Ising model open chain and periodic boundary conditions

In summary, the conversation discusses the treatment of the one dimensional Ising model with external fields. It is noted that the model is often treated as an open chain system with free ends, but when an external field is added, it is treated with cyclic boundary conditions. The question is raised whether these methods are equivalent or not. The main reason for using these methods is the computational cost of a quantum treatment for the transverse field Ising model, which can grow exponentially with the number of spins. However, it is suggested that the classical transverse field Ising model for an infinite 1D lattice can be approximated using the maximal entropy random walk (MERW). Additionally, it is mentioned that the 2D classical Ising
  • #1
LagrangeEuler
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One dimensional Ising model is often treated as open chain system with free ends. Then when external field is added it is treated with cyclic boundary condition. Can someone explain me are those methods equivalent, or not?
 
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  • #2
The main reason is probably computational cost of quantum treatment of transverse-filed Ising model, naively growing exponentially with the number of spins.

However, classical transverse field Ising model for infinite 1D lattice can be well approximated with MERW ( https://en.wikipedia.org/wiki/Maximal_entropy_random_walk ).
Also, 2D classical Ising for finite width approximation: "w x infinity" lattice e.g. with cyclic boundary conditions can be solved analytically with MERW - for both see https://arxiv.org/pdf/1912.13300
 

1. What is the Ising model and how does it relate to open chain and periodic boundary conditions?

The Ising model is a mathematical model used to study the behavior of interacting particles, such as atoms or spins. It is commonly used in statistical physics to understand phase transitions and critical phenomena. The model can be applied to both open chain and periodic boundary conditions, which refer to the arrangement of particles in a linear chain or in a loop, respectively. The differences in boundary conditions can affect the behavior of the system and the results obtained from the model.

2. How do open chain and periodic boundary conditions differ in the Ising model?

In open chain boundary conditions, the particles at the ends of the chain are not connected to each other, while in periodic boundary conditions, the ends of the chain are connected to form a loop. This means that in open chain conditions, the particles at the ends only interact with their nearest neighbors, while in periodic conditions, they interact with all particles in the chain.

3. What are the advantages and limitations of using open chain and periodic boundary conditions in the Ising model?

The advantage of using open chain boundary conditions is that it allows for the study of finite systems, which can be useful in understanding real-world systems. However, this approach does not take into account the effects of long-range interactions that may occur in larger systems. On the other hand, periodic boundary conditions allow for the study of infinite systems, which can provide a more accurate representation of the behavior of the system. However, this approach may not accurately represent the behavior of finite systems.

4. How do boundary conditions affect the critical temperature in the Ising model?

The critical temperature, which is the temperature at which a phase transition occurs, can be affected by the choice of boundary conditions. In open chain conditions, the critical temperature is lower compared to periodic conditions due to the absence of long-range interactions. This means that the behavior of the system may change at a lower temperature in open chain conditions compared to periodic conditions.

5. Can the Ising model with open chain and periodic boundary conditions be applied to real-world systems?

Yes, the Ising model can be applied to real-world systems, but it is important to consider the limitations of the chosen boundary conditions. In some cases, neither open chain nor periodic conditions may accurately represent the behavior of the system, and other boundary conditions may need to be considered. Additionally, the Ising model is a simplified representation of complex systems and should be used in conjunction with other models and experimental data for a more comprehensive understanding.

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