# Bernard Krauth 2011

E. P. Bernard, W. Krauth Two-step melting in two dimensions: First-order liquid-hexatic transition (Physical Review Letters 107, 155704 (2011)) (The paper comes with 12 pages of supplemental material, see below)

# Paper

Abstract: The hard-disk model has exerted outstanding influence on computational physics and statistical mechanics. Decades ago, hard disks were the first system to be studied by Markov-chain Monte Carlo methods and by molecular dynamics. It was in hard disks, through numerical simulations, that a two-dimensional melting transition was first seen to occur even though such systems cannot develop long-range crystalline order. Scores of theoretical, computational, and experimental works have analysed this fundamental melting transition, without being able to settle its nature. The first-order melting scenario between a liquid and a solid (as in three dimensions), and the Kosterlitz, Thouless, Halperin, Nelson and Young (KTHNY) scenario with an intermediate hexatic phase separated by continuous transitions from the liquid and the solid have been mainly focussed upon. Here we show by large-scale simulations using the powerful Event-chain Monte Carlo algorithm that the hard-disk system indeed possesses a narrow hexatic phase, where orientational order is maintained across large samples while positional order is short-ranged. However, in difference with the KTHNY scenario, the liquid-hexatic phase transition is proven to be first-order. In simulations at fixed volume and number of disks, we identify a two-phase region, where the liquid with large but finite orientational correlation length coexists with the hexatic. At higher densities, we reach the pure hexatic phase, and then witness the transition into the solid phase characterised by quasi-long range positional order. Our work closes a crucial gap in the understanding of one of the fundamental models in statistical physics, which is at the basis of a large body of theoretical and experimental work in films, suspensions, and other condensed-matter systems.

Supplemental material The supplemental material at PRL contains two additional files

# Confirmation

When the paper first came out, in 2011, the first-order liquid-hexatic transition was highly controversial. Fortunately, a number of papers were able to confirm our solution, to start with the 2012 paper by Engel et al.

# Illustration

Here are pictures of the local orientations (upper panels) and the local densities (lower panels) for the 1024x1024 systems at various densities around the transition region. Phase separation is easily visible. One also nicely sees the transition from pure phases (at density $\eta=0.700$ and at density $\eta=0.716$), bubble-shaped minority phases (at densities $\eta=0.704$ and $\eta=0.712$) and strip-shaped phases in the middle of the coexistence interval, at density $\eta=0.708$.

# Algorithm

The event-chain algorithm used in this work is from Bernard_Krauth_Wilson_2009. Check out that page for a discussion.