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It is shown in a recent preprint [arXiv:2001.10008] that the central spin model with XX-type qubit-bath coupling is integrable for a central spin $s_0=1/2$. Two types of eigenstates, separable states (dark states) and entangled states (bright states) between the central spin and the bath spins, are manifested. In this work, we show by using an operator product state approach that the XX central spin model with central spin $s_0>1/2$ and inhomogeneous coupling is partially solvable. That is, a subset of the eigenstates are obtained by the operator product state ansatz. These are the separable states and those entangled states in the single-spin-excitation subspace with respect to the fully polarized reference state. Due to the high degeneracy of the separable states, the resulting Bethe ansatz equations are found to be non-unique. In the case of $s_0=1/2$ we show that all the separable and entangled states can be written in terms of the operator product states, recovering the results in [arXiv:2001.10008]. Moreover, we also apply our method to the case of homogeneous coupling and derive the corresponding Bethe ansatz equations., 12 pages, 2 figures, to appear in Physical Review B

We investigate the ground-state energy of a Richardson-Gaudin integrable BCS model, generalizing the closed and open p+ip models. The Hamiltonian supports a family of mutually commuting conserved operators satisfying quadratic relations. From the eigenvalues of the conserved operators we derive, in the continuum limit, an integral equation for which a solution corresponding to the ground state is established. The energy expression from this solution agrees with the BCS mean-field result., Comment: Submission to SciPost, 17 pages, no figures. This version is a minor modification, in response to referee comments

Objective At the time of multiple sclerosis (MS) diagnosis, identifying those at risk for poorer health-related quality of life and emotional well-being can be a critical consideration for treatment planning. This study aimed to test whether adverse childhood experiences predict MS patients’ health-related quality of life and emotional functioning at time of diagnosis and initial course of disease. Methods We recruited patients at the time of new MS diagnosis to complete self-report surveys at baseline and a one-year follow-up. Questionnaires included the Adverse Childhood Experiences (ACEs), as well as the MS Knowledge Questionnaire (MSKQ), the 36-Item Short Form Health Survey (SF-36), and Self-Management Screening (SeMaS). Results A total of n = 31 participants recently diagnosed with relapsing remitting MS (median EDSS = 1.0, age M = 33.84 ± 8.4 years) completed the study measures. The ACEs significantly predicted health-related quality of life (SF-36) at baseline (Adjusted R 2 = 0.18, p = 0.011) and follow-up (Adjusted R 2 = 0.12, p = 0.03), baseline scores on the SeMaS Depression scale (Adjusted R 2 = 0.19, p = 0.008), as well as follow-up scores on the SeMaS Anxiety (Adjusted R 2 = 0.19, p = 0.014) and SeMaS Depression (Adjusted R 2 = 0.14, p = 0.036) scales. Importantly, increased ACEs scores were predictive of increased anxiety at the one-year follow-up assessment, compared to baseline. Conclusions Childhood adversity predicts health-related quality of life and emotional well-being at time of MS diagnosis and over the initial course of the disease. Measured using a brief screening inventory (ACEs), routine administration may be useful for identifying patients in need of increased supportive services.

Consider the statement "Every Yang-Baxter integrable system is defined to be exactly-solvable". To formalise this statement, definitions and axioms are introduced. Then, using a specific Yang-Baxter integrable bosonic system, it is shown that a paradox emerges. A generalisation for completely integrable bosonic systems is also developed., Comment: 16 pages, revised version correcting typographical errors

Using the exact Bethe Ansatz solution, we investigate methods for calculating the ground-state energy for the $p + ip$-pairing Hamiltonian. We first consider the Hamiltonian isolated from its environment (closed model) through two forms of Bethe Ansatz solutions, which generally have complex-valued Bethe roots. A continuum limit approximation, leading to an integral equation, is applied to compute the ground-state energy. We discuss the evolution of the root distribution curve with respect to a range of parameters, and the limitations of this method. We then consider an alternative approach that transforms the Bethe Ansatz equations to an equivalent form, but in terms of the real-valued conserved operator eigenvalues. An integral equation is established for the transformed solution. This equation is shown to admit an exact solution associated with the ground state. Next we discuss results for a recently derived Bethe Ansatz solution of the open model. With the aforementioned alternative approach based on real-valued roots, combined with mean-field analysis, we are able to establish an integral equation with an exact solution that corresponds to the ground-state for this case., 36 pages, 7 figures, 1 table

