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Applications of Density Matrix Renormalization Group to Problems in Magnetism

100%

Gehring G. A., Bursill R. J., Xiang T.

Acta Physica Polonica A

1997

vol. 91

issue 1

105-119

The application of real space renormalization group methods to quantum lattice models has become a topic of great interest following the development of the density matrix renormalization group by White. This method has been used to find the ground and low-lying excited state energies and wave functions of quantum spin models in which the form of the ground state is not clear, for instance because the interactions are frustrated. It has also been applied to fermion problems where the tendency for localization due to the strong Coulomb repulsion is opposed by the lowering of the kinetic energy which occurs as a result of electron transfer. The approach is particularly suitable for one-, or quasi-one-dimensional problems. The method involves truncating the Hilbert space in a systematic and optimised manner. Results for the ground state energy are thus variational bounds. The results for low-lying energies and correlation functions for one-dimensional systems have unprecedented accuracy and the method has become the method of choice for solving one-dimensional quantum spin problems. We review the method and results obtained for the spin-1 chain with biquadratic exchange as well as the spin-1/2 model with competing nearest and next nearest neighbour exchange will be described. More recently, the density matrix renormalization group has been applied to reformulate the coupling constant renormalization group approach which is appropriate for the study of critical properties. This approach has been applied to the anisotropic spin-1/2 Heisenberg chain. Finally, we discuss recent work which has borne promising applications in two dimensions - the Ising model and the two-dimensional Hubbard model.

Bosons and Magnons in Ordered Magnets

80%

Köbler U.

Acta Physica Polonica A

2015

vol. 127

issue 6

1694-1702

In earlier experimental studies we have shown that in accordance with the principles of renormalization group theory the spin dynamics of ordered magnets is controlled by a boson guiding field instead by exchange interactions between nearest magnetic neighbors. In particular, thermal decrease of the magnetic order parameter is given by the heat capacity of the boson field. The typical signature of boson dynamics is that the critical power functions either at T=T_{c} or at T=0 hold up to a considerable distance from critical temperature. The critical power functions of the atomistic models hold asymptotically at T=T_{c} or at T=0 only. In contrast to the atomistic magnons field bosons cannot directly be observed using inelastic neutron scattering. However, for some classes of magnets the field bosons seem to have magnetic moment and thus are able to interact directly with magnons. This interaction, although weak in principle, leads to surprisingly strong functional modifications in the magnon dispersions at small q-values. In particular, the magnon excitation gap seems to be due to the magnon-boson interaction. In this communication we want to show that for small q-values the continuous part of the magnon dispersions can be fitted over a finite q-range by a power function of wave vector. The power function can be identified with the dispersion of the field bosons. It appears that for low q-values magnon dispersions get attracted by the boson dispersion and assume the dispersion of the bosons. This allows for an experimental evaluation of the boson dispersions from the known magnon dispersions. Exponent values of 1, 1.25, 1.5, and 2 have been identified. The boson dispersion relations and the associated power functions of temperature for the heat capacity of the boson fields are now empirically known for all dimensions of the field and for magnets with integer and half-integer spin quantum number. These are two 2× 3 exponent schemes.

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2 Acta Physica Polonica A

1 2015

1 1997

1 Bursill R. J.

1 Gehring G. A.

1 Köbler U.

1 Xiang T.

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Applications of Density Matrix Renormalization Group to Problems in Magnetism

Bosons and Magnons in Ordered Magnets