Questions tagged [ising-model]
The Ising model of ferromagnetism in statistical mechanics consists of discrete bimodal (+1 or −1) "spin" (moment) variables in a simple Hamiltonian interacting with their next neighbors on a lattice. The one-dimensional variant does not evince a phase transition, but the two-dimensional square-lattice one does. Use for analog and generalized discrete models on several lattices and dimensions.
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Spectral gap and Ising model
Consider an instance of the Ising model, with $N$ number of spins on a 2D square lattice (or any other 2D structure) wrapped into a torus to avoid boundary conditions (in other words, periodic ...
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Is flowing with RG in 1D Ising model equivalent to changing the temperature of the system
Let us consider the easiest form of the Ising Hamiltonian:
$$ \beta H(s_i; J) = -J\sum_i^N s_i s_{i+1} $$ ($\beta = 1/k_BT$ so we already defined $J = \tilde{J}/k_BT$ with $\tilde{J}$ constant).
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Reduced state of transverse ising model
Consider the model
$$
H = - \sum_{i=1}^{N} (\lambda \sigma_i^x \sigma_{i+1}^x + \sigma_z)
$$
with periodic boundary conditions $\sigma_1^x=\sigma_{N+1}$.
Equation 33 in this paper says the reduced ...
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Long Range interaction in Ising Model [duplicate]
How do we apply mean Field Approximation to the following Hamiltonian of Ising Model in 1D?
\begin{equation}
\mathcal{H} = -\sum_{i=1}^{N}\sum_{j<i}\frac{J}{\lvert i - j \rvert ^ {\alpha}}\sigma_i \...
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Ising Model in 1D with long range interaction [closed]
I would like to arrive at the critical temperature for an Ising model with a long-range power-law Hamiltonian given by:
$$H = -\sum_{i=1}^{N}\sum_{j<i}\frac{J}{\lvert i - j\rvert^\alpha}\sigma_i \...
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Ising model approximation of partition sum
If we consider the Ising model without magnetic field and without periodic boundary conditions we get for the partition sum $Z_N=2(2\cosh(K))^{N-1}$ ($K$ is dimensionless coupling). Why can we then ...
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Is $\phi^5$ a descendant in $\phi^4$-theory (at the conformal Wilson-Fisher fixed point)?
I'm wondering if $\phi^5$ is a descendant in $\phi^4$-theory in $d = 4 - \epsilon$ at the conformal Wilson-Fisher fixed point, where the coupling constant is $\lambda$. The e.o.m. tells us that $\phi^...
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How to interpret weights of the Principal Component Analysis of the Ising model?
I'm trying to replicate the results obtained in this paper: https://arxiv.org/pdf/1606.00318.pdf
. On page 3 the autors mention that the fact that the weight of the first principal component is ...
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Renormalisation flow and conformal symmetry
In mathematics, a meta-principle is that when a group of symmetries $G$ acts on Euclidean space $\mathbf{R}^n$ and fixes the origin $0$, then if $Y\to \mathbf{R}^n$ is any structure sitting over ...
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Duality relation for the correlation length
In this answer to my previous question, Yvan Velenik mentioned the equality for correlation lengths of dual Ising models on a square lattice
$$
\xi(T) = \xi(T^*)/2.
$$
I have the following questions ...
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Correlation function in the critical two-dimensional Ising model
Fifty years ago, McCoy and Wu in their book The Two-Dimensional Ising Model formulated a hypothesis about the correlation function in the critical two-dimensional Ising model. According to this ...
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Ising model but energy is lower only for up-up states
I understand that in the Ising model, when two consecutive spins align, the energy is lower. I wonder if there is a name for a modified Ising model where only two consecutive spins align in the up-up ...
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"Long-range" 1D Ising model with exponentially decaying interactions
Consider a 1D Ising model with infinite-range but exponentially decaying interactions. For definiteness, say it is defined by the following Hamiltonian on $n$ spins:
\begin{align}
H\left[\left(\...
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Eigenvalue of transfer matrix in Shankar (3.3.4), p.34
In the book of Quantum Field Theory and Condensed Matter written by Shankar, (3.3.4), p.34, there defined a transfer matrix
$$T=\left(
\begin{array}{cc}
1 & e^{-2K} \\
e^{-2K} & 1 \\
\end{...
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Ising model rescaling
Consider the 2D classical Ising model. It's understood that there is a critical temperature $T_c$, and that the correlation length $\xi(T)$ defined by:
$$\langle \sigma_i \sigma_j \rangle_\mathrm{...
