When dealing with phase transitions, the behavior of physical quantities can be described by the theory of critical exponents. For example, let's consider a one-dimensional spin $1/2$ $XY$ Hamiltonian with a field $\Omega$ applied along $x$ $$ H = -J \sum_i ( S^x_i S^x_{i+1} + S^y_i S^y_{i+1}) \ - \Omega \sum_i S^x_i $$ where $J, \ \Omega > 0$. Defining the collective spin as $J^x = \sum_i S_i^x$ it is known that, in the ground state $$ J^x \sim \Omega^{1/\delta}$$ where $\delta$ is one of the notorious critical exponents (which, by the way, in this case should be connected to the Luttinger parameter $K$ as $\delta=8K+1$. My question is: is it possible to obtain a similar scaling relation for the variance of $J^z$?
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