The 2018 static-auxiliary-field Monte Carlo study of the two-dimensional mass- and population-imbalanced attractive Hubbard model (Karmakar, Phys. Rev. A 97, 033617) mapped the thermal phase diagram — superfluid transition temperature TcT_c, pseudogap scales T∗T^*, breached-pair and Larkin-Ovchinnikov (LO) regimes — but deliberately neglected temporal (quantum) fluctuations of the pairing field, restricted the Hubbard-Stratonovich decomposition to the pairing channel, compared only single-modulation-wave-vector LO trial states, and worked near half filling where lattice effects are strong. The open problem is to determine, quantitatively, the finite-temperature phase diagram of this mixture with the neglected physics included: how temporal pairing-field and density/spin-channel fluctuations shift TcT_c, T∗T^*, and the phase boundaries; whether multi-wave-vector modulated (supersolid) states compete with or supplant the single-wave-vector LO state; and how the lattice results extrapolate to the dilute continuum relevant to free-space 6^6Li-40^{40}K mixtures. A 2024 sign-problem study confirms that unbiased quantum Monte Carlo in the population-imbalanced regime remains blocked, and no later work has quantified these corrections.

Background

When two fermion species with unequal masses and unequal populations attract each other, superfluidity competes with several rival orders. This is the setting of the 6^6Li-40^{40}K mixture realized with ultracold atoms, where the mass ratio mL/mH≈0.15m_L/m_H\approx0.15 (light/heavy) and a population difference between the species can be tuned independently. A standard model for this physics is the two-dimensional attractive Hubbard lattice model, in which the two species have unequal hopping amplitudes tσ∝1/mσt_\sigma\propto1/m_\sigma (encoding mass imbalance, with ratio η=mL/mH\eta=m_L/m_H), an onsite attraction UU, and a chemical-potential difference h=(μL−μH)/2h=(\mu_L-\mu_H)/2 (encoding population imbalance). As UU grows, the system crosses over from BCS pairing (weak attraction, overlapping Cooper pairs) to a BEC of tightly bound dimers — the BCS-BEC crossover — and at finite population imbalance the uniform superfluid competes with the breached-pair state (a gapless coexistence of paired and excess fermions), the partially polarized Fermi liquid, and the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) phases: superfluids whose pairing field oscillates spatially, with the Larkin-Ovchinnikov (LO) variant being a real amplitude modulation ∣Δi∣∝Δ0cos⁡(q⋅ri)|\Delta_i|\propto\Delta_0\cos(\mathbf{q}\cdot\mathbf{r}_i) at wave vector q\mathbf{q}. In two dimensions the pairing field can also lose phase coherence above the superfluid transition while pairs survive, producing a pseudogap (a suppression of the single-particle density of states without long-range order) between the pair-formation scale T∗T^* and TcT_c; for mass-imbalanced mixtures Karmakar (2018) found two such regimes, PG-I (both species pseudogapped) and PG-II (only the light species). Her tool was the static auxiliary field (SAF) Monte Carlo method — also called the static path approximation — in which the pairing field is decoupled from the interaction via a Hubbard-Stratonovich transformation (an exact rewrite of the interaction in terms of a fluctuating auxiliary field) and sampled over all spatial configurations at each imaginary time, but frozen in imaginary time: only the zero Matsubara frequency (the zeroth discrete Fourier mode of the field in imaginary time) of the pairing field is retained, so temporal (quantum) fluctuations are neglected. On a lattice this is defensible because the lattice gaps the translational and rotational Goldstone modes (the gapless modes associated with symmetry breaking) that would otherwise destroy two-dimensional order, leaving only the U(1) phase mode, which undergoes a Berezinskii-Kosterlitz-Thouless transition (the vortex-unbinding transition of a two-dimensional phase field). Nevertheless the method is uncontrolled in the temporal sector. The same study used a single-channel decomposition (pairing field only, no density- or spin-channel auxiliary fields), optimized the LO state over single-wave-vector trial modulations only, and worked at a chemical potential near half filling; reaching the dilute continuum regime of free-space gases would require fillings of order 0.001, which the source states is "corresponding to lattice size of ~10410^4" — outside the range of what could be attained at the time (Karmakar 2018, Sec. IV.1.2). The source explains why the required dilution is so extreme: near unitarity the scattering couples states at the upper band edge whose energies are set by the lattice, so even where the Fermi surface is circular the lattice results match the continuum only in the ~0.001-filling limit (Karmakar 2018, Sec. IV.1.2). Cluster dynamical mean-field theory (a method treating correlations exactly inside a finite cluster embedded in a self-consistent medium) on quasi-one-dimensional lattices (Heikkinen et al. 2014) had earlier shown that non-local quantum fluctuations can dramatically impact TcT_c in imbalanced systems while leaving room for FFLO order, so the size of the neglected corrections in the two-dimensional mass-imbalanced case is a genuine quantitative question, not a formality.

Problem Statement

For the two-dimensional attractive Hubbard model with unequal hoppings encoding a mass imbalance ratio η=mL/mH\eta=m_L/m_H (the 6^6Li-40^{40}K value being η=0.15\eta=0.15) and population imbalance h=(μL−μH)/2h=(\mu_L-\mu_H)/2, determine the finite-temperature phase diagram beyond the static (equal-time) auxiliary-field approximation used by Karmakar (2018), with three linked components that jointly constitute one objective, all of which are required to resolve the task — a fluctuation-complete, quantitatively reliable thermal phase diagram: (1) compute the thermal scales — superfluid transition temperature TcT_c and pseudogap temperatures T∗T^* (including the PG-I/PG-II distinction) — and the phase boundaries among the uniform superfluid, breached-pair, LO, and partially polarized Fermi-liquid phases, with temporal (quantum) fluctuations of the pairing field included, and quantify how much they shift relative to the static-auxiliary-field values; the treatment must also address whether instabilities in the density and spin channels (neglected by the pairing-only Hubbard-Stratonovich decomposition) alter these boundaries quantitatively; (2) settle the competition among modulated states by allowing multi-wave-vector LO configurations (supersolid-type pairing fields) alongside the single-wave-vector solution and establishing where, if anywhere, they are favored in the (η,h,U)(\eta,h,U) parameter space; and (3) provide a controlled extrapolation of these lattice results to the dilute continuum limit that connects to the universal behavior of free-space Fermi gases such as the 6^6Li-40^{40}K mixture, replacing the current situation in which the required fillings (~0.001) are numerically out of reach with no systematic alternative identified; for each of the three components the approximation error of the reported result must be quantified before that component counts as resolved. The source paper's parameter conventions (tLt_L energy units, fixed total chemical potential near half filling) should be followed or the relation to them stated.

References

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