# Correlation length for ising-Kitaev chain: Coincidence or are they same?

The correlation length for the two-dimensional classical ising model goes as $$\xi_{ising}(T)\sim |T-T_c|^{-\nu};\qquad \nu=1$$ We can map the classical ising model to its quantum cousin, one-dimensional transverse field ising model or simply TIM. Jordan-Wigner transformation followed by Bogolibouv transformation gives free fermionic theory whose mass goes like $$m\sim |g-g_c|$$ Field theory interpretation (baiscally $$m=\xi^{-1}$$) gives $$\xi_{fermionic}\sim |g-g_c|^{-\nu};\qquad \nu=1$$ Is this coincidence that both critical exponents $$\nu$$ pertaining to correlation length turns out to be same?

With this question, I also give a relation of $$g$$ with $$T$$: $$g(T)=T\frac{e^{-J/T}}{J}$$ when $$T\gg T_c$$ then $$g$$ goes linear as $$g(T)\sim \frac{T}{J}-2$$. But we intrested near criticality, i.e., $$T$$ near $$T_c$$.

• Are you aware of universality and universality classes? en.m.wikipedia.org/wiki/Universality_(dynamical_systems), en.m.wikipedia.org/wiki/Universality_class May 31, 2020 at 5:05
• @bRost03 I am aware of Universality class. Are you hinting 1D Kitaev chain and 2D ising belong to same universality class? May 31, 2020 at 8:38
• It was more just a general comment that if you were unaware of universality classes, then that'd be a good starting point for thinking about why different models might share critical exponents. Here's a paper that shows the two belong to the same universality class for a given symmetry point: arxiv.org/pdf/1710.04716.pdf Jun 1, 2020 at 17:25
• @bRost03 The reference is helpful. I have one general question. When two models belong to two same universality class, they share the same critical exponents. For the Ising model, the characteristic length of the spin-spin correlation function is $\xi$. For the Kitaev chain, what is this correlation length? In my question, I said $m=\xi^{-1}$, keeping momentum space propagator for scalar fields $\frac{1}{p^2-m^2+i\epsilon}$ which decays after length $\xi=m^{-1}$. But what about fermion? Or what is this correlation length about? Jun 2, 2020 at 9:52
• I might make a mistake when I am talking field theory. Feel free to point out if I say something wrong. Jun 2, 2020 at 9:59