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The coding theorem for a class of quantum channels with long-term memory
Get PDFIn this paper we consider the transmission of classical information through a class of quantum channels with long-term memory, which are given by convex combinations of product channels. Hence, the memory of such channels is given by a Markov chain which is aperiodic but not irreducible. We prove the coding theorem and weak converse for this class of channels. The main techniques that we employ, are a quantum version of Feinstein’s Fundamental Lemma [6, 11] and a generalization of Helstrom’s Theorem. [8]
Probing the fuzzy sphere regularisation in simulations of the 3d λϕ^4 model
Get PDFWe regularise the 3d λφ^4 model by discretising the Euclidean time and representing the spatial part on a fuzzy sphere. The latter involves a truncated expansion of the field in spherical harmonics. This yields a numerically tractable formulation, which constitutes an unconventional alternative to the lattice. In contrast to the 2d version, the radius R plays an independent rôle. We explore the phase diagram in terms of R and the cutoff, as well as the parameters m^2 and λ. Thus we identify the phases of disorder, uniform order and non-uniform order. We compare the result to the phase diagrams of the 3d model on a non-commutative torus, and of the 2d model on a fuzzy sphere. Our data at strong coupling reproduce accurately the behaviour of a matrix chain, which corresponds to the c = 1–model in string theory. This observation enables a conjecture about the thermodynamic limit
Calculating the Superconformal Index and Seiberg Duality
Get PDFWe develop techniques to calculate an index for four dimensional superconformal field theories. This superconformal index is counting BPS operators which preserve only one supercharge. To calculate the superconformal index we quantize the field theory on S^3×R and show that the twisted theory has an appropriate mass gap. This allows for the interactions to be switched off continuously without the superconformal index being changed. We test those techniques for theories which go through a non-trivial RG flow and for Seiberg dual theories. This leads to the conjecture of some group/number theoretical identities
Goldstone models in D + 1 dimensions, D = 3, 4, 5, supporting stable and zero topological charge solutions
Get PDFWe study finite energy static solutions to a global symmetry breaking Goldstone model described by an isovector scalar field in D + 1 spacetime dimensions. Both topologically stable multisolitons with arbitrary winding numbers, and zero topological charge soliton–antisoliton solutions are constructed numerically in D = 3, 4, 5. We have explored the types of symmetries the systems should be subjected to, for there to exist multisoliton and soliton–antisoliton pairs in D = 3, 4, 5, 6. These findings are underpinned by constructing numerical solutions in the D ≤ 5 examples. Subject to axial symmetry, only multisolitons of all topological charges exist in even D, and in odd D, only zero and unit topological charge solutions exist. Subjecting the system to weaker than axial symmetries, results in the existence of all the possiblilities in all dimensions. Our findings apply also to finite ’energy’ solutions to Yang–Mills and Yang-Mills–Higgs systems, and in principle also sigma models
Sensitivity of Nonrenormalizable Trajectories to the Bare Scale
Get PDFWorking in scalar field theory, we consider RG trajectories which correspond to nonrenormalizable theories, in the Wilsonian sense. An interesting question to ask of such trajectories is, given some fixed starting point in parameter space, how the effective action at the effective scale, Λ, changes as the bare scale (and hence the duration of the flow down to Λ) is changed. When the effective action satisfies Polchinski’s version of the Exact Renormalization Group equation, we prove, directly from the path integral, that the dependence of the effective action on the bare scale, keeping the interaction part of the bare action fixed, is given by an equation of the same form as the Polchinski equation but with a kernel of the opposite sign. We then investigate whether similar equations exist for various generalizations of the Polchinski equation. Using nonperturbative, diagrammatic arguments we find that an action can always be constructed which satisfies the Polchinski-like equation under variation of the bare scale. For the family of flow equations in which the field is renormalized, but the blocking functional is the simplest allowed, this action is essentially identified with the effective action at Λ = 0. This does not seem to hold for more elaborate generalizations
Particle-like solutions to the Yang-Mills-dilaton system in d=4+1 dimensions
Get PDFWe construct static solutions to a SU(2) Yang–Mills (YM) dilaton model in 4 + 1 dimensions subject to bi-azimuthal symmetry. The YM sector of the model consists of the usual YM term and the next higher order term of the YM hierarchy, which is required by the scaling condition for the existence of finite energy solutions. The basic features of two different types of configurations are studied, corresponding to (multi)solitons with topological charge n^2 , and soliton–antisoliton pairs with zero topological charge
d = 4 + 1 gravitating non-Abelian solutions with bi-azimuthal symmetry
Get PDFWe construct static, asymptotically flat solutions of SU(2) Einstein-Yang-Mills theory in 4 + 1 dimensions, subject to bi-azimuthal symmetry. Both particle-like and black hole solutions are considered for two different sets of boundary conditions in the Yang–Mills sector, corresponding to multisolitons and soliton-antisoliton pairs. For gravitating multi-soliton solutions, we find that their mass per unit charge is lower than the mass of the corresponding unit charge, spherically symmetric soliton
C*-algebraic approach to the Bose-Hubbard model
Get PDFWe give a new derivation of the variational formula for the pressure of the long-range-hopping Bose-Hubbard model, which was first proved in \cite{BD}. The proof is analogous to that of a theorem on noncommutative large deviations introduced by Petz, Raggio and Verbeure \cite{PRV} and could similarly be extended to more general Bose system of mean-field type. We apply this formalism to prove Bose-Einstein condensation for the case of small coupling
Two documents relating to Ó Conchubhair Donn
Get PDFReports discovery of two sixteenth-century Irish deeds in Norfolk Record Office, with edition and translatio
Duality, the Semi-Circle Law and Quantum Hall Bilayers
Get PDFThere is considerable experimental evidence for the existence in Quantum Hall systems of an approximate emergent discrete symmetry, Γ_0(2) ⊂ SL(2, Z). The evidence consists of the robustness of the tests of a suite a predictions concerning the transitions between the phases of the system as magnetic fields and temperatures are varied, which follow from the existence of the symmetry alone. These include the universality of and quantum numbers of the fixed points which occur in these transitions; selection rules governing which phases may be related by transitions; and the semi-circular trajectories in the Ohmic-Hall conductivity plane which are followed during the transitions. We explore the implications of this symmetry for Quantum Hall systems involving two charge-carrying fluids, and so obtain predictions both for bilayer systems and for single-layer systems for which the Landau levels have a spin degeneracy. We obtain similarly striking predictions which include the novel new phases which are seen in these systems, as well as a prediction for semicircle trajectories which are traversed by specific combinations of the bilayer conductivities as magnetic fields are varied at low temperatures