Perturbation theory for strongly frustrated finite spin systems

Nikolaos P. Konstantinidis, Dermot Coffey

We investigate the use of perturbation theory in finite-sized frustrated spin systems by calculating the effect of quantum fluctuations on coherent states derived from the classical ground state. We first calculate the ground- and first-excited-state wave functions as a function of applied field for a 12-site system and compare with the results of exact diagonalization. We then apply the technique to a 20-site system with the same threefold site coordination as the 12-site system. Frustration results in asymptotically convergent series for both systems which are summed with Pad\'e approximants. We find that at zero magnetic field the different connectivity of the two systems leads to a triplet first excited state in the 12-site system and a singlet first excited state in the 20-site system, while the ground state is a singlet for both. We also show how the analytic structure of the Pad\'e approximants at $|\ensuremath{\lambda}|\ensuremath{\simeq}1$ evolves in the complex $\ensuremath{\lambda}$ plane at the values of the applied field where the ground state switches between spin sectors and how this is connected with the nontrivial dependence of the $〈{S}^{z}〉$ number on the strength of quantum fluctuations. We discuss the origin of this difference in the energy spectra and in the analytic structures. We also characterize the ground and first excited states according to the values of the various spin correlation functions.

Kenn Kubo, Tota Nakamura

Tota Nakamura, 中村 統太

Steffen Sykora, A. Hübsch, K. W. Becker

This paper demonstrates that a computer-aided perturbation theory can easily be realized by use of a cumulant approach. In contrast to a recent alternative formulation on the basis of Wegner's flow equation method the present approach can be applied to systems with arbitrary Hilbert space. In particular an equidistant spectrum of the unperturbed part of the Hamiltonian is not needed. The method is illustrated in detail for dimerized and frustrated spin $\frac{1}{2}$ chains for which the ground state energy is calculated up to seventh order perturbation theory.

Karol Gregor

In the first part we study O(N) spins on geometrically frustrated pyrochlore and planar pyrochlore lattices. For classical spins the strong frustration causes a lack of ordering for N 6= 2 at all temperatures. Nevertheless at low temperatures the system is strongly correlated. We find degrees of freedom that are relevant in this regime and find that the correlations are dipolar up to distance ∼ 1/√T and then decay exponentially. We use various approaches — most notably, we show that this holds to any order in the 1/N expansion. Then we look at spin evolution under precessional and Monte-Carlo dynamics and find that in both cases the dynamical correlations decay exponentially in time with a scale τ ∼ 1/T and that the long distance and long time correlations are described by the simple Langevin evolution of the degrees of freedom found in the static case. Finally we look at the effects of quantum fluctuations by studying the quantum rotor model on the pyrochlore lattice. Using quantum Monte-Carlo simulations and analytical arguments we find that for every N there is a region in quantum coupling—temperature plane with long range nematic order. In the second part we study stability of topological order in Kitaev’s model under arbitrary small local perturbations. Using the Fredenhagen-Marcu order parameter [13] we show in quantum perturbation theory that in two or more dimensions, the topologically ordered phase is distinct from more conventional phases. Further we show that this distinction survives to positive temperatures in three or more dimensions and that the topological phase is also distinct from the high temperature phase. We extend this result to odd Ising gauge theories and note that the relation between these and dimer models strongly suggests

Hidetsugu Kitatani, 北谷 英嗣