An exactly solved bosonic tunneling model is studied along a line of the coupling parameter space, which includes a quantum phase boundary line. The entire energy spectrum is computed analytically, and found to exhibit multiple energy level crossings in a region of the coupling parameter space. Several key properties of the model are discussed, which exhibit a clear dependence on whether the particle number is even or odd., Comment: 12 pages, 7 figures

The precise control of quantum systems will play a major role in the realization of atomtronic devices. As in the case of electronic systems, a desirable property is the ability to implement switching. Here we show how to implement switching in a model of dipolar bosons confined to three coupled wells. The model describes interactions between bosons, tunneling of bosons between adjacent wells, and the effect of an external field. We conduct a study of the quantum dynamics of the system to probe the conditions under which switching behavior can occur. The analysis considers both integrable and non-integrable regimes within the model. Through variation of the external field, we demonstrate how the system can be controlled between various switched-on and switched-off configurations., Revised Communications Physics (open access) version; Major revision: 8 pages, 6 figures; Supplementary material: 2 pages, 5 figures

The extended Bose-Hubbard model for a double-well potential with atom-pair tunneling is studied. Starting with a classical analysis we determine the existence of three different quantum phases: self-trapping, phase-locking and Josephson states. From this analysis we built the parameter space of quantum phase transitions between degenerate and non-degenerate ground states driven by the atom-pair tunneling. Considering only the repulsive case, we confirm the phase transition by the measure of the energy gap between the ground state and the first excited state. We study the structure of the solutions of the Bethe ansatz equations for a small number of particles. An inspection of the roots for the ground state suggests a relationship to the physical properties of the system. By studying the energy gap we find that the profile of the roots of the Bethe ansatz equations is related to a quantum phase transition., 10 pages, 10 figures

The Bethe Ansatz solution for the class of rational, sl(2) Richardson– Gaudin systems is presented. Completeness of this solution is discussed for the case where all operators are realised in terms of the spin-1/2 representation. This discussion is based on a set of operator identities. Next, a generalised system with broken u(1)-symmetry is introduced, which admits an analogous set of operator identities. Analysis of this generalised system shows that the Bethe Ansatz solution for it is also complete. The prospects for extending this approach to higher spin systems are mentioned.

We discuss a generalised version of Sklyanin's Boundary Quantum Inverse Scattering Method applied to the spin-1/2, trigonometric sl(2) case, for which both the twisted-periodic and boundary constructions are obtained as limiting cases. We then investigate the quasi-classical limit of this approach leading to a set of mutually commuting conserved operators which we refer to as the trigonometric, spin-1/2 Richardson-Gaudin system. We prove that the rational limit of the set of conserved operators for the trigonometric system is equivalent, through a change of variables, rescaling, and a basis transformation, to the original set of trigonometric conserved operators. Moreover we prove that the twisted-periodic and boundary constructions are equivalent in the trigonometric case, but not in the rational limit., Comment: 29 pages

We obtain the exact solutions for a family of spin-boson systems. This is achieved through application of the representation theory for polynomial deformations of the $su(2)$ Lie algebra. We demonstrate that the family of Hamiltonians includes, as special cases, known physical models which are the two-site Bose-Hubbard model, the Lipkin-Meshkov-Glick model, the molecular asymmetric rigid rotor, the Tavis-Cummings model, and a two-mode generalisation of the Tavis-Cummings model., LaTex 15 pages. To appear in Nonlinearity

We study an exactly solvable model of D(D-3) non-Abelian anyons on a one-dimensional lattice with a free coupling parameter in the Hamiltonian. For certain values of the coupling parameter level crossings occur, which divide the ground-state phase diagram into four regions. We obtain explicit expressions for the ground-state energy in each phase, for both closed and open chain boundary conditions. For the closed chain case we show that chiral phases occur which are characterised by non-zero ground-state momentum. (C) 2010 Elsevier B.V. All rights reserved.

