Abanov, Ar

Kashuba, A

Pokrovsky, VL

Physical Review Letters

English

Texas A&M University

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6 Spin Diffusion in 2D XY Ferromagnet with
Dipolar Interaction
A.Kashubaab, Ar.Abanovb and V.L.Pokrovskyab
March 5, 2018
aLandau Institute for Theoretical Physics, Kosygin 2, Moscow 119740,
Russia
bDepartment of Physics, Texas A&M University, College Station, Texas 77843-
4242, USA
Abstract
In the ordered phase of 2D XY ferromagnet the dipole force in-
duce a strong interaction between spin-waves in the long wavelength
limit. This interaction leads to the transformation of the spin-wave
excitation into a new soft-mode excitation in an intermediate range of
wavelengths limited in magnitude and direction; and into an anoma-
lous anisotropic diffusion mode excitation at long wavelengths. The
dissipation of a spin-wave at short wavelengths is found to be highly
anisotropic.
PACS Numbers: 75.10Hk; 75.30Ds; 75.40Gb
1
The conventional condensed matter theory deals with elementary exci-
tations and their interactions. The excitations such as electrons, phonons,
spin-waves etc. have a propagating, wave-like nature. The momentum p
and the energy ω of a single excitation are related by the dispersion relation
(spectrum): ω = ǫ(p). A weak interaction changes slightly the spectrum and
leads to a finite life-time of the excitations [1]. Effect of a strong interac-
tion is not so universal. Migdal [2] has shown that a strong electron-phonon
interaction renormalizes substantially electron velocities and, if it exceeds a
critical value, destroys the electron quasi-excitations. Recently a strong in-
teraction of electrons in 2 dimensions with gauge fields has been shown to
result in non-Fermi-liquid behavior [3].
In the long wavelength (hydrodynamic) limit quasi-excitations turn into
classical modes, such as sound or spin-wave. Not only the propagating waves,
but also particle, heat and spin diffusion can be considered as hydrodynamic
modes. The interaction between these modes has been shown to be substan-
tial in the critical region [4]; in the dynamics of liquid crystals [5], [6]; and
in the dynamics of a Charge Density Wave interacting with impurities [7].
In all these systems the interaction leads to a drastic reconstruction of the
dispersion relation.
In this Letter we solve the experimentally feasible magnetic model, in
which a strong interaction between spin-waves leads to the replacement of
the propagating spin-wave by a diffusion mode and to appearance of a new
soft-mode in a range of momentum. This model is the two-dimensional XY
ferromagnet with the dipolar interactions between spins.
A spin diffusion mode appears naturally in the paramagnetic phase and
in the vicinity of the Curie point [8]. We consider a low temperature ordered
phase, where no diffusion is expected but rather a propagating and weakly
dissipating spin-wave mode. In a 3D ferromagnet the exchange interaction
between spin-waves vanishes in the long wavelength limit [9]. The dipole
force generates three spin-wave processes [8, 10] and violates the total spin
conservation law. This interaction is dominant in the dissipation of a spin-
wave via decaying it into the two spin-waves or via merging it with other
spin-wave. But dissipation is weak.
In a 2D XY ferromagnet at low temperatures the dipolar interaction is
relevant in the long wavelength limit, even despite of the low density of spin-
waves. It was shown by one of the authors [11] that dipolar force induces an
