Fermi-liquid breakdown in the BCS-BEC crossover normal state: temperature dependence of the vanishing quasiparticle residue, the incoherent spectral background, and the balanced-gas-Fermi-polaron similarity

Momentum-resolved radio-frequency (photoemission) spectroscopy of a nearly homogeneous strongly interacting Fermi gas (Sagi, Drake, Paudel, Chapurin and Jin, Phys. Rev. Lett. 114, 075301 (2015)) showed that the quasiparticle residue $Z$ of the normal state vanishes abruptly at $(k_Fa)^{-1}=0.28\pm0.02$ on the BEC side of the BCS-BEC crossover, at $T\approx0.2\,T_F$ just above $T_c$, while at unitarity $Z$ and the effective mass $m^*$ remain close to Fermi-polaron values. The authors left open the temperature dependence of this Fermi-liquid breakdown, the microscopic origin of the broad incoherent spectral background that replaces the quasiparticle peak, the unexplained quantitative similarity between balanced-gas and Fermi-polaron quasiparticle observables, and a quantitative confrontation of the data with t-matrix and beyond-t-matrix crossover theories. A decade of follow-up — uniform-gas rf spectroscopy, smooth polaron-molecule transition experiments, Gaussian-pair-fluctuation, self-consistent t-matrix, and quantum Monte Carlo calculations — has sharpened but not resolved these questions. The surviving open core is a microscopic, quantitative theory of the crossover normal state, tested against temperature-dependent homogeneous-gas spectroscopy, that identifies the mechanism by which fermionic quasiparticles disintegrate into pair-dominated excitations.

Background

Landau's Fermi-liquid theory describes low-temperature fermionic systems in terms of long-lived quasiparticle excitations that carry the quantum numbers of the bare fermions. Two renormalized parameters characterize each quasiparticle: the effective mass $m^*$ and the quasiparticle residue $Z$, the fraction of the single-particle spectral weight carried by the coherent quasiparticle peak. $Z$ lies between 0 and 1 and shrinks as interactions strengthen; when $Z$ vanishes, the fermionic quasiparticle picture fails even if a degenerate Fermi surface survives — a breakdown of the Fermi-liquid description.

An ultracold two-component Fermi gas with a Feshbach resonance — a magnetically tunable coupling between atomic scattering channels that makes the s-wave scattering length $a$ controllable — provides a clean realization of this physics. The dimensionless parameter $(k_Fa)^{-1}$, where $k_F$ is the Fermi wave number of the noninteracting gas, sweeps from the weakly attractive Bardeen–Cooper–Schrieffer (BCS) side (negative values) through unitarity ($a\to\infty$, $(k_Fa)^{-1}=0$) to the Bose–Einstein-condensate (BEC) side, where atoms form tightly bound diatomic molecules. Below the superfluid transition temperature $T_c$ the ground state is always a condensate of pairs, but the normal state above $T_c$ must evolve from a Fermi liquid to a Bose gas of molecules. How this happens — in particular whether an intermediate pseudogap regime, in which incoherent pairing above $T_c$ suppresses single-particle spectral weight near the Fermi level, intervenes — has been a long-standing controversy. The many-body contenders are pairing-fluctuation schemes, none of which has a small parameter at strong coupling: t-matrix theories, which resum the ladder diagrams of the particle–particle scattering channel either self-consistently or with bare propagators; Gaussian pair fluctuation theory, which includes fluctuations beyond mean field around pair formation; and auxiliary-field quantum Monte Carlo, a stochastic sampling method that is essentially exact for the balanced gas but yields imaginary-time data whose continuation to real frequencies is mathematically ill-posed.

The primary experimental probe is radio-frequency (rf) spectroscopy: an rf photon transfers atoms between hyperfine spin states with negligible momentum, and in its momentum-resolved form — called photoemission spectroscopy (PES) by analogy with angle-resolved photoemission in solids — the measured momentum of the transferred atom together with the rf detuning yields the single-particle energy $E$ and momentum $k$. The signal is $I(k,E)\propto k^2 A(k,E)f(E)$, where $A(k,E)$ is the single-particle spectral function and $f(E)$ is the Fermi function. Early PES measurements on trapped strongly interacting gases reported pseudogap-like behavior, but their interpretation was controversial because averaging over the inhomogeneous trap density can mimic or obscure intrinsic effects.

