Commensurate-Incommensurate Transition in Submonolayer ${}^3$He on Graphite

We report high-precision heat-capacity measurements of submonolayer ${}^3$He adsorbed on highly crystalline graphite, revealing new aspects of the commensurate$-$incommensurate transition. Below 1K, two possible striped domain-wall phases emerge: ${\alpha }_1$ with variable wall spacing and ${\alpha }_2$ with fixed spacing. The $T$-linear heat capacity in ${\alpha }_1$ arises from one-dimensional phonons along the walls. ${\alpha }_2$ melts into ${\alpha }_1$ at a critical density via a second-order transition, consistent with a quantum nematic (quantum liquid-crystal) state in ${\alpha }_1$, and reconciling thermodynamic and prior nuclear-magnetic data.