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Analyzing a Kawasaki-evolving Ising model? (conserved-order-parameter Ising model)
Focusing on 2D in further text
I am struggling to understand how the conserved-order-parameter Ising model (also known/reached through Kawasaki algorithm) shows criticality and also how it can be ...
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Specific Heat Capacity in the 2D Ising model of spin glass
I am studying statistical mechanics on my own and I read that the 2D Ising model can be used to model spin glass thermal properties.
I looked at Onsager's formula for free energy there are some points ...
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Heaviside function in the form of an integral
I am currently reading Optimal storage properties of neural network models by E. Gardner. (DOI 10.1088/0305-4470/21/1/031)
In appendix 1, the Heaviside function is expressed in integral form eq A1.1
$$...
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Classical Heisenberg model and variational free energy for the mean field [closed]
This is a homework question that I'm stuck on for 2 days, we are asked to analyze the mean field method for the Classical Heisenberg model using the variational free energy. The Hamiltonian is given ...
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Extra term in $2+\epsilon$ expansion of sigma model
I'm working through David Tong's notes on Statistical Field Theory, in particular the $2+\epsilon$ expansion of the sigma model with free energy
$$F[\vec{n}]=\int d^dx \frac{1}{2e^2}\nabla\vec{n}\cdot\...
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Metropolis Monte-Carlo for magnetic system with $S > 1/2$ or arbitrary set of quantum systems
A well-known example of classical Monte-Carlo method application is Ising model with $S=1/2$.
As I understood, people there widely use it for any kind of magnetic materials following the same idea
$$
...
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Infrared bound on Ising model
I'm currently trying to understand aspects of Hugo Duminil-Copin's Lectures on the Ising and Potts models on the hypercubic lattice. In section 4.3, he claims that for the Ising model in $\mathbb{Z}^d$...
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Motivation of $p$-spin glass model
In general, the $p$-spin glass model focuses on $p$-body interactions like
$${\displaystyle H({\boldsymbol {\sigma }})=\sum_{i_1,...,i_p} J_{i_{1},\ldots i_{p}} \sigma _{i_{1}}\cdots \sigma _{i_{p}}.}$...
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Di Francesco et al.'s CFT - additional corrections to free-energy for strip geometries on a lattice?
In classical spin systems, there's a nice way to extract the central charge of the model by looking at finite-size corrections to the free energy of strips of length $L$ and width $W$ in the limit of ...
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Status of Approach of constructing Hamiltonians from Transfer Matrix
I am studying this old paper from J.B.Kogut on lattice gauge theories and spin systems [Rev. Mod. Phys. 51, 659(1979)].
This paper discusses about the way of constructing a quantum Hamiltonian using ...
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How does one get the uncertainties for the critical exponents in Metropolis Monte Carlo for the Ising model?
I've recently learned the basics about simulating the Ising model with Metropolis Monte Carlo. In particular, I've seen how to evolve the system, compute the average magnetization, find the critical ...
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Non-translation invariant Gibbs state of the Ising model
In the book "Statistical mechanics of lattice systems" by Sacha Friedli and Yvan Velenik, on page 157 we have the following theorem concerning non translation invariant Gibbs states of the ...
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Ising model has short range correlation (Exercise in Velenik's book)
I'm studying the book "Statistical mechanics of lattice systems" by Sacha Freidli and Yvan Velenik, exercise 3.15 page 109:
Let $\beta\geq 0$ and $h\in\mathbb{R}$, show that $\langle\cdot\...
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How renormalization allows to describe critical point behaviour using the critical fixed point?
As in the title, I am trying to understand how the critical fixed point (CFP) can be used to derive the thermodynamic singular behavior of the physical critical point (PCP). The context I have in mind ...
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Equivalence between $J = 0$ and $h = 0$ in Ising model
This is the hamiltonian for the Ising-model in one dimension
$$H=-J \sum_{i=1}^N \sigma_i \sigma_{i+1}-h \sum_{i=1}^N \sigma_i$$
with $\sigma_i=\pm 1$ and $\sigma_{N+1}=\sigma_1$
$J$ is the ...
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Mean field theory of an Ising antiferromagnet
I am calculating the free energy of an Ising antiferromagnet under the static magnetic field, and trying to get an expansion of free energy near the Neel temperature.
But my calculation leads to a ...
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Difference between two Monte-Carlo methods in Ising model
I was working on a Monte-Carlo simulation of the Ising model. It seems that we have two different way to flip a single spin and I didn't quite understand the difference between them.
