Extracting transport coefficients from local ground-state currents

TL;DR

This work addresses how to extract transport coefficients, notably the local Hall response, from static measurements in gapped quantum systems. It introduces a single-frequency ansatz for the time-dependent current-current correlator $ ext{C}( ext{r}, t)$ and shows that the local Hall conductivity $ olinebreak \sigma_H( ext{r})$ can be determined from static observables via $ olinebreak ext{C}( ext{r}, 0)$, $ olinebreak extomega_0$, and $ olinebreak Gamma$, yielding the local Chern marker ${ extsf{Ch}}( ext{r}) = 2\, extpi\, olinebreak \sigma_H( ext{r})$. It develops a Baker–Campbell–Hausdorff expansion to express the key coefficients $c_m$ as static current expectations, showing that a small set $(c_0,c_1,c_2)$ suffices in gapped, finite-velocity systems. The approach is validated numerically in the Hofstadter Hofstadter Chern insulator, where local Chern markers reconstructed from static currents agree with the expected topology, and the method is argued to extend to fractional Chern insulators and finite-temperature states. Overall, the paper provides a practical, static-measurement framework for probing transport in engineered quantum matter, enabling local probes of topology in cold-atom platforms and beyond.

Abstract

Transport properties are central to characterizing quantum matter, yet their extraction typically requires external forcing and time-resolved measurements. In this work, we propose a scheme to access transport coefficients directly from measurements of local, static ground-state currents -- quantities readily accessible in quantum-engineered platforms. By exploiting the exponential decay of correlations in gapped systems and the finite velocity of correlation spreading, we demonstrate that the local Hall response can be reconstructed from a small set of quasi-local current observables. We derive explicit relations connecting these static observables to a practical local Chern marker, and introduce a scalable digital protocol for measuring the required generalized currents in cold-atom quantum simulators. Numerical simulations of a non-interacting Chern insulator validate our approach. Moreover, the scheme extends naturally to fractional Chern insulators and other strongly correlated systems, even at finite temperature, offering a broadly applicable route to probing transport in engineered quantum matter.

Figures (6)

• Figure 1: (a) We propose to measure ground-state currents (red arrows) within a finite radius (gray shading) around a reference site (blue dot) to extract the local Hall response (Chern marker). (b) In a Chern insulator state, the current-current correlator $\mathcal{C}(\vec{r}, t)$ defined in Eq. \ref{['eq:CorrelatorC_of_t']}, relative to a reference site $\vec{r}$, exhibits damped oscillations, which are well captured by the ansatz in Eq. \ref{['eq:Approximate_C_of_t']}. (c) Upon fitting the damped oscillations, one can use the expression in Eq. \ref{['eq:HallFromDampedOscillations']} to obtain a local Chern marker. The result is in agreement with the Chern number ${\sf Ch}=1$ of the populated band. Here, the calculations were performed by filling the lowest band of the Hofstadter model with non-interacting fermions, at flux $\alpha=1/q$ per plaquette.
• Figure 2: (a-c) Currents $\hat{j}_{\vec{\mu}}(\vec{R}, \Theta)$ to be evaluated at (a) zeroth, (b) first, and (c) second order in $t$ for a Hofstadter model on a square lattice of $13\times 13$ sites with the reference site (blue) chosen in the center of the system. Darker arrows indicate that more than one generalized current with different phases $\Theta$ have to be measured. The necessary currents are found by first applying the iterative construction in Eq. \ref{['eq:mFoldCommutator']}, then extracting the imaginary part of the correlator using Eq. \ref{['Eq:ConnectionRule']}, and integrating over all lattice sites $\vec{R}$. (d) Local Chern marker ${\sf Ch}(\vec{r}) = 2\pi\sigma_{\rm H}(\vec{r})$ [Eq. \ref{['eq:HallFromDampedOscillations']}] as obtained from the coefficients $c_m$ in Eq. \ref{['eq:coefficients_cm']} using the experimental protocol. Here, the coefficients $c_m$ are connected to the parameters of Eq. \ref{['eq:Approximate_C_of_t']} via Eq. \ref{['eq:ConnectionCm_DampedOsci']}. The values for the Chern marker obtained in this manner are in good agreement among each other as well as with the expected ${\sf Ch}=1$ for the lowest Hofstadter band. For additional details see Appendix \ref{['app:HigherCms']}.
• Figure 3: Upper panel: Sketch of the pulse sequence needed to evaluate further-range currents. Lower left: Ground state density (dots) and $m=0$ currents (red arrows) of the Chern insulator in the lowest band of a Hofstadter model at $\alpha=1/5$. The blue arrow indicates a possible path to measure a current between next-nearest neighbors. Lower right: Density difference after the pulse sequence and coefficients $\gamma_i$ with which the respective points along the path are weighted.
• Figure 4: Characteristic frequency $\omega_0$ obtained from a fit of the correlator $\mathcal{C}(\vec{r}, t)$ using Eq. \ref{['eq:Approximate_C_of_t']} (blue dots), and compared to that extracted from the coefficients $c_m$ in Eq. \ref{['eq:ConnectionCm_DampedOsci']} (orange crosses). The cyclotron frequency $\Omega_c$ obtained from the single-particle spectrum is also indicated. The bandwidth $W$ of the lowest Hofstadter band is plotted in the inset. It is shown to agree with the decay rate of the correlations $\Gamma \approx W$. The cyclotron frequency $\Omega_c$ approaches the continuum value $\Omega_c^{\rm cont}=4\pi\alpha$ in the low-flux (continuum) limit.
• Figure 5: Density and local Chern marker as obtained from the fit in Eq. \ref{['eq:Approximate_C_of_t']} for a Hofstadter model with opposite flux $\pm\alpha=1/5$ per plaquette in the left and right half separated by a single column of a local potential. As expected, we find that the local Chern marker ${\sf Ch}(\vec{r})$ changes sign as the reference site $\vec{r}$ is moved from one region to the other.