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

Measuring Entropy and Short-Range Correlations in the Two-Dimensional Hubbard Model PHYSICAL REVIEW X 7, 031025 (2017) 1,2 1,2 1 1 1 1 1 E. Cocchi, L. A. Miller, J. H. Drewes, C. F. Chan, D. Pertot, F. Brennecke, and M. Köhl Physikalisches Institut, University of Bonn, Wegelerstrasse 8, 53115 Bonn, Germany Cavendish Laboratory, University of Cambridge, JJ Thomson Avenue, Cambridge CB3 0HE, United Kingdom (Received 20 February 2017; revised manuscript received 1 June 2017; published 4 August 2017) 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. DOI: 10.1103/PhysRevX.7.031025 Subject Areas: Atomic and Molecular Physics, Quantum Physics, Quantum Information Quantum mechanical correlations between particles give occupation of one particle per lattice site and an energy rise to collective behavior beyond intuitive imagination. gap for the creation of particle-hole excitations of order U [1]. In contrast, for weak interactions and/or low lattice Numerous classes of many-body states whose properties fillings, the fermions delocalize into Bloch waves and occur as result of quantum correlations are known to exist, such as Bose-Einstein condensates, Mott insulators, quan- constitute a metallic state with finite charge compressibility. tum magnets, and superconductors. A general feature of a Recently, antiferromagnetic correlations in the Hubbard model have been studied in several experiments [2–8]; correlated many-body system on a lattice is a strong however, access to different correlation functions sheds correlation between a single lattice site and its surrounding light on other properties of the Hubbard model, such as the environment. These correlations induce the sensitivity and charge degree of freedom. In this work, we measure both the vulnerability of a many-body state to external perturbations mutual information between a single lattice site and its since even a very localized perturbation can destroy the environment and the single-particle density matrix between nonlocal correlations. We explore the two-dimensional Hubbard model of spin- nearest neighbors hc ˆ c ˆ i. The mutual information iþ1;σ σ i;σ 1=2 fermionic atoms in an optical lattice. The Hubbard measures the amount of correlation between different model considers the two elementary processes of tunneling subsystems of an optical lattice and has been measured between neighboring lattice sites with amplitude t and on- with bosonic atoms [9]. In contrast, the single-particle site interaction between two fermions of opposite spin with density matrix is notoriously difficult to measure. In weakly strength U. In a single-band approximation the Hubbard interacting Bose gases, a measurement has been facilitated Hamiltonian reads by interference experiments after releasing the particles from a trap [10]. However, in strongly correlated ensembles, X X interaction effects would severely challenge the interpreta- ˆ ˆ ˆ ˆ H ¼ −t c c þ U n n : ð1Þ iσ jσ i↓ i↑ tion of similar experiments. Our novel approach to meas- hi;ji;σ uring the single-particle density matrix is different: even though the correlations are of microscopic origin, they are Here, c ˆ (c ˆ ) denotes the annihilation (creation) operator iσ iσ macroscopically manifest in the thermodynamic observ- of a fermion on lattice site i in spin state σ ¼f↑; ↓g, the ables of the system. Reversing this argument, the correla- bracket h;i denotes the sum over nearest neighbors, and † tions can be determined from precise thermodynamical n ˆ ¼ c ˆ c ˆ is the number operator. A Mott insulator forms iσ iσ iσ measurements. In the particular case of the single-particle at half filling and strong repulsion, i.e., for n ¼hn ˆ iþ i↑ density matrix, the corresponding thermodynamic quantity hn i¼ 1, and U ≫ t, k T. It is characterized by an i↓ B is the kinetic energy. The measurement of kinetic energy requires knowledge of both pressure and entropy, which we determine from the density profile recorded as a function of Published by the American Physical Society under the terms of chemical potential. Previous measurements of the pressure the Creative Commons Attribution 4.0 International license. and/or entropy in cold gases have focused on continuous Further distribution of this work must maintain attribution to (i.e., nonlattice) systems for noninteracting [11] and the author(s) and the published article’s title, journal citation, and DOI. strongly interacting [12–17] Fermi gases. In a spin-polarized 2160-3308=17=7(3)=031025(7) 031025-1 Published by the American Physical Society E. COCCHI et al. PHYS. REV. X 7, 031025 (2017) (a) (b) gas in an optical lattice, the entropy has been measured site resolved in the atomic limit, i.e., disregarding fluctuations 8 8 from tunneling [18]. Our measurements extend beyond this 6 6 by providing a spatially (and thus filling-)resolved detection of the entropy without the zero-tunneling approximation. 