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The M-G Hamiltonian - A Pedagogic Review

Ghosh, Dipan Kumar

Published Jan 1, 2006• 0 citations
OpenAlex

Abstract

In this review, we discuss Majumdar-Ghosh Hamiltonian which is a one dimensional system of spin 1/2 particles with isotropic interactions between nearest and next-nearest neighbours. The model is exactly solvable when the next-nearest neighbour strength is half that between the nearest neighbours. The ground state is doubly degenerate and consists of dimers of nearest neighbour singlet pairs.

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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.

arXiv:0705.0261•Physical Review B•Aug 30, 2007•11 citations
In this review, we discuss Majumdar-Ghosh Hamiltonian which is a one dimensional system of spin 1/2 particles with isotropic interactions between nearest and next-nearest neighbours. The model is exactly solvable when the next-nearest neighbour strength is half that between the nearest neighbours. The ground state is doubly degenerate and consists of dimers of nearest neighbour singlet pairs.
Hamiltonians of one dimensional spin half models in condensed matter physics are expounded in this chapter. Hamiltonian of spins interacting with an external magnetic field and separate Hamiltonians for the direct exchange spin-spin interactions in X, Y, Z directions, Heisenberg Hamiltonian and the Dzyaloshinskii-Moriya interaction are dealt with in detail. It is emphasized here that the Hamiltonians are not only for the nearest-neighbour interaction but it is for the rth nearest-neighbour interaction, this increases the scope and usability of the recipies. Other important variants such as the Ising model, transverse Ising model, Majumdar-Ghosh model and a plethora of other spin chain models can be easily constructed using our recipes given and it has been highlighted via examples.
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.
Starting from the exact result of Majumdar and Ghosh (1969) for the dimerized ground state of a Heisenberg S=1/2 antiferromagnet with competing next-nearest-neighbour interaction J2+1/2J1 on a linear chain, it is shown that this state is the ground state of the same kind of Hamiltonian, if couplings are extended up to 2n-nearest-neighbours, and the relation J1=2n, J2=2n-1, ..., J2n=1 holds. Furthermore, it is proved rigorously that this state is the ground state when n=2 and J1-2J2+J3=0, J3=2J4, in a finite interval of values of the parameter J4. A rigorous lower limit for the extension of this interval is found to be 0<or=J4<or=1/4J1. A comparison with ground-state configurations of the equivalent classical Heisenberg and Ising model is discussed.
A method to construct various s= 1/2 isotropic spin models with exact dimer ground states is proposed. Constructed models are usually of much simpler forms than those produced by the method using Lowdin's projection operators. They include a series of one-dimensional Heisenberg models which consist of linearly decayed exchange interactions and are natural extensions of the Majumdar-Ghosh model. A two-dimensional spin model with four spin interactions is also constructed to have the same spatial symmetry as a simple square lattice.
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.