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Fluctuations in heat current and scaling dimension

In this work, we theoretically study the heat flow between two $1+1$d chiral gapless systems connected by a point contact. With a small temperature gradient between the two, we find that the ratio between fluctuations of the heat current and the heat current itself is proportional to the scaling dimension -- a universal number that characterizes the distribution of the particles tunneling through the point contact. We adopt two different approaches, scattering theory and conformal field theory, to calculate this ratio and see that their results agree. Our findings are useful for probing not only fractional charge excitations in fractional quantum Hall states but also neutral ones in non-Abelian phases.