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Journal of the Physical Society of Japan

Symmetry Studies of Antiferromagnetic Heisenberg Model

Takashi Ishino, Riichiro Saito, Hiroshi Kamimura

Published Nov 15, 1990• 16 citations
Publisher

Abstract

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.

DOI
10.1143/jpsj.59.3886

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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.
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.
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.
The spin-1/2 antiferromagnetic Heisenberg and X Y models are investigated on the triangular lattice by numerically diagonalizing finite size systems (up to 21 spins). The ground state energies of both models are found to be close to other theoretical predictions except those of Marland and Betts who performed similar analysis to ours. We have numerically constructed a resonating valence bond state of Anderson. It has a relatively large projection on the ground state of the finite Heisenberg model, although the absolute value of the projection decreases linearly with the system size. The trial function proposed by Miyashita for the X Y model does not have an appreciable projection on the numerically exact state. We present evidence for the absence of a sublattice long range order in the Heisenberg model and the existence of the same quantity in the X Y model.