arXiv subjects

Allan John Gerrard

Publications and source records attributed to Allan John Gerrard.

4 recordsLinked to original sources

arXiv2026

Perfect transmission and parallel composition for quantum walks on graphs with two leads

We study scattering for continuous-time quantum walks on finite graphs with two attached leads. We derive explicit formulae for the two-terminal scattering matrix in terms of characteristic polynomials of the finite graph and its vertex-deleted subgraphs. For real-weighted two-terminal graphs, we then introduce three real quantities, $μ_1$, $μ_2$, and $ν$, which are each additive under parallel composition of graphs. In these variables, perfect transmission at fixed momentum is characterized by the condition $μ_1=μ_2$ together with a hyperbola in the corresponding $(μ,ν)$-plane, whose points determine the transmission phase. This turns the search for graphs with prescribed transmission properties into a geometric vector-sum problem for smaller building blocks.

quant-ph↗

arXiv2026

Multiple commutation relations of the quantum affine algebra $U_q(\widehat{\mathfrak{gl}}_N)$, nested Bethe vector and the Gelfand-Tsetlin basis

We study a certain type of multiple commutation relations of the quantum affine algebra $U_q(\widehat{\mathfrak{gl}}_N)$. We show that all the coefficients in the multiple commutation relations between the $L$-operator elements are given in terms of the trigonometric weight functions for the vector representation, independent of the representation of the $L$-operator. For rank one case, our proof also gives a conceptual understanding why the coefficients can also be expressed using the Izergin-Korepin determinants. As a related result, by specializing expressions for the universal nested Bethe vector by Pakuliak-Ragoucy-Slavnov, we also find a construction of the Gelfand-Tsetlin basis for the vector representation using different $L$-operator elements from the constructions by Nazarov-Tarasov or Molev. We also present corresponding results for the Yangian $Y_h(\mathfrak{gl}_N)$.

math.QA↗

arXiv2024

Higher rank elliptic partition functions and multisymmetric elliptic functions

We introduce and investigate a class of $\mathfrak{gl}_{M+1}$ partition functions which is an extension of the one introduced by Foda-Manabe. We characterize the partition functions by a nested version of Izergin-Korepin analysis, and determine the explicit forms, for each of the rational, trigonometric and elliptic versions. The resulting multisymmetric functions can be regarded as extensions of the rational, trigonometric and elliptic weight functions.

math-ph↗

arXiv2024

On the nested algebraic Bethe ansatz for spin chains with simple $\mathfrak{g}$-symmetry

We propose a new framework for the nested algebraic Bethe ansatz for a closed, rational spin chain with $\mathfrak{g}$-symmetry for any simple Lie algebra $\mathfrak{g}$. Starting the nesting process by removing a single simple root from $\mathfrak{g}$, we use the residual $U(1)$ charge and the block Gauss decomposition of the $R$-matrix to derive many standard results in the Bethe ansatz, such as the nesting of Yangian algebras, and the AB commutation relation.

math-ph↗