).
Fig. 4. Graphical representation the equation for the effec-
tive four-point vertex (a) 11,12 1 2 1 2( | ), ,k k k q k qΓ − + and
22,21 1 2 1 2(( , , | )b) k k k q k qΓ − + . Here the dashed line describes
the transverse spin-wave GF G22(q).
1570 Low Temperature Physics/Fizika Nizkikh Temperatur, 2017, v. 43, No. 11
Longitudinal magnetization dynamics in Heisenberg magnets: Spin Green functions approach
makes the spin-waves with p ~ 0 to be dominant ones. A
simple comparison reveals that this channel is a direct ana-
log of the magnetization longitudinal dynamics in a ferro-
magnet found by Izumov et al. [32].
There is also the second channel — the second term in
the right-hand-site of Eq. (12) — with characteristic excita-
tion energy 2 2 ( ) ±ω = ε − εp p qq . It corresponds to a virtual
creation and annihilation of exchange spin-waves with
anti-phase precession of sublattices magnetization. This
channel is also controlled by the related occupation factor
determined through the Bose distribution functions n2(ε2p),
which makes exchange spin-waves with p ~ 0 dominant in
this way of longitudinal spin dynamics.
The last two items in the brace of Eq. (12) represent the
third channel of longitudinal excitations. Namely, there is a
two-spin-wave creation/annihilation process at frequency
ω(q) = ε1p + ε2p ± q which corresponds to creation or anni-
hilation of one acoustic and one exchange transverse modes.
This channel remains in force even in the absence of ther-
mal excitations, i.e., when 1 1( ~) 0n ε p and/or 2 2( ~) 0n ε p .
We note also some important new in comparison with the
case of a ferromagnet feature of the χzz(q,ω). As it follows
already from Eq. (12) [see last two items in Eq. (12)], for a
ferrimagnetic system the uniform longitudinal susceptibil-
ity χzz(0,ω) is nonzero, that reflects the fact that now mag-
netization is not a conservation quantity.
All listed mechanisms of longitudinal spin excitations
remain valid in the high-loops approximation, too. Also, as
in the case of a ferromagnet, there is a strong renormaliza-
tion of the longitudinal spin excitations frequency. As a
result, the energy of longitudinal spin vibration strongly
differs form a simple algebraic sum of acoustic and/or ex-
change transverse modes energy. The analysis reveals [37]
that in the (q,ω) region where the first two channels are
actual, the function ( , )zzχ ωq possesses of two types of re-
sonances. There is a peak which characterizes a preces-
sion-like motion with the frequency res ~ Dq±Ω and
damping ( , ) ~ Dqγ ωq ; both these functions linearly de-
pend on the wave vector (the acoustic branch of longitudi-
nal spin excitations). I.e., in the system the wave-like vi-
brations of spins longitudinal components exist, though
with a strong attenuation. At the same time, there is also a
quasi-diffusive pole at difi± Ω , i.e., there is the quasi-relax-
ation mode connected with diffusion of longitudinal fluc-
tuations, which forms the central peak in the spectral func-
tion 1 Im ( , )zz−ω χ ωq . Thus, in this frequency and wave
vector domains the ferrimagnet spectral function behavior
is similar to that of a ferromagnet [32].
In the (q,ω) region where the third channel (crea-
tion/annihilation of one acoustic and one exchange trans-
verse modes) is actual, the leading part of the longitudinal
spin susceptibility reads
3
01 2
2 2
exc exc1 2
( ) ( , )( , ) ~
2
zz Jb b N
J ib b
− ω
χ ω
±ω+Ω + γq
qq . (15)
The quantity Ωexc = Ωexc(q,T) is the frequency of collec-
tive (exchange type) vibrations of longitudinal magnetiza-
tion components which in this (q,ω) region is described by
the following expression [37]:
2
exc 1 2 0
1 2 0
( , ) ( ) 1 2ln ( )
( ) 2
TT b b J f T q
b b J
π
Ω ≈ − + + +
−
q ,
where
2
0 1 2 0( ) 3( ) 4 ln 2 1
4q
J b b J D Df T
TJ T T
− + ≈ + − +
.
The exchange longitudinal mode is energetically above the
exchange mode of the transverse spin-waves (14) at the
same temperature and wave vector, and linearly depends
on q. The ratio exc exc( , ) / ( , )Tγ Ω Ωq q is approximately
01 2
exc exc
1 2
( , ) / ( , ) ~
Jb bT
b b T
γ Ω Ω
−
q q
and at low temperatures T < J0 is large enough. But at
1 1~ CT T b S<< , 2 2b S<< and exc exc( , ) / ( , ) 1Tγ Ω Ω <<q q .
Thus, in a ferrimagnet wave-like excitations of longitudi-
nal components of magnetization exist, though with a
strong attenuation at low temperatures.
