Attractively Interacting Fermi gases - Pairing Instability

In contrast to bosons, the non-interacting Fermi gas does not show any phase transition down to zero temperature. One might assume that this qualitative fact should not change as interactions are introduced, at least as long as they are weak. This is essentially true in the case of repulsive interactions. For attractive interactions, the situation is, however, dramatically different. Even for very weak attraction, the fermions form pairs and become superfluid, due to a generalized form of pair condensation.

The idea of pairing might be natural, as tightly bound pairs of fermions can be regarded as point-like bosons, which should form a Bose-Einstein condensate. However, for weak attractive interaction -- as is the case for the residual, phonon-induced electron-electron interaction in metals -- it is not evident that a paired state exists. Indeed, we will see in the following that in three dimensions there is no bound state for two isolated particles and arbitrarily weak interaction. However, by discussing exact solutions in 1D and 2D, where bound states exist for weak interactions, we gain insight into how a modified density of states will lead to bound states even in 3D -- this is the famous Cooper instability.

Two-body bound states in 1D, 2D and 3D

Localizing a quantum-mechanical particle of mass $\mu =m/2$ to a certain range $R$ leads to an increased momentum uncertainty of $p\sim \hbar /R$ at a kinetic energy cost of about $E_{R}=p^{2}/m=\hbar ^{2}/mR^{2}$. Clearly, a shallow potential well of size $R$ and depth $V$ with $V/E_{R}\equiv \epsilon \ll 1$ cannot confine the particle within its borders. But we can search for a bound state at energy $|E_{B}|\ll E_{R}$ of much larger size $r_{B}=1/\kappa \equiv {\sqrt {\hbar ^{2}/m|E_{B}|}}\gg R$.

• In 1D and 2D, there is always a bound state for an arbitrarily weak attractive interaction.
• In 3D: For a spherically symmetric well, the Schr\"odinger equation for the wave function transforms into an effective one-dimensional problem for the wave function $u=r\psi$ c). We might now be tempted to think that there must always be a bound state in 3D, asthe case in 1D. However, the boundary condition on $u(r)$ is now to vanish linearly at $r=0$, in order for $\psi (0)$ to be finite. Outside of the potential well, we still have $u\propto e^{-\kappa r}$ for a bound state. Inside the well the wave function must fall off to zero at $r=0$ and necessarily has to change its slope from $-\kappa$ outside to $\sim 1/R$ inside the well over a distance $R$. This costs the large kinetic energy $\sim \hbar ^{2}u''/2mu\approx \hbar ^{2}/mR^{2}=E_{R}$. If the well depth is smaller than a critical depth $V_{c}$ on the order of $E_{R}$, the particle cannot be bound.

The results hold for quite general shapes $V(r)$ of the (purely attractive) potential well.

Applying these results to the equivalent problem of two interacting particles colliding in their center-of-mass frame, we see that in 1D and 2D, two isolated particles can bind for an arbitrarily weak purely attractive interaction. Hence in 1D and 2D, pairing of fermions can be understood already at the two-particle level. Indeed, one can show that the existence of a two-body bound state for isolated particles in 2D is a necessary and sufficient condition for the instability of the many-body Fermi sea (Cooper instability, see below)~\cite{rand89bound}. In 3D, however, there is a threshold interaction below which two isolated particles are unbound. We conclude that if pairing and condensation occur for arbitrarily weak interactions in 3D, then this must entirely be due to many-body effects.

Density of states

We show that the "density of states" in the different dimensions as the decisive factor for the existence of bound states.

