Identification of non-Fermi liquid fermionic self-energy from quantum Monte Carlo data

The study addresses non-Fermi liquid (NFL) behavior of itinerant electrons near metallic quantum critical points (QCPs), where soft bosonic order-parameter fluctuations induce singular fermionic self-energies Σ(ω) ∝ ω^α with α < 1, invalidating Landau Fermi-liquid theory at low energies. Analytical approaches (e.g., Eliashberg-type theories of effective fermion-boson models) predict specific scaling forms, such as ω^(2/3) for 2D nematic or Ising ferromagnetic (Q = 0) QCPs, but higher-loop corrections and superconducting instabilities complicate the accessible scaling regime. Large-scale sign-problem-free QMC has emerged as a powerful unbiased probe, yet practical simulations occur at finite temperature, where Matsubara-frequency data mix thermal (static) and quantum (dynamical) contributions, obscuring the T = 0 NFL signatures. The research question is whether one can systematically disentangle thermal and quantum parts of the fermionic self-energy from finite-T QMC to recover the zero-temperature NFL form and its scaling exponent. The paper proposes and validates a framework to separate Σ into Σ_T (thermal) and Σ_0 (quantum) and to extract Σ_0(ω) at T = 0 from QMC data for a 2D metal at an Ising ferromagnetic QCP, thereby testing the predicted ω^(2/3) scaling and the Eliashberg theory without adjustable parameters beyond a weak coupling constant.

Prior work established effective fermion-boson descriptions of metallic QCPs and predicted NFL self-energies; one-loop results for 2D Q = 0 instabilities yield Σ(ω) ∝ ω^(2/3), while higher-loop analyses indicate multiplicative logarithms that can modify very-low-frequency behavior. Numerical designer models have enabled sign-problem-free QMC studies of various QCPs (nematic, ferromagnetic, antiferromagnetic, gauge-field and Yukawa-SYK). Earlier finite-T QMC analyses at stronger couplings showed partial agreement (∼20% discrepancy) between analytics and numerics but were insufficient to reliably isolate Σ_0(ω). The present work advances this by operating in a weaker-coupling regime, where |Σ| ≪ ω_n and thermal and quantum parts can be cleanly separated.

