Ising-Heisenberg Model Hamiltonian: a>2?

[LagrangeEuler]

Hamiltonian
$H=-J\sum_{\vec{n},\vec{m}}(S_{\vec{n}}^xS_{\vec{m}}^x+S_{\vec{n}}^yS_{\vec{m}}^y+aS_{\vec{n}}^zS_{\vec{m}}^z)$
If $a>2$ is that behaves like Ising model? For example in $2d$ lattice is critical temperature of that system $T_c \approx 2.269$?

 

[Jolb]

Ising and Heisenberg models only have nearest-neighbor interactions. Your model doesn't restrict long-range interactions and thus the model is quite different.

 

[LagrangeEuler]

Ok and for nearest neighbours? I thought of that case when I right.

 

[Andy Resnick]

Hamiltonian
$H=-J\sum_{\vec{n},\vec{m}}(S_{\vec{n}}^xS_{\vec{m}}^x+S_{\vec{n}}^yS_{\vec{m}}^y+aS_{\vec{n}}^zS_{\vec{m}}^z)$
If $a>2$ is that behaves like Ising model? For example in $2d$ lattice is critical temperature of that system $T_c \approx 2.269$?

I don't recognize that Hamiltonian- The Heisenberg model of ferromagnetism is H = -2J $\sum s_{n}\cdot s_{m}$, where the sum is over nearest neighbors. Any Hamiltonian that is invariant under Z_2 has Ising symmetry (the Ising model is equivalent to a Z_2 clock model), but the primary difference between Heisenberg and Ising models is that the Ising model treats spin classically (the spin variables do not obey quantum commutation relations). Both models describe a high-temperature paramagnetic phase and a low-temperature ferromagnetic phase.

Another link is that the classical Heisenberg model is invariant under O_3, while the Ising model is invariant under O_1 (with the caveat that rotations in 1D are not defined).

 

[PJC01]

The Ising-Heisenberg Model Hamiltonian is a mathematical representation of a physical system that combines the Ising model with the Heisenberg model. The Ising model describes a system of interacting spins that can only take on two values, typically denoted as "up" and "down." The Heisenberg model, on the other hand, describes a system of interacting spins that can take on any value and are subject to quantum mechanical effects.

In the Ising-Heisenberg Model Hamiltonian, the parameter "a" represents the strength of the interaction between the spins in the z-direction. If a > 2, this means that the z-direction interaction is stronger than the x and y-direction interactions. In this case, the behavior of the system will be dominated by the Ising model, as the Heisenberg model effects will be negligible.

In a 2D lattice, the critical temperature of the system is an important factor in understanding its behavior. The critical temperature is the temperature at which the system undergoes a phase transition from an ordered state to a disordered state. In the case of the Ising-Heisenberg Model Hamiltonian, the critical temperature is approximately 2.269 for a = 2. This value may change if a > 2, as the behavior of the system will be more Ising-like and less affected by the Heisenberg model.

In conclusion, for a > 2, the Ising-Heisenberg Model Hamiltonian will behave more like the Ising model and the critical temperature may change. Further research and analysis would be needed to fully understand the behavior of the system with a > 2.

 

[PJC01]

I would like to clarify that the Ising-Heisenberg Model Hamiltonian is a theoretical model used in statistical mechanics to describe the behavior of magnetic systems. It is a combination of the Ising model, which describes the interactions between spins in a lattice, and the Heisenberg model, which takes into account the quantum nature of the spins.

In this model, the parameter 'a' represents the strength of the interaction between the spins in the z-direction. If 'a' is greater than 2, then the system will behave more like the Ising model, where the spins align either up or down, rather than the Heisenberg model where the spins can exist in any direction.

In a 2D lattice, the critical temperature of the system is approximately 2.269, as mentioned. This means that at temperatures below this value, the system will exhibit long-range order and the spins will align in a specific direction. However, at temperatures above this value, the system will undergo a phase transition and the spins will become disordered.

Therefore, if 'a' is greater than 2, the system will behave more like the Ising model and the critical temperature will remain around 2.269. However, if 'a' is less than 2, the system will exhibit more Heisenberg-like behavior and the critical temperature may shift. It is important to note that the exact value of the critical temperature also depends on other factors such as the lattice structure and the strength of other interactions present in the system.

In conclusion, the value of 'a' in the Ising-Heisenberg Model Hamiltonian does affect the behavior of the system and can shift the critical temperature, but it is not the only factor at play. Further research and experiments are needed to fully understand the behavior of this model and its implications in real-world magnetic systems.