Thermodynamics of Fermi Gases

Abstract

Recently, ultra-cold atoms have established a very fruitful connection with condensed matter physics, nuclear physics, astrophysics, and high energy physics on many-body problems in strongly correlated systems. Starting from the pioneering work of Popov and Eagles in the 1970s [1], the connection between superfluidity in a fermionic system with attractive interaction and superfluidity of bosonic pairs of fermions has been the subject of intense theoretical activity (See for instance [2, 3] and other contributions in this book). While the two limiting cases, weakly attractive Fermi gas and weakly repulsive Bose gas are well described by mean field theories, the so-called BEC–BCS crossover region where the gas is strongly correlated poses a challenging theoretical problem. A flurry of theoretical works (see for instance [2, 4–9]) have been developed in order to address the properties of this seemingly simple many-body system of a fermionic species with two spin states interacting with purely s-wave contact potential, but with tunable strength.

Appendix

11.1.1 Spin Susceptibility of a Gapped System at Zero Temperature

Let us consider a system containing $N_\uparrow$ and $N_\downarrow$ spin up and spin down particles. We define $M\,{=}\,N_\uparrow-N_\downarrow$ and $N\,{=}\,N_\uparrow+N_\downarrow$ the polarization and the total atom number and we denote E(N,M) as the energy of the system. If one assumes that the energy can be expanded in M then by symmetry the linear term vanishes and one gets $E(N,M)\,{=}\,E(N,0)+M^2/2\chi+...$ (see also below). With this definition, $\chi$ is then the spin susceptibility of the system. Indeed, adding a magnetic field h contributes to a $-hM$ term to the energy and we immediately see that the energy minimum is shifted from $M\,{=}\,0$ to $M\,{=}\,\chi h.$

This argument is no longer true in the case of a gapped system. Indeed, polarizing a spin balanced system costs the binding energy of the broken pairs. This definition applies to any system composed of spin-singlet dimers, from a fermionic superfluid composed of Cooper pairs, or a pure gas of uncondensed molecules, and leads to the following leading order expansion

$$ E(N,M)\,{=}\,E(N,0)+|M|\Updelta+... $$

To evaluate the spin susceptibility, we add as above a magnetic field h changing the energy into $E-h M.$ We see that for $h\,{\neq}\,0,$ the potential is tilted but the energy minimum stays located at $M\,{=}\,0,$ as long as $|h|\,{<}\,\Updelta$ corresponding to the Pauli limit below which pairing can resist spin polarization (see Fig.  11.13).

Fig.  11.13

Energy E of a gapped system versus spin polarization M. Solid line: the spin polarizing field h is 0 and the slope of the energy is given by the gap $\Updelta.$ Dotted line: non zero spin polarizing field. The energy is tilted but as long as $|h|\,{<}\,\Updelta,$ the ground state remains unpolarized

11.1.2 Fermi Liquid Theory in the Grand Canonical Ensemble

At the macroscopic level, the Fermi liquid is characterized by three properties

• 1. The specific heat$C_V$varies linearly with temperature and$C_V/C_V^0\,{=}\,m^*/m,$where$C_V^0$is the specific heat of the ideal gas. By definition, the heat transfer $\delta Q$ at constant volume and atom number reads $\delta Q\,{=}\,C_V dT+h dV,$ and we thus have

  $$ dS\,{=}\,{\frac{\delta Q}{T}}\,{=}\,{\frac{C_V}{T}}dT+{\frac{h}{T}}dV. $$

  (11.58)

  Using the thermodynamic identity $dF\,{=}\,-SdT-PdV+\sum_\sigma \mu_\sigma dN_\sigma$ for the free energy F, we have finally

  $$ C_V\,{=}\,-T\left({\frac{\partial^2 F}{\partial T^2}}\right). $$

  (11.59)

• 2. The compressibility of the system is finite at T = 0. We have by definition of pressure

  $$ P\,{=}\,-\left({\frac{\partial F}{\partial V}}\right)_{T,N,M}, $$

  (11.60)

  hence for a compression at constant temperature and atom number

  $$ dP\,{=}\,-\left({\frac{\partial^2 F}{\partial V^2}}\right)_{T,N,M}dV $$

  (11.61)

  From the definition of the isotherme compressibility, we have

  $$ \kappa_{T}\,{=}\,{\frac{-1}{V}}\left({\frac{\partial V}{\partial P}}\right)_{T,N,M}, $$

  (11.62)

  and we have finally

  $$ \kappa_{T}^{-1}\,{=}\,V\left({\frac{\partial^2 F}{\partial V^2}}\right)_{T,N,M}. $$

  (11.63)

• 3. The magnetic susceptibility is finite at zero temperature. This calculation is very similar to the previous one. Indeed, we have

  $$ h\,{=}\,\left({\frac{\partial F}{\partial M}}\right)_{N,T,V}, $$

  (11.64)

  with $M\,{=}\,(N_\uparrow-N_\downarrow)/2$ and $N\,{=}\,(N_\uparrow+N_\downarrow)/2.$ Polarizing the system at constant atom number, we have

  $$ dh\,{=}\,\left({\frac{\partial^2 F}{\partial M^2}}\right)_{N,V,T}dM. $$

  (11.65)