anomalous anisotropic scaling of spin-spin correlations in the ordered phase.
2
In this article we find an analogous dynamical scaling.
The dipolar force stabilizes the ferromagnetic long-range order [12], sup-
pressing strong XY thermal fluctuation. Therefore, we represent the unit
vector field of magnetization S by the two spin-wave fields – in-plane φ(x, t)
and out-of-plane π(x, t) = Sz:
S =
(
−
√
1− π2 sin φ;
√
1− π2 cosφ; π
)
(1)
where both π and φ are small. The field π(t,x) is canonically conjugated to
the field φ(t,x).
The Hamiltonian of a 2D XY ferromagnet contains three terms: the ex-
change and the anisotropy energies:
HEA =
∑
ω p
(Jp2|φωp|2 + λ|πωp|2)/2,
as well as the dipolar force term (see e.g. [11]):
H = HEA + g
∑
ω p
∣∣∣∣pxφωp + py (φ2/2)ωp
∣∣∣∣2 /2|p|, (2)
with the corresponding couplings being – the exchange constant J , the anisotropy
λ and the dipole constant g [13]; The Hamiltonian (2) is written in terms
of the Fourier transformed fields – φωp and πωp, the abbreviation - (φ
2/2)
pω
- denotes the Fourier transform of φ2(x, t)/2. The anisotropy is assumed
to be weak: λ ≪ J . Therefore the spin S averaged over scales larger than√
J/λ turns into the plane. The bare spin-wave spectrum extracted from the
quadratic part of the Hamiltonian (2) reads:
ǫ2(p) = c2
(
p2 + p0p
2
x/p
)
, (3)
where p = |p|, the characteristic wave-vector p0 = g/J and the spin-wave
velocity c =
√
Jλ. In the region p0 ≪ p≪
√
λ/J spin-waves have the linear,
acoustic-like spectrum. We call this spherical shell of momentum the A-shell.
The non-linear part of the Hamiltonian (2) Hint contains the three-leg
and four-leg vertices. In particular the three-leg vertex is:
f(p1,p2,p3) =
3∑
i=1
pixpiy/ | pi | (4)
3
Since the sum of momenta entering the vertex is zero it depends on two
momenta. Further we denote it as f(p1,p2).
The dynamical equation associated with the Hamiltonian (2) reads:
(
ω2
λ
− ǫ
2(p)
λ
+ i
ω
Γ0
)
φωp =
δHint
δφ−ω−p
+ ηωp. (5)
The thermal noise ηωp and the bare dissipation coefficient Γ
−1
0 generate the
stochastic dynamics in (5). The noise-noise corelation function is related to
the bare dissipation coefficient: 〈|ηωp|2〉 = 2T/Γ0. It vanishes at T = 0. We
do not consider external sources of dissipation, spin-waves dissipate due to
interaction generated by dipole force. Hence, we set Γ−10 = +0, and look for
the generated dissipation: Γ−1(ω,p).
The Green function G(ω,p) is defined as the linear response of the mag-
net to an external magnetic field with the same frequency and wave-vector.
The l.-h.-side of Eq.5 represents the inverse of the bare Green function
λ−1G−10 (ω,p). According to the Fluctuation-Dissipation Theorem, the imag-
inary part of the Green function multiplied by the factor T/ω is the spin-spin
correlation function D(ω,p). For convenience we use the dissipation func-
tion defined as: b(ω,p) = λ/Γ(ω,p). The self-energy term Σ(ω,p) equals
G−10 (ω,p) − G−1(ω,p) by definition. We notify the real and the imaginary
part of the self-energy term as: Σ = a2(ω,p) − iωb(ω,p). Thus, the Green
function reads:
G−1(ω,p) = ω2 − ǫ2(p)− a2(ω,p) + iωb(ω,p), (6)
whereas the spin-spin correlation function reads:
D(ω,p) =
b(ω,p)
[ω2 − ǫ2(p)− a2(ω,p)]2 + ω2b2(ω,p) (7)
(we refer the factor T to vertices). We use the reduced temperature t =
T/4πJ as a small and L = log(
√
Jλ/g) as a large parameters (the latter
means large A-shell).
We employ the standard, so-called Janssen-De-Dominicis Functional method