- Model and parameters: A sign-problem-free lattice model of itinerant fermions coupled to transverse-field Ising ferromagnetic fluctuations on a square lattice with two fermion layers (λ = 1,2). Hamiltonian H = H_t + H_s + H_f, with nearest-neighbor hopping t, chemical potential μ tuning the Fermi surface, Ising exchange J and transverse field h controlling proximity to QCP, and onsite fermion–Ising-spin coupling ξ. Bare dispersion ε(k) = −2t[cos(k_x)+cos(k_y)] − μ, bandwidth W = 8t. Focused parameters: t = 1, μ = 0.5t, J = 1, ξ/t = 1, yielding an Ising ferromagnetic QCP at h_0/J ≈ 3.270(6). Representative Fermiology (θ = 0 and π/4) provided, with E_F ≈ 1.6. Temperatures studied T ≈ 0.05–0.1 (first Matsubara πT ≈ 0.16–0.31). - Analytical framework (Eliashberg/modified Eliashberg theory, MET): Start from coupled self-consistent equations for fermionic self-energy Σ and bosonic self-energy Π using full propagators G and D. In the weak-coupling, finite-T regime with |Σ(ω_n)| ≪ ω_n and g ≪ E_F, vertex corrections are neglected and fermionic dispersion is linearized near the Fermi surface. The bosonic propagator near the QCP includes mass M^2(T), q^2, c^2Ω^2, and Landau-damping contributions. - Thermal–quantum separation: At finite T, decompose Σ(ω_n) = Σ_T(ω_n) + Σ_0(ω_n). For a broad frequency window with ω_n ≫ |Σ|, the thermal part from static fluctuations has a simple form Σ_T(ω_n) ≈ a(T)/ω_n (with only weak/logarithmic ω_n dependence), implying ω_n Σ_T(ω_n) ≈ a(T) ≈ const over Matsubara points. The quantum part Σ_0(ω_n) recovers the T = 0 Eliashberg result: Σ_0(ω_n) ∝ ω_n [ω_n/ω_F]^{2/3} U(ω_n/ω_F), crossing over from ω^{2/3} scaling at low frequencies to a modified high-frequency form governed by a scaling function U(z) and by the presence of a regular q^2 term in D. - Relevant scales: In MET for the chosen parameters (bare g ≈ 0.25), characteristic scales are ω_b ≈ 0.71 and an extremely small ω_F ≈ 2.38×10^−5; most accessible ω_n lie above strict asymptotic ω^{2/3} regime but within the MET validity, leading to observable deviations from pure ω^{2/3} while retaining NFL character. - QMC simulations: Determinantal QMC (sign-problem-free) provides Σ(ω_n) at the QCP and in the disordered Fermi-liquid phase. The measured bosonic propagator fits D(q) = [M^2(T) + q^2 + c^2Ω^2 + η q^4]^{-1} with D_0 = 1, c = 3.16, and M^2(T) ≈ 0.13 T^{1.48}. Data satisfy |Σ(ω_n)| ≪ ω_n throughout. - Data analysis procedure to extract Σ_0: 1) Plot ω_n Σ(ω_n) versus ω_n and fit simultaneously across T to ω_n Σ(ω_n) = a(T) + ω_n Σ_0(ω_n; g, system parameters), treating a(T) and g as fitting parameters while keeping all other system parameters from QMC/Model. This effectively subtracts Σ_T ≈ a(T)/ω_n and compares remaining Σ_0 with the T = 0 MET prediction. 2) Refined fit allowing weak ω_n-dependence in the thermal background: use a full-frequency MET expression and fit ω_n Σ(ω_n) = α′(T) + [MET prediction for ω_n Σ_0(ω_n)] at fixed g ≈ 0.25 (or allowing small variation) to account for finite-size/gap effects; interpret α′(T) ≈ Δ^2(T) as an effective small gap contribution. 3) Independent numerical evaluation: perform Matsubara sums of MET expressions to compute Σ_T(ω_n) and Σ_0(ω_n) with g = 0.25, including finite-mass and first-Matsubara effects, and compare directly to QMC Σ(ω_n) at the QCP and in the FL regime. - Practical considerations: Exclude the first Matsubara frequency in scaling fits due to known deviations from quantum-critical scaling; verify robustness when including/excluding it. Evaluate confidence intervals for fitted parameters and quantify agreement across temperatures.

- Thermal–quantum separation works quantitatively: After subtracting a thermal background Σ_T(ω_n) ≈ a(T)/ω_n, the remaining quantum part Σ_0(ω_n) extracted from QMC at the Ising FM QCP collapses onto the T = 0 MET/Eliashberg prediction across all Matsubara points n > 0. - NFL scaling confirmed: At low frequencies, Σ_0(ω_n) exhibits the expected ω^{2/3} dependence (Eliashberg theory), while deviations at higher ω_n are well described by the MET scaling function U(z) and by the regular q^2 term in the boson propagator. - Parameter estimates and agreement: • Fitted coupling from simultaneous multi-T fit: g = 0.245 ± 0.023 (95% confidence), in excellent agreement with the bare g ≈ 0.25. • The thermal amplitude a(T) is nearly constant across studied T with a(T) ≈ 8×10^−3 for ω_n Σ(ω_n), indicating a dominant 1/ω_n thermal contribution within the accessible frequency window. • Characteristic scales inferred/used: ω_b ≈ 0.71, ω_F ≈ 2.38×10^−5, πT ≈ 0.16–0.31; E_F ≈ 1.6; |Σ| ≪ ω_n throughout. - Refined background and small gap: Allowing a refined fit yields α′(T) consistent with a small effective gap Δ(T) via α′(T) = Δ^2(T), indicating onset of a tiny gap around T ≈ 0.1, likely due to finite-size or slight detuning effects; its magnitude is well below reciprocal lattice spacing and below resolution of standard gap diagnostics. - Cross-validation with MET sums: Direct numerical summations of MET expressions (with g = 0.25) reproduce QMC Σ(ω_n) at the QCP and also in the disordered FL regime (h/J = 3.6), up to a T-dependent constant offset, confirming both the decomposition and the analytical framework. - First Matsubara caveat: As expected from theory, the first Matsubara frequency deviates from ω^{2/3} scaling; excluding it does not change the conclusions. Overall, the study demonstrates that finite-T QMC can yield the zero-T NFL self-energy via a simple subtraction of a 1/ω_n thermal term, quantitatively confirming ω^{2/3} scaling at a 2D Ising ferromagnetic QCP.

