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

TL;DR

The paper develops a dynamical NLCE framework to compute real-time current–current correlations and obtain spin and charge optical conductivities for the 2D square-lattice Fermi-Hubbard model in the thermodynamic limit. By fitting time-domain data and enforcing sum rules, it extracts Drude weights and regular conductivities, enabling DC limits to be compared with optical-lattice experiments and revealing agreement when Drude contributions are included. The results show a linear-in-$T$ charge resistivity in the strange-metal and Fermi-liquid regions, a nonzero spin Drude weight that helps close the theory–experiment gap at half filling in the strong-coupling regime, and overall a coherent dynamical picture across densities and interactions. The method extends to other dynamical properties and models, offering a route to study transport in quantum lattice systems in regimes challenging for other numerical techniques.

Abstract

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.

Figures (11)

• Figure 1: Comparison of dNLCE current correlators of the FHM at different temperatures with those from DMRG in 1D. (a) The real part of the charge current-current correlation functions vs real time for the 1D FHM at half filling with $U/t=4$. Black dashed lines are the DMRG results for a $100$-site chain from Ref. c_karrasch_14. Solid and dashed color lines represent dNLCE results at orders 10 and 9, respectively (no resummations used). Their deviations from each other marks the time at which the series loses convergence. (b) Same as (a) except for the real part of the spin current-current correlation functions vs real time.
• Figure 2: Comparison of dNLCE current correlators of the half-filled FHM for different $U$ values with those from DMRG in 1D. Line types are the same as Fig. \ref{['fig:dmrgcompare1']}, except that the DMRG results are at $T=\infty$ and dNLCE results are at a high temperature of $T/t=11$. The DMRG data are from Ref. c_karrasch_14.
• Figure 3: dNLCE results for the (a)-(b) real and (c)-(d) imaginary parts of the charge (left) and spin (right) current correlators of the half-filled 2D FHM vs real time at different temperatures for $U/t=8$. Solid lines show results after the Wynn resummation and dashed lines are fits based on Eq. \ref{['eq:expcos']}. The dNLCE data are shown up to $\tau \sim 1.5/t-2.0/t$, roughly within their time interval of convergence. The same time interval is used in the fits. The resummations can result in spurious features in some regions, which show up as small dents, mostly at low temperatures.
• Figure 4: (a) Charge and (b) spin optical conductivities of the 2D FHM at half filling for $U/t=8$ and $T/t=1.0$, obtained from Fourier transforms of the corresponding current operators according to Eqs. (\ref{['eq:ImOnly']}) - (\ref{['eq:ReOnly']}). Thick lines in (b) are the regular parts of the spin conductivity (Re $\sigma_{s{\textrm{(}reg)}}$) and thin dotted lines are with the estimate for the Drude weight added (see text).
• Figure 5: The imaginary time (a) charge and (b) spin current-current correlation functions of the FHM for $U/t=8$ at half filling at the midpoint of the inverse temperature range ($\beta/2$) as a function of temperature. The black dashed line is calculated directly in the NLCE and the markers are those obtained from the right hand side of Eq. (\ref{['eq:lambdas']}) with the corresponding optical conductivities from Eqs. (\ref{['eq:ImOnly']}) - (\ref{['eq:ReOnly']}) (see Fig. \ref{['fig:ACCond']}). The agreements serve as a check on the validity of the optical conductivities.