Magnetic response of the J 1 -J 2 Spin Hamiltonian and its Implication for FeAs

Transcription

1 1 Magnetic response of the J 1 -J 2 Spin Hamiltonian and its Implication for FeAs arxiv: Zhihua Yang(SKKU) In collaboration with: Jung Hoon Kim(SKKU), Prof. Jung Hoon Han(SKKU) 5/1/2009

2 Introduction The new Fe-based Superconductors - Magnetic layers of 3d atoms The parent FeAs compound - Multi-band Metal (π, 0) - SDW order phase: T SDW ~137K (P. Dai, Nature(08)) T<T SDW, antiferromagnetic SDW magnetic order T>T SDW, global antiferromagnetism is absent - Structure phase transition ( P. Dai (08)) Orthorhombic-Tetragonal - In mean-field picture, when T>T SDW, normal metallic behavior might be expected. P.Dai(08) Neutron-scattering experiment C.De la Cruz, Q.Huang, J.W.Lynn,J.Li,W.Ratcliff II, J.L.Zarestky, H.A.Mook,G.F.Chen,J.L.LuoN.L.Wang, P.Dai,Nature 453,899(08) 2

3 Introduction 3 Magnetic properties of FeAs parent compound - linear temperature dependent uniform susceptibility T-linear over 150~ K! - contrary to mean-field picture of FeAs parent compound where susceptibility should be Pauli-like (itinerant) or Curie-Weiss-like (local) Klingeler(08),B=1T Microscopic theories based on - Itinerant eletrons - Local moments G. Wu,et al. J. Phys.C.20, (2008); J.-Q. Yan, et al., PRB.78, (2008); X. F. Wang, et. al. arxiv: ; R. Klingeler et. al,arxiv: ; X. F. Wang et. al,arxiv: Wang(08), B=5T

4 Introduction Local spin model H H NN, J αβ ij ~ J ij αβ ij S iα S = J αβ 1 NNN, J αβ ij Disregarding multi-orbital nature = J1 Si S j + J 2 < ij> << ij>> (J 1 =J 2 ~500K) jβ (Si & Abrahams, PRL(08)) S i S = J αβ 2 (Ma, Lu & Xiang, PRB (08)) j As Fe As Fe J 2 Fe As Fe As J 1 Yildirim,PRL(08) & Ma, PRB(08) 4 T.Yildirim,PRL.101,057010(2008); Q. Si and E. Abrahams, PRL.101, (2008); F. Ma, Z. Y. Lu, and T. Xiang, PRB.78, (2008).

5 Introduction J 1 -only model (J 2 =0) H = J S i S 1 < ij> - Schwinger Boson Mean field thoery (Auerbach & Arovas,PRB&PRL(88)) - Modified Spin wave theory (Takahashi,PRB(87,98)) j - Classical Monte Carlo (Hinzke&Nowak&Garanin, EPJB(00)) χ u T - Quantum Monte Carlo (QMC) (Makivic&Ding PRB(91), Kim&Troyer, PRL(98)) χ u T Auerbach and D.P. Arovas, PRL.61, 617 (1988); M. Takahashi, PRB.36, 3791(1987);40, 2494 (1989); D. Hinzke, U. Nowak, and D. A. Garanin, E.Phys.J. B.16, 435 (2000); M.S. Makivic and H.-Q. Ding, PRB.43, 3562 (1991); J.-K. Kim and M. Troyer,PRL.80, 2705 (1998). 5 Kim&Troyer, PRL(98))

6 Introduction 6 The J 1 -J 2 Heisenberg model (J 2 6=0) AF1 AF2 (π, π) (π, 0) (2J 2 /J 1 ) c =1 J 1 J 2 J 1 J 2 Linear-T uniform susceptibility (Zhang (08), Kou(08)) i.e., Zhang et al. χ u = χ 0 (1 + a( T J 1 )) with the Dyson Maleev transformation Kou et al. χ u = χ lo + χ it J 2 model+a coupling between local moments and itinerant electrons Zhang(08) G. M. Zhang, Y. H. Su, Z. Y. Lu, Z. -Y.Weng, D. H. Lee, and T. Xiang, arxiv: v3; S.-P. Kou, T. Li, and Z.-Y. Weng, arxiv: v3.

7 Introduction 7 Others: - ONLY the uniform susecptibility - ONLY for special cases, not the whole range of 2J 2 /J 1 Us: Magnetic response of J 1 -J 2 Heisenberg model (Yang, Kim&Han(09) ) 1. Both uniform and staggered susceptibilities χ (0,0), χ (π,0), χ (π,π) 2. In the whole range 2J2/J1 [0,2] 3. Classical Monte Carlo(CMC) calculation 4. Schwinger Boson mean field theory(sbmft) calculation Z.H. Yang, J. H. Kim, J. H. Han, arxiv: , submitted to PRB(2009)

8 Review of Classical Mean Field Phase diagram Ordering pattern: hs i i = m 1 ( 1) x i + m 2 ( 1) x i+y i µ 2 m 1 +m 2 = tanh T [(J 2m 1 +(J 1 J 2 )m 2 ], µ 2 m 1 m 2 = tanh T [J 2m 1 (J 1 J 2 )m 2 ] I: m 2 6=0,m 1 =0 II: m 1 6=0,m 2 =0 T/J1 2 1 I T MF (π, π) Paramagnetic T MF II (π, 0) 1 st order phase transition J 2 /J 1 8

