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The Matrix Product Formalism and the Generalization of Majumdar-Ghosh Model to Arbitrary Spins

Vahid Karimipour, Laleh Memarzadeh

Published Dec 12, 2007• 0 citations
OpenAlex

Abstract

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.

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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 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.
Using the matrix product formalism, we define a multiparameter family of spin models on one-dimensional chains, with nearest- and next-nearest-neighbor interactions for which exact analytical expressions can be found for its doubly degenerate ground states. The family of Hamiltonians depends on five continuous parameters and the Majumdar-Ghosh model is a particular point in this parameter space. The model can also be represented as a zigzag spin ladder in which there are extra three-particle couplings between spins on vertices of triangles. The doubly degenerate ground states of these models models have a very simple structure, they are the product of entangled (dimer) states on adjacent sites (or consecutive rungs of the ladder in the zigzag ladder picture). At the Majumdar-Ghosh point, these entangled states become the spin-singlets pertaining to this model. We will also calculate, in closed form, the two point correlation functions, both for finite sizes of the chain and in the thermodynamic limit.
Inspired by the exact solution of the Majumdar-Ghosh model, a family of one-dimensional, translationally invariant spin Hamiltonians is constructed. The exchange coupling in these models is antiferromagnetic, and decreases linearly with the separation between the spins. The coupling becomes identically zero beyond a certain distance. It is rigorously proved that the dimer configuration is an exact, superstable ground-state configuration of all the members of the family on a periodic chain. The ground state is twofold degenerate, and there exists an energy gap above the ground state. The Majumdar-Ghosh Hamiltonian with a twofold degenerate dimer ground state is just the first member of the family. The scheme of construction is generalized to two and three dimensions, and illustrated with the help of some concrete examples. The first member in two dimensions is the Shastry-Sutherland model. Many of these models have exponentially degenerate, exact dimer ground states.
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.
Abstract A systematic and compact treatment of arbitrary su (2) invariant spin- s quantum chains with nearest-neighbour interactions is presented. The ground-state is derived in terms of matrix product states (MPS). The fundamental MPS calculations consist of taking products of basic tensors of rank 3 and contractions thereof. The algebraic su (2) calculations are carried out completely by making use of Wigner calculus. As an example of application, the spin-1 bilinear-biquadratic quantum chain is investigated. Various physical quantities are calculated with high numerical accuracy of up to 8 digits. We obtain explicit results for the ground-state energy, entanglement entropy, singlet operator correlations and the string order parameter. We find an interesting crossover phenomenon in the correlation lengths.
We discuss spin-SSS antiferromagnetic Heisenberg chains with three-spin interactions, next-nearest-neighbor interactions, and bond alternation. First, we prove rigorously that there exist parameter regions of the exact dimerized ground state in this system. This is a generalization of the Majumdar-Ghosh model to arbitrary SSS. Next, we discuss the ground-state phase diagram of the models by introducing several effective field theories and the universality classes of the transitions are described by the level-2S2S2S SU(2)\mathrm{SU}(2)SU(2) Wess-Zumino-Witten model and the Gaussian model. Finally, we determine the phase diagrams of S=1S=1S=1 and S=3/2S=3/2S=3/2 systems by using exact diagonalization and level spectroscopy.