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

2

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

3

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

Sz = S ¡ n n = byb hb; by

i= 1

r

1¡ n

2S= 1¡ n

4S+O( 1

S2)

S+ =p2S

r

1¡ n

2Sb

S¡ =p2Sby

r

1¡ n

2S

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)

5

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

6

2.1 Simplifications to relevant physical properties

crystal structure of YIG microscopic Hamiltonian

quantum spin S ferromagnet

dipole-dipoleinteractionsZeeman term

Hmag=¡1

2

X

ij

JijS i ¢ Sj ¡ hX

i

Szi

¡ 1

2

X

i

X

j 6=i

¹2

jrij j3[3(S i ¢ rij)(Sj ¢ rij)¡ S i ¢ Sj ]

AK, Sauli, Bartosch, Kopietz ('09)

7

2.1 Linear Spin Wave Theory

� classical groundstate for stripe geometry� Holstein Primakoff transformation (bosons)

� partial Fourier transformation (quasi 2D)

dipolar tensor

H2 =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 = ¡S2[Dxx

ij ¡ 2iDxyij ¡Dyy

ij ]

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

H =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~kdj j~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

)

16

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

)

17

4 Acknowledgement

Group of Peter Kopietz at ITP in Frankfurt (Germany)

www.itp.uni-frankfurt.de/~kreisel/en

Poster: BEC at finite momentum