  Defining the susceptibility of the gas has

  $$ \chi\,{=}\,{\frac{1}{V}}\left({\frac{\partial M}{\partial h}}\right)_{N,T,V}, $$

  (11.66)

  and we have the identity

  $$ \chi^{-1}\,{=}\,V\left({\frac{\partial^2 F}{\partial M^2}}\right) $$

  (11.67)

At low temperature and spin imbalance, we can expand the free energy with respect to T and M. To second order, we have

$$\begin{aligned}[b] F(T,V,N,M)\,{=}\, & F_0(T,V,N)+T\left({\frac{\partial F}{\partial T}}\right)_0+M\left({\frac{\partial F}{\partial M}}\right)_0\\[4pt] & +{\frac{T^2}{2}}\left({\frac{\partial^2 F}{\partial T^2}}\right)_0+{\frac{M^2}{2}}\left({\frac{\partial^2 F}{\partial M^2}}\right)_0+MT\left({\frac{\partial^2 F}{\partial T\partial M}}\right)_0\!{.}\\ \quad \end{aligned}$$

(11.68)

According to the Third Law of Thermodynamics, entropy is zero at zero temperature. From the previous expansion, we thus have

$$ S\,{=}\,-\left({\frac{\partial F}{\partial T}}\right)_{V,N,M}\,{=}\,\left({\frac{\partial F}{\partial T}}\right)_0+T\left({\frac{\partial^2 F}{\partial T^2}}\right)_0+M\left({\frac{\partial^2 F}{\partial T\partial M}}\right)_0\,{=}\,0. $$

(11.69)

We then obtain readily

$$ \left({\frac{\partial F}{\partial T}}\right)_0\,{=}\,\left({\frac{\partial^2 F}{\partial T\partial M}}\right)_0\,{=}\,0, $$

(11.70)

and thus

$$ F(T,V,N,M)\,{=}\,F_0(T,V,N)+{\frac{T^2}{2}}\left({\frac{\partial^2 F}{\partial T^2}}\right)_0+{\frac{M^2}{2}}\left({\frac{\partial^2 F}{\partial M^2}}\right)_0. $$

(11.71)

Let us set $C_V\,{=}\,V\gamma T$ at low temperature. According to the previous discussion we therefore have

$$ F\,{=}\,F_0-{\frac{\gamma T^2}{2}}+{\frac{M^2}{2\chi}}+... $$

(11.72)

Let us now consider the grand potential $\Upomega\,{=}\,F-\mu N-h M.$ We have using the relation $h\,{=}\,\chi M$ in the weakly polarized limit

$$ \Upomega(\mu,h,T,V)\,{=}\,F_0-\mu N-V{\frac{\gamma T^2}{2}}-V{\frac{\chi h^2}{2}}, $$

(11.73)

which can be recast as

$$ \Upomega(\mu,h,T,V)\,{=}\,\Upomega_0(\mu,V)-\left({\frac{\gamma T^2}{2}}+{\frac{\chi h^2}{2}}\right)V, $$

(11.74)

where $\Upomega_0$ is the value of the grand potential for $T\,{=}\,h\,{=}\,0.$ Let us introduce $P_1 (\mu)$ the pressure of a single species Fermi gas at zero temperature. We then have using $\Upomega\,{=}\,-PV$

$$ {\frac{P}{P_1}}\,{=}\,{\frac{P_0(\mu)}{P_1(\mu)}}\left[1+{\frac{\gamma T^2}{2P_0}}+{\frac{\chi h^2}{2P_0}}\right]. $$

(11.75)

11.1.3 Susceptibility of a Molecular Gas

Let us consider a mixture of spin-1/2 fermions in the BEC side of a Feshbach resonance and in this regime $a\,{>}\,0,$ so two atoms can form a stable dimer of energy $\hbar^2/ma^2.$ Let us consider the free energy $F(N_\uparrow,N_\downarrow,N_b).$ Since the formation of $\delta N_b$ dimers requires the disappearance of a same number of spin-up and down atoms, the free energy varies at fixed temperature as

$$ \delta F\,{=}\,\left(\partial_{N_b}F-\partial_{N_\uparrow}F-\partial_{N_\downarrow}F\right)\delta N_b. $$

(11.76)

