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Symmetry

Application of Symmetry Methods to Low-Dimensional Heisenberg Magnets

Irina Gennad'evna Bostrem, Alexander S. Ovchinnikov, Valentine E. Sinitsyn

Published Apr 9, 2010• 2 citations
Publisher

Abstract

An account of symmetry is very fruitful in studies of quantum spin systems. In the present paper we demonstrate how to use the spin SU(2) and the point symmetries in optimization of the theoretical condensed matter tools: the exact diagonalization, the renormalization group approach, the cluster perturbation theory. We apply the methods for study of Bose-Einstein condensation in dimerized antiferromagnets, for investigations of magnetization processes and magnetocaloric effect in quantum ferrimagnetic chain.

DOI
10.3390/sym2020722

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An account of symmetry is very fruitful in studies of quantum spin systems. In the present paper we demonstrate how to use the spin SU(2) and the point symmetries in optimization of the theoretical condensed matter tools: the exact diagonalization, the renormalization group approach, the cluster perturbation theory. We apply the methods for study of Bose-Einstein condensation in dimerized antiferromagnets, for investigations of magnetization processes and magnetocaloric effect in quantum ferrimagnetic chain.
We present a quantum cluster solver for the spin- S Heisenberg model on a two-dimensional lattice. The formalism is based on the real-space renormalization procedure and uses the lattice point group-theoretical analysis and non-Abelian SU (2) spin symmetry technique. The exact diagonalization procedure is used twice at each renormalization group step. The method is applied to the spin-half antiferromagnet on a square lattice, and a calculation of local observables is demonstrated. A symmetry-based truncation procedure is suggested and verified numerically.
We study quantum ferrimagnets in one, two, and three dimensions by using a variety of methods and approximations. These include: (i) a treatment based on the spin coherent state path-integral formulation of quantum ferrimagnets by taking into account the leading order quantum and thermal fluctuations (ii) a field-theoretical (non-linear σσσ-model type) formulation of the special case of one-dimensional quantum ferrimagnets at zero temperature (iii) an effective description in terms of dimers and quantum rotors, and (iv) a quantum renormalization group study of ferrimagnetic Heisenberg chains. Some of the formalism discussed here can be used for a unified treatment of both ferromagnets and antiferromagnets in the semiclassical limit. We show that the low (high) energy effective Hamiltonian of a (S_1, S_2) Heisenberg ferrimagnet is a ferromagnetic (antiferromagnetic) Heisenberg model. We also study the phase diagram of quantum ferrimagnets in the presence of an external magnetic field h (hc1<h<hc2h_{c1} < h < h_{c2}hc1​<h<hc2​) and show that the low- and the high-field phases correspond respectively to the classical Néel and the fully polarized ferromagnetic states. We also calculate the transition temperature for the Berezinskii-Kosterlitz-Thouless phase transition in the special case of two-dimensional quantum ferrimagnets.
Second-order spin-wave expansions are used to compute the ground-state energy and sublattice magnetizations of the quantum one-dimensional Heisenberg ferrimagnet with nearest-neighbor antiferromagnetic interactions and two types of alternating sublattice spins S1>S2.{S}_{1}>{S}_{2}.S1​>S2​. It is found that in the extreme quantum cases (S1,S2){(S}_{1}{,S}_{2})(S1​,S2​)=(1,1/2), (3/2,1), and (3/2,1/2), the estimates for the ground-state energy and sublattice magnetizations differ less than 0.03% for the energy and 0.2% for the sublattice magnetizations from the recently published density-matrix renormalization-group numerical calculations. The reported results strongly suggest that the quantum Heisenberg ferrimagnetic chains give another example of a low-dimensional quantum spin system where the spin-wave approach demonstrates a surprising efficiency.
Thermodynamic properties of the quantum Heisenberg spin chains with S=1/2,S=1/2,S=1/2, 1, and 3/23/23/2 are investigated using the transfer-matrix renormalization-group method. The temperature dependence of the magnetization, susceptibility, specific heat, spin-spin correlation length, and several other physical quantities in a zero or finite applied field are calculated and compared. Our data agree well with the Bethe ansatz, exact diagonalization, and quantum Monte Carlo results and provide further insight into the quantum effects in the antiferromagnetic Heisenberg spin chains.
Abstract We investigate magnetic and thermodynamic properties of a spin-1 Heisenberg diamond chain in a magnetic field using a combination of analytical and numerical methods including the variational approach, exact diagonalization, density-matrix renormalization group, localized-magnon theory, and quantum Monte Carlo simulations. In the unfrustrated regime, the model exhibits a quantum ferrimagnetic phase that captures key magnetic features of the nickel-based polymeric compound [Ni 3 (OH) 2 (C 4 H 2 O 4 )(H 2 O) 4 ] · 2H 2 O such as a flat minimum in the temperature dependence of the susceptibility times temperature product and an intermediate one-third magnetization plateau. In the frustrated regime, we uncover a rich variety of unconventional quantum phases including uniform and cluster-based Haldane states, fragmented monomer-dimer phase, and bound-magnon crystals. Analysis of the adiabatic temperature change and magnetic Grüneisen parameter reveals an enhanced magnetocaloric effect near field-induced transitions between these exotic quantum phases. Additionally, we demonstrate that the frustrated spin-1 diamond chain can operate as an efficient working medium of a quantum Stirling engine, which approaches near-optimal efficiency when driven into these unconventional quantum states.
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.