Random matrices and quantum spin chains

Random matrix ensembles are introduced that respect the local tensor structure of Hamiltonians describing a chain of $n$ distinguishable spin-half particles with nearest-neighbour interactions. We prove a central limit theorem for the density of states when $n \rightarrow\infty$, giving explicit bounds on the rate of approach to the limit. Universality within a class of probability measures and the extension to more general interaction geometries are established. The level spacing distributions of the Gaussian Orthogonal, Unitary and Symplectic Ensembles are observed numerically for the energy levels in these ensembles.Comment: Updated figures, as accepted in 'Markov Processes and Related Fields' on 3 March 201

Spectra and eigenstates of spin chain Hamiltonians

We prove that translationally invariant Hamiltonians of a chain of $n$ qubits with nearest-neighbour interactions have two seemingly contradictory features. Firstly in the limit $n\rightarrow\infty$ we show that any translationally invariant Hamiltonian of a chain of $n$ qubits has an eigenbasis such that almost all eigenstates have maximal entanglement between fixed-size sub-blocks of qubits and the rest of the system; in this sense these eigenstates are like those of completely general Hamiltonians (i.e. Hamiltonians with interactions of all orders between arbitrary groups of qubits). Secondly in the limit $n\rightarrow\infty$ we show that any nearest-neighbour Hamiltonian of a chain of $n$ qubits has a Gaussian density of states; thus as far as the eigenvalues are concerned the system is like a non-interacting one. The comparison applies to chains of qubits with translationally invariant nearest-neighbour interactions, but we show that it is extendible to much more general systems (both in terms of the local dimension and the geometry of interaction). Numerical evidence is also presented which suggests that the translational invariance condition may be dropped in the case of nearest-neighbour chains.Comment: Updated figures, as accepted in 'Communications in Mathematical Physics' on 5 January 201

Localization and its consequences for quantum walk algorithms and quantum communication

The exponential speed-up of quantum walks on certain graphs, relative to classical particles diffusing on the same graph, is a striking observation. It has suggested the possibility of new fast quantum algorithms. We point out here that quantum mechanics can also lead, through the phenomenon of localization, to exponential suppression of motion on these graphs (even in the absence of decoherence). In fact, for physical embodiments of graphs, this will be the generic behaviour. It also has implications for proposals for using spin networks, including spin chains, as quantum communication channels.Comment: 4 pages, 1 eps figure. Updated references and cosmetic changes for v

Stability of the Period-Doubled Core of the 90-degree Partial in Silicon

In a recent Letter [N. Lehto and S. Oberg, Phys. Rev. Lett. 80, 5568 (1998)], Lehto and Oberg investigated the effects of strain fields on the core structure of the 90-degree partial dislocation in silicon, especially the influence of the choice of supercell periodic boundary conditions in theoretical simulations. We show that their results for the relative stability between the two structures are in disagreement with cell-size converged tight-binding total energy (TBTE) calculations, which suggest the DP core to be more stable, regardless of the choice of boundary condition. Moreover, we argue that this disagreement is due to their use of a Keating potential.Comment: 1 page. Submitted to Comments section of PRL. Also available at http://www.physics.rutgers.edu/~dhv/preprints/rn_dcom/index.htm

Evaluation of co-production processes in a community-based mental health project in Wandsworth

