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About the Author:I live in Vancouver, BC. I've been attending and volunteering with the Vancouver Queer Film Festival for a few years now. I'm not a filmmaker. Earlier this year, I bought a pass for Doxa Documentary Film Festival, a film festival in my city that I've been aware of for years but never attended. I watched a ton of movies and mostly enjoyed the experience. I think they did a good job of moving online. Not the same though as being able to attend in-person events, experience a film in a crowded cinema, and chat with other audience members before and after. I did watch some of the prerecorded Q&amp;As, but I wish there had been some way to make it more interactive. I have a pass for the Vancouver Queer Film Festival which is coming up later this month, and I'm currently considering buying one for the Vancouver Latin American Film Festival, also coming up this month.There had to have been a reason for Mike Comrie's lack of production to start the season, right? After all, he looked pretty solid during the preseason, was even rumored to be a perfect fit on Sidney Crosby's wing and had the makings of a first liner at best, second liner at worst.
It turns out Comrie was playing through a hip injury for most of his 2010 season, an injury he sustained in the Penguins' second game of the season against the Canadiens, as noted in today's Trib Review.
And now, the news is worse.
"He was trying to battle through and work with the injury he did have," Bylsma said. "At one point, it did get worse. That's when we removed him from the ice, settled it down and started the rehab." The rehab apparently isn't working. Comrie is scheduled to see a doctor regarding the hip injury Dec. 27. He produced four goals during that time and displayed instant chemistry with Evgeni Malkin.
I know Comrie has taken a lot of flak on the site for his limited production during the season, but it looks like he was ultimately just trying to keep his ice time intact while struggling through this injury. I think I speak for everyone when I say it's pretty disappointing to see a guy sustain something like that in the second game of the year, but if anyone's upset about it I'm sure it's Comrie himself.Ultracold three-component atomic Fermi gases in one dimension are expected to exhibit rich physics due to the presence of trions and different pairing states. Quantum phase transitions from the trion state into a paired phase and a normal Fermi liquid occur at zero temperature. We derive the analytical thermodynamics of strongly attractive three-component one-dimensional fermions with SU(3) symmetry via the thermodynamic Bethe ansatz method in unequal Zeeman splitting fields H1 and H2. We find explicitly that for low temperature the system acts like either a two-component or a three-component Tomonaga-Luttinger liquid dependent on the system parameters. The phase diagrams for the chemical potential and specific heat are presented for illustrative values of the Zeeman splitting. We also demonstrate that crossover between different Tomonaga-Luttinger liquid phases exhibit singular behaviour in specific heat and entropy as the temperature tends to zero. Beyond Tomonaga-Luttinger liquid physics, we obtain the equation of state which provides a precise description of universal thermodynamics and quantum criticality in three-component strongly attractive Fermi gases.
The ongoing experimental advances in realizing degenerate quantum gases in low dimensions TGexp (); Toshiya (); Moritz05 (); Druten (); Haller (); Hulet () offer a new and compelling motivation for the further study of quantum many-body systems via exact schemes such as the Bethe Ansatz (BA) and low energy effective field theory Giamarchi-b (). Reducing the dimensionality in a quantum system can have striking consequences. The one-dimensional (1D) many-body systems Giamarchi-b (); Takahashi () possess unique many-body correlation effects which are different from their higher dimensional counterparts. These include the phenomena of spin-charge separation, universal thermodynamics and quantum criticality.
A recent scheme for mapping out physical properties of homogeneous systems by using the inhomogeneity of the trap Ho-Zhou () has been successfully applied to experimental measurements on the thermodynamics of interacting fermions with a wide range of tunable interactions Salomon (); Horikoshi (). Moreover, further experimental advances with ultracold atoms allow the exploration of three-component Fermi gases in the entire parameter space of trions, dimers and free atoms Li (); efimov-trimer (); Li2 (). This provides a promising opportunity to experimentally explore universal thermodynamics and quantum critical behaviour of strongly interacting Fermi gases with high spin symmetries in 1D. In this context, the thermodynamics of 1D attractively interacting fermions Takahashi () has been receiving growing interest Guan2007prb (); Mueller (); Bolech (); Erhai ().
For spin-1/2 fermions with attractive interaction there are three quantum phases at zero temperature: the fully paired phase which is a quasi-condensate with zero polarization p, the fully polarized (normal) phase with p=1, and the partially polarized (1D FFLO) phase where 0<p<1 at zero temperature Orso (); Hu (); Guan2007prb (). This theoretical prediction of the phase diagram for 1D fermions was recently confirmed experimentally by R. Hulet’s group at Rice University Hulet (). In addition, it was recently proved Erhai () that at low temperatures, the physics of the gapless phase belongs to the universality class of a two-component Tomonaga-Luttinger liquid (TLL). However, from the theoretical point of view, understanding the thermodynamics of multi-component Fermi gases with higher spin symmetry imposes a number of challenges Ho (); Tsvelik (); Zhang (); Wang (); Guan ().
For multi-component interacting Fermi gases, the phase diagrams become more complicated in the presence of magnetic fields due to the richer number of quantum phases. In contrast to the two-component Fermi gas, Orso (); Hu (); Guan2007prb () three-component ultracold fermions give rise to quantum phase transitions from a three-body bound state of “trions” into the BCS pairing state and a normal Fermi liquid Rapp (); Lecheminant (); Demler (); Guan2008prl (); Thai (); Silva (); Errea (); AnnPhys (); Suga2009arXiv (); Angela (). The zero temperature phase diagrams of the BA integrable 1D three-component Fermi gas with SU(3) symmetry have been worked out from the dressed energy equations Guan2008prl (); Angela (). It was found that Zeeman splittings can drive transitions between exotic phases of trions, bound pairs, a normal Fermi liquid and mixtures of these phases, see Fig. 1. It is thus very worthwhile to map out such zero temperature phase diagrams to the inhomogeneity of the trap at finite temperatures.
In this paper, we investigate the finite temperature thermodynamic properties of 1D three-component fermions with unequal Zeeman splitting by means of the exact thermodynamic Bethe ansatz (TBA) solution. We prove that at low temperatures the system behaves like either a two-component or a three-component TLL in certain regimes. Exact finite temperature phase diagrams are demonstrated for illustrative values of the Zeeman splitting parameters. Quantum criticality with respect to the specific heat and entropy as the temperature tends to zero is discussed. The equation of state obtained provides an exact description of the thermodynamics and quantum critical behaviour of three-component composite fermions which can possibly be tested in experiments with ultracold atoms.
This paper is set out as follows. In Section II, we present the model and the exact BA solution. We also derive the TBA equations for the thermodynamics. In Section III, we derive the low temperature thermodynamics by the Sommerfeld expansion method. The universal multi-component TLL phases are identified. In Section IV, we present the equation of state in terms of polylogarithm functions from which the quantum phase diagrams can be mapped out. Concluding remarks are given in Section V. Detailed working is given in the appendices. The derivation of the TBA equations is presented in detail in Appendix A. In Appendices B and C, the iteration method is used to derive relevant results for the TBA and the thermodynamics.
where we have included the Zeeman energy EZ=∑3i=1NiϵiZ(μiB,B). The spin-independent contact interaction g1D applies between fermions with different hyperfine states so that the number of fermions in each spin state is conserved. The inter-component interaction g1D is positive for repulsive interaction and negative for attractive interaction. For simplicity, we define the interaction strengths as c=mg1D/ℏ2 and the dimensionless parameter γ=c/n, where n=N/L is the linear density, and set ℏ=