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. / Thesis / Doctor of Philosophy (PhD) / Systems composed of many bodies tend to order as their energy is reduced. Steam,
a state characterized by the complete disorder of the constituent water molecules,
condenses to liquid water as the temperature (energy) decreases, wherein the water
molecules are organized enough for insects to walk atop them. Water freezes to ice,
which is so ordered that it can hold sleds and skaters. Quantum mechanics allows for
patterns of organization that go beyond the solid-liquid-gas states. These patterns
are manifest in the smallest degrees of freedom in a solid, the electrons, and are
responsible for fridge magnets and transistors. While quantum systems still tend to
order at lower energies, they are characterized by omni-present fluctuations that can
conceal hidden forms of organization. One can imagine that the states of matter
live in a vast space, where each point represents a different pattern. In this thesis
we show that by probing the geometry of this space, we can detect hidden kinds of
order that would be otherwise invisible to us.
| Identifer | oai:union.ndltd.org:mcmaster.ca/oai:macsphere.mcmaster.ca:11375/28216 |
| Date | January 2022 |
| Creators | Lambert, James |
| Contributors | Sorensen, Erik S., Physics and Astronomy |
| Source Sets | McMaster University |
| Language | English |
| Detected Language | English |
| Type | Thesis |
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