Majumdar-Ghosh Model
Encyclopedia
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 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
.
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.
:
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 1 to 2.
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.
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.
Quantum mechanics
Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...
Heisenberg spin model
Heisenberg model (quantum)
The Heisenberg model is a statistical mechanical model used in the study of critical points and phase transitions of magnetic systems, in which the spin of the magnetic systems are treated quantum mechanically...
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
J1 J2 model
The J1-J2 model is a quantum spin model like the Heisenberg model but also includes a term for the interaction between next-nearest neighbor spins.-Hamiltonian:-See also:*Spin model*Heisenberg model *Hubbard model*t-J model*Majumdar-Ghosh Model...
. The model is named after Indian physicists Chanchal Kumar Majumdar and Dipan Ghosh
Dipan Ghosh
Dipan Ghosh is an Indian theoretical physicist, best known for his exact enumeration of the ground state of a Heisenberg antiferromagnet, known in literature as the Majumdar–Ghosh Model...
.
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 HamiltonianHamiltonian (quantum mechanics)
In quantum mechanics, the Hamiltonian H, also Ȟ or Ĥ, is the operator corresponding to the total energy of the system. Its spectrum is the set of possible outcomes when one measures the total energy of a system...
:
where the S vector is a quantum spin
Spin (physics)
In quantum mechanics and particle physics, spin is a fundamental characteristic property of elementary particles, composite particles , and atomic nuclei.It is worth noting that the intrinsic property of subatomic particles called spin and discussed in this article, is related in some small ways,...
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 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 examplesof what are known as valence-bond solids (VBS). Thus the Majumdar-Ghosh model is related to another famous spin model, the AKLT Model
AKLT Model
The AKLT model is an extension of the one-dimensional quantum Heisenberg spin model. The proposal and exact solution of this model by Affleck, Lieb, Kennedy and Tasaki provided crucial insight into the physics of the spin 1 Heisenberg chain...
, 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 (quantum)The Heisenberg model is a statistical mechanical model used in the study of critical points and phase transitions of magnetic systems, in which the spin of the magnetic systems are treated quantum mechanically...
- Heisenberg model (classical)Heisenberg model (classical)The Classical Heisenberg model is the n = 3 case of the n-vector model, one of the models used in statistical physics to model ferromagnetism, and other phenomena.-Definition:...
- J1 J2 modelJ1 J2 modelThe J1-J2 model is a quantum spin model like the Heisenberg model but also includes a term for the interaction between next-nearest neighbor spins.-Hamiltonian:-See also:*Spin model*Heisenberg model *Hubbard model*t-J model*Majumdar-Ghosh Model...
- Bethe ansatzBethe ansatzIn physics, the Bethe ansatz is a method for finding the exact solutions of certain one-dimensional quantum many-body models. It was invented by Hans Bethe in 1931 to find the exact eigenvalues and eigenvectors of the one-dimensional antiferromagnetic Heisenberg model Hamiltonian...
- Ising modelIsing modelThe Ising model is a mathematical model of ferromagnetism in statistical mechanics. The model consists of discrete variables called spins that can be in one of two states . The spins are arranged in a graph , and each spin interacts with its nearest neighbors...
- t-J modelT-J modelThe t-J model was first derived in 1977 from the Hubbard model by Józef Spałek. The model describes strongly correlated electron systems. It is used to calculate high temperature superconductivity states in doped antiferromagnets.The t-J Hamiltonian is:...