Say we have $N\...
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Factorization of 1d Ising model partition function
If I'm studying a 1-dimensional Ising model such that $\mathcal H = \sum_k J_k\sigma_k\sigma_{k+1}$, where $$J_k=\begin{cases}J&k \in2\mathbb N\\2J&k\in2\mathbb N+1 \end{cases}$$ can I ...
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Compute correlation over time in Montecarlo COP (Kawasaki) Ising simulation
I'm looking at phase transition in a binary alloy following quenching below the critical temperature. This model is equivalent to a conserved ordered parameter Ising model: the constant concentration ...
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The "overlap function" of a $Z_2$ gauge theory
Consider a $Z_2$ gauge theory on a square lattice (Ising spins on edges) with classical degrees of freedom, i.e.
\begin{equation}
E = -\sum_{\square} \sigma_i\sigma_j\sigma_k\sigma_l
\end{equation}
...
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Invertibility of adjacency matrix of nearest neighbor Ising model
In Goldenfled's lectures on phase transitions and the renormalization group exercise 3.3 .
One is asked to consider nearest neighbor Ising model (as the pic)
In the second problem, I let $B_i=S_i$ ...
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Reference request: readable introduction to Landau theory and phase transition
I am doing some self-study on Landau theory and phase transition models in physics. In particular I am looking at how to apply these ideas to opinion dynamics models. I found a really nice set of ...
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Transverse-field Ising model in the presence of a longitudinal field - ferromagnetic phase diagram
I am wondering what is the phase diagram of the transverse-field Ising model in the presence of a longitudinal field, in particular, a one-dimensional spin-1/2 chain with ferromagnetic interactions. ...
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CFT Description of the 2D Ising Model as both a free fermion theory and a $\varphi^4$ Landau theory
There are numerous Stack Exchange answers that explain how to construct a free fermion CFT ($c = 1/2$) which describes the critical point of a 2D Ising model.
However there are also sources that ...
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Under dynamic scaling, Ising model emerge fractals, which implicate what？
In the lecture note of statistic field theory of David Tong,
Fluctuations occur on all length scales, big and small.
he post a link, a youtube video, which describe the Ising model in different ...
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Local statistical model for percolation
I am interested in classical statistical models with degrees of freedom with a finite configuration set on regular lattices, and Boltzmann weights depending on the configurations in a constant-size ...
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Excitation energy in the free-fermion description of TFIM
The transverse-field Ising model has a gapped phase when the field is large and a gapless phase when the field is small. In the gapped phase the gap grows linearly with the field.
One of the ways to ...
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Interpretation of Renormalization Group
For the purposes of this question, I will be talking about systems in statistical mechanics (e.g the Ising Model) so I will assume the system of interest has a natural cutoff frequency $\Lambda$.
For ...
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Is there a known closed-form expression for the susceptibility of the 2-D Ising model at $B = 0$?
The Onsager solution for the 2-D Ising model allows us to find (among other things) complicated expressions for the internal energy of the system (in the thermodynamic limit and in zero magnetic field)...
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Kasteleyn orientation in Dimer model
Some Background: The problem of dimer configurations or perfect matching occurs naturally when a diatomic gas is adsorbed onto a crystalline substrate and was first discussed by Rushbrooke and Fowler. ...
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How to compute correlation function and length for a given lattice configuration?
Focusing on the Ising model, for a given lattice configuration of up and down spins (say square or triangular lattice), and a given interaction type (ferromagnetic or antiferro), how can one compute ...
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Why do fermions anti-commute in Ising model?
In my course fermions are given like a product of spin and (dual to spin) disorder parameter in 2D Ising square lattice. Then, using the properties of disorder parameter I can prove that fermions ...
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How to get mass near the critical point for 2D Ising model on square lattice for fermions?
Let's assume that we have 2D square lattice with spins $\sigma=\pm 1$ and equal vertical and horisontal energy coefficients K:
$$E_{interaction}=e^{K\sigma_i\cdot\sigma_j}$$
where $\sigma_i,\sigma_i$ ...
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How do spin operators work? [closed]
I am currently studying statistics and 2D Ising models and noticed in my lecturer's notes the operators, acting in the spin space
The text says that this is identity $2^N\times 2^N$ matrix. I don't ...
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How does the value $k$ work in MCMC algorithm in the case of Ising model?
Here
they say that $k$ is Boltzmann's constant.
However, here they are saying that $k$ is a variable
Can anyone explain what the catch is here?
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