Bias-controlled nonperturbative benchmarks for the two-dimensional Fermi gas across the BCS-BEC crossover and their quantitative mapping to quasi-2D experiments

Bertaina and Giorgini's 2011 fixed-node diffusion Monte Carlo benchmarks of the two-dimensional Fermi gas across the BCS-BEC crossover leave open several problems: uncontrolled fixed-node bias at strong coupling (visible as a contact inconsistency near $\ln(k_Fa_{2D})\sim1$ ), the many-body validity of the fitted dimer-dimer and atom-dimer scattering lengths, quantification of finite-size and nodal errors, and the mapping of strictly-2D results onto quasi-2D trapped gases. The objective is to establish a quantitatively reliable, systematically controlled nonperturbative description of the uniform 2D Fermi gas across the BCS-BEC crossover at $T=0$ , and to connect it quantitatively to quasi-2D experiments.

Background

The two-component Fermi gas with short-range attractive $s$ -wave interactions is the cleanest realization of the BCS-BEC crossover: as the attraction is increased, the ground state evolves continuously from a Bardeen-Cooper-Schrieffer (BCS) superfluid of large, overlapping Cooper pairs to a Bose-Einstein condensate (BEC) of tightly bound molecular dimers. In two spatial dimensions (2D) the crossover has special features. Low-energy $s$ -wave scattering in 2D is governed by a scattering length $a_{2D}$ through the logarithmic amplitude $f(k)=2\pi/[\ln(2/ka_{2D}e^{\gamma})+i\pi/2]$ , where $\gamma$ is Euler's constant; a two-body bound state exists for arbitrarily weak attraction, and, unlike in 3D, there is no unitary regime in which low-energy observables become independent of the interaction. The natural crossover parameter is $\eta=\ln(k_Fa_{2D})$ , with $k_F$ the Fermi momentum: $\eta\ll0$ is the deep BEC regime of dimers, $\eta\gtrsim3$ the weak-coupling BCS regime, and $\eta\sim0$ - $1$ the strongly correlated crossover window, where no perturbative parameter exists and mean-field theory fails badly, so quantitative information must come from nonperturbative many-body calculation.

The benchmark computation is Bertaina and Giorgini's fixed-node diffusion Monte Carlo (FN-DMC) study. Diffusion Monte Carlo projects a trial wave function onto the exact ground state by imaginary-time evolution; for fermions, the sign problem is circumvented by fixing the nodal surface (the set of many-body configurations where the trial function vanishes) to that of the chosen trial function, which makes the resulting energy a variational upper bound whose residual error — the fixed-node bias — is unknown a priori and is controlled only by the quality of the trial function. Using trial functions interpolating between a Jastrow-Slater Fermi-liquid form and a paired BCS-dimer form, they computed the $T=0$ equation of state (EOS), the pairing gap $\Delta_{\rm gap}$ (extracted from ground-state energy differences of systems differing by one particle), and Tan's contact $C$ — the coefficient of the $k^{-4}$ tail of the momentum distribution, which for this gas must also equal $(2\pi m/\hbar^2)\,d(nE/N)/d(\ln k_Fa_{2D})$ by Tan's adiabatic theorem — across $\eta\simeq-2$ to $+5$ , finding large deviations from mean-field theory. In the BEC regime they fitted the EOS to a dilute gas of composite bosons (dimers) with dimer-dimer scattering length $a_d$ , obtaining $a_d=0.55(4)a_{2D}$ , and extracted an atom-dimer scattering length $a_{ad}=1.7(1)a_{2D}$ from the atom-dimer interaction contribution to the gap; both are dilute-regime fits to effective energy functionals. Their two determinations of the contact disagreed around $\eta\sim1$ , which the authors attributed to the need for a better-optimized nodal wave function; finite-size effects were corrected using Fermi-liquid theory; and they analyzed the conditions under which these strictly-2D results apply to experiments, which realize 2D through tight harmonic transverse confinement of frequency $\omega_z$ (oscillator length $a_z$ ): the kinematic requirement $\hbar\omega_z\gg\varepsilon_F$ , and the dimer requirement $|\varepsilon_b|\ll\hbar\omega_z$ , equivalently $a_{2D}\gg a_z$ , where $\varepsilon_b$ is the dimer binding energy. Subsequent experiments measured the EOS of quasi-2D gases across the crossover, pair condensation, and the anomalous breaking of classical scale invariance (the 2D contact-interacting gas is classically scale invariant; quantization introduces $a_{2D}$ as a scale and shifts, e.g., the breathing-mode frequency from its scale-invariant value $2\omega_\perp$ ), making the strictly-2D to quasi-2D bridge a quantitative theory-experiment question.

Problem Statement

The problem is to establish a quantitatively reliable, systematically controlled nonperturbative description of the uniform 2D Fermi gas across the BCS-BEC crossover at $T=0$ , and to connect it quantitatively to quasi-2D experiments. Its components jointly serve this single objective — (1) supplies the reference benchmarks, (2) fixes the few-body inputs and validity range of the BEC-limit effective theory, and (3) is the bridge to experiment; together they answer whether theory can predict the measured 2D crossover thermodynamics from first principles:

(1) Strong-coupling benchmarks with controlled bias. Determine the $T=0$ EOS, pairing gap $\Delta_{\rm gap}(\eta)$ , and contact $C(\eta)$ of the strictly-2D gas in the crossover window $\eta\sim0$ - $1$ with the fixed-node (or constrained-path) systematic error quantified and controlled — whether by improved nodal ansätze, by computational schemes that relax the fixed-node constraint in 2D many-fermion systems, or by demonstrated agreement of independent methods whose bias sources differ — thereby settling the contact determination near $\eta\sim1$ (where the two original FN-DMC determinations disagreed) and resolving the residual energy differences between fixed-node and auxiliary field methods at strong coupling.

(2) Effective dimer description. Fix the composite scattering parameters $a_d$ and $a_{ad}$ consistently with the exact few-body values, and establish the range of dimer density and $\eta$ over which the dilute-boson energy functional built on these parameters reproduces the many-body EOS beyond the dilute regime in which it was originally fitted. (The equal-mass few-body values themselves are now settled — $a_{ad}=1.26a_{2D}$ from the exact three-body calculation, correcting the original extraction — so the open content is the many-body validity range, including residual finite-size and nodal errors across the full $\eta$ range.)

(3) Strictly-2D to quasi-2D mapping. Provide the quantitative conditions and correction scheme — including the confinement-induced effective range and finite axial width, beyond the original criteria $|\varepsilon_b|\ll\hbar\omega_z$ or $a_{2D}\gg a_z$ — under which the strictly-2D benchmarks describe quasi-2D trapped gases, at a level that reconciles the theoretical EOS and scale-anomaly predictions with the measured EOS and anomaly observables across the crossover.

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