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We consider the one dimensional, periodic spin chain with $N$ sites, similar to the one studied by Haldane \cite{hal}, however in the opposite limit of very large anisotropy and small nearest neighbour, anti-ferromagnetic exchange coupling between the spins, which are of large magnitude $s$.  For a  chain with an even number of sites we show that actually the ground state is non degenerate and given by a superposition of the two Néel states, due to quantum spin tunnelling.  With an odd number of sites, the Néel state must necessarily contain a soliton. The position of the soliton is arbitrary thus the  ground state is $N$-fold degenerate.  This set of  states reorganizes into a band. We show that this occurs at order $2s$ in perturbation theory.  The ground state is non-degenerate for integer spin, but degenerate for half-odd integer spin as is required by Kramer's theorem \cite{kram}. arXiv:1404.6706 , Phys.Lett. A378 (2014) 3066-3069; arXiv:1304.3734 Phys.Rev. B88 (2013) 22, 220403 

I will present recent theoretical work on cluster Mott insulators (CMI) in which interesting physics such as emergent charge lattices, charge fractionalization and quantum spin liquids are proposed. For the anisotropic Kagome system like LiZn2Mo3O8, we find two distinct CMIs, type-I and type-II, can arise from the repulsive interactions. In type-I CMI, the electrons are localized in one half of the triangle clusters of the Kagome system while the electrons in the type-II CMI are localized in every triangle cluster. Both CMIs are U(1) quantum spin liquids (QSL) in the weak Mott regime with a spinon Fermi surface and gapped charge excitations. In type-II CMI, however, the charge fluctuations lead to a long-range plaquette charge order that breaks the lattice symmetry, gives rise to an emergent charge lattice and reconstructs the mean-field spinon band structure of the underlying U(1) QSL. Such a reconstruction gives a consistent prediction of the "fractional spin susceptibility" that is observed in LiZn2Mo3O8.  For the pyrochlore system, the CMI can further support a charge fractionalization with an emergent gauge photon in the charge sector in addition to the spin fractionalization in the spin sector.

One hallmark of topological phases with broken time reversal symmetry is the appearance of quantized non-dissipative transport coefficients, the archetypical example being the quantized Hall conductivity in quantum Hall states. Here I will talk about two other non-dissipative transport coefficients that appear in such systems - the Hall viscosity and the thermal Hall conductivity. In the first part of the talk, I will start by reviewing previous results concerning the Hall viscosity, including its relation to a topological invariant known as the shift. Next, I will show how the Hall viscosity can be computed from a Kubo formula. For Galilean invariant systems, the Kubo formula implies a relationship between the viscosity and conductivity tensors which may have relevance for experiment. In the second part of the talk, I will discuss the thermal Hall conductivity, its relation to the central charge of the edge theory, and in particular the absence of a bulk contribution to the thermal Hall current. I will do this by constructing a low-energy effective theory in a curved non-relativistic background, allowing for torsion. I will show that the bulk contribution to the thermal current takes the form of an "energy magnetization" current, and hence show that it does not contribute to heat transport.  

Two-dimensional interacting electron gas in strong transverse magnetic field forms a collective state -- incompressible electron liquid, known as fractional quantum Hall (FQH) state. FQH states are genuinely new states of matter with long range topological order. Their primary observable characteristics are the absence of dissipation and quantization of the transverse electro-magnetic response known Hall conductance. In addition to quantized electromagnetic response FQH states are characterized by quantized geometric responses such as Hall viscosity and thermal Hall conductance.
  I will show how to derive the effective action for various Abelian and non-Abelian FQH states on a curved space. In particular, I will derive the quantized universal responses to the changes in geometry of space. These responses are described by Chern-Simons-type terms. It will be shown that in order to obtain the responses in a self consistent way one has to take into account the framing anomaly of the quantum Chern-Simons(-Witten) theory. This peculiar phenomenon illustrates the failure of a classically topological theory to remain topological at the quantum level.
   If time permits I will comment on the coupling of non-relativistic systems to the space-time geometry. Using the appropriate geometry I will write an effective action describing the bulk energy and thermal Hall conductances. From this effective action it will be clear that these response functions are neither universal nor topologically protected.

In (relatively) recent years some philosophers of physics have developed and advocated a new view about how to understand spatiotemporal structures posited in theories such as classical mechanics, relativistic theories and GR; it is called the 'dynamical approach' to spacetime (H. Brown 2005, Physical Relativity).  The dynamical approach (DA) holds that spacetime structure should not be thought of as conceptually prior to the laws of nature, or as constraining the forms that the laws may have. Instead, the DA approach says that the laws of nature are prior, and spacetime structures are no more than a reflection, or codification, of facts (especially symmetry facts) about the dynamical laws in our world. In my talk I will explore the motivations and arguments given in support of the dynamical approach, and raise doubts about whether they are coherent and compelling. Although no-one should come away from my talk with a perfect understanding of the nature of spacetime (or even just: spacetime as it appears in classical relativistic theories), I hope that all will come to appreciate the difficulty of deciding what even clear and mathematically well-understood physical theories really tell us about basic aspects of physical reality.