The Hamiltonian of the XY model -- when is it called the XX model?

In summary, the Hamiltonian of the XY model is a mathematical representation of the energy of the system, describing the interactions between spins in a two-dimensional lattice. It differs from the XX model in the type of interactions between spins, resulting in different energy landscapes and behavior of the system. The XY model becomes the XX model when the anisotropy parameter is set to zero, simplifying the interactions. It is commonly used in the study of magnetic systems and other physical systems, as well as in quantum information processing and computing. The Hamiltonian of the XY model can be seen as a generalization of the Ising model, providing a more comprehensive understanding of its behavior in different dimensions.
  • #1
LagrangeEuler
717
20
Hamiltonian of XY model is defined by
##H=J\sum_i (\sigma_i^x \sigma_{i+1}^x+\sigma_i^y \sigma_{i+1}^y)##
and because it is isotropic it is sometimes called XX model. If we do some unitary transformation, and get hamiltonian
##H=J\sum_i (\sigma_i^x \sigma_{i+1}^x-\sigma_i^y \sigma_{i+1}^y)##
what is name of that model? Is it like X-X model?
 
Physics news on Phys.org
  • #2
You have written the same Hamiltonian!
 

FAQ: The Hamiltonian of the XY model -- when is it called the XX model?

What is the Hamiltonian of the XY model?

The Hamiltonian of the XY model is a mathematical representation of the energy of the system, which is used to describe the interactions between spins in a two-dimensional lattice. It is defined as the sum of the interactions between nearest-neighbor spins, taking into account the strength and orientation of each spin.

What is the difference between the XY model and the XX model?

The main difference between the XY model and the XX model lies in the type of interactions between spins. In the XY model, the interactions are described by a cosine function, while in the XX model, they are described by a product of the Pauli matrices. This results in different energy landscapes and behavior of the system.

When does the XY model become the XX model?

The XY model becomes the XX model when the anisotropy parameter, which determines the strength of the interactions between spins in different directions, is set to zero. This means that the system is no longer sensitive to the orientation of spins, resulting in simpler interactions described by the Pauli matrices.

What are the applications of the Hamiltonian of the XY model?

The Hamiltonian of the XY model is commonly used in the study of magnetic systems, such as thin films and spin chains. It can also be used to model other physical systems, such as superconducting Josephson junctions and superfluids. Additionally, the XY model has applications in quantum information processing and quantum computing.

How does the Hamiltonian of the XY model relate to the Ising model?

The Hamiltonian of the XY model can be seen as a generalization of the Ising model, which describes the behavior of spins in a one-dimensional lattice. The Ising model can be recovered from the XY model by setting the anisotropy parameter to infinity, resulting in interactions that only depend on the orientation of neighboring spins. In this way, the XY model provides a more comprehensive understanding of the Ising model and its behavior in different dimensions.

Similar threads

A Physical meaning of terms in the Qi, Wu, Zhang model
Replies
3
Views
2K
I Proof request for best linear predictor
Replies
1
Views
728
I How Does Weiss Molecular Field Theory Model Magnetization Symmetry?
Replies
5
Views
2K
A Fourier Transforming a HgTe 2D Hamiltonian
Replies
1
Views
2K
Partition Function for Spin-1 One Dimensional Ising Model
Replies
3
Views
2K
A Exploring the 2-D Ising Model & Its Continuum Limit
Replies
5
Views
3K
Can the 1D Random XX Spin Chain Model Be Solved Exactly?
Replies
1
Views
1K
Understanding Hamiltonian with Even/Odd Bonds
Replies
2
Views
2K
I Effective Hamiltonian to Rotational Term Values
Replies
0
Views
1K
Partition function of modified Ising model
Replies
1
Views
1K

Hot Threads

Recent Insights

Back
Top