A New Two-Parameter Family of Potentials with a Tunable Ground State

In a previous paper we solved a countably infinite family of one-dimensional Schr\"odinger equations by showing that they were supersymmetric partner potentials of the standard quantum harmonic oscillator. In this work we extend these results to find the complete set of real partner potentials of the harmonic oscillator, showing that these depend upon two continuous parameters. Their spectra are identical to that of the harmonic oscillator, except that the ground state energy becomes a tunable parameter. We finally use these potentials to analyse the physical problem of Bose-Einstein condensation in an atomic gas trapped in a dimple potential.Comment: 15 pages, 5 figure

Unbinding of giant vortices in states of competing order

Funding: EPSRC (UK) via Grants No. EP/I031014/1 and No. EP/H049584/1.We consider a two-dimensional system with two order parameters, one with O(2) symmetry and one with O(M), near a point in parameter space where they couple to become a single O(2+M) order. While the O(2) sector supports vortex excitations, these vortices must somehow disappear as the high symmetry point is approached. We develop a variational argument which shows that the size of the vortex cores diverges as 1/root Delta and the Berezinskii-Kosterlitz-Thouless transition temperature of the O(2) order vanishes as 1/1n(1/Delta), where Delta denotes the distance from the high-symmetry point. Our physical picture is confirmed by a renormalization group analysis which gives further logarithmic corrections, and demonstrates full symmetry restoration within the cores.Publisher PDFPeer reviewe

Non-equilibrium Berezinskii-Kosterlitz-Thouless Transition in a Driven Open Quantum System

The Berezinskii-Kosterlitz-Thouless mechanism, in which a phase transition is mediated by the proliferation of topological defects, governs the critical behaviour of a wide range of equilibrium two-dimensional systems with a continuous symmetry, ranging from superconducting thin films to two-dimensional Bose fluids, such as liquid helium and ultracold atoms. We show here that this phenomenon is not restricted to thermal equilibrium, rather it survives more generally in a dissipative highly non-equilibrium system driven into a steady-state. By considering a light-matter superfluid of polaritons, in the so-called optical parametric oscillator regime, we demonstrate that it indeed undergoes a vortex binding-unbinding phase transition. Yet, the exponent of the power-law decay of the first order correlation function in the (algebraically) ordered phase can exceed the equilibrium upper limit -- a surprising occurrence, which has also been observed in a recent experiment. Thus we demonstrate that the ordered phase is somehow more robust against the quantum fluctuations of driven systems than thermal ones in equilibrium.Comment: 11 pages, 9 figure

Observation of a superconducting glass state in granular superconducting diamond

The magnetic field dependence of the superconductivity in nanocrystalline boron doped diamond thin films is reported. Evidence of a glass state in the phase diagram is presented, as demonstrated by electrical resistance and magnetic relaxation measurements. The position of the phase boundary in the H-T plane is determined from resistance data by detailed fitting to zero-dimensional fluctuation conductivity theory. This allows determination of the boundary between resistive and non-resistive behavior to be made with greater precision than the standard ad hoc onset/midpoint/offset criterion

Fluctuation spectroscopy as a probe of granular superconducting diamond films

We present resistance versus temperature data for a series of boron-doped nanocrystalline diamond films whose grain size is varied by changing the film thickness. Upon extracting the fluctuation conductivity near to the critical temperature we observe three distinct scaling regions -- 3D intragrain, quasi-0D, and 3D intergrain -- in confirmation of the prediction of Lerner, Varlamov and Vinokur. The location of the dimensional crossovers between these scaling regions allows us to determine the tunnelling energy and the Thouless energy for each film. This is a demonstration of the use of \emph{fluctuation spectroscopy} to determine the properties of a superconducting granular system

Vulnerability of horticultural crop production to extreme weather events

The potential impact of future extreme weather events on horticultural crops was evaluated. A review was carried out of the sensitivities of a representative set of crops to environmental challenges. It confirmed that a range of environmental factors are capable of causing a significant impact on production, either as yield or quality loss. The most important of these were un-seasonal temperature, water shortage or excess,and storms. Future scenarios were produced by the LARS-WG1, a stochastic weather generator linked with UKCIP02 projections of future climate. For the analyses, 150 years of synthetic weather data were generated for baseline, 2020HI and 2050HI scenarios at defined locations. The output from the weather generator was used in case studies, either to estimate the frequency of a defined set of circumstances known to have impact on cropping, or as inputs to models of crop scheduling or pest phenology or survival. The analyses indicated that episodes of summer drought severe enough to interrupt the continuity of supply of salads and other vegetables will increase while the frequency of autumns with sufficient rainfall to restrict potato lifting will decrease. They also indicated that the scheduling of winter cauliflowers for continuity of supply will require the deployment of varieties with different temperature sensitivities from those in use currently. In the pest insect studies, the number of batches of Agrotis segetum (cutworm) larvae surviving to third instar increased with time, as did the potential number of generations of Plutella xylostella (diamond-back moth) in the growing season, across a range of locations. The study demonstrated the utility of high resolution scenarios in predicting the likelihood of specific weather patterns and their potential effect on horticultural production. Several limitations of the current scenarios and biological models were also identified

