Measuring Entropy and Short-Range Correlations in the Two-Dimensional Hubbard Model

We measure entropy and short-range correlations of ultracold fermionic atoms in an optical lattice for a range of interaction strengths, temperatures, and fillings. In particular, we extract the mutual information between a single lattice site and the rest of the system from a comparison between the reduced density matrix of a single lattice site and the thermodynamic entropy. Moreover, we determine the single-particle density matrix between nearest neighbors from thermodynamic observables and show that even in a strongly interacting Mott insulator fermions are significantly delocalized over short distances in the lattice.

Popular Summary

Electrons in most everyday objects barely interact with one another. But in some forms of quantum matter, such as high-temperature superconductors, these interactions cannot be ignored. The microscopic physics of these materials, known as “strongly correlated matter,” is very difficult to understand theoretically. The leading theoretical model, known as the Hubbard model, can be solved for simple systems but is much harder to deal with for complex ensembles of particles. Here, we use ultracold atoms to create an experimental realization of the Hubbard model in two dimensions and, using a novel approach, determine microscopic correlations among the particles from precise thermodynamic measurements.

In our experiment, we study ensembles of potassium isotopes (${}^{40}\mathrm{K}$) in the two lowest hyperfine states, which are stored in an optical lattice. We investigate the transition from a metallic phase to a Mott insulator (an insulator that, according to traditional band theory, should be a conductor) by probing the density distribution inside the lattice and determining the pressure, entropy, and kinetic energy for a range of interactions, temperatures, and fillings. The kinetic energy is proportional to the first-order correlation function between neighboring lattice sites and therefore measures the degree of localization of the atoms. We also demonstrate the development of correlations between a lattice site and its environment upon a reduction of temperature.

Our technique provides access to the full thermodynamic entropy without the need for more complex measurements, and it can be used to evaluate new cooling schemes for strongly correlated quantum gases.

Figure 1

Pressure as a function of interaction strength and temperature. (a) Pressure versus chemical potential for different interaction strengths and temperatures. The uncertainties of the pressure and the chemical potential are smaller or equal to the marker size. (b) Pressure versus filling for the same interactions and the same temperatures as in (a). Horizontal error bars display the standard error obtained from averaging the density data over regions of constant chemical potential. The solid lines in (a) and (b) are the predictions from NLCE data [21] with the exception of the purple solid line, which represents the ideal Fermi gas on a lattice; the black, dashed line in (b) is the $T=0$ prediction of the free Fermi gas using the effective mass at the bottom of the lowest band. (c) Pressure at half filling $n=1$ versus interaction strength. The horizontal error bars display the systematic uncertainties of $U/t$, the vertical error bars indicate the uncertainty of determining half filling from the density profiles $n(\mu )$. The dash-dotted line is the infinite-$U$ and zero-temperature prediction $P=U/(2a^2)$.

Figure 2

Entropy per site versus chemical potential and temperature for different interaction strengths. The top row shows in color code the complete entropy data set, and the bottom row shows entropy data at selected temperatures together with the corresponding NLCE data (solid lines). The dotted horizontal line marks the value of log(2). For weak interactions (a) no Mott insulator forms and the entropy per site peaks at half filling (indicated by vertical dashed lines) at all temperatures explored. For intermediate (b) and strong (c) interactions at low temperature, one observes a local minimum of the entropy per site owing to the charge excitation gap of the Mott insulator. Horizontal errors are smaller than the marker size. The vertical error bars display the fit error of the derivative of the polynomial fit to the pressure data versus temperature. Systematic uncertainties of the entropy arising from the chosen polynomial fitting routine reach for the lowest temperatures presented values up to $0.1k_B$.

Figure 3

Mutual information between a single lattice site and its surrounding environment for different fillings, interactions, and temperatures. (a) Data used to extract the mutual information, for the exemplary case of $U/t=8.2(5)$ and $k_{B}T/t=1.09(5)$. The mutual information is given by the difference between the single-site entropy $s_0$ and the thermodynamic entropy per site $s$. Mutual information for $n=0.20(2)$ (b), $n=0.67(2)$ (c), and $n=1.00(2)$ (d). The lines are theory predictions extracted from NLCE data (solid lines) and for the noninteracting Fermi gas on a lattice (dash-dotted lines). The color code is the same for all plots. Error bars display the statistical measurement uncertainty.

Figure 4

Kinetic energy versus chemical potential and temperature. The top row shows in color code the complete kinetic energy data set, and the bottom row shows cuts at selected temperatures together with the corresponding NLCE theory (solid lines). While the atoms are maximally delocalized for weak interactions, for strong interactions the delocalization is only reduced, but not fully suppressed. Error bars display the statistical measurement uncertainty; see Appendix.

Figure 5

Average values (a) and uncertainties (b) of the individual terms contributing to the magnitude of the kinetic energy $|E_k|$. Note that in the calculation of the error of the kinetic energy $\mathrm{\Delta }{E}_k$ the term $n\mathrm{\Delta }\mu$ cancels with the pressure uncertainty ${\mathrm{\Delta }}_{\mu }P$ concerning the uncertainty of the chemical potential. The term ${\mathrm{\Delta }}_{n}P$ denotes the pressure uncertainty due to the uncertainty of the density $n$. The solid lines show the theoretical expectation of the individual terms obtained from NLCE data of the two-dimensional Hubbard model [21].