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

    I Understanding the Derivation of the Ginzburg Criterion for the Ising Model

    For the Ising Model, the Ginzburg Criterion is, for ##m_{0}## the order parameter and ##\delta m## the fluctuations: $$\langle\delta m\left(x\right)\delta m\left(x^{\prime}\right)\rangle << m_{0}^{2}$$. I want to understand how to derive the left hand side of the inequality from ##\langle M^{2}...
  2. D

    How much physics is needed for Bialek's Biophysics book?

    Hello, I am originally a medical doctor and now doing a PhD in neuroscience. I have no formal physics / math training beyond high school level but I self-studied single variable and multivariable calculus as well as differential equations from MIT's OCW website, did examples, exams etc. I also...
  3. C

    I Can the 3D Ising model be considered solved?

    I would like to know if the 3D Ising model can be considered solved. Which book can be recommended about the Ising model?
  4. LCSphysicist

    Partition function of modified Ising model

    $$H = - J ( \sum_{i = odd}) \sigma_i \sigma_{i+1} - \mu H ( \sum_{i} \sigma_i ) $$ So basically, my idea was to separate the particles in this way:: ##N_{\uparrow}## is the number of up spin particles ##N_{\downarrow}## "" down spin particles ##N_1## is the number of pairs of particles close...
  5. user366312

    A What is a "transient" value in the Ising model?

    What is a "transient" value, result, or state in the Ising model? How do we treat these values during the simulation? Do we discard these values? If so, why?
  6. yklee

    Ising model autocorrelation function calculation

    Dear Mr. and Ms., I am trying to measure the autocorrelation functions of 2D ising model based on the equation given by where A(t) denote a measure. I calculate a c(t) of magnetization. I calculated in this way data_path = f"../../trajectory/data.txt" data = np.loadtxt(data_path)...
  7. J

    I Calculating the specific heat capacity for the 2D Ising model

    So I'm looking at the book "Equilibrium Statistical physics" by Plischke and Bergersen. I'm doing the calculation of the specific heat of the 2D Ising model. I can't seen to quite get out the same expression as in the book - there are a coupe of minus signs that are different. I don't know if I...
  8. AndreasC

    I Problems involving combinatorics of lattice with certain symmetries

    I was reading about numerical methods in statistical physics, and some examples got me thinking about what seems to be combinatorics, an area of math I hardly understand at all beyond the very basics. In particular, I was thinking about how one would go about directly summing the partition...
  9. W

    A Ground state of the one-dimensional spin-1/2 Ising model

    Hi, I know that the ground state of the spin-1/2 Ising model is the ordered phase (either all spin up or all spin down). But how do I actually go about deriving this from say the one-dimensional spin hamiltonian itself, without having to solve system i.e. finding the partition function? $$...
  10. Dom Tesilbirth

    How to find the partition function of the 1D Ising model?

    Attempt at a solution: \begin{aligned}Z=\sum ^{N}_{r=0}C\left( N,r\right) e^{-\beta \left[ -NJ+2rJ\right] }\\ \Rightarrow Z=e^{\beta NJ}\sum ^{N}_{r=0}C\left( N,r\right) e^{-2\beta rJ}\end{aligned} Let ##e^{-2\beta J}=x##. Then ##e^{-2\beta rJ}=x^{r}##. \begin{aligned}\therefore Z=e^{\beta...
  11. J

    A 1D transverse-field Ising model - classical vs quantum differences?

    The 1D transverse field Ising model $$ H(\sigma)=-J\sum_{i\in \mathbb{Z}} \sigma^x_i \sigma^x_{i+1} -h \sum_{i \in \mathbb{Z}} \sigma^z_i$$ is usually solved in quantum way, but we can also solve it classically - e.g. parametrize angles of spins ##\sigma^x_i = \cos(\alpha_i)...
  12. L

    A Ising model open chain and periodic boundary conditions

    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?
  13. IreneFerri

    Problems with Landau Wang algorithm in fortran

    I have been struggling for over a month now with a problem I cannot fix. I would really appreciate any comment or guidance. Thank you! I am considering an Ising-like model with N agents that han hold one of the following 3 states, represented by vectors: state + : vector (1 0) state 0 : vector...
  14. J

    I Born rule in classical Ising model: Feynman -> Boltzmann ensemble?

