Absence of $E_{2g}$ nematic instability and dominant $A_{1g}$ response in the kagome metal CsV$_3$Sb$_5$

Ever since the discovery of the charge density wave (CDW) transition in the kagome metal CsV$_3$Sb$_5$, the nature of its symmetry breaking is under intense debate. While evidence suggests that the rotational symmetry is already broken at the CDW transition temperature ($T_{\rm CDW}$), an additional electronic nematic instability well below $T_{\rm CDW}$ has been reported based on the diverging elastoresistivity coefficient in the anisotropic channel ($m_{E_{2g}}$). Verifying the existence of a nematic transition below $T_{\rm CDW}$ is not only critical for establishing the correct description of the CDW order parameter, but also important for understanding low-temperature superconductivity. Here, we report elastoresistivity measurements of CsV$_3$Sb$_5$ using three different techniques probing both isotropic and anisotropic symmetry channels. Contrary to previous reports, we find the anisotropic elastoresistivity coefficient $m_{E_{2g}}$ is temperature-independent, except for a step jump at $T_{\rm CDW}$. The absence of nematic fluctuations is further substantiated by measurements of the elastocaloric effect, which show no enhancement associated with nematic susceptibility. On the other hand, the symmetric elastoresistivity coefficient $m_{A_{1g}}$ increases below $T_{\rm CDW}$, reaching a peak value of 90 at $T^* = 20$ K. Our results strongly indicate that the phase transition at $T^*$ is not nematic in nature and the previously reported diverging elastoresistivity is due to the contamination from the $A_{1g}$ channel.