Bethe Ansatz equations for the classical A^(1)_n affine Toda field theories

We establish a correspondence between classical A(1)n affine Toda field theories and An Bethe Ansatz systems. We show that the connection coefficients relating specific solutions of the associated classical linear problem satisfy functional relations of the type that appear in the context of the massive quantum integrable model.1751-812

Perturbed conformal field theory, nonlinear integral equations and spectral problems

This thesis is concerned with various aspects of perturbed conformal field theory and the methods used to calculate finite-size effects of integrable quantum field theories. Nonlinear integral equations are the main tools to find the exact ground-state energy of a quantum field theory. The thermodyamic Bethe ansatz (TBA) equations are a set of examples and are known for a large number of models. However, it is also an interesting question to find exact equations describing the excited states of integrable models. The first part of this thesis uses analytical continuation in a continuous parameter to find TBA like equations describing the spin-zero excited states of the sine-Gordon model at coupling β(^2) = 16π/3. Comparisons are then made with a further type of nonlinear integral equation which also predicts the excited state energies. Relations between the two types of equation are studied using a set of functional relations recently introduced in integrable quantum field theory. A relevant perturbation of a conformal field theory results in either a massive quantum field theory such as the sine-Gordon model, or a different massless conformal field theory. The second part of this thesis investigates flows between conformal field theories using a nonlinear integral equation. New families of flows are found which exhibit a rather unexpected behaviour. The final part of this thesis begins with a review of a connection between integrable quantum field theory and properties of certain ordinary differential equations of second- and third-order. The connection is based on functional relations which appear on both sides of the correspondence; for the second-order case these are exactly the functional relations mentioned above. The results are extended to include a correspondence between n(^th) order differential equations and Bethe ansatz system of SU(n) type. A set of nonlinear integral equations are derived to check the results

Exact solution of the p+ip pairing Hamiltonian and a hierarchy of integrable models

Using the well-known trigonometric six-vertex solution of the Yang-Baxter equation we derive an integrable pairing Hamiltonian with anyonic degrees of freedom. The exact algebraic Bethe ansatz solution is obtained using standard techniques. From this model we obtain several limiting models, including the pairing Hamiltonian with p+ip-wave symmetry. An in-depth study of the p+ip model is then undertaken, including a mean-field analysis, analytic and numerical solution of the Bethe ansatz equations, and an investigation of the topological properties of the ground-state wavefunction. Our main result is that the ground-state phase diagram of the p+ip model consists of three phases. There is the known boundary line with gapless excitations that occurs for vanishing chemical potential, separating the topologically trivial strong pairing phase and the topologically non-trivial weak pairing phase. We argue that a second boundary line exists separating the weak pairing phase from a topologically trivial weak coupling BCS phase, which includes the Fermi sea in the limit of zero coupling. The ground state on this second boundary line is the Moore-Read state.Comment: 65 pages, 11 figures, 3 table

Rational Solutions of the Fifth Painlev\'e Equation. Generalised Laguerre Polynomials

In this paper rational solutions of the fifth Painlev\'e equation are discussed. There are two classes of rational solutions of the fifth Painlev\'e equation, one expressed in terms of the generalised Laguerre polynomials, which are the main subject of this paper, and the other in terms of the generalised Umemura polynomials. Both the generalised Laguerre polynomials and the generalised Umemura polynomials can be expressed as Wronskians of Laguerre polynomials specified in terms of specific families of partitions. The properties of the generalised Laguerre polynomials are determined and various differential-difference and discrete equations found. The rational solutions of the fifth Painlev\'e equation, the associated $\sigma$-equation and the symmetric fifth Painlev\'e system are expressed in terms of generalised Laguerre polynomials. Non-uniqueness of the solutions in special cases is established and some applications are considered. In the second part of the paper, the structure of the roots of the polynomials are determined for all values of the parameter. Interesting transitions between root structures through coalescences at the origin are discovered, with the allowed behaviours controlled by hook data associated with the partition. The discriminants of the generalised Laguerre polynomials are found and also shown to be expressible in terms of partition data. Explicit expressions for the coefficients of a general Wronskian Laguerre polynomial defined in terms of a single partition are given.Comment: 44 pages, 22 figure