    Quantum mechanics is often said to be equivalent with Feynman path ensemble, which "after Wick rotation" becomes Boltzmann path ensemble, also called euclidean path integrals (popular for numerical calculations), or random walk/diffusion MERW (maximal entropy random walk). But Boltzmann path...
  15. Danny Boy

    A Quantum Ising model correlation function query

    In this paper, on quantum Ising model dynamics, they consider the Hamiltonian $$\mathcal{H} = \sum_{j < k} J_{jk} \hat{\sigma}_{j}^{z}\hat{\sigma}_{k}^{z}$$ and the correlation function $$\mathcal{G} = \langle \mathcal{T}_C(\hat{\sigma}^{a_n}_{j_n}(t_n^*)\cdot\cdot\cdot...
  16. P

    Partition Function for Spin-1 One Dimensional Ising Model

    $$H=-J\sum_{i=1}^{N-1}\sigma_i\sigma_{i+1}$$ There is no external magnetic field, so the Hamiltonian is different than normal, and the spins $\sigma_i$ can be -1, 0, or 1. The boundary conditions are non-periodic (the chain just ends with the Nth spin) $$Z=e^{-\beta H}$$...
  17. J

    How Does the Ising Model Apply to Lattice Gas Systems?

    Hi everyone, even before addressing the following points I have a serious issue in understandig the text of the Exercise.My idea was to model this system with a lattice gas. Given that each site can host 2 atoms I have 3 possibilities for each site: I'll call'em ##S_{11} S_{00}## and ##...
  18. T

    What is the Correct Way to Plot a 2D Ising Model Temperature Chart?

    Hi, I've been given a homework to do based on 2D Ising model. Ive already read plenty of articles bout 2d ising model but I'm not fairly certain about some things. I got to do a chart similar to the one I attached. Correct me if I am wrong, but on the Y axis, I have to put average of...
  19. Danny Boy

    A Correlation functions of quantum Ising model

    I have a single technical question regarding a statement on page 7 of the paper "Dynamical quantum correlations of Ising models on an arbitrary lattice and their resilience to decoherence". The paper up until page 7 defines a general correlation function ##\mathcal{G}## of a basic quantum Ising...
  20. T

    Solving Ising Spin 1 Model w/ Transfer Matrix Method

    I did the first part using the transfer matrix method: $$ Z = Tr(T^{N}) $$ In this case, the transfer matrix will be $$ T(i,i') = \begin{pmatrix} e^{\beta J} & 1 & e^{-\beta J}\\ 1 &1 &1 \\ e^{-\beta J} & 1 & e^{\beta J} \end{pmatrix} $$ To get the trace of $T^N$, you find the...
  21. hilbert2

    A Exploring the 2-D Ising Model & Its Continuum Limit

    I've recently been reading about the 2-dimensional Ising model and its continuum limit from several sources, including https://webhome.weizmann.ac.il/home/fnfal/papers/Ising/lecture1.pdf https://webhome.weizmann.ac.il/home/fnfal/papers/Ising/lecture2.pdf As far as I understood it, the state...
  22. JD_PM

    Dimensional analysis on an Ising model solution

    Homework Statement The mean field solution for the Ising model is: $$m = tanh[\beta (mJz + H)]$$ I wanted to carry out a dimensional analysis in order to verify the equation. Homework Equations $$m = tanh[\beta (mJz + H)]$$ The Attempt at a Solution Knowing that: $$[m] = \frac{A}{L}$$...
  23. G

    A How can I simulate 2D correlated data with continuous-valued variables?

    Not sure this is the right area to post this. Let's say I have particles on a lattice, and they all have some property (ie, color) that is correlated at some known correlation length. I want to simulate this! In 1D I could do something like have color be a random walk in the given dimension...
  24. mastrofoffi

    Weird fluctuations in Ising model

    Not 100% sure this thread belongs in this section, sorry if it's out of place. I was trying to run a simulation of a 2D Ising model; i thought everything was going fine until I started to look at the numerical results and I noticed that I get some wild fluctuations in my results; as an example...
  25. A

