Tadeusz Lulek

The basis of wavelets for a finite Heisenberg magnet

Abstract We propose here a method for an immediate diagonalisation of the XXX Heisenberg magnetic ring for the spin 1 2 with a finite number N of nodes. The key ingredient is the basis of wavelets, i.e. Fourier transforms on orbits of the translation group on the set of all magnetic configurations with a definite number r of spin deviations from ferromagnetic saturation. The method demonstrates the Yang–Baxter structure of the relevant classical configuration space, reduces by N the size of the secular equation, and explains the dynamics of the Heisenberg ring in terms of geometry of orbits. We show in particular, that a κ-tuply rarefied orbit, with κ a divisor of N, stimulates neighbour regular (N-element) orbits to twist in a way to become a κ-tuply cover of the adjacent rarefied orbit.

The duality of Weyl and linear extension of Kostka matrices

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.

Schwinger geometry, Bethe Ansatz, and a magnonic qudit

Schwinger approach of unitary geometry for a finite-dimensional Hilbert space is interpreted in terms of a magnonic qudit — a hypothetic elementary unit of memory of a quantum computer. The space is interpreted within the Heisenberg model for a magnetic ring, its calculational basis as the classical configuration space for a single spin deviation, treated as a Bethe pseudoparticle, and the dual basis corresponds to quasimomenta, so that the classical phase space spans the quantum algebra of observables. Effects of the Schur-Weyl duality and Bethe ansatz exact eigenstates of the Heisenberg Hamiltonian for the XXX model on properties of the magnonic qudit are presented.

Construction of Kostka matrix at the level of bases

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.

Propagators of N anyons on a ℂ plane

Formulae for propagators of the system of N anyons on a plane ℂ, obeying the exoticstatistics v are explicitly derived in terms of v-equivariant anyonic harmonic functions, bihomogeneous with respect to both holomorphic and antiholomorphic positional variables, for the case of free particles and particles in the potential field of a harmonic oscillator. In the latter case, we consider the cases of absence and presence of a homogeneous magnetic field.

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.

Fourier and Schur-Weyl transforms applied to XXX Heisenberg magnet

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.

Algebraic Bethe Ansatz for N = 4

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.

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.