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

Computational Methods in Science and Technology

COMBINATORIAL STRUCTURES IN SPIN MODELS: A METHOD OF OPERATOR MATRICES GENERATION

Sylwia Bucikiewicz, Wojciech Florek

Published Jan 1, 2001• 3 citations
Publisher

Abstract

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.

DOI
10.12921/cmst.2001.07.01.27-39

Related papers

Theoretical modeling of many-electron spin systems

Bryan Conner

The study of quantum models for magnetism requires the diagonalization of large matrices. For electrons occupying a lattice of sixteen sites, the dimension of these matrices is 65,536 x 65,536. I have written a program in Mathematica to exploit symmetries of the model and render the diagonalization of the Hamiltonian matrix tractable. Specifically, the program utilizes four different operators: translational in the x and y directions (Tx and Ty), total spin in one direction (Sz), and total spin squared (S2). As symmetries, all of these operators commute with the Hamiltonian and therefore the eigenstates of the Hamiltonian are simultaneously eigenstates of these operators. In practice, the eigenstates of these symmetry operators are easier to discover then those of the Hamiltonian. The most difficult symmetry to exploit is the S2 symmetry. In order to generate this symmetry, I reference Clebsch-Gordan coefficient tables. These tables exist to aid in the addition of angular momentum under the laws of quantum mechanics. The tables are generally used to couple two spins. When adding sixteen spins, the coefficients are multiplied together iteratively, adding one spin at a time up to the desired number. The products of these coefficients are then employed to construct all 65,536 eigenstates of the S2 operator. Combined with the remaining symmetry operators, these states can be used to put the Hamiltonian into a simpler, block diagonal form that is readily diagonalized.

•Jan 1, 2006•0 citations

Symmetry Studies of Antiferromagnetic Heisenberg Model

Takashi Ishino, Riichiro Saito, Hiroshi Kamimura

A general treatment of the symmetry for quantum spin systems is presented with the use of permutational operators. Symmetry operations which are commutable with a Hamiltonian form the symmetry group. Complete sets of spin functions which are specified by the total spin ( S tot ) are projected onto the irreducible representations of the symmetry group, using the character table. We show from symmetry consideration that a minimum dimension exists for the ground state of the Heisenberg Hamiltonian with finite N spins. The diagonalization of the Hamiltonian with the use of symmetry consideration can be effectively performed for the complete sets of S tot =0 by the method proposed in this paper. Numerical results of the ground state and all the excited states of S tot =0 are presented for spin systems up to N =20.

Journal of the Physical Society of Japan•Nov 15, 1990•16 citations

Three-Diagonal Blocks of Two-Magnon Hamiltonians with Specified Wave Numbers for Heisenberg Rings

M. Łabuz, Jan Milewski

We demonstrate the method of creating blocks of Hamiltonian matrices for Heisenberg rings with N nodes and r = 2 overturned spins, depending on total quasimomentum k.Initial problem of dimension N 2 reduces, approximately, N -tuply, depending on N and k numbers.We consider block Hamiltonians using a particular basis, called wavelet basis.In this basis, these blocks take three-diagonal form.

Acta Physica Polonica A•Jun 1, 2019•1 citation

Symmetry classes connected with the magnetic Heisenberg ring

B Fiedler

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.

arXiv:0709.0104•Journal of Physics Conference Series•Mar 1, 2008•1 citation

Combined use of translational and spin-rotational invariance for spin systems

Tjark Heitmann, Jürgen Schnack

Exact diagonalization and other numerical studies of quantum spin systems are notoriously limited by the exponential growth of the Hilbert space dimension with system size. A common and well-known practice to reduce this increasing computational effort is to take advantage of the translational symmetry ${C}_{N}$ in periodic systems. This represents a rather simple yet elegant application of the group theoretical symmetry projection operator technique. For isotropic exchange interactions, the spin-rotational symmetry $SU(2)$ can be used, where the Hamiltonian matrix is block structured according to the total spin and magnetization quantum numbers. Rewriting the Heisenberg Hamiltonian in terms of irreducible tensor operators allows for an efficient and highly parallelizable implementation to calculate its matrix elements recursively in the spin-coupling basis. When combining both ${C}_{N}$ and $SU(2)$, mathematically, the symmetry projection technique leads to ready-to-use formulas. However, the evaluation of these formulas is very demanding in both computation time and memory consumption, problems which are said to outweigh the benefits of the symmetry-reduced matrix shape. We show a way to minimize the computational effort for selected systems and present the largest numerically accessible cases.

