Yang-Lee edge singularity at $\mathcal{O}({\partial }^4)$ from the functional renormalization group: revised analysis

Abstract

We determine the critical exponents of the Yang-Lee edge singularity as continuous functions of dimension using the functional renormalization group, retaining all operators at second and fourth order in the derivative expansion. At fourth order, we obtain $\eta =-0.153(7)$, $-0.352(9)$ and $-0.575(9)$ in $d=5$, $4$ and $3$, respectively, in agreement with the six-loop $\epsilon$ expansion. In three dimensions, these results give $\sigma =0.076$ and ${\nu }_c=0.359$. We locate the zero of the edge exponent at $d_{\sigma }=2.65(1)$, also consistent with the six-loop estimate. Continuing the fixed point through $d_{\sigma }$, we obtain $\eta =-0.795(21)$ and $\omega =1.90(6)$ in $d=2$, compared with the exact values $-4/5$ and $2$. For the complex Yang-Lee fixed point, convergence of the field expansion depends crucially on its reference point. Expansions about $\varphi =0$ fail to converge with the smooth regulator used here, whereas expansions about the running imaginary field value at which the mass vanishes converge down to $d=1.3$. This representation preserves the exact relation between the thermal eigenvalue and the field dimension, and hence the hyperscaling identity ${\theta }_h+{\theta }_t=d$, at every field-truncation order. We trace the large violations reported in earlier works to the omission of the expansion-point motion when linearizing the flow, rather than to the truncation itself.

arXiv comment16 pages, 8 figures, 2 tables