Analogue of Callan-Symanzik equation for Ising model

[Bobhawke]

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 scale that it is measured at, and f is the function that relates them.

This tells you the renormalisation group flow, but it doesn't tell you what the coupling is at all scales, just at multiples of 2 of your original scale.

My question is this: would it be possible to form a differential equation for the Ising model (or other statistical system), analagous to the Callan-Symanzik equation for QFT, which tells you how the copuling changes with scale for all scales?

 

[genneth]

Do everything in Fourier space. Integrate out thin shells in momenta, then rescale to restore contrast.

 

[sir-pinski]

The Ising model is a statistical system that describes the behavior of a collection of interacting spins on a lattice. It has been extensively studied in the field of statistical mechanics and has been used as a toy model for understanding phase transitions and critical phenomena. One of the key concepts in studying the Ising model is renormalization, which is a method for understanding the behavior of a system at different length scales.

In the context of renormalization, the Callan-Symanzik equation is a powerful tool in quantum field theory that describes how the couplings in a theory change with energy scale. This equation is derived from the renormalization group flow, which tells us how the physics of a system changes as we zoom in or out on different length scales.

In the case of the Ising model, the renormalization group flow can be described by a similar equation, known as the analogue of the Callan-Symanzik equation. This equation relates the coupling at different scales, just like in the case of QFT, but it also takes into account the discrete nature of the Ising model. This means that instead of a continuous change in energy scale, we have a discrete change given by the lattice spacing.

The analogue of the Callan-Symanzik equation for the Ising model can be written as K(L) = f(K(L/a)), where K is the coupling, L is the lattice size and a is the lattice spacing. This equation describes how the coupling K changes as we change the lattice size, and it is analogous to the Callan-Symanzik equation in QFT.

However, unlike in QFT where the Callan-Symanzik equation gives us information about the coupling at all energy scales, the analogue for the Ising model only gives us information at discrete scales. This is because the Ising model is a lattice system and the renormalization group flow is only defined at these discrete scales.

To answer your question, it is possible to form a differential equation for the Ising model that describes the renormalization group flow at all scales. However, this equation would be different from the Callan-Symanzik equation and would take into account the discrete nature of the Ising model. Such an equation would be useful in understanding the behavior of the Ising model at different length scales and could provide insights into the phase transitions and critical phenomena of the system.