arXiv:1902.02093•Physical review. B./Physical review. B•Apr 2, 2019•24 citations

Heisenberg models and a particular isotropic model

Arjan J. van der Sijs

The Heisenberg model, a quantum-mechanical analogue of the Ising model, has a large ground-state degeneracy, due to the symmetry generated by the total spin. This symmetry is also responsible for degeneracies in the rest of the spectrum. We discuss the global structure of the spectrum of Heisenberg models with arbitrary couplings, using group-theoretical methods. The Hilbert space breaks up in blocks characterized by the quantum numbers of the total spin, S and M, and each block is shown to constitute the representation space of an explicitly given irreducible representation of the symmetric group ${\mathit{S}}_{\mathit{N}}$, consisting of permutations of the N spins in the system. In the second part of the paper we consider, as a concrete application, the model where each spin is coupled to all the other spins with equal strength. Its partition function is written as a single integral, elucidating its N dependence. This provides a useful framework for studying finite-size effects. We give explicit results for the heat capacity, revealing interesting behavior just around the phase transition.

arXiv:cond-mat/9306045•Physical review. B, Condensed matter•Sep 1, 1993•10 citations
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.
The study of quantum models for magnetism requires the diagonalization of large matrices. For electrons occupying a lattice of sixteen sites, the dimension of these matrices is 65,536 x 65,536. I have written a program in Mathematica to exploit symmetries of the model and render the diagonalization of the Hamiltonian matrix tractable. Specifically, the program utilizes four different operators: translational in the x and y directions (Tx and Ty), total spin in one direction (Sz), and total spin squared (S2). As symmetries, all of these operators commute with the Hamiltonian and therefore the eigenstates of the Hamiltonian are simultaneously eigenstates of these operators. In practice, the eigenstates of these symmetry operators are easier to discover then those of the Hamiltonian. The most difficult symmetry to exploit is the S2 symmetry. In order to generate this symmetry, I reference Clebsch-Gordan coefficient tables. These tables exist to aid in the addition of angular momentum under the laws of quantum mechanics. The tables are generally used to couple two spins. When adding sixteen spins, the coefficients are multiplied together iteratively, adding one spin at a time up to the desired number. The products of these coefficients are then employed to construct all 65,536 eigenstates of the S2 operator. Combined with the remaining symmetry operators, these states can be used to put the Hamiltonian into a simpler, block diagonal form that is readily diagonalized.
A general treatment of the symmetry for quantum spin systems is presented with the use of permutational operators. Symmetry operations which are commutable with a Hamiltonian form the symmetry group. Complete sets of spin functions which are specified by the total spin ( S tot ) are projected onto the irreducible representations of the symmetry group, using the character table. We show from symmetry consideration that a minimum dimension exists for the ground state of the Heisenberg Hamiltonian with finite N spins. The diagonalization of the Hamiltonian with the use of symmetry consideration can be effectively performed for the complete sets of S tot =0 by the method proposed in this paper. Numerical results of the ground state and all the excited states of S tot =0 are presented for spin systems up to N =20.
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.
Exact diagonalization and other numerical studies of quantum spin systems are notoriously limited by the exponential growth of the Hilbert space dimension with system size. A common and well-known practice to reduce this increasing computational effort is to take advantage of the translational symmetry CN{C}_{N}CN​ in periodic systems. This represents a rather simple yet elegant application of the group theoretical symmetry projection operator technique. For isotropic exchange interactions, the spin-rotational symmetry SU(2)SU(2)SU(2) can be used, where the Hamiltonian matrix is block structured according to the total spin and magnetization quantum numbers. Rewriting the Heisenberg Hamiltonian in terms of irreducible tensor operators allows for an efficient and highly parallelizable implementation to calculate its matrix elements recursively in the spin-coupling basis. When combining both CN{C}_{N}CN​ and SU(2)SU(2)SU(2), mathematically, the symmetry projection technique leads to ready-to-use formulas. However, the evaluation of these formulas is very demanding in both computation time and memory consumption, problems which are said to outweigh the benefits of the symmetry-reduced matrix shape. We show a way to minimize the computational effort for selected systems and present the largest numerically accessible cases.
The Heisenberg model, a quantum-mechanical analogue of the Ising model, has a large ground-state degeneracy, due to the symmetry generated by the total spin. This symmetry is also responsible for degeneracies in the rest of the spectrum. We discuss the global structure of the spectrum of Heisenberg models with arbitrary couplings, using group-theoretical methods. The Hilbert space breaks up in blocks characterized by the quantum numbers of the total spin, S and M, and each block is shown to constitute the representation space of an explicitly given irreducible representation of the symmetric group SN{\mathit{S}}_{\mathit{N}}SN​, consisting of permutations of the N spins in the system. In the second part of the paper we consider, as a concrete application, the model where each spin is coupled to all the other spins with equal strength. Its partition function is written as a single integral, elucidating its N dependence. This provides a useful framework for studying finite-size effects. We give explicit results for the heat capacity, revealing interesting behavior just around the phase transition.