BEC of magnons and spin wave interactions in QAF Andreas Kreisel Institut für Theoretische Physik Johann Wolfgang Goethe Universität Frankfurt am Main July, 18 2007 collaborators: N. Hasselmann, P. Kopietz
Interacting bosons T=0 Bogoliubov theory (1947) euclidean action for interacting Bose gas S[ ¹ª; ª] = S 2 [ ¹ª; ª] + S 4 [ ¹ª; ª] Z S 2 [ ª; ¹ ª] = ( i! + ² k ¹) ª ¹ K ª K S 4 [ ª; ¹ ª] = K Z Z 1 (2!) 2 ± K1 +K 2 ;K 3 +K 4 V (fk i g) ª ¹ K1 ªK2 ¹ ª K3 ª K4 K 1 K 4 Bogoliubov shift ª K = ª 0 K + ª K ª 0 K = ± K;0 p ½0 h ª k i = 0 ) ~ S ¹ ª; ª = N. Bogoliubov (1947) 4X i=0 ~S i ¹ ª; ª
Goldstone bosons constraint to chemical potential (minimum) ~S 1 ¹ ª; ª = 0 diagonalization q E k = ² 2 k + 2½ 0U(k)² k = c 0 jkj + jkj 3 + O(k 5 ) condensate density ½ 0 = ¹ U(0) gapless excitations: velocity c 0 = r ¹ m
Heisenberg ferromagnet Spin waves in Heisenbergmagnets Holstein-Primakoff transformation around classical ground state S i = p r 2Sb y i 1 n i 2S = (S+ i )y n i = b Si z = S n i bosonic hamiltonian ^H = 1 2 X J ij S i S j J ij = J < 0 ij y i b i ; h i b y i ; b j = ± ij H = E cl + H 2 + H 4 + O(b 6 ) E cl = NS[2DJS h] H 2 = S X J ij ³n i + n j + b y i 2 b j b y j b i + h X ij i H 4 = 1 X ³ J ij 2n i n j b y i 4 n ib j b y j n ib j ij N. Hasselmann, P. Kopietz, Europhys. Lett. 74, 1067 (2006) n i
BEC of Holstein-Primakoff bosons in ferromagnets? FT: diagonalizes quadratic part b i = p 1 X e ik r i b k N k E k = ~ J 0 S(1 k k = X ± ~J 0 = 2DJ e ik ± H 2 = X k E k b y k b k interaction vanishes: to weak interaction of ferromagnetic spin waves H 4 = 1 X ± k1 +k 2V 2 ;k 3 +k 4 V (fk i g) b y k 1 b y k 2 b k3 b k4 k 1 k 4 V (fk i g)» k 1 k 2 + k 3 k 4 + O(k 4 ) no BEC!
^H = 1 2 Heisenberg antiferromagnet X J ij S i S j J ij = J > 0 ij conventional spin wave expansion transformations Holstein-Primakoff H = E cl + H 2 + H 4 + O(b 6 ) E cl = DNJS 2 H 2 = S X J ij ³b y i b i + b y j b j + b i b j + b y i by j ij Fourier transformation using reduced BZ r r 2 X b i = e ik r i 2 X A k b i = e ik r i B k N N k diagonalization: Bogoliubov transformation µ µ µ Ak uk v B y = k k ² k = p 1 k 2 k v k u k y k k = 1 DX cos(k i a) D Anderson (1952), Kubo (1952), Oguchi (1960) k i=1
Spin-wave interactions 2 degenerate magnon modes H 2 = X ³ E k y k k + y k k + 1 + E 0 k E k = ~ J 0 S² k spin-wave interactions: complicated H 4 = 1 X ³ 4V Â 0 k 1 k 4 V (1) 1234 interaction vertices s V (1) 1234» 1 jk 1 jjk 2 j 2 jk 3 jjk 4 j y 1 y 2 3 4 + y 3 y 4 1 2 + : : : µ 1 + k 1 k 2 jk 1 jjk 2 j but: all divergences cancel in physical quantities infrared singular in some limits N. Hasselmann, P. Kopietz, Europhys. Lett. 74, 1067 (2006)
BEC of magnons in QAF quantum antiferromagnet in magnetic field h = g¹ B B ^H = 1 X J ij S i S j h X has U(1) symmetry Si z rotational symmetry 2 ij i around magnetic field classical ground states I. h = 0 II. h < h c = 2 ~ J 0 S III. h > h c A. Kreisel, N. Hasselmann, and P. Kopietz, Phys. Rev. Lett. 98, 067230 (2007)
Formulation in terms of 2 Transformations magnons Holstein-Primakoff (fluctuations around FM ground state) S i = p r 2Sb y i 1 n i 2S = h i (S+ i )y y n i = bi b i ; b y i ; b j = ± ij Si z = S n i H = E cl + H 2 + H 4 + O(b 6 ) fourier transformation: red. BZ (2 Modes) H 2 = X k;¾= (² k¾ ¹) b y k¾ b k¾ ² k = k2 2m + O(k4 ) ² k+ = h c + O(k 2 ) h c = 2 ~ J 0 S