At equilibrium, $\delta F\,{=}\,0$ to first order, yielding the condition

$$\mu_b\,{=}\,\mu_\uparrow+\mu_\downarrow\,{=}\,2\bar\mu, $$

(11.77)

where as before $\bar\mu\,{=}\,(\mu_\uparrow+\mu_\downarrow)/2.$ Let us consider the BEC limit, where the binding energy of the dimers is $-\hbar^2/ma^2$ and the atom-atom, atom-dimer and dimer-dimer interactions are negligible. In this case we can describe the system as a mixture of ideal gases. Moreover, if we work above the quantum degenerate regime, we can use classical thermodynamics to describe the system. In this regime the partition function of an ensemble of N non-interacting particles described by a dispersion relation $E(p)\,{=}\,E_\alpha^0+p^2/2m_\alpha,$ where the $\alpha$ encapsulates the nature and the internal state of the particle, is given by

$$ Z\,{=}\,{\frac{1}{N!}}\left[\int {\frac{Vd^3\user2{p}}{(2\pi\hbar)^3}}e^{-\beta E(p)}\right]\,{=}\,{\frac{1}{N!}}\left({\frac{Ve^{-\beta E_0}}{\lambda_{\rm th}^3}}\right)^N, $$

(11.78)

where $\lambda_{\rm th}\,{=}\,\sqrt{2\pi\hbar^2/m_\alpha k_B T}$ is the thermal wavelength and the factor $N!$ was introduced to take into account the classical indiscernibility of the particles. ⁸ We deduce the chemical potential from the definition $\mu\,{=}\,\partial_N F$ with $F\,{=}\,-k_B T\ln Z.$ After a straightforward calculation we obtain

$$ \mu\,{=}\,E_0+k_B T\left(\ln (n\lambda_{\rm th}^3)-1\right). $$

(11.79)

for the gases of atom $(m_\alpha\,{=}\,m, E_\alpha^0\,{=}\,0)$ and molecules $(m_\alpha\,{=}\,2m$ and $E_\alpha^0\,{=}\,-\hbar^2/ma^2),$ we have

$$ \mu_\sigma\,{=}\,k_BT\left[\ln\left(n_\sigma\lambda_T^3\right)-1\right]$$

(11.80)

$$ \mu_b\,{=}\,-{\frac{\hbar^2}{ma^2}}+k_BT\left[\ln\left(n_\sigma\lambda_T^3/2^{3/2}\right)-1\right], $$

(11.81)

where $\lambda_{\rm th}$ is the thermal wavelength of the atoms.

Let us consider first the balanced case, $n_\uparrow\,{=}\,n_\downarrow\,{=}\,n_a.$ Writing the condition (11.75), we recover the law of mass action

$$ {\frac{n^2_a}{n_b}}\,{=}\,{\frac{e^{-T^*/T}}{2^{3/2}\lambda_{\rm th}^3}}, $$

(11.82)

with $k_BT^*\,{=}\,\hbar^2/ma^2.$ As expected, we see that this ratio goes to zero when the temperature becomes much smaller than $T^*.$ In this limit, we have in particular $N\,{=}\, N_\uparrow + N_\downarrow + 2N_b \sim 2N_b,$ hence

$$ n^2_a\sim {\frac{2n e^{-T^*/T}}{2^{3/2}\lambda_{\rm th}^3}}. $$

(11.83)

Let us now consider the imbalanced case. Using Eq.  11.78, we see that

$$ h\,{=}\,{\frac{k_B T}{2}}\ln\left({\frac{n_\uparrow}{n_\downarrow}}\right). $$

(11.84)

Inverting this relation, we see that when h is small we have

$$ n_\uparrow-n_\downarrow\,{=}\,{\frac{2n_a }{k_{\rm B}T}}h, $$

(11.85)

where $n_a$ is given by (11.81) and is exponentially small in the low temperature limit $T/T^*\ll 1.$ The spin susceptibility is thus given by

$$ \chi\,{=}\,{\frac{1}{V}}\left({\frac{\partial M}{\partial h}}\right)_N\propto {\frac{e^{-T^*/2T}}{T}}. $$

(11.86)

In the opposite limit $T/T^*\gg 1,$ we have $n_b\ll n_a$ hence $n_a\sim n/2$ and

$$ \chi\propto {\frac{1}{T}}, $$

(11.87)

where we recover Curie’s law.

11.1.4 Virial Expansion

Let us consider the Grand canonical partition function of a many-body system

$$ Z\,{=}\,\sum_{\alpha\in {\mathcal H}} e^{-\beta (E_\alpha-\mu N_\alpha)}, $$

where the $|\alpha\rangle$ are the eigenstates of the hamiltonian H and span the whole grand canonical Hilbert space $\mathcal H.$ Let us now decompose this sum over Fock states of fixed atom number N. We thus have

$$ Z\,{=}\,\sum_N \left(e^{\beta\mu N}\sum_{\alpha\in {\mathcal H}_N} e^{-\beta (E_\alpha)}\right). $$

We see in this case that the partition function can be expanded with the fugacity $z\,{=}\,\exp (\beta \mu),$ a result known as the Virial expansion. The grand-potential $\Upomega\,{=} \,-PV\,{=}\,-k_BT\ln Z$ is also analytical with fugacity z and can therefore be expanded as

$$ P(\mu,T)\,{=}\,\sum_{n\ge 1} b_n z^n $$

(11.88)

where $b_n$ can be obtained from the knowledge of the spectrum of the Hamiltonian at a number $n^{\prime}\le n$0 of particles.