Topological frustration (or topological mechanics) is the existence of classical zero modes that are robust to many but not all distortions of the Hamiltonian. It arises naturally from locality in systems whose interactions form a set of constraints such as in geometrically frustrated magnets and balls and springs metamaterials. For a magnet whose classical limit exhibits topological frustration, an important question is what happens to this topology when the degrees of freedom are quantized and whether such frustration could lead to exotic quantum phases of matter like a spin liquid. We answer these questions for a geometrically frustrated spin ladder model. It has the feature of having infinitely many conserved quantities that aid the solution. We find classical zero modes all get lifted by quantum fluctuations and the system is left with a unique rung singlet ground state -- a trivial quantum spin liquid. Moreover, we find low-energy eigenstates corresponding to known symmetry protected topological (SPT) ground states, and a special role of
SU(2) symmetry, that it demands the existence of extra dimensions of classical zero modes -- the phenomena we call symmetry-enriched topological frustration (SETF). These results suggest small violations of the conservation laws in the nearly SETF regime could lead to quantum scars. We further study a two-dimensional bilayer triangular lattice model and find a similar SETF phenomena which also leads to suppressed low-energy topological eigenstates in the quantum regime. These results suggest that in the absence of magnetic order, classical topological frustration manifests at finite spin as asymptotically low energy modes with support for exotic quantum phenomena.
We investigate two-dimensional spin models and begin with an introduction to critical phenomena with emphasis on classical systems with boundaries and conformal invariance. Then we introduce the antiferromagnetic quantum Heisenberg model with frustrating interactions. Conformal invariance of a statistical mechanical system at criticality has powerful implications in two dimensions. With the use of conformal field theory, we investigate finite-size and boundary effects for a class of N-critical models that describe the simplest type of multicritical behaviour. In Papers I and II, correlation functions for the order parameter and other operators of these multicritical Ising models are derived for restricted domains (semi-infinite plane, infinite strip, disc and rectangle) with free or fixed boundary conditions. These yield the corresponding universal surface exponents and allow for a detailed analysis of correlations lengths, structure factors and response functions. The interplay between duality and boundary conditions is also made explicit. In Paper III, we analyse structure factors at criticality and show how apparent power- law scaling at intermediate momenta - induced by finite size and boundaries - depends on the boundary conditions applied. The predictions are confirmed by using correlation functions calculated in Papers I and II. Two-dimensional Heisenberg antiferromagnets have attracted much attention because of their relation to high-Tc superconductors. The second part of the thesis (Papers IV and V) treats spin-1/2 antiferromagnets defined on square- and honeycomb lattices with frustrating interactions. By using a Schwinger-boson mean-field theory, we calculate low-temperature quantum corrections to thermodynamic parameters, such as spin-wave velocity and transverse susceptibility. Their dependence on frustration is explicated and used to estimate the stability of the Néel state via a mapping to the nonlinear .sigma. model that describes the low-energy limit of the antiferromagnet.