Physics Stack Exchange
Physics Stack Exchange

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.

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. By dint of simplicity, it is relatively easy to solve explicitly as a function of the interaction strength: The one-dimensional variant does not evince a phase transition, but the two-dimensional square-lattice one does.

744 questions
57
votes
4 answers

Why is it hard to solve the Ising-model in 3D?

The Ising model is a well-known and well-studied model of magnetism. Ising solved the model in one dimension in 1925. In 1944, Onsager obtained the exact free energy of the two-dimensional (2D) model in zero field and, in 1952, Yang presented a…
Marton Trencseni
  • 3,105
42
votes
2 answers

Examples of important known universality classes besides Ising

I am working with RG and have a pretty good idea of how it works. However I have noticed that even though the idea of universality class is very general and makes it possible to classify critical systems, textbooks seem to always end up with the…
Steven Mathey
  • 4,425
29
votes
2 answers

Confusion about duality transformation in 1+1D Ising model in a transverse field

In 1+1D Ising model with a transverse field defined by the Hamiltonian \begin{equation} H(J,h)=-J\sum_i\sigma^z_i\sigma_{i+1}^z-h\sum_i\sigma_i^x \end{equation} There is a duality transformation which defines new Pauli operators $\mu^x_i$ and…
Mr. Gentleman
  • 1,606
27
votes
4 answers

What is the definition of correlation length for the Ising model?

The correlation length $\xi$ is related to critical temperature $T_c$ as $$ \xi\sim|T-T_{c}|{}^{-\nu}, $$ where $\nu$ is the critical exponent. Is this the formal definition of correlation length? If not, what is the formal definition of…
cosmicraga
  • 550
22
votes
6 answers

Mean field theory Vs Gaussian Approximation?

I am getting confused about the distinction between Mean-field theory (MFT) and the Gaussian approximation (GA). I have being told on a number of occasions (in the context of the Ising model) that the Gaussian approximation is at the same level as…
Quantum spaghettification
  • 7,873
22
votes
2 answers

What new features does the Heisenberg Model have compared to the Ising Model?

Both the Ising and the Heisenberg Models describe spin lattices with interaction on first neighbors. The Hamiltonian in each case is quite similar, despite the fact of treating de spins as Ising variables (1 or -1) or as quantum operators. In the…
P. C. Spaniel
  • 4,476
22
votes
2 answers

Ising model for dummies

I am looking for some literature on the Ising model, but I'm having a hard time doing so. All the documentation I seem to find is way over my knowledge. Can you direct me to some documentation on it that can be parsed by my puny undergrad brain? If…
F. P.
  • 213
  • 2
  • 6
21
votes
1 answer

Continuum Field Theory for the Ising Model

My problem is to take the $d$-dimensional Ising Hamiltonian, $$H = -\sum_{i,j}\sigma_i J_{i,j} \sigma_j - \sum_{i} \tilde{h}_i \sigma_i$$ where $J_{ij}$ is a matrix describing the couplings between sites $i$ and $j$. Applying a Hubbard-Stratonovich…
Kai
  • 3,930
  • 1
  • 15
  • 29
19
votes
1 answer

Is the Ising CFT different from the Majorana CFT?

If by 'Ising CFT' I mean the conformal field theory describing the critical quantum Ising chain $ H = \sum_n \left( \sigma^z_n - \sigma^x_n \sigma^x_{n+1} \right)$ and by 'Majorana CFT' I mean the conformal field theory describing its Jordan-Wigner…
Ruben Verresen
  • 9,713
19
votes
2 answers

What is the momentum canonically conjugate to spin in QM?

In Kopec and Usadel's Phys. Rev. Lett. 78.1988, a spin glass Hamiltonian is introduced in the form: $$ H = \frac{\Delta}{2}\sum_i \Pi^2_i - \sum_{i
derpy
  • 341
19
votes
2 answers

Subtleties in the exact solution to the 1D quantum XY model, in particular the Bogoliubov transformation

I am writing programs to construct the spectra of models with known exact solutions, and soon noticed some subtleties that are not often mentioned in most references. These subtleties are not important in the thermodynamic limit, but since I am…
Chenfeng
  • 527
18
votes
3 answers

Zero magnetization of spin model without external magnetic field

For a given Hamiltonian with spin interaction, say Ising model $$H=-J\sum_{i,j} s_i s_j$$ in which there are no external magnetic field. The Hamiltonian is invariant under transformation $s_i \rightarrow -s_i$, so there are always two spin states…
unsym
  • 4,951
17
votes
2 answers

Interpretation of the 1D transverve field Ising model vacuum state in a spin-language

The 1D transverse field Ising model, \begin{equation} H=-J\sum_{i}\sigma_i^z\sigma_{i+1}^z-h\sum_{i}\sigma^x_i, \end{equation} can be solved via the Jordan-Wigner (JW) transformation (for further reference about the explicit form of the JW…
VanillaSpinIce
  • 1,458
17
votes
1 answer

Understanding Periodic and Anti-periodic boundary condition for Jordan-Wigner transformation

In the study of spin chains with periodic boundary condition ($S_{N+1}=S_{1}$) when one applies Jordan-Wigner transformation to map the spin chain to spinless fermion chain, one needs to make sure in the mapping the periodic boundary condition for…
Galilean
  • 1,098
  • 8
  • 23
17
votes
1 answer

Why do spin correlation functions in Ising Models decay exponentially below the critical temperature?

I'm trying to form a better understanding of the 2D Ising Model, in particular the behaviour of the correlation functions between spins of distance $r$. I've found a number of explanatory texts that seem to indicate that at both above and below the…
Suh doh nimh
  • 173
1
2 3
…
49 50