Ordinary Differential Equations and Integrable Models

We review a recently-discovered link between the functional relations approach to integrable quantum field theories and the properties of certain ordinary differential equations in the complex domain.Comment: 12 pages, 5 figures, JHEP proceedings style. Uses epsfig, amssymb. Talk given at the conference `Nonperturbative Quantum Effects 2000', Pari

The asymmetric quantum Rabi model and generalised Pöschl–Teller potentials

Starting with the Gaudin-like Bethe ansatz equations associated with the quasi-exactly solved (QES) exceptional points of the asymmetric quantum Rabi model (AQRM) a spectral equivalence is established with QES hyperbolic Schrödinger potentials on the line. This leads to particular QES Pöschl-Teller potentials. The complete spectral equivalence is then established between the AQRM and generalised Pöschl-Teller potentials. This result extends a previous mapping between the symmetric quantum Rabi model and a QES Pöschl-Teller potential. The complete spectral equivalence between the two systems suggests that the physics of the generalised Pöschl-Teller potentials may also be explored in experimental realisations of the quantum Rabi model

Validation of loci at 2q14.2 and 15q21.3 as risk factors for testicular cancer.

Testicular germ cell tumor (TGCT), the most common cancer in men aged 18 to 45 years, has a strong heritable basis. Genome-wide association studies (GWAS) have proposed single nucleotide polymorphisms (SNPs) at a number of loci influencing TGCT risk. To further evaluate the association of recently proposed risk SNPs with TGCT at 2q14.2, 3q26.2, 7q36.3, 10q26.13 and 15q21.3, we analyzed genotype data on 3,206 cases and 7,422 controls. Our analysis provides independent replication of the associations for risk SNPs at 2q14.2 (rs2713206 at P = 3.03 × 10-2; P-meta = 3.92 × 10-8; nearest gene, TFCP2L1) and rs12912292 at 15q21.3 (P = 7.96 × 10-11; P-meta = 1.55 × 10-19; nearest gene PRTG). Case-only analyses did not reveal specific associations with TGCT histology. TFCP2L1 joins the growing list of genes located within TGCT risk loci with biologically plausible roles in developmental transcriptional regulation, further highlighting the importance of this phenomenon in TGCT oncogenesis

Snapshot PCR surveillance for SARS-CoV-2 in hospital staff in England

Background: Significant nosocomial transmission of SARS-CoV-2 has been demonstrated. Understanding the prevalence of SARS-CoV-2 carriage amongst HCWs at work is necessary to inform the development of HCW screening programmes to control nosocomial spread. / Methods: Cross-sectional ‘snapshot’ survey from April-May 2020; HCWs recruited from six UK hospitals. Participants self-completed a health questionnaire and underwent a combined viral nose and throat swab, tested by Polymerase Chain Reaction (PCR) for SARS-CoV-2 with viral culture on majority of positive samples. / Findings: Point prevalence of SARS-CoV-2 carriage across the sites was 2.0% (23/1152 participants), median cycle threshold value 35.70 (IQR:32.42–37.57). 17 were previously symptomatic, two currently symptomatic (isolated anosmia and sore throat); the remainder declared no prior or current symptoms. Symptoms in the past month were associated with threefold increased odds of testing positive (aOR 3.46, 95%CI 1.38–8.67; p = 0.008). SARS-CoV-2 virus was isolated from only one (5%) of nineteen cultured samples. A large proportion (39%) of participants reported symptoms in the past month. / Interpretation: The point-prevalence is similar to previous estimates for HCWs in April 2020, though a magnitude higher than in the general population. Based upon interpretation of symptom history and testing results including viral culture, the majority of those testing positive were unlikely to be infectious at time of sampling. Development of screening programmes must balance the potential to identify additional cases based upon likely prevalence, expanding the symptoms list to encourage HCW testing, with resource implications and risks of excluding those unlikely to be infectious with positive tests. / Funding: Public Health England