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arXiv:0709.0104 · Journal of Physics Conference Series

Symmetry classes connected with the magnetic Heisenberg ring

B Fiedler

Published Mar 1, 2008• 1 citation
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Abstract

It is well-known (see [1]) that for a Heisenberg magnet symmetry operators and symmetry classes can be defined in a very similar way as for tensors (see e.g. [2, 3, 4]). Newer papers which consider the action of permutations on the Hilbert space Ti of the Heisenberg magnet are [5, 6, 7, 8]. We define symmetry classes and commutation symmetries in the Hilbert space H of the 1D spin-1/2 Heisenberg magnetic ring with N sites and investigate them by means of tools from the representation theory of symmetric groups C[ S N ] such as decompositions of ideals of the group ring C[ S N ], idempotents of C[ S N ], discrete Fourier transforms of S N , Littlewood-Richardson products. In particular, we determine smallest symmetry classes and stability subgroups of both single eigenvectors v and subspaces U of eigenvectors of the Hamiltonian H of the magnet. Expectedly, the symmetry classes defined by stability subgroups of v or U are bigger than the corresponding smallest symmetry classes of v or U , respectively. The determination of the smallest symmetry class for U bases on an algorithm which calculates explicitely a generating idempotent for a non-direct sum of right ideals of C[ S N ]. Let U (r1,r2) μ be a subspace of eigenvectors of a fixed eigenvalue μ of H with weight ( r 1 , r 2 ). If one determines the smallest symmetry class for every v U (r1,r2) μ then one can observe jumps of the symmetry behaviour. For "generic" v U (r1,r2) μ all smallest symmetry classes have the same maximal dimension d and 'structure'. But U (r1,r2) μ can contain linear subspaces on which the dimension of the smallest symmetry class of v jumps to a value smaller than d. Then the stability subgroup of v can increase. We can calculate such jumps explicitely. In our investigations we use computer calculations by means of the Mathematica packages PERMS and HRing.

