State Space Geometry of Low Dimensional Quantum Magnets

Abstract

In recent decades enormous progress has been made in studying the geometrical structure of the quantum state space. Far from an abstraction, this geometric struc- ture is defined operationally in terms of the distinguishability of states connected by parameterizations that can be controlled in a laboratory. This geometry is manifest in the kinds of response functions that are measured by well established experimen- tal techniques, such as inelastic neutron scattering. In this thesis we explore the properties of the state space geometry in the vicinity of the ground state of two paradigmatic models of low dimensional magnetism. The first model is the spin-1 anti-ferromagnetic Heisenberg chain, which is a central example of symmetry pro- tected topological physics in one dimension, exhibiting a non-local string order, and symmetry protected short range entanglement. The second is the Kitaev honeycomb model, a rare example of an analytically solvable quantum spin liquid, characterized by long range topological order. In Chapter 2 we employ the single mode approximation to estimate the genuine multipartite entanglement in the spin-1 chain as a function of the unaxial anisotropy up to finite temperature. We find that the genuine multipartite entanglement ex- hibits a finite temperature plateau, and recove the universality class of the phase transition induced by negative anisotropy be examining the finite size scaling of the quantum Fisher information. In Chapter 4 we map out the zero temperature phase diagram in terms of the QFI for a patch of the phase space parameterized by the anisotropy and applied magnetic field, establishing that any non-zero anisotropy en- hances that entanglement of the SPT phase, and the robustness of the phase to finite temperatures. We also establish a connection between genuine multipartite entanglement and state space curvature. In Chapter 3 we turn to the Kitaev honeycomb model and demonstrate that, while the QFI associated to local operators remains trivial, the second derivative of such quantities with respect to the driving parameter exhibit divergences. We characterize the critical exponents associated with these divergences.