[14] to account for the non-linear terms in the stochastic Langevin equation
(5). This method generates a diagrammatic expansion in powers of the bare
vertices, associated with Hint. The main contribution to the self-energy is
4
given by the one-loop diagram shown in Fig.1a, 1b. Our theory is valid only
if the temperature is small:
t log(
√
Jλ/g) = tL≪ 1.
Under this condition the two-loop corrections are small and the diagram
Fig.1b contributes to a negligible change of the spectrum (3) [15]. Such ne-
glecting of the two-loop diagrams (vertex correction) was a major assumption
in the so-called mode-coupling methods [8]. Later we proove this assumption.
Thus, the Dyson equation for our problem is as follows:
Σ(Ω,q) = 2g2λ3T
∫
d2p
(2π)2
∫
dω
2π
f 2(p,q)
D(ω,p)G(ω + Ω,p+ q), (8)
The functions b(ω,p) and a(ω,p) are even in both arguments [1]. The imag-
inary part of the self-energy is odd in frequency Ω. Hence, the equation for
the dissipation function reads:
b(Ω,q) = g2λ3T
∫ d2p
(2π)2
∫ dω
2π
f 2(p,q)
D(ω,p)D(ω + Ω,p+ q). (9)
The integrand in (9) is positive. Thus, the main contribution to b(Ω,q)
comes from the region, where poles of the two D-functions coincide. The
function D(ω,p) has poles at: ω ≈ ±ǫ(p), in the A-shell [16]. Following the
terminology of the field theory we call the surface ω2 = ǫ2(p) the mass-shell.
The dissipation in the A-shell is small and the D-function can be represented
as a sum of δ-functions:
D(ω,p) ≈∑
±
π
2ǫ2(p)
δ(∆ω±), (10)
where ∆ω± = ω ± ǫ(p) measures the deviation from the mass-shell. After
integrating ω out from the Eq.9 with theD-functions from (10) we recover the
Fermi Golden Rule for the probability of the spin-wave decay and scattering
processes.
Looking for the long wavelength quasi-excitations we need the self-energy
at very small momenta q ≪ p0, which we denote as Σ0. We expect quasi-
excitations to be soft: Ω ≪ cq, and we restrict the wave-vector q of quasi-
excitation to be directed along the magnetization: |qx| ≪ q. The essential
5
contribution to the integral in Eq.8 comes from the internal momentum p
being in the A-shell and the internal frequency ω = ǫ(p). Taking the integral
over ω, with the D-function from Eq.(10), we find [16]:
Σ0 =
c2p2
0
t
4π
∫
c4p3dp
ǫ4(p)
Ω sin2(2ψ)dψ
Ω− cq cosψ + ib1 , (11)
where b1 is the dissipation function b(ω,p) of a spin-wave inside the A-shell
and ψ is the direction of the internal spin-wave: sinψ = px/p. Note, that
we added a Ω-independent contribution from the diagram Fig.1.b to the self-
energy 11. ReΣ0 vanishes in the static limit Ω = 0.
If cq ≫ b1 we make the integral over ψ in Eq.11 to find:
Σ0(χ) = c
2p2
0
tL cos2 χ exp(−2iχ), (12)
where χ, defined by the equation cosχ = Ω/cq, measures the deviation
from the mass shell. If q is so small that cq ≪ b1 the Eq.11 implies the
q-independent dissipation constant:
b0 = c
2p2
0
tL
∫
dψ
4π
sin2(2ψ)
b1(ψ)
. (13)
In this calculation we have used the fact that the dissipation of a spin-wave
in the A-shell b1 depends only on the angle ψ between the direction of mag-
netization and the spin-wave wave-vector p, which we prove below.
Anyway, we need b1(ψ). An unusual feature of our theory is that the
dissipation process in the A-shell is mediated by an off-mass-shell, virtual
spin-wave. Indeed, the Eq.3 does not allow for decay or merging processes.
Alternatively, the dissipation of a spin-wave in the A-shell propagating in the
direction specified with the angle ψ (sinψ = qx/q) is mediated by an internal
