Unitary Fermi Gas, $\epsilon$ Expansion, and Nonrelativistic Conformal Field Theories

Abstract

We review theoretical aspects of unitary Fermi gas (UFG), which has been realized in ultracold atom experiments. We first introduce the $\epsilon$ expansion technique based on a systematic expansion in terms of the dimensionality of space. We apply this technique to compute the thermodynamic quantities, the quasiparticle cum, and the criticl temperature of UFG. We then discuss consequences of the scale and conformal invariance of UFG. We prove a correspondence between primary operators in nonrelativistic conformal field theories and energy eigenstates in a harmonic potential. We use this correspondence to compute energies of fermions at unitarity in a harmonic potential. The scale and conformal invariance together with the general coordinate invariance constrains the properties of UFG. We show the vanishing bulk viscosities of UFG and derive the low-energy effective Lagrangian for the superfluid UFG. Finally we propose other systems exhibiting the nonrelativistic scaling and conformal symmetries that can be in principle realized in ultracold atom experiments.

Appendix: Scaling Dimensions of Three-Body Operators

In this Appendix, we derive the formula to compute the scaling dimensions of three-body composite operators for arbitrary mass ratio $m_\uparrow/m_\downarrow,$ angular momentum l, and spatial dimension d from a field theory perspective. We first consider a three-body operator composed of two spin-$\uparrow$ and one spin-$\downarrow$ fermions with zero orbital angular momentum $l\,{=}\,0{:}$

$$ {\mathcal{O}}_{\uparrow\uparrow\downarrow}^{(l\,{=}\,0)}(\user2{x}) \equiv Z_\Uplambda^{-1}\phi\psi_\uparrow(\user2{x}), $$

(7.131)

where $Z_\Uplambda$ is a cutoff-dependent renormalization factor. We study the renormalization of the composite operator $\phi\psi_\uparrow$ by evaluating its matrix element $\langle0|\phi\psi_\uparrow(\user2{x})|p,-p\rangle.$ Feynman diagrams to renormalize $\phi\psi_\uparrow$ is depicted in Fig.  7.15. The vertex function $Z(p_0,\user2{p})$ in Fig.  7.15 satisfies the following integral equation:

$$ \begin{aligned} Z(p_0,\user2{p}) \,{=}\,& 1 - i\int {\frac{dk_0d\user2{k}}{(2\pi)^{d+1}}} G_\uparrow(k)G_\downarrow(-p-k)D(-k)Z(k_0,\user2{k}) \\ \,{=}\,& 1 - \int {\frac{d\user2{k}}{(2\pi)^d}} \left.G_\downarrow(-p-k)D(-k)Z(k_0,\user2{k})\right|_{k_0\,{=}\,{\frac{\user2{k}^2}{2m_\uparrow}}}, \end{aligned} $$

(7.132)

where we used the analyticity of $Z(k_0,\user2{k})$ on the lower half plane of $k_0.$ $G_\sigma(p)\equiv\left(p_0-{\frac{\user2{p}^2}{2m_\sigma}}+0^+\right)^{-1}$ is the fermion propagator and $D(p)$ is the resumed propagator of $\phi$ field given in Eq. 7.78. If we set $p_0\,{=}\,{\frac{\user2{p}^2}{2m_\uparrow}},$ Eq.  7.132 reduces to the integral equation for $z(\user2{p})\equiv Z \left({\frac{\user2{p}^2}{2m_\uparrow}},\user2{p}\right){.}$

Fig.  7.15

Feynman diagrams to renormalize three-body composite operators. The solid lines are the propagators of $\psi_\uparrow$ and $\psi_\downarrow$ fields while the dotted lines are the resumed propagators of $\phi$ field. The shaded bulbs represent the vertex function $Z(p)$