We propose a systematic procedure for the construction of exactly solvable kN-body systems which are natural generalisations of Calogero models. As examples, we present two new 3N-body models and determine explicit expressions for their eigenvalues and eigenfunctions.

The exact solution for the energy spectrum of a one-dimensional Hamiltonian with local two-site interactions and periodic boundary conditions is determined. The two-site Hamiltonians commute with the symmetry algebra given by the Drinfeld double D(D-3) of the dihedral group D-3. As such the model describes local interactions between non-Abelian anyons, with fusion rules given by the tensor product decompositions of the irreducible representations of D(D-3). The Bethe ansatz equations which characterise the exact solution are found through the use of functional relations satisfied by a set of mutually commuting transfer matrices. Crown Copyright (C) 2010 Published by Elsevier B.V. All rights reserved.

We study the ground-state properties of an atomic-molecular Bose-Einstein condensate model through an exact Bethe Ansatz solution. For a certain range of parameter choices, we prove that the ground-state Bethe roots lie on the positive real-axis. We then use a continuum limit approach to obtain a singular integral equation characterising the distribution of these Bethe roots. Solving this equation leads to an analytic expression for the ground-state energy. The form of the expression is consistent with the existence of a line of quantum phase transitions, which has been identified in earlier studies. This line demarcates a molecular phase from a mixed phase. Certain correlation functions, which characterise these phases, are then obtained through the Hellmann-Feynman theorem.

A general formulation of the Boundary Quantum Inverse Scattering Method is given which is applicable in cases where R-matrix solutions of the Yang-Baxter equation do not have the property of crossing unitarity. Suitably modified forms of the reflection equations are presented which permit the construction of a family of commuting transfer matrices. As an example, we apply the formalism to determine the most general solutions of the reflection equations for a solution of the Yang-Baxter equation with underlying symmetry given by the Drinfeld double D(D-3) Of the dihedral group D-3. This R-matrix does not have the crossing unitarity property. In this manner we derive integrable boundary conditions for an open chain model of interacting non-Abelian anyons. Crown Copyright (C) 2008 Published by Elsevier B.V. All rights reserved.

It is known that the first two-variable Links–Gould quantum link invariant LG ≡ LG2,1 is more powerful than the HOMFLYPT and Kauffman polynomials, in that it distinguishes all prime knots (including reflections) of up to 10 crossings. Here we report investigations which greatly expand the set of evaluations of LG for prime knots. Through them, we show that the invariant is complete, modulo mutation, for all prime knots (including reflections) of up to 11 crossings, but fails to distinguish some nonmutant pairs of 12-crossing prime knots. As a byproduct, we classify the mutants within the prime knots of 11 and 12 crossings. In parallel, we learn that LG distinguishes the chirality of all chiral prime knots of at most 12 crossings. We then demonstrate that every mutation-insensitive link invariant fails to distinguish the chirality of a number of 14-crossing prime knots. This provides 14-crossing examples of chiral prime knots whose chirality is undistinguished by LG.

