Critical points and symmetries of a free energy function for biaxial nematic liquid crystals

We describe a general model for the free energy function for a homogeneous medium of mutually interacting molecules, based on the formalism for a biaxial nematic liquid crystal set out by Katriel {\em et al.} (1986) in an influential paper in {\em Liquid Crystals} {\bf 1} and subsequently called the KKLS formalism. The free energy is expressed as the sum of an entropy term and an interaction (Hamiltonian) term. Using the language of group representation theory we identify the order parameters as averaged components of a linear transformation, and characterise the full symmetry group of the entropy term in the liquid crystal context as a wreath product $SO(3)\wr Z_2$. The symmetry-breaking role of the Hamiltonian, pointed out by Katriel {\em et al.}, is here made explicit in terms of centre manifold reduction at bifurcation from isotropy. We use tools and methods of equivariant singularity theory to reduce the bifurcation study to that of a $D_3\,$-invariant function on ${\bf R}^2$, ubiquitous in liquid crystal theory, and to describe the 'universal' bifurcation geometry in terms of the superposition of a familiar swallowtail controlling uniaxial equilibria and another less familiar surface controlling biaxial equilibria. In principle this provides a template for {\em all} nematic liquid crystal phase transitions close to isotropy, although further work is needed to identify the absolute minima that are the critical points representing stable phases.Comment: 74 pages, 17 figures : submitted to Nonlinearit

Causal Fermion Systems: A Quantum Space-Time Emerging from an Action Principle

Causal fermion systems are introduced as a general mathematical framework for formulating relativistic quantum theory. By specializing, we recover earlier notions like fermion systems in discrete space-time, the fermionic projector and causal variational principles. We review how an effect of spontaneous structure formation gives rise to a topology and a causal structure in space-time. Moreover, we outline how to construct a spin connection and curvature, leading to a proposal for a "quantum geometry" in the Lorentzian setting. We review recent numerical and analytical results on the support of minimizers of causal variational principles which reveal a "quantization effect" resulting in a discreteness of space-time. A brief survey is given on the correspondence to quantum field theory and gauge theories.Comment: 23 pages, LaTeX, 2 figures, footnote added on page

Automorphic Equivalence within Gapped Phases of Quantum Lattice Systems

Gapped ground states of quantum spin systems have been referred to in the physics literature as being `in the same phase' if there exists a family of Hamiltonians H(s), with finite range interactions depending continuously on $s \in [0,1]$, such that for each $s$, H(s) has a non-vanishing gap above its ground state and with the two initial states being the ground states of H(0) and H(1), respectively. In this work, we give precise conditions under which any two gapped ground states of a given quantum spin system that 'belong to the same phase' are automorphically equivalent and show that this equivalence can be implemented as a flow generated by an $s$-dependent interaction which decays faster than any power law (in fact, almost exponentially). The flow is constructed using Hastings' 'quasi-adiabatic evolution' technique, of which we give a proof extended to infinite-dimensional Hilbert spaces. In addition, we derive a general result about the locality properties of the effect of perturbations of the dynamics for quantum systems with a quasi-local structure and prove that the flow, which we call the {\em spectral flow}, connecting the gapped ground states in the same phase, satisfies a Lieb-Robinson bound. As a result, we obtain that, in the thermodynamic limit, the spectral flow converges to a co-cycle of automorphisms of the algebra of quasi-local observables of the infinite spin system. This proves that the ground state phase structure is preserved along the curve of models $H(s), 0\leq s\leq 1$.Comment: Updated acknowledgments and new email address of S

Why must we work in the phase space?

We are going to prove that the phase-space description is fundamental both in the classical and quantum physics. It is shown that many problems in statistical mechanics, quantum mechanics, quasi-classical theory and in the theory of integrable systems may be well-formulated only in the phase-space language.Comment: 130 page