Linked cluster series expansions for two-particle bound states

Abstract

We develop strong-coupling series expansion methods to study two-particle spectra of quantum lattice models. At the heart of the method lies the calculation of an effective Hamiltonian in the two-particle subspace. We explicitly consider an orthogonality transformation to generate this block diagonalization, and find that maintaining orthogonality is crucial for systems where the ground state and the two-particle subspace are characterized by identical quantum numbers. We discuss the solution of the two-particle Schrödinger equation by using a finite lattice approach in coordinate space or by an integral equation in momentum space. These methods allow us to precisely determine the low-lying excitation spectra of the models at hand, including all two-particle bound/antibound states. Further, we discuss how to generate series expansions for the dispersions of the bound/antibound states. These allow us to employ series extrapolation techniques, whereby binding energies can be determined even when the expansion parameters are not small. We apply the method to the ͑1ϩ1͒-dimensional transverse Ising model and the two-leg spin-1 2 Heisenberg ladder. For the latter model, we also calculate the coherence lengths and determine the critical properties where bound states merge with the two-particle continuum.

Series expansion analysis of a frustrated four-spin tube

“…Apart from generating a continuum of states at fixed total momentum with respect to the relative momentum, interactions may lead to additional collective (anti)bound states which split off from the continuum. The existence of such collective states is know for simpler 1D quantum magnets, such as chains and two-leg ladders [28][29][30][31][32][33][34] . Here we show that similar states exist also on the FFST.…”

“…The effective Hamiltonian, eqn. (5), will mix states at different d. The two-particle spectrum is obtained from the sum of the one-and two-particle irreducible Hamiltonians (see 31,33 and appendix VII), labeled by H 1 and H 2 , with analytic matrix elements calculated by SE from the states of eqn. (11).…”

Statics and Dynamics of Weakly Coupled Antiferromagnetic Spin-1/2 Ladders in a Magnetic Field

“…It has been seen previously that in this system both single-triplet excitations and two-triplet excitations play an important role. [12][13][14][15][16] We discuss these excitations in the following, focusing on the ones that can be created by the symmetric S αα 0 = 2S ±∓ 0 and the antisymmetric S αα π = 2S ±∓ π correlations (see Fig. 11) for the BPCB parameters (39).…”

“…16 The dispersion relation of the bound states has been calculated using a linked cluster series expansion. 14 The first terms of the expansion have an inverse cosine form and the bound state only exists in an interval at approximately q = π (cf. Refs.…”

“…This typical behavior can be observed in spin ladder systems in the presence of a magnetic field. Although such systems have been studied theoretically intensively for many years in both zero [8][9][10][11][12][13][14][15][16] and finite magnetic field, 2,17-27 a quantitive description of the LL low-energy physics remained to be performed specially for a direct comparison with experiments.…”