# Majumdar–Ghosh model

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The Majumdar–Ghosh model is an extension of the one-dimensional quantum Heisenberg spin model in which an extra interaction is added coupling spins two lattice spacings apart such that the second-neighbor coupling is half as strong as the first. It is therefore a special case of the J1 J2 model. The model is named after Indian physicists Chanchal Kumar Majumdar and Dipan Ghosh.[1]

The Majumdar–Ghosh model is notable because its ground states (lowest energy quantum states) can be found exactly and written in a simple form, making it a useful starting point for understanding more complex spin models and phases.

## Definition

The Majumdar–Ghosh model is defined by the following Hamiltonian:

${\displaystyle {\hat {H}}=J\sum _{j=1}^{N}{\vec {S}}_{j}\cdot {\vec {S}}_{j+1}+{\frac {J}{2}}\sum _{j=1}^{N}{\vec {S}}_{j}\cdot {\vec {S}}_{j+2}}$

where the S vector is a quantum spin operator with quantum number S = 1/2.

Other conventions for the coefficients may be taken in the literature, but the most important fact is that the ratio of first-neighbor to second-neighbor couplings is 2 to 1.

## Ground states

It has been shown that the Majumdar–Ghosh model has two minimum energy states, or ground states, namely the states in which neighboring pairs of spins form singlet configurations. The wavefunction for each ground state is a product of these singlet pairs. This explains why there must be at least two ground states with the same energy, since one may be obtained from the other by merely shifting, or translating, the system by one lattice spacing. It should be noted, however, that this ground state degeneracy only appears when the system is taken to be infinite in size (the so-called thermodynamic limit). Otherwise, there is a unique ground state and a second, higher energy state whose energy approaches the first exponentially quickly with increasing system size.

## Generalizations

The Majumdar–Ghosh model is one of a small handful of realistic quantum spin models that may be solved exactly. Moreover, its ground states are simple examples of what are known as valence-bond solids (VBS). Thus the Majumdar–Ghosh model is related to another famous spin model, the AKLT model, whose ground state is the unique one dimensional spin one (S=1) valence-bond solid.

The Majumdar–Ghosh model is also a useful example of the Lieb–Schultz–Mattis theorem which roughly states that an infinite, one dimensional, half-odd-integer spin system must either have no energy spacing (or gap) between its ground and excited states or else have more than one ground state. The Majumdar–Ghosh model has a gap and falls under the second case.