    Four-spin interactions the Ising model

    Homework Statement Let us consider a fully connected Ising model with four-spin interaction in a transverse field Write the mean field Hamiltonian. Homework Equations none The Attempt at a Solution I know that the mean value of the product of two spins is equal, in the thermodynamic limit...
  26. MathematicalPhysicist

    Cluster Approximation for the Two-Dimensional Ising Model

    Homework Statement In the attachments there is the question and its solution, it's problem 3.5. Homework EquationsThe Attempt at a Solution My question is how did they get the dimensionless Hamiltonian in both cases, and how did they explicitly calculated ##m## in both cases? I assume it's...
  27. MathematicalPhysicist

    One-Dimensional Ising Model in Bethe Approximation

    Homework Statement The following question and its solution is from Bergersen's and Plischke's: Equation (3.38) is: $$m = \frac{\sinh (\beta h)}{\sqrt{\sinh^2(\beta h) + e^{-4\beta J}}}$$ Homework EquationsThe Attempt at a Solution They provide the solution in their solution manual which I...
  28. Leonardo Machado

    A Convergence of lattice Ising model

    Hello everyone. I'm working on a program to solve 2D Ising model of magnetic materials, using a system with 10x10 spins for simplicity at a temperature of 1E-8 K. I'm using this parameters to get a faster result of m=1 and guarantee it is correct. but... For now i already pass 300 Monte Carlo's...
  29. F

    Monte Carlo Simulation, Ising model in 2D

    Homework Statement Lo,Im stuck on how to retrieve the specific heat capacity from an MC simulation, with the metropolis algorithm. I want my graph to look something like this: https://i.stack.imgur.com/NXeXs.png Homework Equations C_v = ((<E^2>-<E>^2)/T^2 The Attempt at a Solution My code is...
  30. A

    I Understanding autocorrelations in 2D Ising model

    I wasn't sure where to post this, I hope this was the right section. I've been struggling quite a bit with implementing an autocorrelation code into my current project. The autocorrelation as it is now, is increasing exponentially from 1 at the start of my MC run, and hitting 2 halfway through...
  31. D

    Evaluating partition function (stuck for weeks)

    Hi, I'm trying to calculate the partition function for a certain system and I arrived at an expression for the partition function $Z$, and have been stuck here for two weeks at the least. This is not a homework problem. If this is the wrong place to post a question like this, could you please...
  32. L

    A Ising model and Hamilton function

    In case of Ising model we are working with effective Hamiltonian. So let's look to spins which interact. In a case of feromagnet energy function is defined by ## H=-JS_1S_2 ## We have two possibilities. ##S_1## and ##S_2## has different values. And ##S_1## and ##S_2## has the same value. In...
  33. domainwhale

    Insights High Temperature Low Temperature Duality for the Ising Model on an Infinite Regular Tree - Comments

    domainwhale submitted a new PF Insights post High Temperature Low Temperature Duality for the Ising Model on an Infinite Regular Tree Continue reading the Original PF Insights Post.
  34. C

    Ising Model using renormalisation group

    Homework Statement The Hamiltonian of the 1D Ising model without a magnetic field, is defined via: $$\mathcal H = − \sum_{ i=1}^N K\sigma_i \sigma_{i+1},$$ where ##K ≥ 0## and ##\sigma_i## are the Ising spins (i.e. ##\sigma_i= \pm 1##). A) Set up a decimation procedure with decimation...
  35. C

    Ising Model Spin configurations

    Homework Statement Consider the below network of spins, the spins numbered on the nodes of the diagram. The spins interact via the following Hamiltonian ##\mathcal H = \sum_{\langle i j \rangle} \sigma_i \sigma_j##, where the sum ##\langle i j \rangle## is over nearest neighbours and ##\sigma_i...
  36. B

    Origin of number of bounds in ising model

    what is the origin of the number of bound (N z /2 ) in the calculating of average in ensemble in Mean Field for the Ising model?
  37. P

    Finite T transverse magnetization of transverse Ising chain

    Homework Statement Consider the transverse field Ising model, with $$H=-J\sum_i\left(\sigma^x_i\sigma^x_{i+1}+g\sigma^z_i\right)$$ I have to calculate the magnetization $$\langle\sigma_z\rangle$$ at finite temperature. Homework EquationsThe Attempt at a Solution I have to say, I'm a bit lost.
  38. A

