# Critical exponents for the Heisenberg AFM spin-1/2 chain

## Main Question or Discussion Point

Hi everybody!

I'm looking for the critical exponent ν (i.e. the one of the correlation length) of the Heisenberg (i.e. equal coupling in all directions) antiferromagnetic spin-1/2 model in 1D...
Furthermore, do you know to which universality class it belongs? Is it true that it's the Kosterlitz-Thouless class? Do you know a good (review-) article about this topic? I couldn't find anything useful neither in Sachdev's book nor in Takahashi's one, but maybe I didn't check carfully enough...

One more question: there are two possible ways to move away from criticality in the case of the Heisenberg model, namely altering one of the coulpling s.t. you end up with the XXZ-model, or adding an external homogenous field, say in Z-direction. Do they both yield the same critical behaviour?

Many thanks!

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i remember the <S_z(x)S_z(0)>=x^(1/2)

For these 1d spin problems, such as xxz model and the heisenberg model in magnetic field, the standard method is bosonization.

Hope these help!

i remember the <S_z(x)S_z(0)>=x^(1/2)

For these 1d spin problems, such as xxz model and the heisenberg model in magnetic field, the standard method is bosonization.

Hope these help!
hmmm... i guess you mean x^(-1/2) for the power law decay at criticality... this would imply for the critical exponent η from Γ(x)~x^-(d-2+η) that η=3/2...
however, what i need is the critical exponent ν of the exponential decay when the system is not critical i.e. Γ(x,ε)~x^-(d-2+η)*exp(-x/ξ(ε)) where ξ(ε)=|ε|^-ν is the correlation length...