We study a three-mode Hamiltonian modelling a heteronuclear molecular Bose--Einstein condensate. Two modes are associated with two distinguishable atomic constituents, which can combine to form a molecule represented by the third mode. Beginning with a semi-classical analogue of the model, we conduct an analysis to determine the phase space fixed points of the system. Bifurcations of the fixed points naturally separate the coupling parameter space into different regions. Two distinct scenarios are found, dependent on whether the imbalance between the number operators for the atomic modes is zero or non-zero. This result suggests the ground-state properties of the model exhibit an unusual sensitivity on the atomic imbalance. We then test this finding for the quantum mechanical model. Specifically we use Bethe ansatz methods, ground-state expectation values, the character of the quantum dynamics, and ground-state wavefunction overlaps to clarify the nature of the ground-state phases. The character of the transition is smoothed due to quantum fluctuations, but we may nonetheless identify the emergence of a quantum phase boundary in the limit of zero atomic imbalance., Comment: 23 pages, 10 figures

Recently, generalized Bose–Fermi statistics was studied in a category theoretic framework and to accommodate this endeavor the notion of a pre-monoidal category was developed. Here we describe an algebraic approach for the construction of such categories. We introduce a procedure called twining which breaks the quasi-bialgebra structure of the universal enveloping algebras of semi-simple Lie algebras and renders the category of finite-dimensional modules pre-monoidal. The category is also symmetric, meaning that each object of the category provides representations of the symmetric groups, which allows for a generalized boson-fermion statistic to be defined. Exclusion and confinement principles for systems of indistinguishable particles are formulated as an invariance with respect to the actions of the symmetric group. We apply the procedure to suggest that the symmetries which can be associated to color, spin and flavor degrees of freedom lead to confinement of states.

We study the exact Bethe Ansatz solution of the p+ip Hamiltonian in a form whereby quantum numbers of states refer to hole-pairs, rather than particle-pairs used in previous studies. We find an asymmetry between these approaches. For the attractive system states in the strong pairing regime take the form of a quasi-condensate involving two distinct hole-pair creation operators. An analogous feature is not observed in the particle-pair picture., 19 pages, 2 figures, 2 tables

The Perk--Schultz model may be expressed in terms of the solution of the Yang--Baxter equation associated with the fundamental representation of the untwisted affine extension of the general linear quantum superalgebra $U_q[sl(m|n)]$, with a multiparametric co-product action as given by Reshetikhin. Here we present analogous explicit expressions for solutions of the Yang-Baxter equation associated with the fundamental representations of the twisted and untwisted affine extensions of the orthosymplectic quantum superalgebras $U_q[osp(m|n)]$. In this manner we obtain generalisations of the Perk--Schultz model., 10 pages, 2 figures

In this work we investigate the energy gap between the ground state and the first excited state in a model of two single-mode Bose-Einstein condensates coupled via Josephson tunneling. The energy gap is never zero when the tunneling interaction is non-zero. The gap exhibits no local minimum below a threshold coupling which separates a delocalised phase from a self-trapping phase which occurs in the absence of the external potential. Above this threshold point one minimum occurs close to the Josephson regime, and a set of minima and maxima appear in the Fock regime. Analytic expressions for the position of these minima and maxima are obtained. The connection between these minima and maxima and the dynamics for the expectation value of the relative number of particles is analysed in detail. We find that the dynamics of the system changes as the coupling crosses these points., 12 pages, 5 .eps figures + 4 figs, classical analysis, perturbation theory

In this work we investigate the quantum dynamics of a model for two single-mode Bose--Einstein condensates which are coupled via Josephson tunneling. Using direct numerical diagonalisation of the Hamiltonian, we compute the time evolution of the expectation value for the relative particle number across a wide range of couplings. Our analysis shows that the system exhibits rich and complex behaviours varying between harmonic and non-harmonic oscillations, particularly around the threshold coupling between the delocalised and self-trapping phases. We show that these behaviours are dependent on both the initial state of the system as well as regime of the coupling. In addition, a study of the dynamics for the variance of the relative particle number expectation and the entanglement for different initial states is presented in detail., 15 pages, 8 eps figures, accepted in J. Phys. A

We consider the asymptotic solutions to the Bethe ansatz equations of the integrable model of interacting bosons in the weakly interacting limit. In this limit we establish that the ground state maps to the highest energy state of a strongly-coupled repulsive bosonic pairing model.

