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Europhysics Letters (EPL)

Matrix product representations for spin-(1/2) and spin-(3/2) spontaneous quantum ferrimagnets

S. Alipour, Sima Baghbanzadeh, Vahid Karimipour

Published Dec 1, 2008• 6 citations
Publisher

Abstract

Using the matrix product formalism, we find exact ground states for two new spin-(1/2) and spin-(3/2) quantum chains. On a lattice of 3 N sites, these ground states have a total spin of N /2, and hence have ferrimagnetic character. The novel point of this construction is that we construct a matrix product representation, not for a state which is invariant under rotation as has been hitherto considered, but for a state which transforms to other states under this symmetry operation. The method can be directly generalized to other symmetry groups.

DOI
10.1209/0295-5075/84/67006

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Using the matrix product formalism, we find exact ground states for two new spin-(1/2) and spin-(3/2) quantum chains. On a lattice of 3 N sites, these ground states have a total spin of N /2, and hence have ferrimagnetic character. The novel point of this construction is that we construct a matrix product representation, not for a state which is invariant under rotation as has been hitherto considered, but for a state which transforms to other states under this symmetry operation. The method can be directly generalized to other symmetry groups.
We present a method of constructing matrix-product (MP) states and apply it to the ground state of the Heisenberg spin chain with alternating spins 1 and and antiferromagnetic exchange interactions between nearest neighbors (the simplest example of a quantum ferrimagnet). The elementary matrix state is constructed in a way that ensures given transformational properties under rotations, which allows one to fix the total spin, and its z projection, of the entire MP wave function. We compare the variational MP results with the numerical results obtained through a quantum Monte Carlo method; the agreement is found to be within 0.4% for the ground-state energy and 5% for the correlation functions.
We introduce a simple representation for irreducible spherical tensor operators of the rotation group of arbitrary integer or half integer rank and use these tensor operators to construct matrix product ground states for quantum spin chains of arbitrary spin with nearest and next-nearest neighbor interactions. The ground states of these quantum spin chains are doubly degenerate and are in the form of juxtaposition of dimers (maximally entangled singlets) on adjacent sites. Hence each ground state breaks the translation invariance of the Hamiltonian. In this way, we unravel the matrix product structure of the well-known Majumdar-Ghosh model, with particular emphasis on its symmetry properties and generalize it to arbitrary integer or half integer spin models. We also make a comparison with the AKLT construction of such spin models.
We construct exact nontrivial ground states of spin-2 quantum antiferromagnets on the hexagonal lattice. Using the optimum ground state approach we determine the ground state in different subspaces of a general spin-2 Hamiltonian consistent with some realistic symmetries. These states, which are not of simple product form, depend on two free parameters and can be shown to be only weakly degenerate. We find ground states with different types of magnetic order, i.e., a weak antiferromagnet with finite sublattice magnetization and a weak ferromagnet with ferrimagnetic order. For the latter it is argued that a quantum phase transition occurs within the solvable subspace.
We study ground-state properties of the Heisenberg frustrated spin chain with interactions up to fourth nearest neighbors by the exact-diagonalization method and the density matrix renormalization group method. We find that ferrimagnetism is realized not only in the case of S=1/2 but also S=1 despite that there is only a single spin site in each unit cell determined from the shape of the Hamiltonian. Our numerical results suggest that a "multi-sublattice structure" is not required for the occurrence of ferrimagnetism in quantum spin systems with isotropic interactions.
We have found the exact (factorized) ground state of a general class of ferrimagnets in the presence of a magnetic field which includes the anisotropic and long-range interactions for arbitrary dimensional space. In particular cases our model represents many spin chains with bond alternation of antiferromagnetic-ferromagnetic coupling, ferrimagnetic spin ladders, and also homogeneous spin-sss models. The factorized ground state is a product of single-particle kets on a bipartite lattice composed of two different spins $(\ensuremath{\rho},\ensuremath{\sigma})$ which is characterized by two angles, a biangle state. The spin-wave analysis around the exact ground state shows two branch of excitations which are the origin of two dynamics of the model. The signature of these dynamics is addressed as a peak and a broaden bump in the specific heat.
We introduce a simple representation for irreducible spherical tensor operators of the rotation group of arbitrary integer or half integer rank and use these tensor operators to construct matrix product states corresponding to all the variety of valence bond states proposed in the Affleck--Kennedy--Lieb--Tasaki (AKLT) construction. These include the fully dimerized states of arbitrary spins, with uniform or alternating patterns of spins, which are ground states of Hamiltonians with nearest and next-nearest-neighbor interactions, and the partially dimerized or AKLT/valence bond solid states, which are constructed from them by projection. The latter states are translation-invariant ground states of Hamiltonians with nearest-neighbor interactions.