51ÁÔÆæÈë¿Ú

Abstract:

Abstract Ideas similar to that of the Kauffman bracket introduced in chapter 2 can be used to generate valid anyon theories. We begin with a planar diagram algebra defined only by the value of a loop (the loop weight). One wants to build new particle types by grouping together multiple strands of loops. However, to do so one needs so-called Jones-Wenzl projectors to orthogonalize between species resulting in the so-called Temperly-Lieb Algebra. We demonstrate this construction by building a planar diagram algebra corresponding to the Z2 loop gas (bosons, fermions, semions), and then Ising Anyons. We show how this generalizes to a much bigger set of possible theories. We show how these theories can be made unitary, and how the F-matrices may be calculated. Finally we discuss how the R-matrices can be simply obtained by returning to the Kauffman bracket rules.

Abstract:

Abstract The Toric Code can also be interpreted as a phase of matter. The vertex and plaquette operators become terms of the Hamiltonian, and the ground state space becomes the code space. The various types of errors become the various types of quasiparticle excitations. We explore the braiding properties of these excitations and identify properties of the TQFT described by this phase of matter such as the S- and T-matrices. We also describe this anyon theory with a simple charge-flux model. We repeat the exercise for the ZN toric code.

Abstract:

Abstract If we partition a Hilbert space we can define the entanglement entropy (both von Neumann and Renyi) between the two pieces. We consider breaking up a system into two or more distinct spatial regions and ask about the entanglement between these two regions. 51ÁÔÆæÈë¿Úing the toric code ground state as an initial example, we show that there is an entanglement that scales as the length of the cut between the two regions, but there is also a sub-leading term, known as the topological entanglement entropy, which depends only on the topological properties of the system. While the leading term is nonuniversal, the topological term is robust and can be generalized to any other TQFT. We give three different derivations of the topological entanglement entropy.

Abstract:

Abstract An important process (discussed already in chapter 2) is the process by which a particle twists around its own axis. For a quantum mechanical particle this accumulates a phase associated with the particle’s spin. The particle spin is related in several ways to the R-matrix, and satisfies a number of important identities, including the ribbon identity.

Abstract:

The energy dissipation and entropy production by self-propelled microswimmers differ profoundly from passive particles pulled by external forces. The difference extends both to the shape of the flow around the swimmer, as well as to the internal dissipation of the propulsion mechanism. Here we derive a general theorem that provides an exact lower bound on the total, external and internal, dissipation by a microswimmer. The problems that can be solved include an active surface-propelled droplet, swimmers with an extended propulsive layer and swimmers with an effective internal dissipation. We apply the theorem to determine the swimmer shapes that minimize the total dissipation while keeping the volume constant. Our results show that the entropy production by active microswimmers is subject to different fundamental limits than the entropy production by externally driven particles.Comment: 16 pages, 6 figure