计算物理ComputationalPhysics计算物理 (11).pdf
Computational PhysicsPhysical problem IMagnetic excitation in a one-dimensional Heisenberg spin chain.Extending reading:1.R.W.Gerling and D.P.LandauStatic and dynamic properties of the classical XY chainPhysical Review B 27,1719-1724(1983)2.Z.F.Hou,D.P.Landau,G.M.Stocks,G.BrownSpin-wave multiple excitations in nanoscale classical Heisenberg antiferromagnetsPhysical Review B 91,064417(2015)David P.Landau Distinguished Research Professor of Physics and founding Director of the Center for Simulational Physics at the University of Georgia.He is a Fellow of the American Physical Society.He won the Aneesur Rahman Prize for Computational Physics,the highest award in computational physics given by the American Physical Society.1941-Basic knowledge of magnetism Magnetism Disordered:Paramagnetic,Diamagnetic Ordered:Ferromagnetic,Antiferromagnetic,Ferrimagnetic If J0,the ground state is antiferromagnetic:all nearest-neighbor spins aligned antiparallel.If J1,znew=2-znew,j=j+p else if znew2p,j=j-2p else if j0,j=j+2pMeasurement =Mx,My,Mz C=-2/NkBT2=d/dT Spin correlation function=-SC(0)=1;-SC(r)=SC(L-r)Spin structure factor=*exp(-ik*r)=1/L*Sumr*cos(k*r)-SumkSK(k)=1;-SK(k)=SK(2p-k)k=0,1,2,3,.,L-1,in unit of 2p/L FM:SK(0)=1 AFM:SK(L/2)=1Code10.1.spinwave.cppSpin dynamics dx/dt=x,H/i x=Si Si,H=Si,J*SumSi*Sj=JSi,Si*(Si-1+Si+1)Sx,Sx=Sy,Sy=Sz,Sz=0 Sx,Sy=iSz,Sy,Sz=iSx,Sz,Sx=iSy dSix/dt=JSiz(Si-1y+Si+1y)-Siy(Si-1z+Si+1z)dSiy/dt=JSix(Si-1z+Si+1z)-Siz(Si-1x+Si+1x)dSiz/dt=JSiy(Si-1x+Si+1x)-Six(Si-1y+Si+1y)dSi/dt=-Si x dH/dSi=-J Si x(Si-1+Si+1)4th-order Runge-KuttadSi/dt=-Si x dH/dSi=-J Si x(Si-1+Si+1)d2Si/dt2=-J dSi/dt x(Si-1+Si+1)+Si x(dSi-1/dt+dSi+1/dt)d3Si/dt3=-J d2Si/dt2 x(Si-1+Si+1)+2dSi/dt x(dSi-1/dt+dSi+1/dt)+Si x(d2Si-1/dt2+d2Si+1/dt2)d4Si/dt4=-J d3Si/dt2 x(Si-1+Si+1)+3d2Si/dt2 x(dSi-1/dt+dSi+1/dt)+3dSi/dt x(d2Si-1/dt2+d2Si+1/dt2)+Si x(d3Si-1/dt3+d3Si+1/dt3)Procession:Energy-timedSi/dt=-Si x dH/dSi=-J Si x(Si-1+Si+1)dEi/dt=J(dSi/dt)*(Si-1+Si+i)=0E=a constant10.1.spinwave.cppSpin wave dispersion SK(k,w)=Sumr,wSC(r,t)*exp(-ikr)*exp(-iwt)w(k):dispersion10.1.spinwave.cppw(k)=2Jcos(k)Space-time Correlation function SC(r,t)=SC(0,0)=110.1.spinwave.cppAntiferromagnetic spin wave10.1.spinwave.cppGood Luck in the final exam!