We study the exact solution for a two-mode model describing coherent coupling between atomic and molecular Bose-Einstein condensates (BEC), in the context of the Bethe ansatz. By combining an asymptotic and numerical analysis, we identify the scaling behaviour of the model and determine the zero temperature expectation value for the coherence and average atomic occupation. The threshold coupling for production of the molecular BEC is identified as the point at which the energy gap is minimum. Our numerical results indicate a parity effect for the energy gap between ground and first excited state depending on whether the total atomic number is odd or even. The numerical calculations for the quantum dynamics reveals a smooth transition from the atomic to the molecular BEC., 5 pages, 4 figures

In this paper we investigate the construction of state models for link invariants using representations of the braid group obtained from various gauge choices for a solution of the trigonometric Yang–Baxter equation. Our results show that it is possible to obtain invariants of regular isotopy (as defined by Kauffman) which may not be ambient isotopic. We illustrate our results with explicit computations using solutions of the trigonometric Yang–Baxter equation associated with the one-parameter family of minimal typical representations of the quantum superalgebra Uq[gl(2|1)]. We have implemented MATHEMATICA code to evaluate the invariants for all prime knots up to 10 crossings.

Two different types of integrable impurities in a spin ladder system are proposed. The impurities are introduced in such a way that the integrability of the models is not violated. The models are solved exactly with the Bethe ansatz equations as well as the energy eigenvalues obtained. We show for both models that a phase transition between gapped and gapless spin excitations occurs at a critical value of the rung coupling J. In addition, the dependence of the impurities on this phase transition is determined explicitly. In one of the models the spin gap decreases by increasing the impurity strength A. Moreover, for a fixed A, a reduction in the spin gap by increasing the impurity concentration is also observed.

A pairing model for nucleons, introduced by Richardson in 1966, which describes proton-neutron pairing as well as proton-proton and neutron-neutron pairing, is re-examined in the context of the Quantum Inverse Scattering Method. Specifically, this shows that the model is integrable by enabling the explicit construction of the conserved operators. We determine the eigenvalues of these operators in terms of the Bethe ansatz, which in turn leads to an expression for the energy eigenvalues of the Hamiltonian., Comment: 14 pages, latex, no figures

This is a reply to the comment by P Schlottmann and A A Zvyagin. PACS numbers: 71.10.Fd, 71.27, 75.10.Jm In their comment [1], Schlottmann and Zvyagin raise several issues regarding the nature of integrable impurities in one-dimensional quantum lattice models, and claim to expose false statements in our recent paper [2]. In order to address these issues in a pedagogical manner, we feel that it is appropriate to discuss these questions in terms of models based on the gl(2|1) invariant solution of the Yang–Baxter equation. However, it is important from the outset to make it clear that the arguments we present below are general and apply to other classes of models. The first point that we would like to make is that it is claimed in [1] that there are two approaches to the algebraic Bethe ansatz. However, it appears to us that approach (i) described in point (2) of [1] is the coordinate Bethe ansatz and we do not understand why this should be referred to as an algebraic Bethe ansatz. In the coordinate Bethe ansatz approach, one starts with a prescribed Hamiltonian and then solves the Schrodinger equation to obtain the two-particle scattering matrices and the particle–impurity scattering matrix. Together these scattering matrices form the monodromy matrix. It is impossible to infer the Hamiltonian from such a monodromy matrix. In this context we do not feel that point (1) of [1] answers our query about the existence of the impurity monodromy matrix in the algebraic approach (ii) of point (2) in [1]. Hereafter, we focus our attention on this case. The solution of the Yang–Baxter equation R12(u− v)R13(u)R23(v) = R23(v)R13(u)R12(u− v) (1) associated with the Lie superalgebra gl(2|1) is an operator R(u) ∈ End (V ⊗ V ), where V is a three-dimensional Z2-graded space with one bosonic and two fermionic degrees of freedom. Explicitly, this operator takes the form R(u) = u · I ⊗ I + P (2) where P is the Z2-graded permutation operator. For the purposes of constructing integrable one-dimensional quantum systems on a closed lattice, it is usual to introduce the Yang–Baxter 0305-4470/02/296197+05$30.00 © 2002 IOP Publishing Ltd Printed in the UK 6197

