For a 1D transverse Ising model the Hamiltonian can be expressed as $$H = -J \sum_{i} S_{i}^{x}S_{i+1}^{x} - h \sum_{i} S_{i}^{z}$$ According to my understanding this undergoes a 2nd order phase transition at some temperature $T = T_c \neq 0$. Also I have heard close to a critical point one can describe the transverse Ising model using a 2D conformal field theory. How can that be done ? Most descriptions I found dealt with CFT for XY model, and that too only mentioned the central charge and how they can be related to Majorana fermions. The procedure was not mentioned. So my questions are:

  1. How to derive CFT model for 1d transverse Ising model?
  2. What is the central charge for the above derived cft?

2 Answers 2


The Jordan-Wigner transformation can be used to exactly transform the 1D quantum transverse Ising model into a system of noninteracting fermions. Near the transition point, we can take the long-wavelength limit and describe the system by the free fermion CFT, whose central charge is $c = 1/2$. These slides describe the steps in a fair amount of detail.


As a comment to tparker's answer, everything is correct except for the value of the central charge. Non-interacting fermions, as obtained from Jordan-Wigner from the 1D quantum transverse Ising model, have a central charge c=1, as they are complex fermions. Not to be confused with a real (i.e. Majorana) fermion, for which c=1/2, as in the Ising model for instance.

  • 2
    $\begingroup$ I voted -1: critical Ising chain has c=1/2, as does its JW-dual critical Majorana chain description. $\endgroup$ Jun 3, 2022 at 22:28

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