Negative capacitance transistors are a unique class of switches capable of operation beyond the Boltzmann limit to realize subthermionic switching. To date, the negative capacitance effect has been predominantly attributed to devices employing an unstable insulator with ferroelectric properties, exhibiting a two-well energy landscape, in accordance with the Landau theory. The theory and operation of a solid electrolyte field effect transistor (SE-FET) of subthreshold swing less than 60 mV/dec in the absence of a ferroelectric gate dielectric are demonstrated in this work. Unlike ferroelectric FETs that rely on a sudden switching of dipoles to achieve negative capacitance, we demonstrate a distinctive mechanism that relies on the accumulation and dispersion of ions at the interfaces of the oxide, leading to a subthreshold slope (SS) as low as 26 mV/dec in these samples. The frequency of operation of these unscaled devices lies in a few millihertz because at higher or lower frequencies, the ions in the insulator are either too fast or too slow to produce voltage amplification. This is unlike Landau switches, where the SS remains below 60 mV/dec even under quasi-static sweep of the gate bias. The proposed FETs show a higher on-current with a thicker oxide in the entire range of gate voltage, clearly distinguishing their scaling laws from those of ferroelectric FETs. Our theory, validated with experiment, demonstrates a new class of devices capable of negative capacitance that opens up alternate methods of steep switching beyond the traditional approach of ferroelectric or memristive FETs.