1
Magnon BEC at finite momentum
From textbook knowledge towards BEC of magnons
Andreas Kreisel, Francesca Sauli,Johanes Hick, Lorenz Bartosch,Peter KopietzInstitut für Theoretische PhysikGoethe Universität FrankfurtGermany
SFB TRR 49
European Physical Journal B 71, 59 (2009)Rev. Sci. Instrum. 81, 073902 (2010)arXiv:1007.3200
Christian Sandweg, Matthias Jungfleisch,Vitaliy Vasyucka, Alexander Serga, Peter
Clausen, Helmut Schultheiss,Burkard HillebrandsFachbereich Physik
Technische Universität KaiserslauternGermany
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1. Introduction:Spin-wave theory
� Heisenberg model
� determine ordered classical groundstate ferromagnet
classical groundstate=quantum groundstate
anti-ferromagnet(2 sublattices, Néel groundstate)
triangular anti-ferromagnet(3 sublattices, frustration)Chernychev, Zhitomirsky ('09)Veillette et al. ('05)AK, Kopietz (in preparation)
H =X
ij
Jij ~Si ¢ ~Sj
AK, Hasselmann, Kopietz, '07AK, Sauli, Hasselmann, Kopietz, '08
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1. Spin wave theory
� expand in terms of bosons (1/S expansion), Holstein-Primakoff transformation
� determine properties of resulting interacting theory of bosons
0
1
2
−1
−2
Ŝz = S ¡ n̂ n̂ = b̂yb̂ hb̂; b̂y
i= 1
r
1¡ n̂2S
= 1¡ n̂4S
+O( 1S2)
Ŝ+ =p2S
r
1¡ n̂2Sb̂
Ŝ¡ =p2Sb̂y
r
1¡ n̂2S
H=X
ij
Jij ~Si ¢~Sj
H =X
~k
E~kby~kb~k +
X
~k1;~k2;~k3
¡3(~k1;~k2; ~k3)by~k1b~k2b~k3 +
X
1;2;3;4
¡4(1; 2; 3; 4)by1by2b3b4 + : : :
Holstein, Primakoff, Phys. Rev. 58, 1098 (1940)
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1. Spin wave theory:General results
� ferromagnet
quadratic excitation spectrum vanishing interaction vertices
� antiferromagnet linear spectrum
(Goldstone mode) two modes in magnetic field
(2 sublattices) divergent interaction vertices
¡4 »s
j~k1jj~k2jj~k3jj~k4j
Ã
1§~k1 ¢ ~k2j~k1jj~k2j
!
¡4 » ¡(~k1 ¢ ~k2 + ~k3 ¢ ~k4)
Hasselmann, Kopietz ('06)
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2.1 Spin-wave theory for thin film ferromagnets
� Motivation: Experiments on YIG Crystal structure:
space group: Ia3dY: 24(c) whiteFe: 24(d) greenFe: 16(a) brownO: 96(h) red
low spin wave damping good experimental control
Gilleo et al. 58
Magnetic system:40 magnetic ions in elementary cell40 magnetic bands
Elastic system:160 atoms inelementary cell3x160 phonon bands
103
104
105
2.5
3
3.5
4
4.5
Parametric pumping of magnons at high k-vectors creates magnetic excitations
Observation of the occupation number using microwave antennas or Brillouin Light Scattering (BLS)Sandweg, et al., Rev. Sci. Instrum. 81, 073902 (2010)
BEC of magnons at room temperature!Demokritov et al. Nature 443, 430 (2006)
Question:Time evolution of magnons:Non-equilibrium physics of interacting quasiparticles
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2.1 Simplifications to relevant physical properties
crystal structure of YIG microscopic Hamiltonian
quantum spin S ferromagnet
dipole-dipoleinteractionsZeeman term
Ĥmag=¡1
2
X
ij
JijS i ¢ Sj ¡ hX
i
Szi
¡ 12
X
i
X
j 6=i
¹2
jrij j3[3(S i ¢ r̂ij)(Sj ¢ r̂ij)¡ S i ¢ Sj ]
AK, Sauli, Bartosch, Kopietz ('09)
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2.1 Linear Spin Wave Theory
� classical groundstate for stripe geometry� Holstein Primakoff transformation (bosons)
� partial Fourier transformation (quasi 2D)
dipolar tensor
Ĥ2 =X
ij
·Aijb
yibj +
Bij
2
³bibj + b
yibyj
´¸
Aij = ±ijh+ S(±ijX
n
Jin ¡ Jij) + S"
±ijX
n
Dzzin ¡Dxxij +D
yyij
2
#
;
Bij = ¡S
2[Dxxij ¡ 2iDxyij ¡D
yyij ]
Filho Costa et al. Sol. State Comm. 108, 439 (1998)
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2.1 Linear Spin-wave theory: Numerical approach
102
103
104
105
106
4
4.5
5
5.5
ky [cm
−1]
Ek
y
[G
Hz] Θk = 90◦
E0
Es
1) numerical diagonalization of 2Nx2N matrix
E0 =ph(h+ 4¼¹Ms)
Es = h+ 2¼¹Ms
hybridization: surface mode
2) evaluation of dipol sums(Ewald summation technique)
d = 400a ¼ 0:5¹m N = 400
minimum for BEC
Demokritov et al. 06
Bartosch et al. 06
He = 700 Oe
H2 =
ÃA ~k B ~k
¡B T ~k ¡A ~k
!
