Dorota Jakubczyk

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.

Ground-state and magnetocaloric properties of a coupled spin–electron double-tetrahedral chain (exact study at the half filling)

Ground-state and magnetocaloric properties of a double-tetrahedral chain, in which nodal lattice sites occupied by the localized Ising spins regularly alternate with triangular clusters half filled with mobile electrons, are exactly investigated by using the transfer-matrix method in combination with the construction of the Nth tensor power of the discrete Fourier transformation. It is shown that the ground state of the model is formed by two non-chiral phases with the zero residual entropy and two chiral phases with the finite residual entropy S=NkBln2. Depending on the character of the exchange interaction between the localized Ising spins and mobile electrons, one or three magnetization plateaus can be observed in the magnetization process. Their heights basically depend on the values of Lande g-factors of the Ising spins and mobile electrons. It is also evidenced that the system exhibits both the conventional and inverse magnetocaloric effect depending on values of the applied magnetic field and temperature.

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.

On the SU(2)×SU(2) symmetry in the Hubbard model

We discuss the one-dimensional Hubbard model, on finite sites spin chain, in context of the action of the direct product of two unitary groups SU(2)×SU(2). The symmetry revealed by this group is applicable in the procedure of exact diagonalization of the Hubbard Hamiltonian. This result combined with the translational symmetry, given as the basis of wavelets of the appropriate Fourier transforms, provides, besides the energy, additional conserved quantities, which are presented in the case of a half-filled, four sites spin chain. Since we are dealing with four elementary excitations, two quasiparticles called “spinons”, which carry spin, and two other called “holon” and “antyholon”, which carry charge, the usual spin-SU(2) algebra for spinons and the so called pseudospin-SU(2) algebra for holons and antiholons, provide four additional quantum numbers.

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.

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.

Combinatorial approach to the representation of the Schur-Weyl duality in one-dimensional spin systems

We propose combinatorial approach to the representation of Schur-Weyl duality in physical systems on the example of one-dimensional spin chains. Exploiting the Robinson-Schensted-Knuth algorithm, we perform decomposition of the dual group representations into irreducible representations in a fully combinatorial way. As representation space, we choose the Hilbert space of the spin chains, but this approach can be easily generalized to an arbitrary physical system where the Schur-Weyl duality works.

An alternative approach to the construction of Schur-Weyl transform

We propose an alternative approach for the construction of the unitary matrix which performs generalized unitary rotations of the system consisting of independent identical subsystems (for example spin system). This matrix, when applied to the system, results in a change of degrees of freedom, uncovering the information hidden in non-local degrees of freedom. This information can be used, inter alia, to study the structure of entangled states, their classification and may be useful for construction of quantum algorithms.

On Application of the Jucys–Murphy Operators in the Hubbard Model of Solids

The operator techniques based on the Jucys–Murphy operators are applied in the procedure of an immediate diagonalization of the one-dimensional Hubbard model of solids. The Young orthogonal basis is given by the irreducible basis of the symmetric group acting on the set of nodes of the magnetic chain. An example of the attractive Hubbard model at the half-filled magnetic rings case is considered where the group SU(2)×SU(2) acts within the spin and pseudo-spin space. These techniques significantly reduce the size of eigenproblem of the Hubbard Hamiltonian.