Charge order breaks time-reversal symmetry in CsV$_3$Sb$_5$

The recently discovered vanadium-based kagome metals $A$V$_{3}$Sb$_{5}$ ($A$~=~K,~Rb,~Cs) exhibit superconductivity at low-temperatures and charge density wave (CDW) order at high-temperatures. A prominent feature of the charge ordered state in this family is that it breaks time-reversal symmetry (TRSB), which is connected to the underlying topological nature of the band structure. In this work, a powerful combination of zero-field and high-field muon-spin rotation/relaxation is used to study the signatures of TRSB of the charge order in CsV$_3$Sb$_5$, as well as its anisotropic character. By tracking the temperature evolution of the in-plane and out-of-plane components of the muon-spin polarization, an enhancement of the internal field width sensed by the muon-spin ensemble was observed below $T_{\rm TRSB}=T_{\rm CDW}\simeq95$~K. Additional increase of the internal field width, accompanied by a change of the local field direction at the muon site from the $ab$-plane to the $c$-axis, was detected below $T^\ast\simeq30$~K. Remarkably, this two-step feature becomes well pronounced when a magnetic field of 8~T is applied along the crystallographic $c-$axis, thus indicating a field-induced enhancement of the electronic response at the CDW transition. These results point to a TRSB in CsV$_3$Sb$_5$ by charge order with an onset of ${\simeq}~95$~K, followed by an enhanced electronic response below ${\simeq}~30$~K. The observed two-step transition is discussed within the framework of different charge-order instabilities, which, in accordance with density functional theory calculations, are nearly degenerate in energy.