Title: Two liquid states of distinguishable helium-4: the existence of another non-superfluid frozen by heating

Abstract: We demonstrate that there can exist two liquid states in distinguishable helium-4 ($^4$He) obeying Boltzmann statistics. This is an indication of quantum liquid polyamorphism induced by nuclear quantum effect. For 0.08-3.3 K and 1-500 bar, we extensively conducted the isothermal-isobaric path integral centroid molecular dynamics simulations to explore not only possible states and state diagram but the state characteristics. The distinguishable $^4$He below 25 bar does not freeze down to 0.08 K even though it includes no Bosonic exchange effect and therefore no Bose condensation. One liquid state, low quantum-dispersion liquid (LQDL), is nearly identical to normal liquid He-I of real $^4$He. The other is high quantum-dispersion liquid (HQDL) consisting of atoms with longer quantum wavelength. This is another non-superfluid existing below 0.5 K or the temperatures of LQDL. The HQDL is also a low-entropy and fragile liquid to exhibit, unlike conventional liquids, rather gas-like relaxation of velocity autocorrelation function, while there the atoms diffuse without noticeable contribution from quantum tunneling. The LQDL-HQDL transition is not a thermodynamic phase transition but a continuous crossover accompanied by the change of the expansion factor of quantum wavelength. Freezing of HQDL into the low quantum-dispersion amorphous solid occurs by heating from 0.2 to 0.3 K at 40-50 bar, while this $P$-$T$ condition coincides with the Kim-Chan normal-supersolid phase boundary of real $^4$He. It is suggested that HQDL has relevance to the non-superfluid states of confined subnano-scale $^4$He systems which are semi-Boltzmann-like owing to the suppression of Bosonic correlation.