Ansatz solutions of magnetic pentagonal ring B. Lulek,1 T. Lulek,2 M. Łabuz,3 and J. Milewski4 1East European State Higher School, Tymona Terleckiego 6, 37-700 Przemyśl, Poland 2Faculty of Physics, Adam Mickiewicz University, Umultowska 85, 61-614 Poznan, Poland 3Department of Theoretical Physics, Faculty of Mathematics and Natural Sciences, University of Rzeszow, Pigonia 1, 35-310 Rzeszow, Poland 4Institute of Mathematics, Poznan University of Technology, Piotrowo 3A, 60-965 Poznan, Poland

The exact Bethe eigenfunctions for the heptagonal ring within the isotropic XXX model exhibit a doubly degenerated energy level in the three-deviation sector at the centre of the Brillouin zone. We demonstrate an explicit construction of these eigenfunctions by use of algebraic Bethe Ansatz, and point out a relation of degeneracy to parity conservation, applied to the configuration of strings for these eigenfunctions. Namely, the internal structure of the eigenfunctions (the 2-string and the 1-string, with opposite quasimomenta) admits generation of two mutually orthogonal eigenfunctions due to the fact that the strings which differ by their length are distinguishable objects.

In this paper the field generated by the Bethe parameters related to the XXX model for the Heisenberg pentagon is considered. For the interior of the Brillouin zone, the Galois group of the Bethe number field over the rationals is determined. This Galois group is recognized as the group of arithmetic symmetries of the Bethe parameters.

We propose a modification of the classification of exact Bethe Ansatz eigenstates for a finite Heisenberg magnetic ring in terms of rigged string configurations of Kerov, Kirillov and Reshetikhin. The key change consists in replacing riggings by appropriate quasimomenta resulting from translational symmetry of the ring, and yields an interpretation of strings as objects which are kinematically independent each of another. The size of admissible and forbidden part of the Brillouin zone for strings of the length l is pointed out.

In the present paper we explore some useful devices for discussing the exact solutions of XXX isotropic Heisenberg Hamiltonian for the single node spin equal to 1/2 as the tensor product states. Our aim is to presents the monodromy matrix and the Lax operator in the contex of the Bethe Ansatz, signed from now on BA. We construct the former, for N equal to 4 nodes of the magnet, using the so called auxiliary space which is taken as a copy of 2. The form of this matrix in the basis of orbits of the translation group C4 reveals its block structure of all possible nonzero elements. Each block has its meaning in the language of creation and anihilation of the magnon. This fact implies, that one can think about appropriate operators and create the theory very similar to that of the quantum oscillator.

A magnonic qudit is proposed as the memory unit of a register of a quantum computer. It is the N-dimensional space, extracted from the 2(N)-dimensional space of all quantum states of the magnetic Heisenberg ring of N spins 1/2, as the space of all states of a single magnon. Three bases: positional, momentum, and that of Weyl duality are described, together with appropriate Fourier and Kostka transforms. It is demonstrated how exact Bethe Ansatz (BA) eigenfunctions, classified in terms of rigged string configurations, can be coded using a collection of magnonic qudits. To this aim, the algebraic BA is invoked, such that a single magnonic qudit is prepared in a state corresponding to a magnon in one of the states provided by spectral parameters emerging from the corresponding BA equations.

Similarities and differences between Fourier and Schur-Weyl transforms have been discussed in the context of a one-dimensional Heisenberg magnetic ring with N nodes. We demonstrate that main difference between them correspond to another partitioning of the Hilbert space of the magnet. In particular, we point out that application of the quantum Fourier transform corresponds to splitting of the Hilbert space of the model into subspaces associated with the orbits of the cyclic group, whereas, the Schur-Weyl transform corresponds to splitting into subspaces associated with orbits of the symmetric group.

We demonstrate that, the. seminal one-dimensional model of the Heisenberg magnet, consisting of N spins 1/2 with the nearest-neighbour isotropic interaction, solved exactly by Bethe ansatz, admits an interpretation of a system of r = N/2 - M pseudoparticles (spin deviations) which are indistinguishable, have hard cores and move on the chain by local hoppings. Such an approach allows us to construct a manifold with some boundaries, which is genericly r-dimensional, and whose F-dimensional regions, 0 < F < r, point out all l-strings. The latter classify, in terms of rigged string configurations of Kerov, Kirillov and Reshetikhin, all exact Bethe eigenfunctions. In this way, we interpret these eigenfunctions in terms of the classical configuration space, in particular on the structure of islands of adjacent spin deviations, in a way independent of the size N.