Completeness Results for Parameterized Space Classes

The parameterized complexity of a problem is considered "settled" once it has been shown to lie in FPT or to be complete for a class in the W-hierarchy or a similar parameterized hierarchy. Several natural parameterized problems have, however, resisted such a classification. At least in some cases, the reason is that upper and lower bounds for their parameterized space complexity have recently been obtained that rule out completeness results for parameterized time classes. In this paper, we make progress in this direction by proving that the associative generability problem and the longest common subsequence problem are complete for parameterized space classes. These classes are defined in terms of different forms of bounded nondeterminism and in terms of simultaneous time--space bounds. As a technical tool we introduce a "union operation" that translates between problems complete for classical complexity classes and for W-classes.Comment: IPEC 201

Expanding the expressive power of Monadic Second-Order logic on restricted graph classes

We combine integer linear programming and recent advances in Monadic Second-Order model checking to obtain two new algorithmic meta-theorems for graphs of bounded vertex-cover. The first shows that cardMSO1, an extension of the well-known Monadic Second-Order logic by the addition of cardinality constraints, can be solved in FPT time parameterized by vertex cover. The second meta-theorem shows that the MSO partitioning problems introduced by Rao can also be solved in FPT time with the same parameter. The significance of our contribution stems from the fact that these formalisms can describe problems which are W[1]-hard and even NP-hard on graphs of bounded tree-width. Additionally, our algorithms have only an elementary dependence on the parameter and formula. We also show that both results are easily extended from vertex cover to neighborhood diversity.Comment: Accepted for IWOCA 201

Polynomial Kernels for Weighted Problems

Kernelization is a formalization of efficient preprocessing for NP-hard problems using the framework of parameterized complexity. Among open problems in kernelization it has been asked many times whether there are deterministic polynomial kernelizations for Subset Sum and Knapsack when parameterized by the number $n$ of items. We answer both questions affirmatively by using an algorithm for compressing numbers due to Frank and Tardos (Combinatorica 1987). This result had been first used by Marx and V\'egh (ICALP 2013) in the context of kernelization. We further illustrate its applicability by giving polynomial kernels also for weighted versions of several well-studied parameterized problems. Furthermore, when parameterized by the different item sizes we obtain a polynomial kernelization for Subset Sum and an exponential kernelization for Knapsack. Finally, we also obtain kernelization results for polynomial integer programs

Vertex Cover Kernelization Revisited: Upper and Lower Bounds for a Refined Parameter

An important result in the study of polynomial-time preprocessing shows that there is an algorithm which given an instance (G,k) of Vertex Cover outputs an equivalent instance (G',k') in polynomial time with the guarantee that G' has at most 2k' vertices (and thus O((k')^2) edges) with k' <= k. Using the terminology of parameterized complexity we say that k-Vertex Cover has a kernel with 2k vertices. There is complexity-theoretic evidence that both 2k vertices and Theta(k^2) edges are optimal for the kernel size. In this paper we consider the Vertex Cover problem with a different parameter, the size fvs(G) of a minimum feedback vertex set for G. This refined parameter is structurally smaller than the parameter k associated to the vertex covering number vc(G) since fvs(G) <= vc(G) and the difference can be arbitrarily large. We give a kernel for Vertex Cover with a number of vertices that is cubic in fvs(G): an instance (G,X,k) of Vertex Cover, where X is a feedback vertex set for G, can be transformed in polynomial time into an equivalent instance (G',X',k') such that |V(G')| <= 2k and |V(G')| <= O(|X'|^3). A similar result holds when the feedback vertex set X is not given along with the input. In sharp contrast we show that the Weighted Vertex Cover problem does not have a polynomial kernel when parameterized by the cardinality of a given vertex cover of the graph unless NP is in coNP/poly and the polynomial hierarchy collapses to the third level.Comment: Published in "Theory of Computing Systems" as an Open Access publicatio