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The '''Majumdar-Ghosh model''' is an extension of the one-dimensional [[quantum mechanics|quantum]] [[Heisenberg model (quantum)|Heisenberg spin model]] in which an extra interaction is added coupling spins two lattice spacings apart such that second-neighbor coupling half as strong 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]].<ref>{{cite journal|last=Dattagupta|first=Sushanta|title=Chanchal Kumar Majumdar (1938–2000) – An obituary|journal=Current Science|year=2000|volume=79|issue=1|pages=115–116|accessdate=16 September 2010}}</ref>
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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 (quantum mechanics)|Hamiltonian]]:
 
:<math>\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} </math>
 
where the S vector is a quantum [[Spin (physics)|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 1 to 2.
 
==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 state|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.
 
[[File:Majumdar-ghosh gs.png]]
 
==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 solid|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.
 
==See also==
*[[Heisenberg model (quantum)]]
*[[Heisenberg model (classical)]]
*[[J1 J2 model]]
*[[Bethe ansatz]]
*[[Ising model]]
*[[t-J model]]
 
== References ==
* C K Majumdar and D Ghosh, ''[http://link.aip.org/link/?JMP/10/1388 On Next‐Nearest‐Neighbor Interaction in Linear Chain]''. J. Math. Phys. '''10''', 1388 (1969); {{doi|10.1063/1.1664978}}
* C K Majumdar, ''[http://dx.doi.org/10.1088/0022-3719/3/4/019 Antiferromagnetic model with known ground state]''. J. Phys. C: Solid State Phys. 3 911-915 (1969)
* Assa Auerbach, ''[http://books.google.com/books?id=tiQlKzJa6GEC&lpg=PP1&dq=assa%20auerbach%20interacting%20electrons&pg=PA83#v=onepage&q&f=false Interacting Electrons and Quantum Magnetism]'', Springer-Verlag New York (1992) p.&nbsp;83
{{Reflist}}
 
{{DEFAULTSORT:Majumdar-Ghosh Model}}
[[Category:Spin models]]
[[Category:Statistical mechanics]]
[[Category:Quantum magnetism]]
[[Category:Lattice models]]

Latest revision as of 06:09, 19 December 2014

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