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physica status solidi (b)

Operator techniques in the Hopf algebra for a symmetric group within Racah calculus for Bethe Ansatz eigenfunctions

Paweł Jakubczyk, Dorota Jakubczyk, Tadeusz Lulek

Published Apr 3, 2009• 0 citations
Publisher

Abstract

Abstract We propose some operator techniques, applicable in the procedure of an immediate diagonalization for the Heisenberg chain, such as projection operators for subspaces with definite quasimomentum, total or partial spin, quasimomenta of strings etc. These techniques yield, in particular, matrices of appropriate Clebsch–Gordan coefficients, and reduce significantly the size of an effective Hamiltonian matrix. We apply the basis of wavelets to exploit the translational symmetry, along the Schwinger scheme of unitary geometry, applied to a single N‐dimensional magnon space (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

DOI
10.1002/pssb.200881558

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Abstract We propose some operator techniques, applicable in the procedure of an immediate diagonalization for the Heisenberg chain, such as projection operators for subspaces with definite quasimomentum, total or partial spin, quasimomenta of strings etc. These techniques yield, in particular, matrices of appropriate Clebsch–Gordan coefficients, and reduce significantly the size of an effective Hamiltonian matrix. We apply the basis of wavelets to exploit the translational symmetry, along the Schwinger scheme of unitary geometry, applied to a single N‐dimensional magnon space (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)
A method of separation of subspaces with definite values of the total spin, its z-projection and quasimomentum, within the state space of the XXX model, i.e. the N-th tensor power of a single qubit, is presented. We construct a complete system of orthogonal projectors onto such subspaces by use of the basis of wavelets and chosen measuring operators based on the total spin. We also demonstrate, on examples of N=6 and 8 magnetic spin rings, that the constructed projectors can be further decomposed into the orthogonal sum of density matrices of eigenstates of the XXX model in the corresponding subspaces. This decomposition exhibits a Galois symmetry, stemming from roots of indecomposable factors of characteristic polynomial of the Heisenberg Hamiltonian. In this way, we propose an alternative for Bethe Ansatz for moderate lengths of spin chains, by a return to the original spectral problem, avoiding the highly nonlinear system of Bethe Ansatz equations. Such results are easy to adapt in quantum information processing.
We have derived an explicit formula for a complete set of commuting operators for the XXX model of the Heisenberg magnetic ring, using an algebraic Bethe ansatz approach of Faddeev and Takhtajan. Each operator turns out to be the sum of increasing l -cycles in the symmetric group acting on the set of nodes of the ring. It is demonstrated that the resulting algebra of operators encloses the total spin S , the Hamiltonian and the quasimomentum. We point out that it is a maximal Abelian subalgebra in the algebra of the symmetric group, associated with the basis of exact Bethe ansatz eigenstates, the latter classified by rigged string configurations of Kerrov, Kirillov and Reshetikhin. This algebra is also conjugated to the Jucys–Murphy algebra, responsible for the Young orthogonal basis of standard tableaux along the Schur–Weyl duality.
The study of quantum models for magnetism requires the diagonalization of large matrices. For electrons occupying a lattice of sixteen sites, the dimension of these matrices is 65,536 x 65,536. I have written a program in Mathematica to exploit symmetries of the model and render the diagonalization of the Hamiltonian matrix tractable. Specifically, the program utilizes four different operators: translational in the x and y directions (Tx and Ty), total spin in one direction (Sz), and total spin squared (S2). As symmetries, all of these operators commute with the Hamiltonian and therefore the eigenstates of the Hamiltonian are simultaneously eigenstates of these operators. In practice, the eigenstates of these symmetry operators are easier to discover then those of the Hamiltonian. The most difficult symmetry to exploit is the S2 symmetry. In order to generate this symmetry, I reference Clebsch-Gordan coefficient tables. These tables exist to aid in the addition of angular momentum under the laws of quantum mechanics. The tables are generally used to couple two spins. When adding sixteen spins, the coefficients are multiplied together iteratively, adding one spin at a time up to the desired number. The products of these coefficients are then employed to construct all 65,536 eigenstates of the S2 operator. Combined with the remaining symmetry operators, these states can be used to put the Hamiltonian into a simpler, block diagonal form that is readily diagonalized.
The problem of construction of projection operators on eigensubspaces of symmetry operators is considered. This problem arises in many approximate methods for solving time-independent and time-dependent quantum problems, and its solution ensures proper physical symmetries in development of approximate methods. The projector form is sought as a function of symmetry operators and their eigenvalues characterizing the eigensubspace of interest. This form is obtained in two steps: (1) identification of algebraic structures within a set of symmetry operators (e.g., groups and Lie algebras) and (2) construction of the projection operators for individual symmetry operators. The first step is crucial for efficient projection operator construction because it allows for use of information on irreducible representations of the present algebraic structure. The discussed approaches have potential to stimulate further developments of variational approaches for electronic structure of strongly correlated systems and in quantum computing.
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)