These results are applicable for a system with antiparal-
lel sublattice alignment. As is known, the ground state of
a ferrimagnet can be changed by strong enough magnetic
field and by temperature [38]. If external magnetic field
reaches some critical value Hsf, a ferrimagnet undergoes a
spin-flop transition from antiparallel sublattice order to non-
collinear one (the spin-flop state). The canting angles be-
tween the magnetic sublattices can be further controlled by
the magnetic field. Increasing the magnetic field beyond Hsf
both sublattice magnetizations will tilt towards the external
magnetic field. When the saturation field Hs has to over-
come the inter-sublattice exchange field both magnetiza-
tions align collinearly along the external field (the spin-flip
state). In addition, if the larger (at T = 0) sublattice magnet-
ization decreases faster than the smaller one with growing
temperature, then at certain temperature the total magneti-
zation can become equal to zero. That is the so-called mag-
netic compensation point TM. If the sublattices g factors
differ, g1 ≠ g2, the mechanical (angular momentum) com-
pensation point TL does not coincide with the magnetic
compensation point TM.
The high-field ultrafast laser spectroscopy has just only
started to investigate of such type noncollinear ferrimag-
netic structures. Recent experimental investigations [39]
reveal a strong impact of the magnetic field on the laser-
induced magnetization dynamics in the ferrimagnetic alloy
GdFeCo. Upon reaching the spin-flop field, the laser-in-
duced magnetization excitations change in amplitude and
timescale. The magnetization dynamics in this high-field
Low Temperature Physics/Fizika Nizkikh Temperatur, 2017, v. 43, No. 11 1571
V.N. Krivoruchko
regime is shown to be gradually suppressed in comparison
with the antiparallel sublattice alignment.
Theoretically, features of transverse magnetization dyna-
mics for a two-sublattice ferrimagnet near the compensation
points TM and TL have been recently discussed in Ref. 40.
Yet, the longitudinal energy spectrum and susceptibility of
a ferrimagnet near the compensation points, in a spin-flop
phase, and for the parallel sublattice alignment state, to our
knowledge, has not been still analyzed in details even for
the equilibrium state of the system.
5. Antiferromagnet
Antiferromagnetic compounds are also promising can-
didates for extremely rapid spin manipulation and switch-
ing since, unlike ferromagnets, for this type of magnetic
order there is no requirement of the angular momentum
conservation. Even in the equilibrium state the dynamical
properties of antiferromagnet essentially differ from those
of ferro- and ferrimagnets. Existing experimental results
[41–44] point that the ultrafast response of an antiferro-
magnetic system to intense optical excitation is also dis-
tinctly different from that of a ferromagnetic system.
It should be noted, however, that up to our knowledge,
the longitudinal dynamics for an antiferromagnet has not
been still investigated theoretically in such details as for a
ferromagnet or a ferrimagnet. Here a few characteristic
features should be taken into account: a total macroscopic
magnetization is zero, the anisotropy plays an important
role, and interaction of spin-waves in the longitudinal dy-
namics must be taken into account. In general, as in the case
of a ferrimagnet, there are two processes that determine the
longitudinal energy spectrum and susceptibility function:
(i) absorption of one magnon and excitation of another
one, and (ii) two-magnon excitation (absorption) [45–48].
Taking into account that the overall scattering weight of
the simultaneous creation/annihilation of two magnons re-
lative to one-magnon processes is related to the relative
strength of the zero-point longitudinal quantum fluctua-
tions in the ground state, which reduces the amount of the
ordered spin moment compared to the full spin value S, it
is expected the two-magnon channel has to be the strongest
for antiferromagnets.
The dynamics of the longitudinal spin components in
the isotropic quasi-2D antiferromagnet near and below the
critical temperature T < TN (TN is the Néel temperature)
has been investigated thoroughly theoretically in Ref. 48
using renormalization group methods. It was predicted that
the longitudinal spin components behave rather differently
from transverse spin fluctuations, and while crossing over
from the hydrodynamical to the critical regime, the coeffi-
cient of a spin diffusion becomes divergent. The authors
supposed that such a behavior for T < TN could be de-
scribed by taking into account all loop diagrams built from
transverse spin GFs. Regarding the ferro- and ferrimagnetic
case, it is reasonable to expect that summation of just this
kind of diagrams is of vital importance to properly describe
the longitudinal spin dynamics. Yet, as already mentioned,
an in-depth analysis of the longitudinal spin dynamics in
antiferromagnetic compounds within the microscopic spin
GF approach is still not done.
