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Journal of Mathematical Physics

Combinatorial approach to the representation of the Schur-Weyl duality in one-dimensional spin systems

Dorota Jakubczyk, Paweł Jakubczyk

Published Feb 1, 2018• 6 citations
Publisher

Abstract

We propose combinatorial approach to the representation of Schur-Weyl duality in physical systems on the example of one-dimensional spin chains. Exploiting the Robinson-Schensted-Knuth algorithm, we perform decomposition of the dual group representations into irreducible representations in a fully combinatorial way. As representation space, we choose the Hilbert space of the spin chains, but this approach can be easily generalized to an arbitrary physical system where the Schur-Weyl duality works.

DOI
10.1063/1.5006328

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We propose combinatorial approach to the representation of Schur-Weyl duality in physical systems on the example of one-dimensional spin chains. Exploiting the Robinson-Schensted-Knuth algorithm, we perform decomposition of the dual group representations into irreducible representations in a fully combinatorial way. As representation space, we choose the Hilbert space of the spin chains, but this approach can be easily generalized to an arbitrary physical system where the Schur-Weyl duality works.
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.
The Schur-Weyl states belong to a special class of states with a symmetry described by two Young and Weyl tableaux. Representation of physical systems in Hilbert space spanned on these states enables to extract quantum information hidden in nonlocal degrees of freedom. Such property can be very useful in a broad range of problems in Quantum Computations, especially in quantum algorithms constructions, therefore it is very important to know exact form of these states. Moreover, they allow to reduce significantly the size of eigenproblem, or in general, diminishing the representation matrix of any physical quantities, represented in the symmetric or unitary group algebra. Here we present a new method of Schur-Weyl states construction in a spin chain system representation. Our approach is based on the fundamental shift operators out of which one can build Clebsch-Gordan coefficients for the unitary group U(n) and then derive appropriate Schur-Weyl state probability amplitudes.
We present a detailed analysis of certain quantum spin systems with inhomogeneous (non-random) mean-field interactions. Examples include, but are not limited to, the interchange- and spin singlet projection interactions on complete bipartite graphs. Using two instances of the representation theoretic framework of Schur–Weyl duality, we can explicitly compute the free energy and other thermodynamic limits in the models we consider. This allows us to describe the phase transition, the ground-state phase diagram, and the expected structure of extremal states.