Searching for a shallow bound state of energy $E=-{\frac {\hbar ^{2}\kappa ^{2}}{m}}$ ($m/2$ is the reduced mass), we start by writing the Schrodinger equation for the relative wave function momentum space ($n$-dimensional):

$\psi _{\kappa }({\vec {q}})=-{\frac {m}{\hbar ^{2}}}{\frac {1}{q^{2}+\kappa ^{2}}}\int {\frac {d^{n}q'}{(2\pi )^{n}}}V({\vec {q}}-{\vec {q}}')\psi _{\kappa }({\vec {q}}')\,.$

For a short-range potential, $V({\vec {q}})$ is practically constant $V_{0}$ for all relevant $q$, and falls off to zero on a large $q$-scale of </amth>\approx 1/R</math>. Thus

$\psi _{\kappa }({\vec {q}})\approx -{\frac {mV_{0}}{\hbar ^{2}}}{\frac {1}{q^{2}+\kappa ^{2}}}\int _{q'\lesssim {\frac {1}{R}}}{\frac {d^{n}q'}{(2\pi )^{n}}}\psi _{\kappa }({\vec {q}}')$

We integrate once more over ${\vec {q}}$, applying the same cut-off $1/R$, and then divide by the common factor $\int _{q\lesssim {\frac {1}{R}}}{\frac {d^{n}q}{(2\pi )^{n}}}\psi _{\kappa }({\vec {q}})$. We obtain the following equation for the bound state energy $E$:

$-{\frac {1}{V_{0}}}\;=\;{\frac {m}{\hbar ^{2}}}\int _{q\lesssim {\frac {1}{R}}}{\frac {d^{n}q}{(2\pi )^{n}}}{\frac {1}{q^{2}+\kappa ^{2}}}\;=\;{\frac {1}{\Omega }}\int _{\epsilon <E_{R}}d\epsilon {\frac {\rho _{n}(\epsilon )}{2\epsilon +\left|E\right|}}$

with the density of states in $\rho _{n}(\epsilon )$,the energy cut-off $E_{R}=\hbar ^{2}/mR^{2}$ and the volume $\Omega$ of the system. The question on the existence of bound states for arbitrarily weak interaction has now been reformulated: As $|V_{0}|\rightarrow 0$, the left hand side diverges. This equation has a solution for small $|V_{0}|$ only if the right hand side also diverges for vanishing bound state energy $|E|\rightarrow 0$, and this involves an integral over the density of states. In 1D and 2D, there is always bound states. However, in 3D the integral is finite for vanishing $|E|$, and there is a threshold for the interaction potential to bind the two particles.

These results give us an idea why there might be a paired state for two fermions immersed in a 3D medium, even for arbitrarily weak interactions: It could be that the density of available states to the two fermions is altered due to the presence of the other atoms.

Pairing of fermions -- The Cooper problem

Consider now two weakly interacting spin 1/2 fermions not in vacuum, but on top of a (non-interacting) filled Fermi sea, the Cooper problem. For weak interactions, the particles' momenta are essentially confined to a narrow shell above the Fermi surface. The density of states at the Fermi surface is $\rho _{\rm {3D}}(E_{F})$, which is a constant just like in two dimensions. We should thus find a "bound state" for the two-particle system for "arbitrarily weak attractive interaction".

In principle, the two fermions could form a pair at any finite momentum. However, considering the discussion in the previous section, the largest binding energy can be expected for the pairs with the largest density of scattering states. For zero-momentum pairs, the entire Fermi surface is available for scattering. If the pairs have finite center-of-mass momentum ${\vec {q}}$, the number of contributing states is strongly reduced, as they are confined to a circle. Consequently, pairs at rest experience the strongest binding. In the following, we will calculate this energy.

We can write the Schr\"odinger equation for the two interacting particles as before, but now we need to search for a small binding energy $E_{B}=E-2E_{F}<0$ on top of the large Fermi energy of the two particles. The equation for $E_{B}$ is

$-{\frac {1}{V_{0}}}={\frac {1}{\Omega }}\int _{E_{F}<\epsilon <E_{F}+E_{R}}d\epsilon {\frac {\rho _{\rm {3D}}(\epsilon )}{2(\epsilon -E_{F})+\left|E_{B}\right|}}$

The effect of Pauli blocking of momentum states below the Fermi surface is explicitly included by only integrating over energies $\epsilon >E_{F}$.

