What is Ising model: Definition and 90 Discussions

The Ising model (; German: [ˈiːzɪŋ]), named after the physicist Ernst Ising, is a mathematical model of ferromagnetism in statistical mechanics. The model consists of discrete variables that represent magnetic dipole moments of atomic "spins" that can be in one of two states (+1 or −1). The spins are arranged in a graph, usually a lattice (where the local structure repeats periodically in all directions), allowing each spin to interact with its neighbors. Neighboring spins that agree have a lower energy than those that disagree; the system tends to the lowest energy but heat disturbs this tendency, thus creating the possibility of different structural phases. The model allows the identification of phase transitions, as a simplified model of reality. The two-dimensional square-lattice Ising model is one of the simplest statistical models to show a phase transition.The Ising model was invented by the physicist Wilhelm Lenz (1920), who gave it as a problem to his student Ernst Ising. The one-dimensional Ising model was solved by Ising (1925) himself in his 1924 thesis; it has no phase transition. The two-dimensional square-lattice Ising model is much harder and was only given an analytic description much later, by Lars Onsager (1944). It is usually solved by a transfer-matrix method, although there exist different approaches, more related to quantum field theory.
In dimensions greater than four, the phase transition of the Ising model is described by mean field theory.
The Ising problem without an external field can be equivalently formulated as a graph maximum cut (Max-Cut) problem that can be solved via combinatorial optimization.

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  1. T

    Doubts on 2D and 3D Ising Model

    Considering d=2 or d=3, the Ising model exhibits a second order phase transition at the critical temperature T_c, where the system goes from an ordered phase (spins preferably aligned in a certain direction) to a disordered one. This is reflected by the behaviour of the susceptibility, similar...
  2. C

    Dependence of exchange interaction on system size in Ising model

    In the well known Ising model, without any external field (H=0), the energy (E), spins (s) and exchange interaction (J) are related as in the following equation $$ E = -\sum_{<ij>}J_{ij}s_{i}s_{j} $$ Jij is site dependent and consists of three components JAA, JBB and JABwhere A is say up...
  3. U

    Why is the Ising model so important?

    I've been reading through some of the literature on solutions of the Ising model, but I can't help but notice it doesn't provide that good a model for actual ferromagnetic systems. It seems that these models get a lot of attention and I'm just curious as to why? Also, why is an exact solution to...
  4. K

    Nucleation and growth in a 2D Ising Model with Monte Carlo Simulation

    I'm currently working on a project for a code that does umbrella sampling of a 2-D Ising model [size LxL of a magnet (analyzing up or down spins)]. The next step is to take my code to analyze a nucleation region and its growth by varying the temperature above critical. Before I even attempt to...
  5. L

    Ising model canonical partition function

    Why in case of Ising model ##H=-J\sum S_iS_{i+1}## we calculate canonical partition function?
  6. L

    Classical spin system. Ising model.

    Energy function ##E=-S_1S_2##. I took ##J=1##. If spin are oriented parallel energy is negative. How could energy be negative?
  7. W

    Using RGT to Find Critical Temp/Point of 2D Ising Model

    is it possible to use renormalization group theory to get the critical temperature of the 2d ising model? or even, is it possible to show that there is a critical point?
  8. W

    Where Can I Find Exact Results for the Ising Model?

    is there any reference on ising model i would like to see all the exact results on ising model for 2d ising model, the correlation length diverge as t to -1 around the critical temperature how is this result derived?
  9. L

    Calculating Magnetization for 2x2 Ising Model Lattice

    Homework Statement Calculate magnetisation for partition function ##Z=12+4\cosh (8\beta J)## for Ising model 2x2 lattice.Homework Equations F=-k_BTln Z M(H,T)=-\frac{\partial}{\partial H}(\frac{F}{k_BT}) The Attempt at a Solution For me it looks that magnetisation is zero. By just doing the...
  10. G

    Heath bad method for Ising model

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  11. maverick280857

    Ising Model Monte Carlo - interpretations

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  12. maverick280857

    C/C++ Visualizing the 2D Ising Model with Monte Carlo Algorithm

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  13. M

    How is Ising Model a Markov Chain?

    The title says it all. It looks like the configuration probability only depends on where you want to go, not what state you are in now. Yet when I watch simulations, there is clearly a dependence on the previous state. Is there something pretty basic I'm misunderstanding about configuration...
  14. D

    Effect of sample size when using periodic boundary conditions in 2D Ising model

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  15. H

    How Does the Ising Model Exhibit Diamagnetism to Paramagnetism Transition in 2D?