Sagi, Drake, Paudel, Chapurin and Jin performed the first PES of a nearly homogeneous Fermi gas (the uniform central region of a trapped $^{40}$K cloud) at $T\approx0.2\,T_F$ — where $T_F$ is the Fermi temperature, the temperature scale set by the Fermi energy of the noninteracting gas — just above $T_c$, across the crossover. They fit the spectra with a two-mode form $I(k,E)=Z\,I_{\text{coherent}}(k,E)+(1-Z)\,I_{\text{incoherent}}(k,E)$: a narrow, positively dispersing quasiparticle peak plus a broad incoherent background with negative dispersion, modeled phenomenologically as short-range pairs with a Gaussian distribution of center-of-mass energies. They found that $Z$ is finite at unitarity (close to $0.3$) but vanishes abruptly at $(k_Fa)^{-1}=0.28\pm0.02$ on the BEC side; that the effective mass grows to $m^*=1.21\pm0.03$ at unitarity; that the contact — a universal short-range-correlation parameter that fixes the $1/k^4$ tail of the momentum distribution and the $1/\nu^{3/2}$ tail of the rf line shape in detuning $\nu$ — extracted from the fits agrees with zero-temperature theories; and that the fitted effective pair temperature $T_p=1.09\pm0.08\,T_F$ in the example near-unitarity fit far exceeds the gas temperature $T=0.24\pm0.02\,T_F$, so the phenomenological pairs are not molecules in thermal equilibrium with the atoms.

The Fermi polaron — the quasiparticle formed by a single impurity atom interacting attractively with a Fermi sea of majority atoms, i.e. the extreme spin-imbalance limit of the same gas — is the natural reference point. There, theory and experiment establish a transition from a polaron to a dressed molecular state as attraction increases, with $Z$ acting like an order parameter, whereas the balanced gas is expected to show a continuous crossover. Against this background, the abrupt vanishing of $Z$ within a smooth crossover, and the quantitative similarity of balanced-gas quasiparticle observables ($Z\approx0.3$ and $m^*\approx1.2$ at unitarity) to the polaron values $Z=0.39\pm0.09$ and $m^*=1.20\pm0.02$ measured by Schirotzek et al., were both flagged by the authors as surprising and unexplained.

Problem Statement

Achieve a quantitative microscopic understanding of the normal state of the three-dimensional, balanced, two-component Fermi gas with tunable s-wave interactions across the BCS-BEC crossover, above the superfluid transition temperature $T_c$, sufficient to explain the momentum-resolved spectra of homogeneous gases — specifically the data of Sagi et al. (2015). This comprises four mutually linked parts of one objective:

(1) Determine the temperature dependence of the interaction strength $(k_Fa)^{-1}$ at which the quasiparticle residue $Z$ vanishes and of the sharpness of that vanishing — that is, map the boundary between the Fermi-liquid regime ($Z>0$) and the pair-dominated incoherent regime in the interaction-temperature plane above $T_c$; the 2015 data provide only the single-temperature slice at $T\approx0.2\,T_F$.

(2) Derive the measured incoherent spectral background $I_{\text{incoherent}}(k,E)$ from a microscopic many-body spectral function, rather than from the thermal-pair phenomenology, thereby establishing or excluding a pseudogap in the three-dimensional normal state and clarifying the physical meaning of the fitted pair temperature $T_p$ far above the gas temperature.

(3) Establish whether the quantitative similarity between balanced-gas quasiparticle observables ($Z$ and $m^*$ at and near unitarity) and the corresponding Fermi-polaron measurements reflects a deeper microscopic connection or is coincidental.

(4) Confront t-matrix and beyond-t-matrix crossover theories quantitatively with the nearly homogeneous photoemission data — quasiparticle residue, effective mass, contact, and full line shapes — across the crossover, establishing where they succeed and where and why they fail, so that the mechanism of the Fermi-liquid breakdown is identified rather than fitted.

A complete answer must address all four parts jointly: it consists of a microscopic theory of the crossover normal state, tested against temperature-dependent homogeneous-gas spectroscopy, that yields the temperature-dependent $Z(T,(k_Fa)^{-1})$ boundary, derives the incoherent spectral background $I_{\text{incoherent}}(k,E)$ from a microscopic many-body spectral function, adjudicates the balanced-gas/Fermi-polaron similarity, and confronts the t-matrix and beyond-t-matrix theories quantitatively with the data, so that these jointly settle the mechanism by which fermionic quasiparticles disintegrate into pair-dominated excitations across the crossover above $T_c$; a strict subset of the four parts does not resolve the task.

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