9 Susceptibility 9 The suceptibility χ α k χ α k = S α k S α k TN S α k S α k α = x, y, z k =any vector!! S α k =P i Sα i e ik r i hs α k i=p i hsα i ie ik r i N = L L is lattice size k =(0, 0), (π, 0), (0, π), (π, π)

10 Uniform susceptibility (classical Monte Carlo(CMC)) 10 T CO Linear-T behavior T (0,T CO )

11 Uniform susceptibility (classical Monte Carlo(CMC)) 11 T CO The crossover temperature T CO (0.1J 1, 0.8J 1 ) The slope B of the linear parts χ z = B(T/J 1 )

12 12 Staggered susceptibility (CMC) In (π, π) order Dominant susceptibility diverges In (π, π) order(2j 2 /J 1 1), χ z (π,π) In (π, 0) order (2J 2 /J 1 1), χ z (π,0) Non-dominant, non-divergent susceptibilities In (π, π) order(2j 2 /J 1 < 1), In (π, 0) order (2J 2 /J 1 > 1), χ z (π,0) χ z (π,π) In (π, 0) order A linear-t behavior with slope B

13 Review of CMC 13 Linear-T region stronger when away from criticality The expected crossover temperature T CO ~0.8J 1 ~400K in fair proximity to the FeAs system: K

14 Schwinger Boson mean field theory(sbmft) Spin operators Schwinger Boson operators (b 1,b 2 ), S = 1 2 b α~σ αβ b β. 1: 2: where ~σ αβ sarepaulimatrices. The J 1 J 2 Hamiltonian in the mean-field form H MF = J 1 2 X hiji ij A ij J 2 2 X hhijii ij A ij + h.c. +X i λ i ³b i1 b i1 + b i2 b i2 2S, With the constraint, b 1 b 1 + b 2 b 2 =2S ~ S ~S = S(S +1). where the operators are A ij = b i1 b j2 b i2 b j1 = A ji, ij = ha ij i. 14 A. Auerbach and D.P. Arovas, PRL.61, 617 (1988); D.P. Arovas and A. Auerbach, PRB.38, 316 (1988).

15 Schwinger Boson mean field theory(sbmft) 15 The distribution of ij i,i±ˆx = 2 ( 1) x i+y i + 1x ( 1) x i, i,i±ŷ = 2 ( 1) x i+y i + 1y ( 1) x i. i,i±ˆx±ŷ = A+ B 2 ( 1) y i + A B 2 ( 1) x i. 1 1x D1 k Gap parameters D2 (π, π) 2 DA (π, 0) 1,A,B DB

16 Gap parameters (SBMFT) I II In I and II regions I : 1 6= 2 6=0 II : A = B 6=0, 2 6=0 -The system in I region ~ J 1 model A = B =0 J 2 =0 1,2 = constant Critical point S = 1/2, η c =1.2 S = 1, η c =1.1 S 2, η c = L. Capriotti, A. Fubini, T. Roscilde, and V. Tognetti, PRL.92, (2004)

17 Gap parameters (SBMFT) I II T SBMFT MF The mean-field transition temperature scale, T>T SBMFT MF,allthegapparameters=0 ( 1,2 6=0, i.e. 2J 2 /J 1 < η c 1,2 =0, T < TMF SBMFT T > TMF SBMFT The MF transition temperature scale grows with spin values, T SBMFT MF S(S +1)J 1 17

18 Uniform susceptibility (SBMFT) 18 Linear-T behavior, In the J1 only limit χ (0,0) A + BT T [0 (0.3J 1 0.9J 1 )] (S = 1 2 ) T [0 (0.4J 1 1.0J 1 )] (S =1) A = 2 C 2,C 3.2, A 0.19;. B = ln(l/a) π 2 2

19 CMC vs. SBMFT: uniform susceptibility CMC SBMFT Linear-T behavior CMC: Linear-T 0.8J 1 T 400K SBMFT: Linear-T J 1 T 500K Consistent with experimental results of FeAs: up to K 19 (J 1 ~500K), F. Ma, at.al, PRB.78, (2008).

20 Phase diagram (CMC vs. SBMFT) T SBMFT CO T CMC CO T-Linear region CMC SBMFT T CMC CO 20 T SBMFT CO

21 CMC vs. SBMFT: staggered susceptibilities 21 CMC SBMFT in (π, π) order χ (π,0) in (π, 0) order χ (π,π)

22 Staggered susceptibility (SBMFT) 22 At low temperature, In (π, π) order,χ (π,π) > χ (π,0) In (π, 0) order, χ (π,0) > χ (π,π) χ (π,0) N T, and χ (π,π) N T Near BEC, the gap δ 0, never 0 in 2D J 1 model, χ (π,π) T Nδ,whereδ C T 4 L Non-dominant susceptibilities, i.e. χ (0,π) 1 T At high temperature χ (π,0) or χ (π,π) S(S +1) T

23 Survey of theoretical and experimental results local moments and itinerant electrons 23 Localized moment picture Itinerant picture Kou(08), Local moments Zhang(08), J 1 -J 2 model (S=1,DM) Korshunov (09) Our results, J 1 -J 2 model (CMC&SBMFT) Experimental results Wang(08)

24 Summary 24 We calculated the susceptibilities of J 1 -J 2 model in 0<2J 2 /J 1 <2 -- classical Monte Carlo(CMC) -- Schwinger Boson mean field theroy(sbmft) The uniform susceptibilities by the two methods are consistent In whole range of 2J 2 /J 1, the uniform susceptibility : linear T dependence -- away from the critical point, T~0.8J 1 -J 1 -- which is in agreement with results of the parent FeAs compound! Future: local moments and itinerant electrons Thank you!