virtual spin-wave in (9), with a momentum p ≪ p0 ≪ q and a frequency
ω < cp. The integration over ω with one of the D-functions in (9) taken in
the form (10), leads to a following equation:
b1 =
c2p2
0
tf 2(0,q)
8πq2
∫
d2pD(ǫ(p+ q)− ǫ(q),p) (14)
Since ω = ǫ(p+ q)− ǫ(q) we conclude that ω = cp cosΦ, where Φ = ψ−ϕ is
the angle between the vectors q and p. Invoking the definition of the angle
6
χ for virtual spin-wave, we find that χ = Φ. Now, we look onto Eq.14 in
more details:
b1(ψ) =
c2p2
0
t
2π
Im
∫
sin2 ψ cosψ dpdϕ
p2 sin2 ψ + p0pϕ2 + Σ0(ψ)/c2
. (15)
Note, that ϕ2 ∼ p/p0 ≪ 1 in (15). Thus, χ = ψ. In other words, the
dissipation of a short wavelength spin-wave propagating in the direction ψ is
determined by the scattering on the long wavelength virtual spin-wave, which
lies on the specific distance off the mass-shell: ω/cp = cosψ. The integration
over p in (15) is confined towards the crossover region: p ∼ pc = p0
√
tL.
Substituting Σ0(ψ) from Eq.12 into Eq.15 we find the anisotropic dissi-
pation of a spin-wave mode in the A-shell:
b1(ψ) = β1t
3/4cp0
sin3/2(2ψ) sin(ψ/2)
L1/4 cosψ
, (16)
where the direction of the spin-wave is limited to the fundamental quadrant:
0 < ψ < π/2. We found: β1 = Γ
2(1/4)/4
√
2π ≈ 1.31.
Let us return to very low momenta p≪ b1/c. Plugging Eq.16 into Eq.13
one finds: b0 = β0cp0t
1/4L5/4, where β0 ≈ 1.24 was found numerically. The
condition cpDM ∼ b1 defines the crossover wave-vector: pDM ∼ p0t3/4/L1/4,
between the self-energies Eq.12 and Eq.13.
The dissipation functions (12,13 and 16) conclude the self-consistent so-
lution of the Dyson equation (8, 9).
We verify that the two-loop correction (see Fig.1c) is negligible. Note,
that the main contribution to the diagram Fig.1c comes, if the two internal
momenta p1 and p2 are restricted to theA-shell. Inside theA-shell the Green
and the D-functions live on the mass-shell. But, the prohibition of the three
spin-wave processes leaves the Green function G(ω1 − ω2,p1 − p2) to be off
the mass-shell. The Green function off the mass-shell is small in temperature.
A simple counting shows, then, that the two-loop dissipation function is
b′
0
= b0(1 + t
1/4O(t)). The function b′
1
(ψ, q) − b1(ψ), which represents the
two-loop corrections for b1, is small in t
1/4, and is also small in the ratio
p0/q.
Having the explicit expression for the self-energy we can analyse the dis-
persion relation – ω2 = ǫ2(p)+Σ(ω,p) – in the range of small ω and p. New
results are expected for the region p < pc = p0
√
tL in which Σ0 becomes
7
comparable with ǫ2(p). In a range of momentum pDM ≪ p≪ pc and angles
ψ ≪
√
p0tL/p we find a new propagating soft mode with the dispersion:
ω = cp(p2 + p0pψ
2)1/2/p0
√
tL (17)
The dissipation of the soft mode grows to the boundary of the region and
becomes of the order of its energy at ψ ∼
√
p0tL/p or p ∼ pDM . There is
no soft mode beyond the indicated range. The spin-wave mode persists at
p > p0
√
tL. In a range p ≪ pDM and small angles a new diffusion mode
occurs with the dispersion:
ω = −iǫ2(p)t−1/4L5/4/β0cp0 (18)
The angular range of the diffusion mode increases with decreasing p and
captures the entire circle at p < p0tL.
At even smaller wavelengths p < pA ≪ pDM the interaction between dif-
fusion modes should be taken into account ( pA is the anomalous diffusion
set up wave-vector [17]). The problem can be solved by the renormalization
group method [17]. The growing interaction, although leaving invariant the
diffusive nature of the spin propagation, changes the dispersion. For the prop-