    I'm doing a computational physics project where I'm looking at the Ising model in 2D and external magnetics field. I start with all the spins in same directions my grid is 30x30 and I use periodic boundry conditions. I start with all the spins in the same direction (+1). I do about 150000...
  16. B

    MATLAB Matlab Ising model: Anti-ferromagnet

    Hi I'm working on a MATLAB simulation of the 2D Ising model, and would like to verify my code and its results. One thing I'd like to try and observe is the transition from anti-ferromagnet to ferromagnet, but I'm not sure how to create the initial lattice in Matlab. I've already made a...
  17. M

    Diagonalizing Hamiltonian for Multi-Qubit Ising Model

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  18. O

    2D Ising Model (analytical expressions)

    Hi all, I am doing a program to simulate the 2D Ising Model under the metropolis algorithm. In order to check my results I would like to compare them with the analytical expressions for the mean energy, magnetization, specific heat and magnetic susceptibility. I already found the...
  19. O

    2D Ising Model (analytical expressions)

    Hi all, I am doing a program to simulate the 2D Ising Model under the metropolis algorithm. In order to check my results I would like to compare them with the analytical expressions for the mean energy, magnetization, specific heat and magnetic susceptibility. I already found the...
  20. W

    Level degeneracy of the transverse ising model

    the hamiltonian is H=-J \sum_{l=1}^N \sigma_l^z \sigma_{l+1}^z+ g \sigma_l^x here we assume periodic boundary condition my problem is, what is the highest possible degeneracy of the levels? initially i expected 2 but numerically i find that it is 4 i cannot understand it
  21. A

    Renormalisation in 1D plaquette like ising model.

    Hi guys, I'm working through past papers and I have a problem with deriving the renormalised scaling of the following: [PLAIN]http://dl.dropbox.com/u/16658950/helpme.JPG I'm doing the rescaling as I would for a 1D ising model decimated with l = 2 (so every other spin, but N=4 in this...
  22. V

    Using metropolis algorithm for 2D ising model

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  23. A

    Ising model and more complex interactions between particles

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  24. R

    Simple ising model: Magnetic susceptibility derivation

    I'm stuck on a question about deriving an expression for the magnetic susceptibility in terms of the variance of the magnetisation for a simple 2d square ising model. I get the derivation of the specific heat, and I know am supposed to do something similar to get to the expression for...
  25. I

    Understanding Energy and Spin Interactions in the Ising Model

    Hello, actually, I am not a physicist, I am a Computer Science Graduate Student, Anyways, I am doing a research related to Ising Model My question is about the energy in Ising Model the following statement is written in "History of the Lenz-Ising Model" paper: "when two neighboring spins...
  26. D

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  27. J

    Solving 2D Ising Model Issue with Java

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  28. C

    Applying Ising Model to High Temp Superconductors

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  29. Q

    Searching the phenomenological renormalization group equation for 2d ising model

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  30. S

    Calculating Average Spins at Site i for Ising Model via Transfer Matrix

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  31. G

    Magnetic susceptibility (Ising model)

    Hi, I'm slightly confused about how to prove that: \chi=\vartheta<M>/\varthetaH is equal to... \chi=(<M2>-<M>2 )/ T I've expressed <M> as \sumMsexp(-E/kBT) / \sumexp(-E/kBT) and know that E=-J\sumSiSj-H\sumSi But seem to get lost in the...
  32. K

    Learning 1D Ising Model With Nearest Site Interaction

    I am learning the 1D ising model (spin 1/2), without external field and considering the nearest site interaction, the hamiltonian for 1D chain is simple H = -J\sum_i S_iS_{i+1} Since each spin can only take either +1 or -1, we can write the transition matrix as T = \left(...
  33. B

    Analogue of Callan-Symanzik equation for Ising model

    When I studied renormalisation for the Ising model the procedure was to sum over every second spin and then find a new coupling which produced the same physics. This leads to a relation between the coupling at different scales of the form K(2s)=f(K(s)) Where K is the coupling, s is the...
  34. P

    Why Can't the 1D Ising Model Have a Phase Transition?

    Homework Statement Can someone please explain to me why there can never be a phase transition in the 1D Ising model? Homework Equations The Attempt at a Solution I have read the argument that if we start at T=0, all spins along the 1D chain are aligned (say up). Then if we...
  35. J

    What drives spin alignment in some materials?

    The Ising model starts with an assumption that the nearby spins reach lower potential by pointing at the same direction. However, two magnets would reach lower potential by pointing at different directions. What is the mechanism, that in the first place makes spins attempt to align themselves in...
  36. B

    Coupling in the Ising model

    I was trying to understand why for every spin configuration of a ferromagnetic system, there exists a corresponding isoenergetic state of an antiferromagnetic system. Can I treat an antiferromagnetically coupled 1-D ising model as a combination of two interpenetrating sublattices which are...
  37. C

    Phase transitions from ising model

    I'm having trouble understanding how changing parameters in the ising model leads to first order phase transitions. I understand how the intersection of \frac{kT}{2nJS} (\eta - \frac{g \mu_o H_o}{kT} ) = B_s(\eta) where B_s(\eta) is the Brouillion func leads (in the absence of a field H) to...
  38. D

    Finding Resources on Ising Model

    Hi, I'm looking for something (books or other), about Ising Model. Can anyone of you help me? Thank you
  39. W

    What is the Hamiltonian for a 2D Ising model on a square lattice?

    Homework Statement Consider a 2 dimensional three state Ising model with ferromagnetic coupling J, where si = −1, 0, 1 in a magnetic field h on a triangular lattice. Construct a mean field equation for the magnetization m. The Attempt at a Solution I don't know how to start. 2d ising...
  40. H

    What Does It Mean for 1D RFIM to Be Frustrated?

    I've read in an article that the 1D Ising Model in a random field is a frustrated system. But what does it mean for the 1D RFIM to be frustrated? The internet is very unhelpful. I'd appreciate it if someone could answer this for me. Thanks.
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