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As helium-4 is cooled below 2.17 K in undergoes a phase transition to a state of matter known as a superfluid which supports flow without viscosity. This type of dissipationless transport can be observed by forcing helium to travel through a narrow constriction that the normal liquid could not penetrate. Recent advances in nanofabrication techniques allow for the construction of smooth pores with nanometer radii, that approach the truly one dimensional limit. In one dimension, it is believed that a system of bosons (like helium-4) may have startlingly different behavior than in three dimensions. The one dimensional system is predicted to have a linear hydrodynamic description known as Luttinger liquid theory, where no type of long range order can be sustained. In the limit where the pore radius is small,  helium inside the channel would behave as a sort of quasi-supersolid with all correlations decaying as power-laws at zero temperature.  We have performed large scale quantum Monte Carlo simulations of helium-4 inside nanopores of varying radii at low temperatures with realistic helium-helium and helium-pore interactions. The results indicate that helium inside the nanopore forms concentric cylindrical layers surrounding a core that can be fully described via Luttinger liquid theory and provides insights towards the exciting possibility of the experimental detection of a Luttinger liquid.

I will discuss some of the most basic questions in fermionic and bosonic transport, such as the conditions for the existence of a steady-state current, its uniqueness, the role of interactions and spin statistics, its entanglement entropy, etc. This will lead me to introduce an alternative viewpoint to conduction - the micro-canonical formalism of transport - which is ideal to study the above issues [1]. I will point out the similarities and differences with the widely used Landauer formalism, and advance a series of predictions that can be verified by loading ultra-cold atoms into artificial optical lattices.

[1] M. Di Ventra, Electrical Transport in Nanoscale Systems (Cambridge University Press, 2008).