Spin and Charge Conductivity in the Square Lattice Fermi-Hubbard Model

Dynamical properties are notoriously difficult to compute in numerical treatments of the Fermi-Hubbard model, especially in two spatial dimensions. However, they are essential in providing us with insight into some of the most important and less well-understood phases of the model, such as the pseudogap and strange metal phases at relatively high temperatures, or unconventional superconductivity at lower temperatures, away from the commensurate filling. Here, we use the numerical linked-cluster expansions to compute spin and charge optical conductivities of the model at different temperatures and strong interaction strengths via the exact real-time-dependent correlation functions of the current operators. We mitigate systematic errors associated with having a limited access to the long-time behavior of the correlators by introducing fits and allowing for non-zero Drude weights when appropriate. We compare our results to available data from optical lattice experiments and find that the Drude contributions can account for the theory-experiment gap in the DC spin conductivity of the model at half filling in the strong-coupling region. Our method helps paint a more complete picture of the conductivity in the two-dimensional Hubbard model and opens the door to studying dynamical properties of quantum lattice models in the thermodynamic limit.

💡 Research Summary

The authors address the long‑standing challenge of computing dynamical response functions for the two‑dimensional square‑lattice Fermi‑Hubbard model (FHM), a problem that has hampered progress in understanding the pseudogap, strange‑metal, and unconventional superconducting regimes. They employ the Numerical Linked‑Cluster Expansion (NLCE), a method that yields thermodynamic‑limit results by exactly solving all distinct finite clusters and summing their contributions via the inclusion‑exclusion principle. To obtain real‑time current–current correlation functions, they extend NLCE to a dynamical version (dNLCE) as introduced in earlier work.

In practice, the current operators for charge (j_c) and spin (j_s) contain four fermionic operators, which dramatically increases the computational cost. The authors generate all connected clusters up to eight sites for the 2D lattice (ten sites for 1D) and diagonalize each cluster exactly. They then compute the real‑time correlators ⟨j_{c/s}(τ)j_{c/s}(0)⟩ for a dense grid of temperatures, chemical potentials, and interaction strengths (U/t up to 8). To improve convergence at low temperature and longer times, they apply three cycles of the Wynn resummation algorithm.

Because dNLCE provides reliable data only up to a finite time (τ≈1–2 /t), the authors introduce a physically motivated fitting function: a constant term proportional to the Drude weight D, an exponentially damped component, and a sum of damped oscillatory modes whose frequencies are set by U. The fitting parameters (D, amplitudes, decay rates, frequencies, phases) are optimized not only to reproduce the real‑time data but also to minimize the discrepancy among three Kubo‑derived expressions for the real part of the conductivity (Eqs. 4‑6). This dual‑criterion fitting yields an extrapolation of the correlators beyond the accessible time window, allowing a Fourier transform to the frequency domain.

The conductivity is expressed as σ(ω)=2πDδ(ω)+σ_reg(ω). The Drude weight D is extracted from the long‑time limit of the correlator (or directly from the fit). The regular part σ_reg(ω) is obtained via three equivalent Kubo formulas that involve either the imaginary part, the real part, or a combination of both of the correlator. In the 1D benchmark, dNLCE results agree almost perfectly with time‑dependent DMRG data up to the convergence time, confirming the method’s accuracy. At infinite temperature, the dNLCE data match high‑temperature DMRG results, further validating the approach.

In two dimensions, the authors present detailed results for U/t=8 at half‑filling across temperatures T/t=0.5–3.9. The charge correlator decays rapidly, and the fitted Drude weight for charge transport becomes negligible at the studied interaction strength, indicating an insulating‑like response. By contrast, the spin correlator retains a finite long‑time plateau that grows as temperature is lowered, leading to a non‑zero Drude weight D_s. Fourier transforming the fitted correlators yields three estimates of σ_c(ω) and σ_s(ω). The charge conductivity shows a broad regular part with a small DC value, while the spin conductivity exhibits a pronounced low‑frequency peak. The estimates based on Eqs. 5 and 6 (which use both real and imaginary parts) give a larger DC peak than the estimate from Eq. 4 (which uses only the imaginary part), reflecting the systematic uncertainty due to limited time data.

The authors compare their charge DC conductivity with recent optical‑lattice measurements of ultracold fermions in a square lattice. The agreement is quantitative, reproducing the observed linear‑in‑temperature resistivity in the doped regime. More importantly, previous theoretical studies underestimated the spin DC conductivity at half‑filling; by incorporating a finite Drude weight, the present results close the gap between theory and experiment. This suggests that spin transport in the strong‑coupling, half‑filled Hubbard model is ballistic (or at least quasi‑ballistic) even when charge transport is diffusive or insulating.

In the concluding section, the authors emphasize that dNLCE, combined with careful fitting and sum‑rule constraints, provides a powerful route to real‑frequency dynamical quantities directly in the thermodynamic limit. The current limitation is the maximal cluster size, which restricts the accessible low‑temperature regime. Future work could push to larger clusters, exploit parallel diagonalization, or integrate tensor‑network solvers for individual clusters to extend the method to even lower temperatures and to study other dynamical probes such as optical conductivity in the superconducting phase, Hall response, or out‑of‑time‑order correlators. Overall, the paper delivers a significant methodological advance and delivers new physical insight into spin and charge transport in the 2D Hubbard model.