This volume comprises the proceedings of the Ninth Summer School on Theoretical Physics under the leading title `Symmetry and Structural Properties of Condensed Matter' (SSPCM 2007). The school, organised by Rzeszów University of Technology, Poland, together with AGH University of Science and Technology, Cracow, Poland, in 5–12 September 2007 in Myczkowce. The meeting aimed to continue the series of biannual SSPCM schools (since 1990), and focused on the promotion of some advanced mathematical methods within the physics of condensed matter, with an emphasis on quantum information aspects.

We develop a method of construction of transformation matrix between two bases of the model of Heisenberg magnet. The first one is a natural basis of magnetic configurations while the second is adjusted to the irreducible basis of the duality of Weyl. Proposed method allows us to calculate each matrix element separately, so it does not depend on the dimension of the system. Calculation of a matrix elements is given by ladder construction of consecutive letters of magnetic configurations along the well known Robinson-Schensted algorithm. In this way we obtain a graph with vertices given by Gelfand patterns and edges labelled by insertion algorithm. This graph allows us to read off all Clebsch-Gordan coefficients for a unitary group U(n) and then to calculate the matrix element.

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(ΣN:(ΣN-1×Σ1) 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.

A method for evaluation of Kostka matrices at the level of bases, and determination of related irreducible basis of the Weyl duality is proposed. The method bases on Jucys–Murphy operators which constitute a complete set of commuting Hermitian operators along the general Dirac formalism of quantum mechanics, applied to the algebra of a symmetric group. The way of construction of appropriate projection operators is pointed out, and the combinatorial meaning of the path on the Young graph, corresponding to a standard Young tableau, is made transparent.

Within Bethe ansatz approach, the system of r Bethe pseudoparticles on a magnetic ring of N nodes is characterised by a finite set of magnetic configurations which can be interpreted as a classical configuration space. This space can be embedded into r ‐dimensional manifold. Here we present the fibre bundle structure of this model manifold. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Composition of two combinatoric algorithms, Robinson–Schensted (RS) and Kerov–Kirillov–Reshetikhin (KKR), defines the bijection which maps the set of all magnetic configurations of the Heisenberg ring of N nodes with the spin 12 onto that of all exact Bethe Ansatz (BA) eigenstates of the nearest neighbour isotropic Hamiltonian. We point out here that this bijection allows one to predict all quantum numbers of an exact BA eigenstate (encoded as rigged string configurations of KKR) from each magnetic configuration. Moreover, this bijection provides completeness of solutions on each orbit of the symmetric group of N nodes, acting on the set of all magnetic configurations. Each such an orbit can be interpreted as the classical configuration space of the system of r Bethe pseudoparticles—spin deviations which are hard-core objects, moving on the magnetic ring by jumps to non-occupied nearest neighbours. Such an interpretation provides a transparent and combinatorially unique description of all exact BA eigenstates in terms of magnetic configurations—the initial basis for quantum calculations. Within this picture, an l-string introduced by Bethe corresponds to an extended object, consisting of 2l consecutive nodes: first l spin deviations, and then l nodes with the spin projection +12, all bounded together and put somewhere inside the magnetic chain. Each BA solution is a rigged string configuration, i.e. a distribution of a number q, 0⩽q⩽r, of such objects inside the chain. Allowed distributions are subjected to certain combinatoric restrictions (rules of navigation), expressed in terms of l-holes and riggings, and presented graphically as paths, associated with schemes of consecutive coupling of N spins 12 along the RS algorithm. We discuss here thoroughly some implications of existence of such a bijection. In particular, we demonstrate the way in which structure of orbits of the translation group CN on the classical configuration space for r Bethe pseudoparticles imposes the corresponding arrangement of rigged string configurations. Essentially, within a single cycle along a CN-orbit, each l-string moves from the left to the right with increase of its rigging by one unit until reaching the last node N of the ring, then shortens its length to zero, and next arises at the first (leftmost) node and elongates up to its previous length l. Such considerations allow us also to discuss the geography of rigged string configurations on the classical configuration space. In particular, we put emphasis on the fact that—within this combinatoric picture—the l-strings originate from corresponding sizes of islands of consecutive Bethe pseudoparticles in the classical configuration space. Thus, the set of all magnetic configurations with r spin deviations acquires the interpretation of an r-dimensional manifold with F-dimensional boundaries, the integer F, 1⩽F⩽r, being the number of islands of consecutive Bethe pseudoparticles. Each magnetic configuration belonging to the generic part F=r yields only a 1-string under considered here bijection, whereas l-strings with l>1 arise from islands located in appropriate boundaries.

