A low-energy effective Hamiltonian for Landau quasiparticles

TL;DR

This work builds a comprehensive low-energy description for Landau quasiparticles in Fermi liquids by introducing an energy cutoff $Λ$ and a unitary dressing that decouples quasi-resonant couplings. The resulting $H_{\rm eff}$ unifies Landau’s forward-scattering function $f$ and the collision amplitude into a single, Λ-regularized amplitude $\mathcal{A}$, enabling a Boltzmann treatment of transport and collisions without resorting to Green’s functions. The authors apply the framework to a homogeneous contact Fermi gas, deriving explicit expressions for quasiparticle energies, residues, and collision amplitudes to $O((k_{F}a)^2)$, and obtaining a kinetic equation that is solved across the collisionless-to-hydrodynamic crossover; notably, zero-sound corrections and damping are captured, and the Gor'kov–Melik-Barkhudarov correction to $T_c$ emerges from the dressing. The approach provides a unified, microscopic route to thermodynamics, transport, and pairing in Fermi liquids, with clear relevance to ultracold atomic gases and beyond, offering a controlled way to access corrections to Landau theory in both normal and superfluid phases.

Abstract

We introduce a new renormalisation scheme to construct the Landau quasiparticles of Fermi fluids. The scheme relies on an energy cutoff $Λ$ which removes the quasi-resonant couplings, enabling the dressing of the particles into quasiparticles via a unitary transformation. The dynamics of the quasiparticles is then restricted to low-energy transitions and is fully determined by an effective Hamiltonian which unifies the Landau interaction function $f$ and the collision amplitude in a single amplitude $\mathcal{A}$ regularized by $Λ$. Our effective theory captures all the low-energy physics of Fermi fluids that support Landau quasiparticles, from the equation of state to the transport properties, both in the normal and in the superfluid phase. We apply it to an atomic Fermi gas with contact interaction to compute the speed of zero sound in function of the scattering length $a$. We also recover the Gork'ov-Melik Barkhudarov correction to the superfluid gap and critical temperature as a direct consequence of the dressing of particles into Landau quasiparticles.

Figures (16)

• Figure 1: Construction of the quasiparticle states within quasi-degenerate perturbation theory. An unpertubed Fock state $|i\rangle_0$ is dressed via the operator $\hat{S}$ by its interactions with the off-resonant states $|f\rangle_0$, whose unperturbed energy verifies $|E_{|f\rangle_0}-E_{|i\rangle_0}|>\Lambda$ (here $E_{|\psi\rangle_0}={}_0\langle \psi|\hat{H}_0|\psi\rangle_0$). The dressed state $|i\rangle$ can then be followed adiabatically as the interaction strength $V$ increases. However, due to its incomplete dressing, it is not an eigenstate of $\hat{H}$, and it remains coupled to the nearly degenerate states $|f'\rangle$ of energies $|E_{|f'\rangle}-E_{|i\rangle}|<\Lambda$, (here $E_{|\psi\rangle}=\langle \psi|\hat{H}|\psi\rangle$). This construction applies in particular to the particle Fermi sea $|{\rm FS}\rangle_0$, which evolves into a quasiparticle Fermi sea $|{\rm FS}\rangle$ (red curve). In general the quasiparticle Fermi sea is not the ground state of $\hat{H}$, and therefore not the ground state of our effective Hamiltonian.
• Figure 2: Angular dependence of the function $I_\Lambda$ appearing in $\mathcal{A}_{\uparrow\downarrow}$ and $f_{\uparrow\downarrow}$. For $\epsilon_\Lambda=0$ (black curve), the function displays a logarithmic divergence $\sim \text{ln}(\pi-\theta)+1-\text{ln}\,4$ when $\theta\to\pi$ (black dashed curve). For $\epsilon_\Lambda\neq0$ (red curve) the divergence is regularized, and the function saturates at $\text{ln}(\epsilon_\Lambda/2)+2+O(\epsilon_\Lambda)$ in $\theta=\pi$ (red dashed curve).
• Figure 3: Angular dependence of the function $J_\Lambda$ appearing in both $\mathcal{A}_{\uparrow\downarrow},$$f_{\uparrow\downarrow}$ and $\mathcal{A}_{\sigma\sigma},$$f_{\sigma\sigma}$. As $\epsilon_\Lambda\to0$, the function converges pointwise to $J(\theta)$ (black curve) on $(0,\pi]$. It is however cancelled in an interval of width $\simeq2\epsilon_\Lambda$ about $\theta=0$ (red curve).
• Figure 4: (Main pannel) Difference between the particle momentum distribution $n_p^{|{\rm FS}\rangle}=\langle {\rm FS}|\hat{a}_{\textbf{p}\sigma}^\dagger \hat{a}_{\textbf{p}\sigma}|{\rm FS}\rangle$ (see Eq. \ref{['momentumdistrib']}) and the zero-temperature Fermi-Dirac distribution $n_p^0$ as a function of $p/p_{\rm F}$. The difference is scaled to $(k_{\rm F} a/\pi)^2$ so as to be independent of $a$ in the weak-coupling limit. The discontinuity across the Fermi surface, i.e. between the asymptotic values in $p_{\rm F}-0^-$ and $p_{\rm F}+0^+$ (black dashed lines), is given by $1-Z_{p_{\rm F}}$. At large momenta, the distribution follows a $1/p^4$ behavior, from which the contact $C = 4 \overline{a}^2 / 9\pi^2$ can be extracted (orange dashed curve). (Inset) The bare distribution $n_p^{|{\rm FS}\rangle}$ in function of $p/p_{\rm F}$, evaluated in second-order perturbation theory at $k_{\rm F} a=-1$. The distribution displays the familiar shape of a depleted Fermi sea, with $n_{p}<1$ down to $p=0$.
• Figure 5: (a) The angular parametrization where $\textbf{p}_1+\textbf{p}_2$ is chosen as the polar axis of the spherical frame. This parametrization is used for $W_{E}$ in Eq. \ref{['wethetaphi']}. (b) The angular parametrization where $\textbf{p}_1-\textbf{p}_3$ is chosen as the polar axis of the spherical frame. This parametrization is used for $W_{S}$ in Eq. \ref{['wsthetaphi']} . The last parametrization where $\textbf{p}_1-\textbf{p}_4$ is chosen as the polar axis is not shown here.