The Majumdar–Ghosh model is a one-dimensional quantum Heisenberg spin model in which the nearest-neighbour antiferromagnetic exchange interaction is twice as strong as the next-nearest-neighbour interaction. It is a special case of the more general J 1 {\displaystyle J_{1}} - J 2 {\displaystyle J_{2}} model, with J 1 = 2 J 2 {\displaystyle J_{1}=2J_{2}} . The model is named after Indian physicists Chanchal Kumar Majumdar and Dipan Ghosh. 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:
H ^ = J ∑ j = 1 N S → j ⋅ S → j + 1 + J 2 ∑ j = 1 N S → j ⋅ S → j + 2 {\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. As a result of this ratio, it is possible to express the Hamiltonian (shifted by an overall constant) equivalently in the form
H ^ = J 4 ∑ j = 1 N ( S → j − 1 + S → j + S → j + 1 ) 2 {\displaystyle {\hat {H}}={\frac {J}{4}}\sum _{j=1}^{N}({\vec {S}}_{j-1}+{\vec {S}}_{j}+{\vec {S}}_{j+1})^{2}}
The summed quantity is none other than the quadratic Casimir operator for representation of the spin algebra on the three consecutive sites j − 1 , j , j + 1 {\displaystyle j-1,j,j+1} , which in turn can be decomposed into a direct sum of spin 1/2 and 3/2 representations. It has the eigenvalues 1 2 ( 1 2 + 1 ) = 3 4 {\displaystyle {\tfrac {1}{2}}({\tfrac {1}{2}}+1)={\tfrac {3}{4}}} for the spin 1/2 subspace and 3 2 ( 3 2 + 1 ) = 15 / 4 {\displaystyle {\tfrac {3}{2}}({\tfrac {3}{2}}+1)=15/4} for the spin 3/2 subspace.
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. Furthermore, it has been shown that these (and linear combinations of them) are the unique ground states.
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