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This paper reports the first experimental observation of phonons and theirsoftening on single crystalline LaPt$_2$Si$_2$ via inelastic neutronscattering. From the temperature dependence of the phonon frequency in closeproximity to the charge-density wave (CDW) $q$-vector, we obtain a CDWtransition temperature of T$_{CDW}$ = 230 K and a critical exponent $\beta$ =0.28 $\pm$ 0.03. This value is suggestive of a non-conventional criticalbehavior for the CDW phase transition in LaPt$_2$Si$_2$, compatible with ascenario of CDW discommensuration (DC). The DC would be caused by the existenceof two CDWs in this material, propagating separately in the non equivalent(Si1-Pt2-Si1) and (Pt1-Si2-Pt1) layers respectively, with transitiontemperatures T$_{CDW-1}$ = 230 K and T$_{CDW-2}$ = 110 K. A strong$q$-dependence of the electron-phonon coupling has been identified as thedriving mechanism for the CDW transition at T$_{CDW-1}$ = 230 K while a CDWwith 3-dimensional character, and Fermi surface quasi-nesting as a drivingmechanism, is suggested for the transition at T$_{CDW-2}$ = 110 K. Our resultsclarify some aspects of the CDW transition in LaPt$_2$Si$_2$, which have beenso far misinterpreted by both theoretical predictions and experimentalobservations, and give direct insight into its actual temperature dependence.

Non-Conventional Critical Behavior and Q-dependent Electron-Phonon Coupling Induced Phonon Softening in the CDW Superconductor LaPt2Si2