A new integrable model which is a variant of the one-dimensional Hubbard model is proposed. The integrability of the model is verified by presenting the associated quantum R-matrix which satisfies the Yang-Baxter equation. We argue that the new model possesses the SO(4) algebra symmetry, which contains a representation of the $\eta$-pairing SU(2) algebra and a spin SU(2) algebra. Additionally, the algebraic Bethe ansatz is studied by means of the quantum inverse scattering method. The spectrum of the Hamiltonian, eigenvectors, as well as the Bethe ansatz equations, are discussed.

Form factors are derived for a model describing the coherent Josephson tunneling between two coupled Bose-Einstein condensates. This is achieved by studying the exact solution of the model within the framework of the algebraic Bethe ansatz. In this approach the form factors are expressed through determinant representations which are functions of the roots of the Bethe ansatz equations.

Exactly solvable models provide a unique method, via qualitative changes in the distribution of the ground-state roots of the Bethe Ansatz equations, to identify quantum phase transitions. Here we expand on this approach, in a quantitative manner, for two models of Bose--Einstein condensates. The first model deals with the interconversion of bosonic atoms and molecules. The second is the two-site Bose--Hubbard model, widely used to describe tunneling phenomena in Bose--Einstein condensates. For these systems we calculate the ground-state root density. This facilitates the determination of analytic forms for the ground-state energy, and associated correlation functions through the Hellmann--Feynman theorem. These calculations provide a clear identification of the quantum phase transition in each model. For the first model we obtain an expression for the molecular fraction expectation value. For the two-site Bose--Hubbard model we find that there is a simple characterisation of condensate fragmentation., Comment: 16 pages, 5 figures, 2 tables

Integrable Kondo impurities in two cases of one-dimensional q-deformed t-J models are studied by means of the boundary Z2-graded quantum inverse scattering method. The boundary K matrices depending on the local magnetic moments of the impurities are presented as nontrivial realizations of the reflection equation algebras in an impurity Hilbert space. Furthermore, these models are solved by using the algebraic Bethe ansatz method and the Bethe ansatz equations are obtained.

Two integrable quantum spin ladder systems will be introduced associated with the fundamental su(2 \2) solution of the Yang-Baxter equation. The first model is a generalized quantum Ising system with Ising rung interactions. In the second model the addition of extra interactions allows us to impose Heisenberg rung interactions without violating integrability. The existence of a Bethe ansatz solution for both models allows us to investigate the elementary excitations for antiferromagnetic rung couplings. We find that the first model does not show a gap whilst in the second case there is a gap for all positive values of the rung coupling.

A new model for a spin 1/2 ladder system with two legs is introduced. It is demonstrated that this model is solvable via the Bethe ansatz method for arbitrary values of the rung coupling J. This is achieved by a suitable mapping from the Hubbard model with appropriate twisted boundary conditions. We determine that a phase transition between gapped and gapless spin excitations occurs at the critical value J_c=1/2 of the rung coupling., Comment: 8 pages, LaTex

[Formula: see text]We introduce and study in detail an invariant of (1, 1) tangles. This invariant, derived from a family of four dimensional representations of the quantum superalgebra Uq[gl(2|1)], will be referred to as the Links–Gould invariant. We find that our invariant is distinct from the Jones, HOMFLY and Kauffman polynomials (detecting chirality of some links where these invariants fail), and that it does not distinguish mutants or inverses. The method of evaluation is based on an abstract tensor state model for the invariant that is quite useful for computation as well as theoretical exploration.

A quantum algebra invariant integrable closed spin 1 chain is introduced and analysed in detail. The Bethe ansatz equations as well as the energy eigenvalues of the model are obtained. The highest weight property of the Bethe vectors with respect to U_q(sl(2)) is proved., Comment: 13 pages, LaTeX, to appear in J. Math. Phys