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physica status solidi (b)

Linear extension of the Robinson–Schensted algorithm

Paweł Jakubczyk, Barbara Lulek

Published Jan 24, 2005• 3 citations
Publisher

Abstract

Abstract The Robinson–Schensted (RS) algorithm demonstrates a bijection between the set of magnetic configurations f and the set of pairs of tableaux: a semistandard Weyl tableau P(f) accompanied by a standard Young tableau Q(f). We show that it is not only a bijection between sets, but it can be extended to a linear unitary transformation within the space of all quantum states of the magnet. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

DOI
10.1002/pssb.200460054

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Abstract The Robinson–Schensted (RS) algorithm demonstrates a bijection between the set of magnetic configurations f and the set of pairs of tableaux: a semistandard Weyl tableau P(f) accompanied by a standard Young tableau Q(f). We show that it is not only a bijection between sets, but it can be extended to a linear unitary transformation within the space of all quantum states of the magnet. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)
Quantum kinematics of the linear Heisenberg ring, consisting of N crystal nodes, each with spin s, is presented in terms of the Weyl duality between actions of the symmetric and unitary groups in the space of quantum states of the magnet. This space is spanned on magnetic configurations which gives rise to an application of the combinatorial Robinson–Schensted–Knuth algorithm for a unique classification of irreducible basis of the duality of Weyl in terms of a pair of tableaux: a standard Young tableau in the alphabet of nodes, accompanied by a semistandard Weyl tableau in the alphabet of spins. Similarities and distinctions between various group-theoretic and combinatorial objects are discussed within the context of the magnetic interpretation. In particular, the role of the spectrum of Jucys–Murphy operators in the clasification and construction of magnetic eigenstates corresponding to Young tableaux is illustrated.
Abstract Bethe Ansatz provides an exact classification of eigenstates of the Heisenberg Hamiltonian for a finite magnetic ring, consisting of N nodes, each with the spin s = ½ (with some extensions to an arbitrary spin s) in terms of rigged string configurations. The latter are some combinatorial objects which serve as classification labels for solutions of Bethe equations. An astonishing feature is existence of Robinson–Schensted (RS) and Kerov–Kirillov–Reshetikhin (KKR) bijections between sets of (i) all magnetic configurations, (ii) all pairs of standard Young and Weyl tableaux of N boxes and n = 2s + 1 rows, (iii) all rigged string configurations, for given N and s. These bijections allow to point out an exact correspondence between physically admissible solutions of highly nonlinear Bethe Ansatz equations and the initial basis of quantum calculations – magnetic configurations which are just possible distributions of spins over the nodes. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)
The Robinson-Schensted and Kerov-Kirillov-Reshetikhin (RSKKR) bijections allow us to confirm the completeness of solutions of the eigenproblem of the one-dimensional Heisenberg Hamiltonian in a purely combinatorial manner, by studying the structure of the classical configuration space of the system. The combined bijection relates two sets, namely the basis of magnetic configurations and the set of combinatorial objects called rigged string configurations. The former serve as the initial basis for quantum computations, whereas the latter classify the exact Bethe Ansatz (BA) eigenstates. We discuss in this report the application of this bijection in the procedure of construction of two-particle states within Clebsch-Gordan scheme. This bijection provides the irreducible bases for the decomposition of the tensor product of transitive representations R ( Σ N :(Σ N-1 × Σ 1 ) within the scheme of a Hopf algebra of symmetric groups. We point out the role of true physical BA eigenstates, as well as fictituous configurations corresponding to doubly occupied sites (the diagonal of the cartesian square of the one-magnon classical configuration space) and to antisymmetric states.
We construct an efficient quantum algorithm to compute the quantum Schur-Weyl transform for any value of the quantum parameter q∈[0,∞]q \in [0,\infty]q∈[0,∞]. Our algorithm is a qqq-deformation of the Bacon-Chuang-Harrow algorithm, in the sense that it has the same structure and is identically equal when q=1q=1q=1. When q=0q=0q=0, our algorithm is the unitary realization of the Robinson-Schensted-Knuth (or RSK) algorithm, while when q=∞q=\inftyq=∞ it is the dual RSK algorithm together with phase signs. Thus, we interpret a well-motivated quantum algorithm as a generalization of a well-known classical algorithm.