Jordan–Wigner transformation

Using the Jordan -Wigner transformation, various one-dimensional quantum mechanical systems can be mapped onto each other. Specifically, it is possible with the transformation mapping one-dimensional spinless fermions on Spin-1/2-Ketten on a chain.

The Jordan -Wigner transformation maps the Spin-1/2-Operatoren from on creation and annihilation operators for spinless fermions. Using the transformation, the equivalence between the one-dimensional Heisenberg model and spinless fermions are shown on a one-dimensional lattice with nearest neighbor interaction.

Transformation was published in 1928 by Pascual Jordan and Eugene Wigner in the Journal of Physics.

The basic idea

Considering Spin-1/2-Operatoren of place i, we find that these obey the basic canonical (anti) commutation (anti- commutation ) for fermions:

The idea is therefore to consider the Spin-1/2-Operatoren as fermionic operators. However, the Spin-1/2-Operatoren meet no anti- commutation relations at different lattice sites and:

Jordan and Wigner have realized that this can be remedied, however, with the introduction of a phase operator before Spin-1/2-Operatoren. An Wegorientierung defined by a phase factor that depends on the number of up- spin before the considered spin.

Is at the position i is a spin - up, a phase factor (-1) " picked ", for a down - spin nothing happens ( phase factor 1):

The so-defined fermionic operators satisfy the anti- commutation relations at different courses and:

Especially helpful are the following relationships for the mapping between different models:

Applications

1D Heisenberg model

To illustrate the Jordan -Wigner transformation it is applied to the one-dimensional Heisenberg model. The necessary products of the different operators are listed in the previous section. The Hamiltonian of the 1D Heisenberg model can therefore be written as:

The transition thus showing the equivalence of the 1D Heisenberg model with spinless fermions on the lattice with periodic boundary conditions and only nearest neighbor interaction. The first term describes non-interacting fermions and the second term is the interaction term with an interaction given by the coupling constant of the Heisenberg model.

1D XY model

Another example is the one-dimensional model as a special case XY of the 1D Heisenberg model. The Hamiltonian of the XY - model can be written as:

The Jordan - Wigner transformation thus forms the spin system onto noninteracting spinless fermions. For this system, you can specify the partition function exactly.

Swell

• Quantum mechanics
• Statistical Physics
• Transformation