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Homework

Homework. Ferromagnetic spin waves. Consider a ferromagnet with all the spins line up in equilibrium. Consider small deviation from it. Write S i =S 0 +  S i ,

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Homework

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  1. Homework

  2. Ferromagnetic spin waves • Consider a ferromagnet with all the spins line up in equilibrium. Consider small deviation from it. Write Si=S0+ Si, •  Si=Ak exp(ik t-kr), Ak=A(1, i, 0) and ~k = 2J|S0| (1-cos{k }). For k small, k~Dk2 where D=JzS02

  3. Ferromagnetic spin waves  Si=Ak exp(ik t-kr), Ak=A(1, i, 0). Take the real part. At t=0,  S is along x at r=0 and along y at k r=/2. When t=/2, S is along y at r=0 and along –x at k r=/2

  4. Magnon: Quantized spin waves • a=S+/(2Sz)1/2, a+=S-/(2Sz)1/2. • [a,a+]~[S+,S-]/(2Sz)=1. • aa+=S-S+/(2Sz)=(S2-Sz2-Sz)/2Sz=[S(S+1)-Sz2+Sz]/2Sz=[(S+Sz)(S-Sz)+S-Sz] /2Sz. • S-Sz~aa+ • Hexch=-J (S-ai+ai)(S-aj+aj)+(Si+Sj-+Si-Sj+)/2 ~ constant-JS (-ai+ai-aj+aj+aiaj++ai+aj) =kk nk。 • ~k = 2J|S0| (1-cos{k })+K

  5. Quantization: Magnons are Bosons • Eigenvalue of n=a+a is quantized with eigenfunction |n>=a+|n-1>/n0.5. (the conjugate is a|n>=n0.5|n>. • First prove that the normalization is correct: <n|n>=<n-1|aa+|n-1>/n=<n-1|(a+a+1)|n-1>/n =[(n-1)<n-2|n-2>+1]/n=1. Finally a+a|n>=a+n0.5|n-1>=n|n>. Thus the energy of the system changes by integer multiples of k

  6. Magnon heat capacity • <E>=kk<nk>=kk /(e/kBT-1) • For T<J, only magnons with small k is excited. If T>K, can neglect the gap. <E>=[V/(2)3] d3k Dk2/(eDk2/kBT-1). • <E>/V=[(kBT)5/2/(D3/242)]0xm dx x3/2/(ex-1). • At low T aprroximate xm by . Then <E>/V T5/2; C T3/2

  7. Refresher for Bose Statistics • <n>=k=0 e-kx k/Z where x= /kBT. • Z=k e-kx =1/(1-e-x). • <n>=-x lnZ=e-x/(1-e-x)=1/(ex-1).

  8. Antiferromagnetic magnons: physics related to superconductivity • H=J  SjSj+ -2BHASajz +2BHASbjz. • a=Sa+/(2Sz)1/2, a+=Sa-/(2Sz)1/2 ; b+=Sb+/(2Sz)1/2, b=Sb-/(2Sz)1/2 ; Sajz=S-aj+aj, Sblz=-S+bl+bl. • H=ek[k( ak+bk++akbk)+(ak+ak+bk+bk)]+ a(ak+ak+bk+bk)]; e=2JzS, k= exp(ik)/z, a= 20Ha. • H involves products of two creation operators!

  9. AF magnons: [ak+,H]=ek[ak+,akbk] + (e+a)[ak+,ak+ak] = -ekbk -(e+a)ak+ ; [bk,H]= ekak+ +(e+a)bk; • Define k= ukak-vkbk+ ; k=ukbk-vkak+. Look for solutions of the form k+ exp(i t). itk+=[k+,H]=-k+. • [k+,H]= uk [-ekbk -(e+a)ak+ ]-vk [ekak+ +(e+a)bk]=- ( ukak +-vkbk ). Get uk (e+a) +vkek = uk ;uk ek +vk (e+a)=- vk .

  10. uk (e+a) +vkek = uk ;uk ek +vk (e+a )= - vk • (e+a )2-2 =(ek )2 ; k2 = (e+a )2-(ek )2 • Long wavelength limit k= [(e+a+e) (e (1-k )+a)]0.5 ; • k=0= [(2e+a ) a ] 0.5 >> a; (FMR) • k (a=0)=e (1-k 2) 0.5 k. (For F, k2)

  11. Normalization [k,k+]=uk2[ak,ak+]+vk2[bk+,bk]=uk2-vk2=1. • Write u=cosh , v=sinh  • Homework : Is it true that tanh 2=-(e+a)/[e(1-k)]?

  12. Superconductivity and antiferromagnet • Superconductivity • k=ukck-vkc-k+; -k=ukc-k+vkck+ • AF-Magnon: • k= ukak-vkbk+ ; k=ukbk-vkak+.

  13. Ground state magnetization • ak=ukk+vkk+; bk=ukk+vkk+ • <Saz>=NS- ak+ak=NS- (uk2k+k+vk2kk++ off-diagonal terms). • At T=0, nk=0, NS-<Saz>= vk2=k sinh2(k)  ddk/k. Fluctuation is infinite in 1 dimension.

  14. Magnons: Holstein-Primakoff transformation • Define spin wave operators a, a+ by S+/(2S)1/2=(1-a+a/2S)1/2a; S-/(2S)1/2= a+(1-a+a/2S)1/2 a; Sz=S-a+a • Assume a+a/2S<<1, Sz~S; then [S+,S-] =2Sz=2S[a,a+]=2S if [a,a^+]=1. a behaves like a boson destruction operator.

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