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# Dr Stephen MᶜAteer

## A diagrammatic approach to factorizing F-matrices in XXZ and XXX spin chains

Stephen Gerard Mc Ateer (supervised by Omar Foda)

Abstract: The aim of this thesis is a better understanding of certain mathematical structures which arise in integrable spin chains. Specifically, we are concerned with XXZ and XXX Heisenberg spin-$\frac{1}{2}$ chains and their generalizations.

The mathematical structures in question are the $F$-matrix (a symmetrising, change-of-basis operator) and the Bethe eigenvectors (the eigenvectors of the transfer matrix of integrable spin chains). A diagrammatic tensor notation represents these operators in a way which is intuitive and allows easy manipulation of the relations involving them. The sun F-matrix is a representation of a Drinfel’d twist of the $R$-matrix of the quantum algebra $U_q(\mathfrak{su}_n)$ and its associated Yangian $Y(su_n)$. The $F$-matrices of these algebras have proven useful in the calculation of scalar products and domain wall partition functions in the spin-$\frac{1}{2}$ XXZ model. In this thesis we present a factorized diagrammatic expression for the $\mathfrak{su}_2$ $F$-matrix equivalent to the algebraic expression of Maillet and Sanchez de Santos. Next we present a fully factorized expression for the $\mathfrak{su}_n$& $F$-matrix which is of a similar form to that of Maillet and Sanchez de Santos [18] for the $\mathfrak{su}_2$ $F$-matrix and equivalent to the unfactorized expression of Albert, Boos, Flume and Ruhlig [2] for the $\mathfrak{su}_n$ $F$-matrix.

Using a diagrammatic description of the nested algebraic Bethe Ansatz, we present an expression for the eigenvectors of the $\mathfrak{su}_n$ transfer matrix as components of appropriately selected $\mathfrak{su}_n$ $F$-matrices. Finally, we present expressions for the sun elementary matrices (and therefore the local spin operators in the case of $\mathfrak{su}_2$) in terms of components of the $\mathfrak{su}_n$ monodromy matrix.

(PhD thesis, available here, submitted October 2015.)

## Usage of wharf space in Cairns and Darwin by RAN vessels (DSTO client report)

Justin D. Beck, Stephen G. Mc Ateer and Timothy J. Surendonk

Summary: In this report, navigational and operational data is used to provide estimates on Royal Australian Navy (RAN) vessel numbers in Cairns and Darwin.

(DSTO client report: DSTO-CR-2013-0238, not to be cited without approval, title and summary approved for public release, image credit: Ken Hodge)

## Supporting Defence decision making with big data: less talk, more data! (DSTO discussion paper)

Timothy J. Surendonk, Justin D. Beck and Stephen G. Mc Ateer

Summary: In this paper, we discuss big data in the context of Defence science.

(DSTO discussion paper: DSTO-DP-1255, not to be cited without approval, title and summary approved for public release, image credit: U.S. Navy photo by Journalist 1st Class James Pinsky)

## Going around in circles: a heuristic method for path categorisation

S.G. Mc Ateer and J.D. Beck

Abstract: In this paper, we describe a heuristic method for the categorisation of paths in a timed geospatial data set. Based on their shape and speed, paths are categorised as ‘slow’, ‘straight’, ‘miscellaneous’ or ‘overlapping’. The method is demonstrated using Global Positioning System (GPS) data obtained from HMAS GLENELG during patrols and transits near Christmas Island. The method enables analysis and categorisation of paths without reference to an external data source (such as ship’s logs) which explicitly indicates activities over time.

(Australian Defence Operations Research Review 2012, 94-109, title and abstract approved for public release, image credit: Davidarichter)

## HMAS Glenelg speed profiling using operational categorisation: analysis details (DSTO technical report)

S.G. Mc Ateer and J.D. Beck

Abstract: The Defence White Paper 2009 instructed Defence to develop proposals to rationalise Australia's small vessel capability by replacing several classes of vessel with a single rerolable class—the future Offshore Combatant Vessel (OCV) (Project SEA 1180). Global Positioning System (GPS) readings from current Armidale Class Patrol Boat HMAS GLENELG were combined with operational data and a heuristic categorisation method to provide speed profiles broken down by type of operation. The results of this study inform the future requirements through the generation of speed profiles using operational categorisation.

(DSTO Report Series, 2013, report number: DSTO-TR-2896, title and abstract approved for public release, image credit: Ken Hodge)

## On factorizing $F$-matrices in $\mathcal{Y}(sl_n)$ and $\mathcal{U}_q(\widehat{sl}_n)$ spin chains

S.G. Mc Ateer and M. Wheeler*

Abstract: We consider quantum spin chains arising from $N$-fold tensor products of the fundamental evaluation representations of $\mathcal{Y}(sl_n)$ and $\mathcal{U}_q(\widehat{sl}_n)$. Using the partial $F$-matrix formalism from the seminal work of Maillet and Sanchez de Santos, we derive a completely factorized expression for the $F$-matrix of such models and prove its equivalence to the expression obtained by Albert, Boos, Flume and Ruhlig. A new relation between the $F$-matrices and the Bethe eigenvectors of these spin chains is given.

(Journal of Statistical Mechanics: theory and experiment, also available from arXiv:1112.0839 [math-ph], 30 April 2012, *alphabetical)

## Factorizing $F$-matrices and the XXZ spin-$\frac{1}{2}$ chain: A diagrammatic perspective

S.G. Mc Ateer and M. Wheeler*

Abstract: Using notation inherited from the six-vertex model, we construct diagrams that represent the action of the factorizing $F$-matrices associated to the finite length $XXZ$ spin-$\frac{1}{2}$ chain. We prove that these $F$-matrices factorize the tensor $R^{\sigma}_{1... n}$ corresponding with elements of the permutation group. We consider in particular the diagram for the tensor $R^{\sigma_c}_{1... n}$, which cyclically permutes the spin chain. This leads us to a diagrammatic construction of the local spin operators $S_i^{\pm}$ and $S_i^{z}$ in terms of the monodromy matrix operators.

(Nuclear Physics B, also available from arXiv:1103.4488 [math-ph], 30 May 2011, *alphabetical)

## Unjamming as structural instability

S.G. Mc Ateer

Abstract: The transition from solid-like to liquid-like behaviour of granular materials is investigated through analysis of mechanical stability beyond the scale of the individual contact: i.e on the scale of clusters of particles in the mesoscopic domain. This solid-liqiud like phase transition, otherwise known as "unjamming" in the physics community or "failure" in the engineering community, nucleates from inside the material. Mesoscopic structures emerge inside the material during deformation: like any structure, these are subject to instabilities that ultimately lead to failure. The failure of such structures on the mesoscale serves as "trigger events" or precursors to unjamming of the granular material on the macroscopic scale.

(honours thesis, Available here in PDF format, Submitted 23 November 2007)

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