The Heisenberg Antiferromagnet and the Lanczos Algorithm Abstract

BSc Final Year Report - Matthew HolmesIf only the ground or lower levels are required the Lanczos method enables much larger systems to betreated than would otherwise be the case. That said, the treatment of 1D chains is still limited to N~20spins [19] even on powerful systems [20]. At first glance this may seem small but it presents a~10 6 x10 6 matrix, which is large!Initiating the ProcedureThe initial vector v 1 , must have some projection along the desired eigenstate v. This is usually satisfiedby randomising the coefficients [20,19] of v 1 ,CALL RANDOM_NUMBER(v1)v1= v1/SQRT(DOT_PRODUCT(v1,v1))Remember, v 1 is a vector in the original basis { i } and has coefficients. Generally then,v 1 =c 1 1 +c 2 2 +…c ,. It is tempting to choose equal coefficients (as the author did) in the hope someprojection along the desired state. The Lanczos method will not work if this is done [20] as v 2 will bezero and the iteration fails. A 'proof' is proposed below.Consider the 'symmetric vector' (of equal projection, c, in all basis states) v, = c{ ψ i}; i = 1. (29)Normalising, we obtain the initial vector v 1 ,c= { ψ } = { ψ } { ψ }1 i2 i=cηc1 , (30)ηias the i are themselves normalised such that |v|=[c 2 |{ i }| 2 ] 1/2 =c 1/2 . Substituting Eq. (30) into Eq. (20),we have for 0 ,α1η{ } H { }ψ iψ i0 = (31i)εεα0=ii=ηη2{ ψ } { ψ } = { ψ } εi(31ii)Where ε is the sum along any row or column of H. Substituting (Eq. 31ii) into Eq. (18), we obtain,ε1{ ψ } −ε{ ψ } 02=ii= . (32)ηηFrom Eq. (32) a new vector v 2 cannot be produced and we conclude that an initial symmetric vector isunsuitable when representing a symmetric matrix. Such a vector is orthogonal to all eigenstates. Thisresult is physically intuitive, as a symmetric vector represents a state of zero magnetisation.Application & AnalysisOrthogonalityWhen using the Lanczos method it is vital that the basis is orthogonal. Fig. 4 shows the increasingoverlap between successive states in the absence of any remedial action. Without such action, rogueeigenvalues are introduced to the spectra [14]. This makes it difficult to check for accuracy with anexact routine when setting up the Lanczos method.9