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2.1 Linear spin-wave theory: Analytical approach
� dispersion via Bogoliubov transformation
� no dipolar interaction:
� uniform mode approximation
� lowest eigenmode approximation
¢ = 4¼¹MS
¢ = 0
Ĥ =X
~k
hA~kb
y~kb~k +
B~k2b~kb~k +
B¤~k2by~kby~k
i
E~k =
q[h+ ½ex~k2 +¢(1¡ f~k) sin
2£~k][h+ ½ex~k2 +¢f~k]
E~k = h+ ½ex~k2
f~k =1¡ e¡j~kjd
j~kjd
f~k = 1¡j~kdjj~kdj3+j~kdj¼2+2¼2(1+e¡j~kdj)
(~k2d2 + ¼2)2
compare: Kalinikos et al. 86Tupitsyn et al. 08
AK, Sauli, Bartosch, Kopietz ('09)
10
102
103
104
105
106
2.5
3
3.5
4
4.5
5
5.5
kz [cm
−1]
Ek
z
[G
Hz]
Θk = 0◦
E0
2.1 Comparison
blue: analytical (uniform)green: analytical (eigenmode)black: numerical
small deviations
ferromagnetic resonance
E~k =
q[h+ ½ex~k2 +¢(1¡ f~k) sin
2£~k][h+ ½ex~k2 +¢f~k]
d = 400a ¼ 0:5¹mHe = 700 Oe
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2.2 Condensate in YIG:BEC at finite momentum
� Hamiltonian
� new features for YIG system condensate at finite wave-vectors
possible 2 condensates
explicitly symmetry breaking term
parallel pumping
H2 =X
~k
²~kby~kb~k +
1
2
X¡°byby + °¤bb
¢
Ák = ±k;kminÁ0
²~k = ²¡~k
Ák = ±k;kminÁ+0 + ±k;¡kminÁ
¡0
103
104
105
2.5
3
3.5
4
4.5
−1
−0.5
0
0.5
1
−1
−0.5
0
0.5
1
−0.04
−0.03
−0.02
−0.01
0
Napoleons hat potential
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2.2 BEC at finite momentum
� Euclidean action
� Bogoliubov shift� Gross-Pitaevskii equation from stationary
point
� two component BEC does not solve GPE
S[©] = S2[©]+ S3[©]+ S3[©]
S2[©] =1
2
Z ¯
0
d¿X
k
(©¹a¡k ;©a¡k)
µ@¿ + ²k ¡ ¹ °k
°k ¡@¿ + ²k ¡ ¹
¶Ã©ak©¹ak
!
Á¾k = ±k;qþn + ±k;¡qÃ
¾n
©~k(¿) = Á~k + ±©~k(¿)
0 =±S[©]
±©¾k (¿)
¯̄¯̄©=Á
= (²k ¡ ¹)Á¹¾¡k + °kÁ¾¡k
+1
2pN
X
k+k1+k2=0
X
¾1¾2
¡3(k¾;k1¾1;k2¾2)Á¾1k1Á¾2k2
+1
3!N
X
k+k1+k2+k3=0
X
¾1¾2¾3
¡4(k¾;k1¾1;k2¾2;k3¾3)Á¾1k1Á¾2k2Á
¾3k3
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2.2 BEC at finite momentum
� interactions provoke condensation at integer multiples of
� discrete Gross-Pitaevskii equationfor Fourier components
� condensate density
~kmin Á¾k =pN
1X
n=¡1
±k;nqþn
½(r) = jÁa(r)j2
= 4X
n
jÃnj2 cos2(nq ¢ r)
¡(²nq ¡ ¹)ù¾n ¡ °nþn =1
2
X
n1n2
X
¾1¾2
±n;n1+n2V¾¾1¾2nn1n2
þ1n1þ2n2
+1
3!
X
n1n2n3
X
¾1¾2¾3
±n;n1+n2+n3U¾¾1¾2¾3nn1n2n3
þ1n1þ2n2þ3n3 :
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2.2 Interaction Vertices for YIG
� no symmetry
0 0.02 0.04 0.06 0.08 0.1
−0.4
−0.2
0
0.2
0.4
0 0.002 0.004 0.006 0.008 0.01−0.2
−0.1
0
0.1
0.2
0.3
0.4
10 20 30 40 50 60
0
0.1
0.2
0.3
0.4
finite size effects¡ »Ms Rezende 09ferromagnetic magnons k=kmin¡ » ¡Jk2
F. Sauli (in preparation)U(1)
H4 =1
N
X
~k1¢¢¢~k4
³ 1(2!)2
¡(2;2)bybybb+1
3!
n¡(3;1)bybbb+ h.c.
o+1
4!
n¡(4;0)bbbb+ h.c.
o´
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2.2 Solution of GPE
� solve discrete Gross-Pitaevskii equation
¡(²nq ¡ ¹)ù¾n ¡ °nþn =1
2
X
n1n2
X
¾1¾2
±n;n1+n2V¾¾1¾2nn1n2
þ1n1þ2n2
+1
3!
X
n1n2n3
X
¾1¾2¾3
±n;n1+n2+n3U¾¾1¾2¾3nn1n2n3
þ1n1þ2n2þ3n3 :
−4−2
02
4x 10
4
01
2x 105
2.5
3
3.5
4
4.5
ky
(cm−1
)
Ek
(GH
z)
kz
(cm−1
)
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3 Summary
� description of magnetic insulators:Spin-wave theory
� development of interactingspin-wave theory withdipole dipole interactions
� interesting properties of theenergy dispersion
� interactions: possiblecondensation of bosonsat finite wave-vectorsand integer multiples
−4−2
02
4x 10
4
01
2x 105
2.5
3
3.5
4
4.5
ky
(cm−1
)
Ek
(GH
z)
kz
(cm−1
)
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4 Acknowledgement
Group of Peter Kopietz at ITP in Frankfurt (Germany)
www.itp.uni-frankfurt.de/~kreisel/en
Poster: BEC at finite momentum