DOI
10.1088/1742-6596/104/1/012035

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It is well-known (see [1]) that for a Heisenberg magnet symmetry operators and symmetry classes can be defined in a very similar way as for tensors (see e.g. [2, 3, 4]). Newer papers which consider the action of permutations on the Hilbert space Ti of the Heisenberg magnet are [5, 6, 7, 8]. We define symmetry classes and commutation symmetries in the Hilbert space H of the 1D spin-1/2 Heisenberg magnetic ring with N sites and investigate them by means of tools from the representation theory of symmetric groups C[ S N ] such as decompositions of ideals of the group ring C[ S N ], idempotents of C[ S N ], discrete Fourier transforms of S N , Littlewood-Richardson products. In particular, we determine smallest symmetry classes and stability subgroups of both single eigenvectors v and subspaces U of eigenvectors of the Hamiltonian H of the magnet. Expectedly, the symmetry classes defined by stability subgroups of v or U are bigger than the corresponding smallest symmetry classes of v or U , respectively. The determination of the smallest symmetry class for U bases on an algorithm which calculates explicitely a generating idempotent for a non-direct sum of right ideals of C[ S N ]. Let U (r1,r2) μ be a subspace of eigenvectors of a fixed eigenvalue μ of H with weight ( r 1 , r 2 ). If one determines the smallest symmetry class for every v U (r1,r2) μ then one can observe jumps of the symmetry behaviour. For "generic" v U (r1,r2) μ all smallest symmetry classes have the same maximal dimension d and 'structure'. But U (r1,r2) μ can contain linear subspaces on which the dimension of the smallest symmetry class of v jumps to a value smaller than d. Then the stability subgroup of v can increase. We can calculate such jumps explicitely. In our investigations we use computer calculations by means of the Mathematica packages PERMS and HRing.
Abstract. It is well-known (see [1]) that for a Heisenberg magnet symmetry operators and symmetry classes can be defined in a very similar way as for tensors (see e.g. [2, 3, 4]). Newer papers which consider the action of permutations on the Hilbert space H of the Heisenberg magnet are [5, 6, 7, 8]. We define symmetry classes and commutation symmetries in the Hilbert space H of the 1D spin-1/2 Heisenberg magnetic ring with N sites and investigate them by means of tools from the representation theory of symmetric groups SN such as decompositions of ideals of the group ring C[SN], idempotents of C[SN], discrete Fourier transforms of SN, Littlewood-Richardson products. In particular, we determine smallest symmetry classes and stability subgroups of both single eigenvectors v and subspaces U of eigenvectors of the Hamiltonian H of the magnet. Expectedly, the symmetry classes defined by stability subgroups of v or U are bigger than the corresponding smallest symmetry classes of v or U, respectively. The determination of the smallest symmetry class for U bases on an algorithm which calculates explicitely a generating idempotent for a non-direct sum of right ideals of C[SN]. Let U (r1,r2)
Abstract. In [1] we defined symmetry classes, commutation symmetries and symmetry operators in the Hilbert space H of the 1D spin-1/2 Heisenberg magnetic ring with N sites and investigated them by means of tools from the representation theory of symmetric groups SN such as decompositions of ideals of the group ring C[SN], idempotents of C[SN], discrete Fourier transforms of SN, Littlewood-Richardson products. In the present paper we consider symmetry operators a ∈ C[SN] on H which fulfil a ∗ = a, where a denotes the complex conjugate of a and the element a ∗ ∈ C[SN] of a = P p app is defined by a ∗ = P p app−1. Such symmetry operators are self-adjoint because of the relation 〈au|v 〉 = 〈u|a ∗ v 〉 which holds true for all a ∈ C[SN] and u, v ∈ H. They yield consequently observables of the quantum mechanical Heisenberg model. We prove the following results:
Symmetries of finite Heisenberg groups represent an important tool for the study of deeper structure of finite-dimensional quantum mechanics. This short contribution presents extension of previous investigations to composite quantum systems comprised of k subsystems which are described with position and momentum variables in n i , i = 1,..., κ . Their Hilbert spaces are given by κ -fold tensor products of Hilbert spaces of dimensions n 1 ,..., n κ . Symmetry group of the corresponding finite Heisenberg group is given by the quotient group of a certain normalizer. We provide the description of the symmetry groups for arbitrary multipartite cases. The new class of symmetry groups represents very specific generalization of finite symplectic groups over modular rings.
Finite spin models, applicable to investigations of mesoscopic rings, give rise to eigenproblems of very large dimensions.Efficient, and as exact as possible, solutions of such eigenproblems are very difficult.A method leading to block diagonalization of Hamiltonian matrix is proposed in this paper.For a given symmetry group of a Heisenberg Hamiltonian commuting with the total spin projection (i.e. with the total magnetization being a good quantum number) appropriate combinatorial and group-theoretical structures (partitions, orbits, stabilizers etc.) are introduced and briefly discussed.Generation of these structures can be performed by means of algorithms being modifications of standard ones.Main ideas are presented in this paper, whereas the actual form of algorithms will be discussed elsewhere.
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 extend a recent GPU implementation of the finite-temperature Lanczos method (FTLM) for Heisenberg spin Hamiltonians, which uses only total-magnetization symmetry, to the full permutation symmetry, including non-Abelian point and space groups. For highly symmetric clusters such as icosahedral polyhedra, the multidimensional irreducible representations (irreps) of the permutation group reduce the block dimensions substantially further than any Abelian subgroup. Here, we provide a matrix-free formalism (avoiding explicit storage of the projected Hamiltonian) for using such symmetries. The implementation applies equally to Abelian groups, retaining the advantages of GPU acceleration. Several cluster topologies are built in; beyond these, symmetry groups can be supplied as site-permutation generators, with irrep matrices computed automatically, so that user-defined clusters run without any code changes. We demonstrate the approach in production runs on a single NVIDIA B200 accelerator for the s=3/2 Heisenberg antiferromagnet on the dodecahedron (Hilbert-space dimension 4^20 ~ 1.1x10^12) and for the s=1/2 square lattice of 5x8 sites (M=0 dimension 1.4x10^11). With R=24 random vectors per symmetry block and 60 Lanczos steps, the dodecahedron campaign (with the largest iterated symmetry block having a dimension of 3.6x10^9) requires about 50 GPU-hours. The code is openly available under the Apache-2.0 license at https://github.com/ghasdeke/ftlm-pg-gpu, archived at DOI: 10.5281/zenodo.21445872.