We sketch a reciprocal space analogue of the combinatorial bijection of Robinson- Schensted and Kerov-Kirillov-Reshetikhin (RSKKR) between magnetic configurations (the initial basis for quantum calculations of the eigenproblem of the Heisenberg Hamiltonian for a one-dim finite Heisenberg chain), and rigged string configurations (the classification labels for the exact results of Bethe Ansatz). Existence of such a bijection admits an interpretation of the exact quantum numbers of riggings as quasimomenta of l-strings. The extended size of an l-string results in selection rules for these quasimomenta, and thus in a division of the Brillouin zone into compact subzones of forbidden and allowed states of the system of coupled Bethe pseudoparticles. The forbidden Brillouin subzone for a particular l-string is evidently the effect of kinematical restrictions for motions of constituent Bethe pseudoparticles. These restrictions can be easily predicted in a combinatorially unique way due to completness of the RSKKR bijection.

Application of the Robinson-Schensted algorithm to the basis of magnetic configurations of the one-dimensional Heisenberg magnet with an arbitrary spin gives an efficient way for a classification of the irreducible basis of the Weyl duality. The plactic monoid is shown to be an adequate tool for describing this irreducible basis in a way consistent with the Schensted insertion procedure, i.e. the creation of a new single-particle state (a letter of the single-node spin) in already constructed Young and Weyl tableaux. Schensted insertion is interpreted in terms of Gelfand triangles - combinatoric analogues of Weyl tableaux with exposed occupation numbers, consistent with canonical chains of subgroups of both the symmetric and the unitary group. A transition matrix between these two bases should exist due to the linear structure of the Hilbert space. This matrix can be looked at as the linear extension of the famous Kostka matrix. We show how to obtain this matrix and give an interpretation of its elements as coefficients of certain wave packet with exactly defined symmetry.

The main purpose of this report is a thorough analysis of completeness of solutions of the one-dimensional Heisenberg Hamiltonian through the hypothesis of strings. A somehow astonishing conclusion emerges from studying of the structure of the classical configuration space of this system. Namely, all allowed information concerning quantum states, which are exact solutions of the Bethe equations, encoded in quantum numbers, are predictable via a bijection between the set of the magnetic configurations and the string configurations. This startling and beautiful observation constitutes the proof of the completeness of the eigenstates of the Heisenberg Hamiltonian, deduced in a purely combinatorial way. We interpret the set of all magnetic configurations with a fixed number r of spin deviations as the classical configuration space of a hypothetic system of r Bethe pseudoparticles, which move, in a stroboscopic manner, on the magnetic ring. The geometry of this configuration space, induced by the action of Heisenberg Hamiltonian and the translation symmetry group of the ring, implies the structure of a locally r-dimensional hypercubic lattice with well defined F-dimensional boundaries, 1 ⩽ F ⩽ r. We demonstrate that rigged string configurations originate from these boundaries, depending upon the island structure of spin deviations. We show that a relatively simple combinatoric definition of rigged strings reproduces completely exact results of Bethe Ansatz. It is expressed in terms of a combined bijection: Robinson-Schensted with Kerov- Kirillov-Reshetikhin (RSKKR) which produces a geography of exact Bethe Ansatz solutions on the classical configuration space.

In this work we want to amplify the RS algorithm in the language of paths, Thus, it is a continuation of the previous work [Lulek et al., to appear in Mol. Phys.]. where the magnetic interpretation of RS algorithm was given. We also want to present the intermediative role of paths in bijection between standard Young tableaux and rigged configurations. We will treat the paths as a "bridge" which allow us to move between this two objects. (c) 2005 WILEY-VCH Verlag GmbH & Co, KGaA Weinheim.

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)

The Bethe Ansatz can be treated using two approaches: The Co-ordinate Bethe Ansatz (CBA) and The Algebraic Bethe Ansatz (ABA). In this paper the relation between the solutions given by these two methods is addressed. The considerations are based on the finite Heisenberg magnet with N = 8 nodes and for r = 4 reversed spins s = 1/2. Both methods provide consistent solutions in the entire Brillouin zone (BZ). Moreover there is a special point - a border of the BZ - for both approaches, in which the solutions are not well-defined.