Graduate Ising model open chain and periodic boundary conditions

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SUMMARY

The discussion focuses on the one-dimensional Ising model, specifically comparing open chain systems with free ends to cyclic boundary conditions when an external field is applied. It establishes that while these methods are not equivalent, the classical transverse field Ising model for an infinite 1D lattice can be effectively approximated using Maximal Entropy Random Walk (MERW). Additionally, the 2D classical Ising model for a finite width lattice with cyclic boundary conditions can also be solved analytically using MERW, as detailed in the referenced arXiv paper.

PREREQUISITES
  • Understanding of the Ising model in statistical mechanics
  • Familiarity with boundary conditions in physical systems
  • Knowledge of Maximal Entropy Random Walk (MERW)
  • Basic concepts of quantum treatment in statistical physics
NEXT STEPS
  • Research the application of Maximal Entropy Random Walk (MERW) in statistical mechanics
  • Study the differences between open chain and cyclic boundary conditions in the Ising model
  • Explore the computational methods for the quantum treatment of the transverse-field Ising model
  • Read the arXiv paper on the analytical solutions for the 2D classical Ising model with cyclic boundary conditions
USEFUL FOR

Physicists, researchers in statistical mechanics, and students studying quantum models who seek to understand the implications of boundary conditions in the Ising model.

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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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
 

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