agation along the spontaneous magnetization it is: ω ∝ −ip47/27, whereas for
the propagation in the perpendicular direction it is: ω ∝ −ip47/36. This is a
dynamic analog of the non-Gaussian fixed point found in [11].
In conclusion, we discuss the experimental feasibility of the above consid-
ered effects in the epitaxial magnetic films. The main difficulty is that all of
them are confined to rather long wavelengths. Therefore even weak in-plane
anisotropy can suppress them. One need to use substrates with the six-fold
symmetry axis. The six-fold anisotropy is much weaker than the tetragonal
one. Moreover, it was predicted in [18] that there exists a temperature inter-
val in which the hexagonal anisotropy vanishes at large distances. A proper
substrate is, for example, a (111) face of Cu, Au etc. Unfortunately the
epitaxial growth of ferromagnetic films on this faces has not been realized so
far. In a recent work [19] a growth of a ultrathin Ru/C(1000) film has been
reported. It is the in-plane ferromagnet. We hope that further sophistication
of the experimental technics will allow to observe the predicted dynamical
behavior.
We are indebted to M.V. Feigelman, E.I. Kats and V.V. Lebedev for
useful discussions and indicating some references.
8
References
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(1958)].
[3] D.V.Khveshchenko and P.C.E.Stamp, Phys.Rev.Lett. 71, 2118 (1993);
B.L.Altshuler, L.B.Ioffe, and A.Millis, Phys. Rev. B 51, (1994); A.Stern
and B.I.Halperin, Phys.Rev.B, 52, 5890 (1995);
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[7] G.Gru¨ner, Rev.Mod.Phys., 60,1129 (1988).
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(1987).
[11] A.Kashuba, Phys.Rev.Lett., 73, 2264 (1994)
[12] S.V.Maleev, ZhETF 71, 2375 (1976);[Soviet Physics JETP 43, 1240
(1976)]; V.L.Pokrovsky and M.V.Feigelman, ZhETF 72, 557 (1977);
[Soviet Physics JETP 45, 291 (1977)].
[13] g = 2π(γµBSσ)
2, where γ is the Lande-factor of magnetic ions, S is its
spin, µB is the Bohr magneton and σ is the area density of magnetic
ions in the film.
9
[14] P.C.Martin, E.D.Siggia and H.A.Rose, Phys.Rev.A, 8, 423 (1973);
C.De Dominicis and L.Peliti, Phys.Rev.B, 18, 353 (1978); H.K.Janssen,
Z.Phys.B, 23, 372 (1976).
[15] An important property of the Langevin equation is that a diagram in
statics equals to the corresponding diagram in dynamics at the coincid-
ing external times (such as Fig.1b). See e.g. J. Zinn-Justin, Quantum
Field Theory and Critical Phenomena (Oxford, 1993). But the static
corrections to the Green function are small [11], [17].
[16] The a-function is negligible in the A-shell. If one assumes, that
a2(ω,p) ≪ ǫ2(p) inside the A-shell, then from the Eq.8 one finds:
a2(ǫ(q),q) ∼ tLc2p0√p0q, justifying the assumption.
[17] Ar.Abanov, A.Kashuba, and V.L.Pokrovsky, to be published.
[18] V.L. Pokrovsky and G.V. Uimin, Phys.Lett. 45A, 467, 1973;J.V.Jose et
al., Phys.Rev.B, 16, 1217 (1977).
[19] R. Pfandzelter, G. Steierl, and C.Rau, Phys. Rev. Lett. 74, 3467 (1995).
Figure Caption
• FIG.1. a) Main one-loop self-energy diagram; b) One-loop four-leg
vertex diagram; c) Two-loop self-energy diagram. Momenta of internal
lines are indicated.
10
g,p)ωG(
D(ω+Ω,p+q)
g
p
2
p
1
p-p
1 2
g
D


Spin diffusion in 2D XY ferromagnet with dipolar interaction

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Spin diffusion in 2D XY ferromagnet with dipolar interaction

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