Because of the scale and rotational invariance of the system, we can assume the form of $z(\user2{p})$ to be $z(\user2{p})\propto\left({\frac{|\user2{p}|} {\Uplambda}}\right)^{\gamma},$ where $\Uplambda$ is a momentum cutoff. Accordingly the renormalization factor becomes $Z_\Uplambda\propto\Uplambda^{-\gamma}$ with $\gamma\,{=}\,-{\partial}\hbox{ln} Z_\Uplambda/{\partial}\hbox{ln}\Uplambda$ being the anomalous dimension of the composite operator $\phi\psi_\uparrow.$ In terms of $\gamma,$ the scaling dimension of the renormalized operator ${\mathcal{O}}_{\uparrow\uparrow\downarrow}^{(l\,{=}\,0)}$ is given by

$$ \Updelta_{\uparrow\uparrow\downarrow}^{(l\,{=}\,0)} \,{=}\, \Updelta_{\phi}+\Updelta_{\psi_\uparrow}+\gamma \,{=}\, 2+{\frac{d}{2}}+\gamma. $$

(7.133)

Substituting the expression of $z(\user2{p})$ into Eq. 7.132 and performing the integration over $|\user2{k}|$ at $\Uplambda\,{\to}\,\infty,$ we obtain the following equation to determine $\gamma{:}$

$$ 1 \,{=}\, {\frac{2\pi^{1/2} \left[{\frac{m_\downarrow(2m_\uparrow{+}m_\downarrow)}{(m_\uparrow{+}m_\downarrow)^2}}\right]^{1-d/2}} {\Upgamma \left(1-{\frac{d}{2}}\right) \Upgamma \left({\frac{d-1} {2}}\right)\sin[(\gamma+1)\pi]}} \int\limits_0^\pi d\theta \sin^{d-2}\theta {\frac{\sin[(\gamma+1)\chi]}{\sin\,\chi}} $$

(7.134)

with $\cos\,\chi\equiv{\frac{m_\uparrow}{m_\uparrow+m_\downarrow}}\cos\,\theta.$ The integration over $\theta$ can be done analytically in $d\,{=}\,3,$ but otherwise, has to be done numerically.

Similarly, for general orbital angular momentum l, we can derive the equation satisfied by the anomalous dimension $\gamma_l{:}$

$$ 1 \,{=}\, {\frac{2\pi^{1/2} \left[{\frac{m_\downarrow(2m_\uparrow{+}m_\downarrow)}{(m_\uparrow{+}m_\downarrow)^2}}\right]^{1-d/2}} {\Upgamma\!\left(1-{\frac{d}{2}}\right) \Upgamma \left({\frac{d-1} {2}}\right)\sin[(\gamma_l+l+1)\pi]}} \int\limits_0^\pi d\theta \sin^{d-2}\theta \tilde{P}_l(\cos\,\theta) {\frac{\sin[(\gamma_l+l+1)\chi]}{\sin\,\chi}}, $$

(7.135)

where $\tilde{P}_l(z)$ is a Legendre polynomial generalized to d spatial dimensions. ⁷ The scaling dimension of the renormalized operator ${\mathcal{O}}_{\uparrow\uparrow\downarrow}^{(l)}$ with orbital angular momentum l is now given by

$$ \Updelta_{\uparrow\uparrow\downarrow}^{(l)} \,{=}\, \Updelta_{\phi}+\Updelta_{\psi_\uparrow}+l+\gamma_l \,{=}\, 2+{\frac{d} {2}}+l+\gamma_l. $$

(7.136)

$\Updelta_{\uparrow\uparrow\downarrow}^{(l)}$ for $l\,{=}\,0,1$ in $d\,{=}\,3$ are plotted as functions of the mass ratio $m_\uparrow/m_\downarrow$ in Fig.  7.12, while $\Updelta_{\uparrow\uparrow\downarrow}^{(l)}$ for $l\,{=}\,0,1$ with equal masses $m_\uparrow\,{=}\,m_\downarrow$ are plotted in Fig.  7.14 as functions of the spatial dimension d.