Continuum formulation neglect gapped mode for simplicity ª k = p Na D b k interacting bosonic fields H = Z d D k (2¼) D µ k 2 2m ¹ ª y k ª k + 1 2 Z k Z k 0 Z q U q ª y k+ q ªy k 0 q ª k 0 ª k effective interaction: constant for small momenta U q = ( 0 jqj)â 1 0 Â 0 = 1 2 ~J 0 a D m = 1 2JSa 2
Hermitian parametrization hermitian operators k = y k k = y k [ k ; q ] = i(2¼) D ±(k q) transformation: split up field r s ª k = 2 µ k + p i k 2sµ s = S a D transverse longitudinal advantage: physical interpretation of operators k» X i k» X i ³ i e ik r i S x i ³ i e ik r i S y i ³ i = h = he z ½ 1 i 2 A 1 i 2 B P. W. Anderson (1952) N. Hasselmann, P. Kopietz, Europhys. Lett. 74, 1067 (2006)
Symmetry breaking impose nonzero expectation value k = k + (2¼) D ±(k)á 0 H = E 0 + H 1 + H 2 + H 3 + H 4 fix chemical potential ¹ = U 0sÁ 0 2 diagonalization: Bogoliubov transformation H 2 = 1 Z d D k ³ 2 (2¼) ² D k y k k + y k k + E 0 r ¹ ² k = c 0 jkj + jkj 3 + O(k 5 ) c 0 = m = 1 8 p m 3 ¹
Quasiparticle decay decay in two or more quasiparticles energy conservation momentum conservation E k = E k1 + E k2 k = k 1 + k 2 consider energy dispersion E k = c 0 jkj + jk 3 + O(k 5 ) stable quasiparticles > 0 unstable quasiparticles concerning spontaneous decay < 0 M. E. Zhitormirsky, A. L. Chernyshev, Phys. Rev. Lett. 82, 4536 (1999)
Correlation functions hermitian fields ª k = r s 2 µ k + i p 2sµ k transverse correlations in gaussian approximation (linear spinwave theory)  1 0 h K K 0i = ± K; K 0! 2 + c 2 0 k2! h K K 0i = ± K; K 0! 2 + c 2 0 k2 h K K 0i = ± K; K 0 C. Castellani, C. Di Castro, F. Pistolesi, and G. C. Strinati, Phys. Rev. Lett. 78, 1612 (1997) F. Pistolesi, C. Castellani, C. Di Castro, and G. C. Strinati, Phys. Rev. B 69, 024513 (2004) K = (!; k) longitudinal  0 c 2 0k 2! 2 + c 0 k 2 quantitatively wrong in gaussian approximation
Beyond Bogoliubov divergences in pertubation theory (contribution to anomal Propagator) cancellations controlled by Ward identities due to U(1) - symmetry (2 correlations correct) critical continuum in longitudinal correlation function h K K i = Z 2 k contribution cancels in physical quantities of the Bose gas! 2! 2 + c 2 k 2 + K (mc) 3 D+1 Z½ 2½ 0 observable? Yes, BEC of magnons! fields: staggered magnetization 8 >< >: ln 2 3 D h (mc) 2! 2 =c 2 +k 2 i h i D 3! 2 c + k 2 2 2 D = 3 1 < D < 3
Possible measurement neutron scattering might detect anomalous dimension in correlation function longitudinal 2 staggered structure factor 3 S k (k;!) = Âs2 M 2 s 4 Z2 k 2 cjkj ± (! cjkj) + C D(mc) 2 (! cjkj) 5 Z½½ 3 3 D! 2 0 c k2 2 2 critical continuum! only valid at (Ginzburg-scale) c k G
Restrictions Ginzburg-scale! c k G 2 contributions to abnormal propagator symmetry breaking divergent diagrams k G ¼ (mc)3 ½ 0 ¼ 4p 2 Sa µ» µ D = 2 k G ¼ mc e ½ 0 (mc) 3 ¼ p 3 S a µe 6 p 3µ D = 3 energy integrated structure factor spectral weight I ± : ± -function I critical continuum ³ c : ¼ k G k jkj ln G jkj D = 2 I c I ± I c I ± ¼ (mc)2 ½ 0 k G jkj D = 3
Summary not-so-well known facts about the Bose gas large corrections to Bogoliubov approximation in 1 < D 3 Spin-wave interactions in QAF interaction vertices: divergent (usual spin-wave expansion) hermitian field parametrization weak interactions between Goldstone modes fields: physical quantities QAF in uniform magnetic field: BEC of magnons measurable quantities via neutron scattering but only at D=2