The unitary limit of a Fermi gas is described here as when the scattering length is comparable or exceeds the interparticle distance. For $ak_F<0$, this is the BCS limit of a weakly interacting Fermi gas. When $0<ak_F<1$, the interaction is stronger and we are in the BEC limit.

My question is how well can we describe the unitary limit of the Fermi gas with a Fermi liquid description? It is my understanding that Fermi liquid theory is just the phenomenological approach to understanding the physical model of a unitary Fermi gas, but if we remain far from the BCS and BEC limits and remain firmly in the world of unitarity, when will this Fermi liquid description fail? I have seen studies such as this that use a Fermi liquid theory to describe the unitary Fermi gas, but I have yet to see a reference which tells me when, exactly, this Fermi liquid description fails.


This is a question on which I did my PhD thesis. The question of Fermi liquid theory describing Unitary Fermions can only be asked above the supefluid transition temperature since as mentioned in the post above below critical temperature the physics is described by a superfluid phonon.

In unbroken or normal phase, Fermi liquid theory fails to describe unitary fermions since the additional requirement of Schrodinger invariance at unitarity is incompatible with Fermi liquid theory. Basically since scale and boost invariance are spontaneously broken by a finite chemical potential we need Goldstone bosons to non-linearly realize these broken symmetries and presence of scalar bosons destroys Fermi liquid behavior.

Here are two papers I wrote on this if you need the mathematical proofs:

  1. https://arxiv.org/abs/1712.07795

  2. https://arxiv.org/abs/1712.07797

The first paper addresses the question of more general symmetries and second one is exclusively for unitary fermions.


1) I would not call the Landau Fermi Liquid theory "just phenomenological". It is a rigorous description of a cold Fermi liquid that is continuously connected to a free Fermi gas. In particular, the excitations have the same quantum numbers (spin, charge, etc) as the excitations of a free Fermi gas. Of course, the theory can be used for phenomenology, and the parameters are often fitted to experiments.

2) The unitary Fermi gas is not a Fermi liquid, because it is a high $T_c$ superfluid, the fermionic excitations acquire large gaps, and the only low energy mode is a Goldstone phonon.

3) The weakly attractive Fermi gas (the BCS limit) is also a superfluid, but in this case the gap is exponentially small. This means that there is a regime $T_c\ll T\ll T_F$ in which the Landau Fermi-liquid description is valid. Indeed, this theory can be used to compute $T_c$.

4) This does not mean that one cannot try to use the Landau theory as an approximate phenomenological theory to understand thermodynamics and quasi-particle properties at $T\sim T_c \sim T_F$. This has indeed been done, see, for example, http://www.nature.com/nature/journal/v463/n7284/abs/nature08814.html .


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