Nonergodic dynamics of the extended anisotropic Heisenberg chain

Abstract

The issue of ergodicity is often underestimated. The presence of zero-frequency excitations in bosonic Green's functions determine the appearance of zero-frequency momentum-dependent quantities in correlation functions. The implicit dependence of matrix elements make such quantities also relevant in the computation of susceptibilities. Consequently, the correct determination of these quantities is of great relevance and the well-established practice of fixing them by assuming the ergodicity of the dynamics is quite questionable as it is not justifiable a priori by no means. In this manuscript, we have investigated the ergodicity of the dynamics of the z-component of the spin in the 1D Heisenberg model with anisotropic nearest-neighbor and isotropic next-nearest-neighbor interactions. We have obtained the zero-temperature phase diagram in the thermodynamic limit by extrapolating Exact and Lanczos diagonalization results computed on chains with sizes L = 6 ÷ 26. Two distinct non-ergodic regions have been found: one for J ′ /Jz 0.3 and |J ⊥ |/Jz < 1 and another for J ′ /Jz 0.25 and |J ⊥ |/Jz = 1. On the contrary, finite-size scaling of T = 0 results, obtained by means of Exact diagonalization on chains with sizes L = 4 ÷ 18, indicates an ergodic behavior of dynamics in the whole range of parameters.

Ergodicity of the extended anisotropic 1D Heisenberg model: Response at low temperatures

“…1 in Ref. [4]). A transition zone exists at finite sizes, which probably would become a transition line in the bulk system.…”

“…1 the Kubo susceptibility is plotted as a function of temperature for the ergodic (panel a) and non-ergodic (panel b) regions from the phase diagram obtained in Ref. [4]. While at high temperatures w 0 ðTÞ$1=T, as it should be after the Curie law, the most interesting behavior is concentrated at low temperatures.…”

“…Previously [3,4] we have already studied the question of ergodicity of the response of S z i in the Hamiltonian (6). We have constructed a phase diagram of (6) in the plane J 0 À J ?…”

Frustration-Driven Quantum Phase Transition in the 1D Extended Anisotropic Heisenberg Model

“…The starting point for our analysis is the doubly degenerate completely polarized state with all spins either "up" or "down", located on the phase diagram of the system within the limits 0 < J ⊥ < J z and J < ∼ 0.31J z , as found in Ref. [5].…”

XXZ-like phase in the F-AF anisotropic Heisenberg chain

“…The very rich phase diagram, the competition between frustration and anisotropy and the absence of evident small parameters has attracted the attention of many researchers in the field. When J z , J ⊥ and J are all of the same order of magnitude, exact (or quasi-exact) treatments, such as Bosonization and Renormalization Group, are inapplicable and numerical techniques are the only possible tool of investigation [9]. In this manuscript, we have used the Density Matrix Renormalization Group (DMRG) technique [10,11] in order to obtain the ground state properties of the Hamiltonian (1).…”