    Ising Model and Partition FUnction

    Hi all This is probably a naïve question to ask, but I am puzzled by it and need an answer. The first time I encountered the term 'partition function' that was in context of Boltzmann distribution. But the same formulas of manipulating a partition function ( to obtain free energy, temperature...
  39. N

    Please explain the Ising model in a simpler way, thank you

    Hello, can anyone please explain to me in a different approach, rather than repeating a book definition, what is the Ising model? and what conclusions were made from it? I would sincerely appreciate it, since I can't seem to grasp the concept even after reading numerous articles online and in...
  40. N

    Ising model, Hiblert space, Hamiltonian

    Can anyone please explain to me what is the Ising model, Hilbert space, and Hamiltonian ? However, please explain it as simple as possible because I am a freshman. I have looked up all three things. I've tried my best to make some sense of it, but I am, honestly, still confused on what any of...
  41. Matt atkinson

    1D ising model - Helix-coil transistion

    Homework Statement A simple model of a polymer undergoing a helix-coil transition is to describe the polymer in terms of N equal length segments, each of which can be in either a coil or a helix state. A more realistic model also takes into account the energy cost associated with a boundary...
  42. C

    Solving Antiferromagnetic Ising Model on Square Lattice

    Hello, I am trying to work out a mean field theory for an antiferromagnetic Ising model on a square lattice. The Hamiltonian is: ## H = + J \sum_{<i,j>} s_{i} s_{j} - B \sum_{i} s_{i} ## ## J > 0 ## I'm running into issues trying to use ## <s_{i}> = m ## together with the self-consistency...
  43. T

    Generalising the Ising model to multiple spin values

    My tutor asked us today to consider the partition function of the following model as an aside to our topic at the moment. I went to work out the maths of it today and I'm quite stuck for how the calculation can proceed. It's a 1d closed chain with some number, n, points. Each point has some...
  44. M

    Critical Exponents in the 1D Ising Model

    Homework Statement Obtain the critical exponents for specific heat, susceptibility, and the order parameter (magnetization). Homework Equations $$A = -k_B T N \ln \left[e^{\beta J} \cosh (\beta h) +\sqrt{ e^{2\beta J}\sinh^2 \beta h + e^{-2\beta J} }\right]$$ $$\left<m \right> \propto...
  45. fluidistic

    Ising model without spin-spin interaction

    Homework Statement Hey guys! I'm not convinced by what I obtained so far, please tell me whether I'm in the right or wrong direction. The Hamiltonian has the form ##H=-h\sum _i s_i## where ##s_i=\pm 1## and ##h=kT##. 1)Calculate the partition function and the Gibbs free energy. 2)Calculate the...
  46. U

    Flipping state: Metropolis in Ising model

    Homework Statement Hi everyone, I'm having some difficulties with my code: 1. Randomly Choose a lattice at position (x, y) within the NxN lattice 2. Why by writing "int x = int(drand48()*L);" and "int y = int(drand48()*L);" it doesn't extract the value stored at that location (x, y)? 3...
  47. maajdl

    Ising model, same interaction with all spins

    Hello, I just read a question about the Ising model, and this reminds me of an old interrogation I had long ago. It is simply that: The Ising model deals with spins interacting only with close neighbors. I would be interested in a model where all spins interact with each other in exactly the...
  48. A

    Ising Model for Spins: External Magnetic Field Effect

    I think most of you are familiar with this model (sum runs over nearest neighbours): H = -J ∑S_iz * S_jz It demonstrates one of the succeses of meanfield theory as one can succesfully introduce: S_iz = <S_iz> + S_iz - <S_iz> = <S_iz> + δS_iz Such that: S_iz*S_jz ≈ 2S_iz<S_jz> +...
  49. G

    Applicability of Ising model on real materials

    Hi I chose a Monte Carlo simulation of the 2D Ising model as my Computational Physics course project. Unfortunately, I ran intro problems when formulating the exact problem since my professor probably wants me to simulate a real life material and extract magnetization curve M(T,H) out of it...
  50. W

    Summing over spins in Ising model.

    "Summing over spins" in Ising model. I don't understand this concept very well. Is it like taking the energy each possible configuration and adding them together? For example, if we had a 2D lattice gas and wanted to sum over the "white" the spins in a "checkerboard pattern": What are we...