The notion of partnerships and co-production has been introduced in the latest public services policies, suggesting that the key to reforming them is to encourage users to design and deliver services in equal partnerships with professionals. It is argued that co-production has the potential to deliver a major shift in the way we provide health, education, policing and other services in ways that make them much more effective, more efficient, and therefore more sustainable. This report presents findings from an evaluation study of the co-production processes in a community-based mental health project at the London Borough of Wandsworth. The evaluation sought to describe actions, changes, and functions that brought about a co-productive way of offering Improve Access to Psychological Therapies (IAPT) services in this locality. The study aimed at producing transferable knowledge about a novel model of public service provision, which was developed by Wandsworth Community Empowerment Network (WCEN) in association with the South West London and St George’s Mental Health Trust. The ‘Wandsworth Model’ entails canvassing partnerships with local faith-based and other community groups, who got engaged in co-producing responsive mental health services, in an attempt to address issues such as access and effectiveness of service delivery. The study applied a participatory research approach to capture the co-production processes that took place in establishing the partnership between the mental health services and WCEN and the impact of such initiatives in reaching out to local BME communities. Our main method of gathering evidence was narrative interviews which were conducted with key informants from the three groups involved in delivering co-produced services: IAPT professionals, WCEN workers, and community/religious leaders. The thematic interview areas were: the participants’ involvement in the co-produced services, views about co-production, benefits and challenges of co-production for all stakeholders, and suggestions for improvement. The findings for this study suggest that co-production can be very rewarding for both public agencies and communities, if supported and implemented with a view to empower people instead of making false economies for the welfare services. The ultimate goal should be that service users become partners in managing their own health however this is a major shift that requires a lot of experience and commitment in the co-production of services and, perhaps, it can only be possible when systemic barriers at community, public agency and state levels are brought down. Nonetheless, the ‘Wandsworth model’ of co-production appears to be a promising approach and should be further explored and supported to achieve its full potential

A new correlator in quantum spin chains

We propose a new correlator in one-dimensional quantum spin chains, the $s-$Emptiness Formation Probability ($s-$EFP). This is a natural generalization of the Emptiness Formation Probability (EFP), which is the probability that the first $n$ spins of the chain are all aligned downwards. In the $s-$EFP we let the spins in question be separated by $s$ sites. The usual EFP corresponds to the special case when $s=1$, and taking $s>1$ allows us to quantify non-local correlations. We express the $s-$EFP for the anisotropic XY model in a transverse magnetic field, a system with both critical and non-critical regimes, in terms of a Toeplitz determinant. For the isotropic XY model we find that the magnetic field induces an interesting length scale.Comment: 6 pages, 1 figur

Autocorrelation of Random Matrix Polynomials

We calculate the autocorrelation functions (or shifted moments) of the characteristic polynomials of matrices drawn uniformly with respect to Haar measure from the groups U(N), O(2N) and USp(2N). In each case the result can be expressed in three equivalent forms: as a determinant sum (and hence in terms of symmetric polynomials), as a combinatorial sum, and as a multiple contour integral. These formulae are analogous to those previously obtained for the Gaussian ensembles of Random Matrix Theory, but in this case are identities for any size of matrix, rather than large-matrix asymptotic approximations. They also mirror exactly autocorrelation formulae conjectured to hold for L-functions in a companion paper. This then provides further evidence in support of the connection between Random Matrix Theory and the theory of L-functions

What is the probability that a random integral quadratic form in nn variables has an integral zero?

We show that the density of quadratic forms in $n$ variables over $\mathbb Z_p$ that are isotropic is a rational function of $p$, where the rational function is independent of $p$, and we determine this rational function explicitly. When real quadratic forms in $n$ variables are distributed according to the Gaussian Orthogonal Ensemble (GOE) of random matrix theory, we determine explicitly the probability that a random such real quadratic form is isotropic (i.e., indefinite). As a consequence, for each $n$, we determine an exact expression for the probability that a random integral quadratic form in $n$ variables is isotropic (i.e., has a nontrivial zero over $\mathbb Z$), when these integral quadratic forms are chosen according to the GOE distribution. In particular, we find an exact expression for the probability that a random integral quaternary quadratic form has an integral zero; numerically, this probability is approximately $98.3\%$.Comment: 17 pages. This article supercedes arXiv:1311.554

Sustainable care: theorising the wellbeing of caregivers to older persons

The term ‘care crisis’ is invoked to denote chronic system failures and bad outcomes for the people involved. We present a comprehensive wellbeing framework and illustrate its practicality with evidence of negative outcomes for those who provide care. We find evidence of substantial material and relational wellbeing failures for family carers and for care workers, while there has been little interest in carers’ views of their ability to live the life that they most value. Understanding and improving wellbeing outcomes for carers is an essential component of sustainable care, which requires the wellbeing of the different actors in care arrangements