The Heisenberg Antiferromagnet and the Lanczos Algorithm Abstract

BSc Final Year Report - Matthew Holmesalways positive and converges to the ferromagnetic ground state. The pair are always well separatedthen.The modified method works by iteratively improving the initial trial wavefunction v 1 , such that it hasincreasing projection along the ground state. Although the eigenvalue pairs are not to blame, theconstruction of new trial wavefunctions with less ground state overlap could still be the cause.Oscillatory convergence is not believed to be solely, or even at all, responsible for any inaccuracyhowever. The eigenvalues of the N=4,5 systems readily converge, and are more accurate than largersystems (Fig. 8) yet fail to show the same accord as the N3 systems. Another factor was sought.Eigenvalue Spacing & DegeneracyHaving maintained an overlap of approximately zero, ~10 -20 , it was hoped that the variational estimateswould improve on the apparent N=3 limit of the regular method (Fig. 6). Whilst accuracy is clearlyincreased in Fig. 8, ground state estimates of a different physical model, the Mathieu equation [8], arevirtually concurrent. This suggested that the overlap is a feature of the 1D spin chain system.Although no finely spaced eigenvalue pairs ever appear in the 2x2 matrices of the modified Lanczosscheme, it is suggested [20,18,9] that eigenvalue spacing, or degeneracy, in the full spectra may inhibitaccuracy for N>3.As eigenvalue separations generally decrease with size in quantum systems [20], the loss of accuracywith N suggests that poor spacing may be a factor.Spin Chains in Magnetic FieldsIf poorly spaced eigenvalues are responsible for the discrepancies, this should be resolved by placingthe chain in an external magnetic field B, pulling apart the eigenvalues in a similar fashion to Zeemansplitting. By aligning the field along the z-axis, as described by Eq. (33) [8], we simplify thecomputation.zH = −JSˆ ⋅ Sˆ− gµB Sˆ(33)i< i,j>jBiiHere g is the Landé splitting factor, ~2 for pure spin, and µ B is the Bohr magneton, ~58µeVT -1 . Thefield term is summed over i, it being independent of exchange interactions. From Eq. (33), magneticordering via the exchange interaction now competes with the tendency toward dipole-field alignmentvia µ B B. For positive B, the arbitrary negative sign of the field term favours spin-up states in thechosen coordinate system. For strong enough fields, the exchange coupling should be sufficientlyfrustrated that we observe an antiferromagnetic, fully polarised spin-up ground state, of oppositepolarity to the ferromagnetic ground state in the field.Ground State Energy, J0-1-2-3-4Variational Estimates at B~10THamiltonian Ground StateFig. 10 Concurrentground stateenergies. Thevariational estimatesapproach the truevalues as the fieldstrength is increasedtoward ~10T, thelowest field at whicha complete accordexists.-5Modified Lanczos GroundState-62 3 4 5 6 7 8System Size, NAs the field strength is increased, the variational eigenvalues approach the true values. Completeagreement up to N=8, first occurs at a field strength of ~10T as shown in Fig. 10. This comparative13