Superconductivity near two-dimensional Van Hove singularities: a determinant quantum Monte Carlo study

The gist The superconducting transition temperature Tc of the two-dimensional attractive Hubbard model is computed in the vicinity of both ordinary (logarithmic) and higher-order (power-law) Van Hove singularities using determinant quantum Monte Carlo simulations.

Model and Methods

** The study utilizes the attractive Hubbard model, defined by the Hamiltonian H = X kσ ϵkc†kσckσ + U X i ni↑ni↓, where U = −U < 0. For the ordinary VHS case, t′′ = 0 and a second neighbor hopping t′ is needed to move the logarithmic VHS away from perfect nesting of the Fermi surface. The non-interacting bandwidth is defined as W = 8t in both cases. The model is analyzed for interaction strengths U ≤ 8t. **

Numerical Techniques

The researchers employed determinant quantum Monte Carlo (DQMC) simulations to study the model. The simulation was performed on square lattices with N = L2/2 sites, up to a linear system size L = 22 with periodic boundary conditions, down to temperatures T /t ≈ 0.03. Finite-size effects were reduced by threading a small magnetic flux through the system. The superconducting transition temperature Tc is extracted from the superfluid density ρs via the BKT criterion, where ρs(T − c) = 2πTc. The superfluid density is obtained from DQMC data according to rho s = 1/4 [⟨Kxx⟩ − Λxx(q → 0, iνm = 0)].

Perturbative Analysis

** Perturbative calculations were performed to interpret the DQMC results in the regime of weak to intermediate coupling, using the dimensionless coupling U ρ0 as a standard measure. The perturbative self-energy is given by Σ(k) = 1/2 Un − U2/T N X k′ χ0(k − k′)G0(k′). The pairing vertex Λ is expressed in terms of the irreducible vertex I as Λ(k) = 1 − 1/βN X k′ I(k, k′)G(k′)G(−k′)Λ(k'). The superconducting transition occurs when the maximal eigenvalue λmax of the equation-1/βN X k′ I(k, k′)G(k′)G(−k′)Λ(k') = λ(T)Λ(k) goes to unity, i.e., λmax(Tc) = 1. The perturbative treatment including both self-energy and vertex corrections is in satisfactory agreement with the DQMC calculations for U ≲ 1/4. However, both self-energy and vertex corrections tend to reduce Tc by smearing the DOS and decreasing the effective interaction strength. The VHS also exacerbates these effects. **

Results for Weak to Intermediate Coupling

** For U ≲ 2t, Tc is maximal near the VHS density for both the ordinary and higher-order VHS. The maximum becomes broader with increasing strength of the interaction. The asymmetry of the HOVHS divergence leads to a shift in the Tc maximum with varying U, but it continues to track the density feature. Comparison between Figs. 3 and 4 shows that the enhancement of Tc upon going from ordinary to higher-order VHS is relatively weak and diminishes as interaction strength increases. The DQMC Tc is compared with second-order perturbation theory, showing satisfactory agreement over much of the density range. **

Results for Strong Coupling

** Beyond a critical coupling strength of order U ∼ W/3, the effects of the VHS are rapidly washed out, and the position of the Tc maximum moves away from the VHS density with increasing U. For values of U ≫ W, this evolution is expected as the system crosses over into a regime of preformed local pairs and Bose-Einstein condensation (BEC) where Tc ∼ t2/U. What is surprising is how sudden the crossover is from “Fermi-surface pairing” to “local pairing”. For the model parameters used, the maximal Tc occurs at intermediate U and at a density far from the VHS, not tied to any features of the non-interacting band structure. **

Conclusion

** For interaction strengths up to U ≲ W/3, proximity to a VHS enhances Tc, albeit more weakly than expected from weak-coupling BCS theory. For U ≳ W/3, the influence of the VHS is rapidly suppressed. The maximal Tc occurs at intermediate coupling U ≈ −6t and at a density away from the VHS, consistent with strong-coupling theory. The results suggest that the enhancement of superconductivity by a VHS appears to be essentially a weak-coupling phenomenon, with interactions rapidly limiting its effectiveness. Concurrent observation of a sharp VHS in the DOS is required to establish this connection. **

Further Context

The paper notes that the enhancement of Tc near VHSs is likely a general phenomenon, with only modest enhancement found in Migdal-Eliashberg theory for electron-phonon systems. DQMC studies of superconductivity near an antiferromagnetic quantum critical point found that Tc is insensitive to the presence of a VHS. The findings suggest that strategies for achieving higher-Tc in strongly correlated systems should not rely solely on engineering DOS singularities, but must account for the interplay between interactions and band structure. The authors conclude that a maximum in Tc at a given density is not sufficient evidence for a VHS mechanism; concurrent observation of a sharp VHS in the DOS is required to establish this connection. The results are obtained within the attractive Hubbard model, which is at best a crude approximation to real materials. The limited effectiveness of VHSs in enhancing Tc at both strong and intermediate coupling is likely a general phenomenon. A key question is whether specific features of these systems enhance their sensitivity to VHSs, or whether they are effectively “marginal,” lying just on the weak-coupling side of the weak-to-strong coupling crossover. The enhancement of Tc in some systems may arise from mechanisms unrelated to VHSs. The paper also provides a strong-coupling expansion, where the mean-field Tc is given by Eq. (H15). A global maximum in Tc at n ≈ 1.35 and U ≈ 6t is found. The paper concludes that for sufficiently weak coupling, proximity to the VHS density yields the optimal Tc. With increasing interaction strength, however, the influence of the VHS is rapidly suppressed.

1. Bold header: Enhanced Superconductivity Prediction in Correlated Systems

The improved AI system can accurately predict superconducting transition temperatures (Tc) in attractive Hubbard models by systematically tracking their evolution from weak to strong coupling, specifically identifying that the maximal Tc as a function of U and density n occurs at the intermediate value U ≈ −6t and at a density far from the VHS.

2. Bold header: Van Hove Singularity Feature Extraction

The system can precisely locate Van Hove singularities in interacting systems by using proxies like the imaginary-time proxy [72]ρβ, allowing it to determine if the location of the VHS density for U ≠ 0 was identified using the imaginary-time observable ρβ.

3. Bold header: Crossover Dynamics Mapping

The AI can model and predict the nature of phase transitions by identifying that the crossover is abrupt, with a strongcoupling expansion already providing a qualitatively accurate description for U ≈ W, distinguishing between Fermi-surface pairing and local pairing based on interaction strength.

4. Bold header: Self-Energy and Vertex Correction Analysis

The system can quantify the impact of many-body effects by showing how the inclusion self-energy and vertex corrections, both of which tend to reduce Tc, thereby understanding why the enhancement becomes severely limited at already moderate interactions strengths.

The superconducting transition temperature T c of the two-dimensional attractive Hubbard model is computed in the vicinity of both ordinary (logarithmic) and higher-order (power-law) Van Hove singularities using determinant quantum Monte Carlo simulations. For interaction strengths U W/3, where W is the electronic bandwidth, T c is enhanced in the neighborhood of the Van Hove point, albeit more weakly than expected from weak-coupling BCS theory. Enhancing the Van Hove singularity from logarithmic to power-law yields only a minor additional enhancement of T c. For U W/3, the maximum T c shifts away from the Van Hove point and instead occurs at a density unrelated to any features in the non-interacting density of states, consistent with a strong-coupling interpretation. We find that the maximal T c in the model is achieved at intermediate U and at a density away from the Van Hove point.