Longitudinal spin fluctuations in the antiferromagnets
experimentally was extensively studied by polarized neutron
scattering [46,47,49]. In experiments Ref. 46 using polariza-
tion analysis, the spectra of transverse and longitudinal
magnetic vibrations in the anisotropic antiferromagnet MnF2
have been separated. It was found that indeed while trans-
verse modes are related to single-magnon scattering, the
longitudinal part is essentially due to two-magnon scatter-
ing. The dynamic magnetic response due to the two-mag-
non creation or annihilation is separated by a gap centered
near the spin-wave frequency from the central peak corre-
sponding to neutron-magnon scattering (creation of one
magnon and annihilation of another). The longitudinal en-
ergy spectrum extends to about twice the frequency of the
zone boundary modes. The observed longitudinal spectra
are in qualitative agreement with the theory for two-mag-
non processes. The neutron scattering cross section below
the Néel temperature in RbMnF3 has been studied in
Ref. 49 with the aid of neutron spin polarization analysis.
In addition to the spin-wave scattering, a small central
component was observed and found to be longitudinal in
character. This longitudinal scattering is quasi-elastic, with
an intensity that decreases with increasing wave vector and
with decreasing temperature below TN.
6. Concluding remarks
Our understanding of spin dynamics in solid-state sys-
tems is largely based on experimental and theoretical re-
sults for the equilibrium states, when the timescale of dy-
namical process is determined by the resonance frequency
of the underlying excitation. Laser-induced femtosecond
magnetism opens a new frontier for magnetic dynamics by
exploring new regimes of the magnetic resonances in out-
of-equilibrium state. But probing such a fast magnetization
change is a big challenge for experiment and theory, and
complete microscopic understanding of magnetization dy-
namics that involves correlated interactions of spins, elec-
trons, photons, and phonons on femtosecond timescales
has yet to be developed. Apart from the complexity of the
problem itself, among the reasons for the lack of funda-
mental understanding of ultrafast magnetism at the micro-
scopic scale is the absence of the answer to the question
how intrinsic magnetic properties can control the ultrafast
dynamics of spin subsystems in magnetic materials. In
other words, can ultrafast magnetization dynamics be pri-
marily determined by the intrinsic magnetic properties of
the material rather than the result of laser-induced ultrafast
transient changes in the material, e.g., hot electrons or
1572 Low Temperature Physics/Fizika Nizkikh Temperatur, 2017, v. 43, No. 11
Longitudinal magnetization dynamics in Heisenberg magnets: Spin Green functions approach
phonons? Naturally, the answer to this question requires
understanding the role of different magnetic interactions in
the ultrafast demagnetization process, and, first of all, the
role of such fundamental quantum mechanical magnetic
interaction as the exchange interaction. That is why, as was
noted in Introduction, the author consciously restricted
himself by consideration of the basic microscopic models
of magneto-dielectrics — the Heisenberg model — and has
not discussed roles of other subsystems (electrons, pho-
nons, etc.) in these processes. (The reader interested in
these questions is referred to reports [50–56] and refer-
ences therein.)
At present, it is accepted that within the Heisenberg
model approach, the dynamics of longitudinal spin compo-
nents implies a different physical picture than transverse
spin oscillations and reflects both the specifics of a mag-
netic system dynamic variables — the spin operators —
and the character of the magnetic order. It is ascertained
that the longitudinal magnetization dynamics realizes, in
general case, through three different channels: (i) a process
of creation and annihilation of in-phase sublattice magneti-
zation precessions, (ii) a process of creation and annihila-
tion of out-of-phase sublattice magnetization precessions,
and (iii) by a creation/annihilation of one in-phase and one
out-of-phase sublattice magnetization excitations. The first
two channels are controlled by the occupation factors de-
termined by the Bose distribution function and provide
zero contribution into the uniform susceptibility. The third
channel remains in force even in the absence of thermal
spin-waves excitations and makes finite the uniform sus-
ceptibility χzz(0,ω). Collective vibrations of magnetization
longitudinal components energetically are above the trans-
verse spin-waves frequency at the same temperature and
wave vector. These results have direct relation to a final
phase of the equilibration processes when after a nanosec-
ond evolution magnetization reproduces longitudinal dy-
namics of the equilibrium system.
Acknowledgments
The author dedicates this article to the memory of
Anatolii Illarionovich Zvyagin; in 1984–1990 I had the
opportunity to participate in the work of scientific group
led by A.I. Zvyagin on searching and investigating new
types of magnetic structures and resonances in low-dimen-
sional multiple sublattice antiferromagnets. The author is
grateful for valuable discussions with V.G. Bar’yakhtar
and B.A. Ivanov; I also extend appreciation to M. Belo-
golovskii for reading the manuscript and critical remarks.
This work is partly supported by the Ukrainian State Fund
for Fundamental Research (project No F71/59-2017 “Mul-
tifunctional Photonic Structures”) and the European Union’s
Horizon 2020 research and innovation program under
Marie Skłodowska-Curie (project “MagIC”, Grant Agree-
ment No. 644348).
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1. Introduction
2. Longitudinal spin dynamics. Spin Green functions approach
3. Ferromagnet
4. Ferrimagnet
5. Antiferromagnet
6. Concluding remarks
Acknowledgments
References
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