Recent experimental results [1] indicate that phosphorus, a single-component system, can have two liquid phases: a high-density liquid (HDL) and a low-density liquid (LDL) phase. A firstorder transition between two liquids of different densities [2] is consistent with experimental data for a variety of materials [3,4], including singlecomponent systems such as water [5][6][7][8], silica [9] and carbon [10]. Molecular dynamics simulations of very specific models for supercooled water [2,11], liquid carbon [12] and supercooled silica [13], predict a LDL-HDL critical point, but a coherent and general interpretation of the LDL-HDL transition is lacking. Here we show that the presence of a LDL and a HDL can be directly related to an interaction potential with an attractive part and two characteristic short-range repulsive distances. This kind of interaction is common to other single-component materials in the liquid state (in particular liquid metals [14-21,2]), and such potentials are often used to decribe systems that exhibit a density anomaly [2]. However, our results show that the LDL and HDL phases can occur in systems with no density anomaly. Our results therefore present an experimental challenge to uncover a liquid-liquid transition in systems like liquid metals, regardless of the presence of the density anomaly.Several explanations have been developed to understand the liquid-liquid phase transition. For example, the two-liquid models [4] assume that liquids at high pressure are a mixture of two liquid phases whose relative concentration depends on external parameters. Other explanations for the liquid-liquid phase transition assume an anisotropic potential [2,[11][12][13]. Here we shall see that liquid-liquid phase transition phenomena can arise solely from an isotropic pair interaction potential with two characteristic lengths.For molecular liquid phosphorus P 4 (as for water), a tetrahedral open structure is preferred at low pressures P and low temperatures T , while a denser structure is favored at high P and high T [1,6,8]. The existence of these two structures with different densities suggests a pair interaction with two characteristic distances. The first distance can be associated with the hard-core exclusion between two particles and the second distance with a weak repulsion (soft-core), which can be overcome at large pressure. Here we will use a generic three dimensional (3D) model composed of particles interacting via an isotropic soft-core pair potential. Such isotropic potentials can be regarded as resulting from an average over the angular part of more realistic potentials, and are often used as a first approximation to understand the qualitative behavior of real systems [14][15][16][17][18][19][20][21][22]2]. For Ce and Cs, Stell and Hemmer proposed a potential with nearest-neighbor repulsion and a weak long-range attraction [15]. By means of an exact analysis in 1D, they found two critical points, with the high-density critical point interpreted as a solid-solid transition. Then analytic calc...