Ordering against confinement

The development of order in a many-particle system is the prime example when, in the words of Aristotle, "the whole is greater than the sum of its parts". Consider for instance a block of iron. Below its melting temperature at around 1500°C, the atoms occupy well-defined positions on a crystalline lattice, similar to the audience on their seats in a sold-out concert hall. By knowing the position of a few atoms one can predict the position of the others due to the periodic structure: The atoms are ordered over a long distance. On heating above the melting temperature, however, this long-ranged order is lost and the crystalline structure gives away to a less structured liquid state through a phase transition. The detection and characterization of ordered states and the corresponding phase transitions is a key focus of current many-body physics research -- in a variety of system, often much more complicated than simple iron -- with huge technological applications. In our work we theoretically studied the ordered superfluid state of an ensemble of atoms confined to a two-dimensional geometry at very low temperatures. The analysis was inspired by recent experiments in collaboration with the present authors covered in Murthy et al., Phys. Rev. Lett. 115, 010401 (2015), where the associated ordering had been measured for the first time. A tricky element in these experiments is the presence of an additional trapping potential, which makes the atoms want to sit in the center of the trap. Think of a football game with free choice of seats: The central regions close to the game will be seated first, whereas the crowd will thin out towards the outer regions. In the present article we could show theoretically that the presence of the trap still allows for superfluid order, although with a very characteristic power law decay of correlation towards the outer regions, which arises from the interplay between ordering and confinement. Such correlations have indeed been observed in the mentioned experiments. We thereby also confirm that the superfluid ordered state was reached in the experiments and give a guidance to the conception of future cold atom experiments.

Thermodynamics close to the absolute zero of temperature

One of the few formulas probably everybody remembers from their physics or chemistry classes in school is the equation of state of an ideal gas given by PV=NkT. It relates the volume V and pressure P of a gas of N atoms or molecules to its temperature T; the conversion factor k is Boltzmann's constant. The formula finds application in a vast range of fields from engineering and life sciences to meteorology and ecology. Despite this remarkable success, its validity is restricted to situations where quantum mechanical and interaction effects between the particles can be neglected - which is typically guaranteed at high enough temperatures. However, for electrons in many novel and technologically promising solid state materials, such as high-temperature superconductors or graphene, both conditions are not satisfied. This is because in these materials the electrons (which are fermions) interact strongly with each other and, in addition, are constrained to move in a two-dimensional plane. How does the formula PV=NkT change when quantum, interaction, and reduced dimension effects become important? To answer this question, we have confined a gas of fermionic atoms in a two-dimensional plane and cooled it to temperatures only a few billionths of a Kelvin above absolute zero. With this setup we have measured for the first time the equation of state of the gas as a function of temperature and atomic interaction strength. The obtained formula can be applied for phenomenological treatments of not only cold atom systems but also other quantum systems with similar conditions. Our results may form the basis for further theoretical and experimental studies of novel quantum materials with possibly groundbreaking technological applications.