Probing the magnetic origin of the pseudogap using a Fermi-Hubbard quantum simulator

TL;DR

The study uses a spin- and density-resolved quantum gas microscope to probe the 2D Fermi-Hubbard system across wide doping and temperature ranges, revealing a doping-dependent energy scale Delta that governs the growth of spin correlations and matches the pseudogap onset temperature T*. By analyzing second-, third-, fourth-, and fifth-order correlations, the work demonstrates extended magnetic polarons and significant higher-order spin-charge correlations near dopants, signaling a strongly correlated regime associated with the pseudogap. Comparisons with dQMC, METTS, and geometric-string theory show qualitative and quantitative agreement in many observables, supporting a magnetic origin for the pseudogap while highlighting the limits of current theoretical approaches at finite doping. The results establish a link between spin stiffness-like scales, pseudogap physics, and dopant-induced polaron structures, paving the way for exploring pairing and collective phenomena at lower temperatures.

Abstract

In strongly correlated materials, interacting electrons are entangled and form collective quantum states, resulting in rich low-temperature phase diagrams. Notable examples include cuprate superconductors, in which superconductivity emerges at low doping out of an unusual ``pseudogap'' metallic state above the critical temperature. The Fermi-Hubbard model, describing a wide range of phenomena associated with strong electron correlations, still offers major computational challenges despite its simple formulation. In this context, ultracold atoms quantum simulators have provided invaluable insights into the microscopic nature of correlated quantum states. Here, we use a quantum gas microscope Fermi-Hubbard simulator to explore a wide range of doping levels and temperatures in a regime where a pseudogap is known to develop. By measuring multi-point correlation functions up to fifth order, we uncover a novel universal behaviour in magnetic and higher-order spin-charge correlations. This behaviour is characterized by a doping-dependent energy scale that governs the exponential growth of the magnetic correlation length upon cooling. Accurate comparisons with determinant Quantum Monte Carlo and Minimally Entangled Typical Thermal States simulations confirm that this energy scale agrees well with the pseudogap temperature $T^{*}$. Our findings establish a qualitative and quantitative understanding of the magnetic origin and physical nature of the pseudogap and pave the way towards the exploration of pairing and collective phenomena among charge carriers expected to emerge at lower temperatures.

Figures (9)

• Figure 1: Quantum simulation of the Fermi-Hubbard model.a. Examples of experimental snapshots using a quantum gas microscope with spin and charge resolution. b. Averaged atomic density, depicting the central region ($\Omega$) over which the analysis is carried out, and the surrounding reservoir with chemical potential $\mu_\mathrm{r}$ experimentally adjusted using a DMD (see SI). c. Conjectured phase diagram of the FHM. The hatched region approximately depicts the regime accessed by our experimental apparatus. AFM: region with sizeable antiferromagnetic correlations. d-SC: conjectured superconducting phase. d. Theoretical methods employed in this work: dQMC, METTS, and geometric strings (see text).
• Figure 2: Thermometry.a. The temperature of the experimental datasets is extracted by finding the best fit between dQMC numerical calculations (solid lines) and the measured spin correlations $C_{\mathrm{ss}}^{(2)}(|\bm{d}|)$ (data points), with the temperature being the only fitting parameter. Experimental errorbars in this figure and in all figures correspond to the standard deviation of a bootstrap analysis. b. Residuals between dQMC and experiment (solid line) and comparison to METTS (dashed line), taking the experimental data points as a reference, for half-filling (top) and at $|\delta| = 7.5%$ doping (bottom). Here, METTS is computed on a $32 \times 4$ cylinder, and dQMC on a square with periodic boundary conditions of size $16 \times 16$ (half-filling) or $10 \times 10$ (finite doping). Numerical errorbars (see SI) in this figure and in all figures are indicated by shaded regions.
• Figure 3: Magnetic correlations in the pseudogap.a. Spin correlation map at $T/t = 0.25(1)$ and $\delta \approx -2.5%$. b. Sign-corrected spin correlations as a function of distance. The inset shows the same data in log scale, and the solid line is an exponential decay extracted from the spin structure factor. c. Symmetrized spin structure factor $\mathcal{S}(\bm q)$ evaluated from a subregion of the spin correlation map (gray square in (a)). d. Spin structure factor evaluated over a path $\bm{0} \to (\pi, 0) \to \bm{q}_{\mathrm{AFM}} \to \bm{0}$. The solid line is an Ornstein-Zernike fit to the peak at $\bm{q} = \bm{q}_{\mathrm{AFM}}$, from which the correlations length $\xi$ is extracted. The grey area denotes the background used for the fit. e. Correlation length as a function of doping $\delta$ and temperature $T/t$. Each vertex is a data point, and each triangle is coloured according to the average of its vertices. The black line corresponds to the experimentally extracted doping-dependent energy scale $\Delta(\delta)$ (see panel (f)). The inset shows the comparison between the scale $\Delta$ and the pseudogap onset $T^*$ (stars) as determined from METTS (see text and panel (g)). f. Correlation length as a function of the rescaled temperature $T/\Delta$ (see text), with the orange line showing the exponential dependence defined by Eq. \ref{['eq:weak_coupling']}. Both insets show the same data, without rescaling (top) and in semi log-scale (bottom). g. Magnetic susceptibility extracted from $\mathcal{S}(\bm q \to \bm 0)$ as a function of temperature at half-filling, 5% doping and 12.5% doping. The solid and dashed lines are obtained from dQMC and METTS (only $T/t \geq 0.27$ dQMC data are shown for $|\delta| > 0$). The shaded region indicates the regime where $T \lesssim \Delta$, and the star corresponds to the maximum of susceptibility in the METTS data, used to define $T^*$.
• Figure 4: Emergence of extended polarons in the pseudogap.a. Example of polaron correlations $C_{\mathrm{dss}}(\bm r, \bm d)$ at low temperature ($T/t \approx 0.25$, first row) and slightly larger temperature ($T/t \approx 0.4$, second row) and for different doping levels. The maps are symmetrized, and the circle is a guide to the eye, indicating the distance over which correlations are sizeable. b. Distance-averaged, sign-corrected polaron correlations associated to the coldest datasets (first row in (a)), for the same doping levels. Different spin bonds ($|\bm d| = 1, \sqrt{2}, 2$ for NN, diagonal and NNN bonds, respectively) are represented by different symbols. Solid, dashed, and dotted lines correspond to numerical simulations (dQMC, METTS, and geometric strings, respectively). c. Polaronic correlations on selected bonds at short distances as a function of doping. Data points in dark (light) blue correspond to a temperature $T/t \approx 0.25$ ($T/t \approx 0.4$). d. Strength of the polaron $\mathcal{C}$ (see text) as a function of the rescaled temperature $T/\Delta$. The inset depicts the same data in semi-log scale.
• Figure 5: Higher-order correlations.a, b.$4^\mathrm{th}$-order spin correlations with 4 spins arranged in a T-shape (a) or in a diamond shape (b). The connected and bare correlations are represented by green circles and grey squares, respectively, while the colour scale indicates the doping and is plotted against the rescaled temperature $T/\Delta$. c, d.$5^\mathrm{th}$-order spin-charge correlations with 4 spins arranged in a T-shape close to a dopant (a) or in a diamond shape (b) around the dopant. The conditioned correlations (blue circles) are compared to the bare correlations (grey squares), and plotted against doping. Lighter colours indicate higher temperatures. In all panels, the dashed lines and the dots correspond to METTS and geometric string calculations, respectively.