The findings resolve a central challenge in identifying NFL behavior from finite-temperature QMC: thermal fluctuations mask quantum-critical signatures in Σ(ω_n). By establishing and validating a simple and general decomposition Σ(ω_n) = a(T)/ω_n + Σ_0(ω_n), the work enables direct extraction of the zero-temperature quantum self-energy and rigorous comparison with Eliashberg/MET predictions. The quantitative agreement—including a fitted coupling matching the bare value and consistent scaling across temperatures—substantiates the theoretical description of dynamical quantum criticality in metals and confirms ω^{2/3} scaling at an Ising FM QCP within the accessible frequency window. The approach bridges numerical and analytical methods, offering a pathway to test subtle predictions (e.g., crossover functions, deviations from pure ω^{2/3} scaling, finite-frequency modifications) and to map energy-scale hierarchies in quantum critical metals. The method is broadly applicable to other itinerant QCPs (nematic, AFM, gauge-field-coupled metals), and can be extended to probe effects beyond one-loop Eliashberg theory (e.g., logarithmic higher-loop corrections) and to study the flow of the dynamical exponent z, thereby deepening understanding of NFL physics and its relation to competing superconductivity.

This work introduces and validates a practical framework to extract the zero-temperature non-Fermi-liquid fermionic self-energy from finite-temperature QMC data by subtracting a universal thermal 1/ω_n background. Applied to a 2D Ising ferromagnetic QCP, the method yields a quantum self-energy in excellent agreement with Eliashberg theory and exhibits ω^{2/3} scaling at low frequencies. The fitted coupling agrees with the bare model parameter, and full MET calculations quantitatively reproduce QMC data at the QCP and in the Fermi-liquid regime. The approach provides a general tool for QMC studies of quantum critical metals, enabling stringent tests of analytical theories, exploration of crossover regimes and higher-order corrections, and potential determination of dynamical critical properties. Future work includes applying the method to nematic and antiferromagnetic QCPs, investigating higher-loop/logarithmic corrections and their relevance relative to superconducting scales, assessing finite-size and first-Matsubara effects more systematically, and tracking the flow of the dynamical exponent z.

- Validity relies on a weak-coupling regime with g ≪ E_F and |Σ(ω_n)| ≪ ω_n; outside this regime vertex corrections and non-Eliashberg effects may become important. - Finite temperature and discrete Matsubara frequencies limit access to the deepest asymptotic low-frequency ω^{2/3} regime, especially since ω_F is extremely small; most data lie in crossover regions where deviations from pure scaling occur. - The subtraction assumes ω_n Σ_T(ω_n) ≈ a(T) with weak frequency dependence; residual ω_n-dependence and finite-size/detuning effects introduce a T-dependent offset (interpretable as a small gap), potentially complicating precise extraction near the first few Matsubara points. - The first Matsubara frequency does not follow quantum-critical scaling and must be treated with care or excluded from fits. - Analysis focuses on specific Fermi-surface points (e.g., along k_x) and a particular model; momentum anisotropies or other QCP classes may require additional considerations. - One-loop Eliashberg/MET forms neglect potential higher-loop logarithmic corrections that can modify Σ at extremely low frequencies; resolving these may require lower T and larger system sizes to push below superconducting or finite-size scales.