Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1501.07801 · Journal of Statistical Mechanics Theory and Experiment

Heisenberg model and rigged configurations

Pulak Ranjan Giri, Tetsuo Deguchi

Published Jul 8, 2015• 3 citations
arXivHTMLPDF

Abstract

We show a correspondence of all the solutions of the Bethe ansatz equations of the one-dimensional spin-1/2 isotropic Heisenberg model with periodic boundary conditions for N = 12 to the rigged configurations. This mapping is based on the comparison of the set of Takahashi quantum numbers arranged in lexicographical order with the set of riggings of the rigged configurations arranged in co-lexicographical order.

DOI
10.1088/1742-5468/2015/07/p07007

Related papers

Bethe's quantum numbers and rigged configurations

Anatol N. Kirillov, Reiho Sakamoto

We propose a method to determine the quantum numbers, which we call the rigged configurations, for the solutions to the Bethe ansatz equations for the spin-1/2 isotropic Heisenberg model under the periodic boundary condition. Our method is based on the observation that the sums of Bethe's quantum numbers within each string behave particularly nicely. We confirm our procedure for all solutions for length 12 chain (totally 923 solutions).

arXiv:1509.02305•Nuclear Physics B•Feb 27, 2016•1 citation

Singular solutions to the Bethe ansatz equations and rigged configurations

Anatol N. Kirillov, Reiho Sakamoto

We provide a conjecture for the following two quantities related to the spin- isotropic Heisenberg model defined over rings of even lengths: (i) the number of solutions to the Bethe ansatz equations which correspond to non-zero Bethe vectors; (ii) the number of physical singular solutions of the Bethe ansatz equations in the sense of Nepomechie and Wang 2013 J. Phys. A: Math. Theor. 46 325002. The conjecture is based on a natural relationship between the solutions to the Bethe ansatz equations and the rigged configurations.

arXiv:1402.0651•Journal of Physics A Mathematical and Theoretical•May 6, 2014•24 citations

Rigged configurations approach for the spin-1/2 isotropic Heisenberg model

Reiho Sakamoto

We continue the rigged configurations analysis of the solutions to the Bethe ansatz equations for the spin-1/2 isotropic Heisenberg model. We analyze the non self-conjugate strings of Deguchi–Giri and the counter example for the string hypothesis discovered by Essler–Korepin–Schoutens. In both cases clear discrete structures appear.

arXiv:1501.00532•Journal of Physics A Mathematical and Theoretical•Apr 2, 2015•1 citation

Geometry and rigged strings in Bethe Ansatz

Dorota Jakubczyk, Tadeusz Lulek, Paweł Jakubczyk +1 more

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.

Journal of Physics Conference Series•Feb 28, 2006•4 citations

A combinatoric description of Bethe Ansatz solutions for nanoscopic systems

Tadeusz Lulek

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)

physica status solidi (b)•Jan 17, 2005•4 citations

Bethe Ansatz and Geometry of the Classical Configuration Space

Barbara Lulek, Tadeusz Lulek, Jan Milewski

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 .

Acta Physica Polonica A•Jan 1, 2009•4 citations
We show a correspondence of all the solutions of the Bethe ansatz equations of the one-dimensional spin-1/2 isotropic Heisenberg model with periodic boundary conditions for N = 12 to the rigged configurations. This mapping is based on the comparison of the set of Takahashi quantum numbers arranged in lexicographical order with the set of riggings of the rigged configurations arranged in co-lexicographical order.
We provide a conjecture for the following two quantities related to the spin- isotropic Heisenberg model defined over rings of even lengths: (i) the number of solutions to the Bethe ansatz equations which correspond to non-zero Bethe vectors; (ii) the number of physical singular solutions of the Bethe ansatz equations in the sense of Nepomechie and Wang 2013 J. Phys. A: Math. Theor. 46 325002. The conjecture is based on a natural relationship between the solutions to the Bethe ansatz equations and the rigged configurations.
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.
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)
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 .