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 $T_c$.

  • $\begingroup$ Are you aware of universality and universality classes? en.m.wikipedia.org/wiki/Universality_(dynamical_systems), en.m.wikipedia.org/wiki/Universality_class $\endgroup$
    – bRost03
    May 31, 2020 at 5:05
  • $\begingroup$ @bRost03 I am aware of Universality class. Are you hinting 1D Kitaev chain and 2D ising belong to same universality class? $\endgroup$ May 31, 2020 at 8:38
  • $\begingroup$ 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 $\endgroup$
    – bRost03
    Jun 1, 2020 at 17:25
  • $\begingroup$ @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? $\endgroup$ Jun 2, 2020 at 9:52
  • $\begingroup$ I might make a mistake when I am talking field theory. Feel free to point out if I say something wrong. $\endgroup$ Jun 2, 2020 at 9:59


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