Rigged String Configurations Research Articles

Analytical and Combinatorial Aspects of the Eigenproblem for the Two-Magnon Sector of XXX Heisenberg Rings

In this paper we study both analytical and combinatorial properties of solutions of the eigenproblem for the Heisenberg s-1/2 model for two deviations. Our analysis uses Chebyshev polynomials, inverse Bethe Ansatz, winding numbers and rigged string configurations. We show some combinatorial aspects of strings in a geometric way. We discuss some exceptions from the connection between the combinatorial nature of an eigenstate and the analytical type of a solution of the eigenproblem. In particular, as an illustration of the aforementioned exceptions, we analyze the singularities of Bethe parameters for bound states at the border of the Brillouin zone.

Galois symmetry of string structure of quanta of phase in the XXX heptagonal ring

An arithmetic qubit, separated from the state space of the heptagonal magnetic ring within the XXX model in the three-magnon sector at the center of the Brillouin zone, is shown to have a Galois symmetry, stemming from a polynomial f of degree 6 indecomposable over the prime field Q, with admissible quanta of phase as its roots. The Galois group is shown to be isomorphic with the dihedral group D6, and the main Galois theorem for this case is demonstrated in detail, including generators for each subfield, the corresponding minimal polynomials, as well as intepretation in terms of rigged string configurations. The splitting number field for the polynomial f proves to be the arena for exact solutions of Bethe Ansatz equations for this case.

Classification of entanglement via rigged string configurations

We investigate the structure of bipartite entanglement in one-dimensional spin systems. The standard approach is based on a reduced two-qubit density matrix and uses concurrence as the measure of entanglement. We show that for a spin system consisting of four qubits the structure of bipartite entanglement can be classified by rigged string configurations in a strictly combinatorial manner.

Computer implementation of the Kerov–Kirillov–Reshetikhin algorithm

The Kerov–Kirillov–Reshetikhin (KKR) algorithm establishes a bijection between semistandard Weyl tableaux of the shape λ and weight μ and rigged string configurations of type (λ,μ). This algorithm can be applied to Heisenberg magnetic chains and their generalisations by use of Schur–Weyl duality, and gives a way to classify all eigenstates of the chain in a methodological manner. Program summaryProgram title: KKR algorithmCatalogue identifier: AELQ_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AELQ_v1_0.htmlProgram obtainable from: CPC Program Library, Queenʼs University, Belfast, N. IrelandLicensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.htmlNo. of lines in distributed program, including test data, etc.: 974No. of bytes in distributed program, including test data, etc.: 11 118Distribution format: tar.gzProgramming language: MapleComputer: PCOperating system: Windows, LinuxRAM: 1 GBClassification: 2.7, 4.2Nature of problem: The method of classification of all eigenstates and eigenvalues is given for the generalised model of the one-dimensional Heisenberg magnet consisting of N nodes.Solution method: We use a prolongation of the RSK bijection [1] to the set of all eigenstates of an integrable model, in terms of rigged string configurations. More specifically, one selects from the irreducible basis birr of the Schur–Weyl duality all highest weight states |λthwy〉 with respect to the unitary group U(n) and generates a rigged string configurations |ν→,{J}〉=KKR(|λthwy〉) for each y∈SYT(λ).Running time: The running time is about 1 ms.Reference:[1]S. Veigneau, Algebraic Combinatorics Environment for the computer algebra system, Userʼs Reference Manual, Version 3.0 98-11, Université de Marne-la-Vallé, Institut Gaspard-Monge, CNRS, 1998.

Algebraic and geometric properties of Bethe Ansatz eigenfunctions on a pentagonal magnetic ring

The exact solution of the eigenproblem of the Heisenberg Hamiltonian for the XXX model in the case of a magnetic ring with N=5 nodes is presented in a closed algebraic form. It is demonstrated that the eigenproblem itself is expressible within the extension of the prime field Q of rationals by the primitive fifth root of unity, whereas the associated Bethe parameters, i.e. pseudomomenta, phases of scattering, and spectral parameters, require some finite field extensions, such that the nonlinearity remains algebraic rather than transcendental. Classification of exact Bethe Ansatz eigenstates in terms of rigged string configurations is presented.

A complete set of commuting operators for the Bethe ansatz

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.

Rigged strings and quasimomenta in Bethe Ansatz

Abstract 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.

Magnonic qudit and algebraic Bethe Ansatz

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 2N-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.

Bethe Ansatz and Geometry of the Classical Configuration Space

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 .

Bethe ansatz, quantum computers, and unitary geometry

The one-dimensional ring of N magnetic nodes, each node with the spin 1/2, can be seen as a faithful prototype of a quantum computer with N qubits. We point out here some analogies between typical problems and known solutions of the Heisenberg model of a magnet on this ring, and the related questions posed within the midst of quantum information processing. In particular, we relate such notions as the memory, gates, or entanglement of quantum states with exact results of Bethe Ansatz, expressed in combinatoric terms of rigged string configurations. A promising and transparent interpretation of relation between Bethe Ansatz and quantum computation can be provided by the unitary geometry of Schwinger, presented in our previous SSPCM schools by prof. Vourdas.

The role of Robinson-Schensted and Kerov-Kirillov-Reshetikhin bijections in Bethe ansatz

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.

An explicit demonstration of integrability for the linear Heisenberg chain of N = 9 nodes

AbstractWe propose the system of N – 1 = 8 mutually commuting operators which classify all eigenstates of the Heisenberg Hamiltonian for the linear magnetic chain of N = 9 nodes, each with the spin 1/2. These eigenstates, determined exactly by Bethe Ansatz and classified combinatorially by rigged string configurations of Kerov, Kirillov and Reshetikhin, can be also associated with the spectrum of this set of commuting operators. (© 2007 WILEY‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

Rigged strings, Bethe Ansatz, and the geometry of the classical configuration space of the Heisenberg magnetic ring

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 1 2 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 2 l consecutive nodes: first l spin deviations, and then l nodes with the spin projection + 1 2 , 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 1 2 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 C N on the classical configuration space for r Bethe pseudoparticles imposes the corresponding arrangement of rigged string configurations. Essentially, within a single cycle along a C N -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.

Quasimomenta of string configurations

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.

Geometry and rigged strings in Bethe Ansatz

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.

A combinatoric description of Bethe Ansatz solutions for nanoscopic systems

AbstractBethe 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 &amp; Co. KGaA, Weinheim)