In conventional superconductors, the natural cut-off energy $E_{R}$ is given by the Debye frequency $E_{R}=\hbar \omega _{D}$, corresponding to the highest frequency at which ions in the crystal lattice can respond to a bypassing electron. Since we have $\hbar \omega _{D}\ll E_{F}$, we can approximate $\rho _{\rm {3D}}(\epsilon )\approx \rho _{\rm {3D}}(E_{F})$ and find:

Failed to parse (syntax error): {\displaystyle E_B = - 2 \hbar \omega_D e^{-2 \Omega/\rho_{\rm 3D}(E_F) \left|V_0\right|}\\ }

In the case of an atomic Fermi gas, we should replace $1/V_0$ by the physically relevant scattering length $a < 0$. The equation for the bound state becomes

$-{\frac {m}{4\pi \hbar ^{2}a}}={\frac {1}{\Omega }}\int _{E_{F}}^{E_{F}+E_{R}}d\epsilon {\frac {\rho _{\rm {3D}}(\epsilon )}{2(\epsilon -E_{F})+\left|E_{B}\right|}}-{\frac {1}{\Omega }}\int _{0}^{E_{F}+E_{R}}d\epsilon {\frac {\rho _{\rm {3D}}(\epsilon )}{2\epsilon }}$

The right hand expression is now finite as we let the cut-off $E_{R}\rightarrow \infty$, the result being (one assumes $\left|E_{B}\right|\ll E_{F}$)

$-{\frac {m}{4\pi \hbar ^{2}a}}={\frac {\rho _{\rm {3D}}(E_{F})}{2\Omega }}\left(-\log \left({\frac {\left|E_{B}\right|}{8E_{F}}}\right)-2\right)$

Inserting $\rho _{\rm {3D}}(E_{F})={\frac {\Omega mk_{F}}{2\pi ^{2}\hbar ^{2}}}$ with the Fermi wave vector $k_{F}={\sqrt {2mE_{F}/\hbar ^{2}}}$, one arrives at

$E_{B}=-{\frac {8}{e^{2}}}E_{F}\,e^{-\pi /k_{F}\left|a\right|}\,.$

The binding energies can be compared with the result for the bound state of two particles in 2D. The role of the constant density of states $\rho _{\rm {2D}}$ is here played by the 3D density of states at the Fermi surface, $\rho _{\rm {3D}}(E_{F})$.

We, therefore, see that two weakly interacting fermions on top of a Fermi sea form a bound state due to Pauli blocking. However, in this artificial problem we neglected the interactions between particles "inside" the Fermi sea. As we ``switch on the interactions for all particles from top to the bottom of the Fermi sea, the preceding discussion indicates that the gas will reorder itself into a completely new, paired state. The Fermi sea is thus unstable towards pairing (Cooper instability). The full many-body description of such a paired state, including the necessary anti-symmetrization of the full wave function, was achieved by Bardeen, Cooper and Schrieffer (BCS) in 1957. As we will see in the next section, the self-consistent inclusion of all fermion pairs leads to more available momentum space for pairing. The effective density of states is then twice as large, giving a superfluid gap $\Delta$ that differs from above result by a factor of 2 in the exponent.

It should be noted that the crucial difference to the situation of two particles in vacuum in 3D is the "constant density" of states at the Fermi energy (and not the 2D character of the Fermi surface). Therefore, if we were to consider the Cooper problem in higher dimensions and have two fermions scatter on the $(n-1)$ dimensional Fermi surface, the result would be similar to the 2D case (due to the constant density of states), and not to the case of $(n-1)$ dimensions.

The conclusion of this section is that Cooper pairing is a many-body phenomenon, but the binding of two fermions can still be understood by two-body quantum mechanics, as it is similar to two isolated particles in two dimensions. To first order, the many-body physics is not the modification of interactions, but rather the modification of the density of states due to Pauli blocking.