4 4 In our experiment, we prepare a spin-balanced quantum 2 2 degenerate mixture of the two lowest hyperfine states jF ¼ 9=2;m ¼ −9=2i and jF ¼ 9=2;m ¼ −7=2i of fer- F F 0 0 mionic Katoms [19,20]. We load the quantum gas into an -20 -16 -12 -8 -4 0 4 0 0.5 1 1.5 2 - U/2 (t) anisotropic, three-dimensional optical lattice in which tun- (c) neling is suppressed along the vertical direction. Hence, the 8 k T/t U/t dynamics is restricted to two-dimensional planes within -) 0.2(3) 1.35(4 which we choose a lattice depth of 5.2ð1ÞE ≤ V ≤ 6 rec xy 2 2 2 2.4(2) 0.78(3) 6.6ð1ÞE , where E ¼ ℏ π =ð2ma Þ denotes the recoil rec rec 8.2(5) 0.63(2) energy, a ¼ 532 nm is the lattice period, and m is the atomic 12.0(7) 0.70(4) mass. The Hubbard interaction parameter U is controlled by 2 utilizing a Feshbach resonance near 202 G, which provides us 19.5(1.3) 1.41(5) with access to the parameter range from weak to strong 048 12 interactions, 0 ≲ U=t ≲ 20. The temperature of the gas is U/t adjusted by heating due to a hold time in the optical lattice potential or periodic modulation of the trapping potential FIG. 1. Pressure as a function of interaction strength and temperature. (a) Pressure versus chemical potential for different followed by a thermalization time. Thereby, we prepare interaction strengths and temperatures. The uncertainties of the equilibrium systems with well-defined parameters t, U, pressure and the chemical potential are smaller or equal to the and k T. By combining radio-frequency spectroscopy and marker size. (b) Pressure versus filling for the same interactions absorption imaging, we simultaneously detect the in and the same temperatures as in (a). Horizontal error bars display situ density distributions of singly occupied lattice sites the standard error obtained from averaging the density data over (“singles”), n ¼hn ˆ − n ˆ n ˆ i, and doubly occupied lattice S i↑ i↑ i↓ regions of constant chemical potential. The solid lines in (a) and sites (“doubles”), n ¼hn ˆ n ˆ i, in a single two-dimensional D i↑ i↓ (b) are the predictions from NLCE data [21] with the exception of layer of the optical lattice. Our technique gives direct access to the purple solid line, which represents the ideal Fermi gas on a lattice; the black, dashed line in (b) is the T ¼ 0 prediction of the the density distribution nðμÞ as a function of the chemical free Fermi gas using the effective mass at the bottom of the lowest potential μ. We perform thermometry by fitting the measured band. (c) Pressure at half filling n ¼ 1 versus interaction strength. density profile nðμÞ with numerical linked cluster expansion The horizontal error bars display the systematic uncertainties of (NLCE) calculations of the two-dimensional Hubbard model U=t, the vertical error bars indicate the uncertainty of determining [21] and the ideal (U ¼ 0) Fermi gas on a square lattice. half filling from the density profiles nðμÞ. The dash-dotted line is To access the thermodynamics of the Hubbard model, we the infinite-U and zero-temperature prediction P ¼ U=ð2a Þ. first determine the pressure from the measured density profile nðμÞ [22] by employing the Gibbs-Duhem relation determined pressures are nearly independent of interaction SdT − AdP þ Ndμ ¼ 0, where S denotes the entropy, A strength and agree well with the theoretical prediction of the area, P the pressure, and N the total particle number. the free Fermi gas. We attribute this behavior to the Expressing all extensive quantities in units per lattice site, suppression of interaction effects at low filling and pressure and density are related to each other in thermal the nearly harmonic dispersion relation at the bottom of equilibrium and at constant temperature by the band. However, for n ≳ 0.5, we observe deviations from 1 the free Fermi gas behavior. For weak interactions, 0 0 Pðμ;TÞ¼ nðμ ;TÞdμ : ð2Þ U=t ≲ 3, the pressure is smaller than that of the free −∞ Fermi gas since for n ≳ 0.5 the particles experience the In order to limit the accumulation of technical noise in the nonharmonic dispersion, which affects the pressure versus density relation. For strong interactions, U=t ≳ 8, the numerical integration of the experimental data, we choose a pressure increases over that of the free Fermi gas and, in lower bound of the integration region μ corresponding to min particular, develops a near-vertical slope at half filling when an average lattice site occupation of nðμ Þ¼ 0.01. The min the lattice gas enters into a Mott insulator. This behavior is resulting systematic uncertainty of the pressure is compa- associated with the opening of the charge gap of the Mott rable to or below the statistical uncertainty of our data. In insulator, and one can understand the pressure at half filling Fig. 1, we show the measured pressure as a function of the in the limit of zero temperature and infinite interactions by chemical potential μ [Fig. 1(a)] and as a function of n [Fig. 1(b)]. We find that, for low filling, experimentally considering the internal energy E ¼hHi¼ 0, which leads 031025-2 -2 -2 Pressure P (ta ) Pressure P (ta ) -2 Pressure P (t a ) MEASURING ENTROPY AND SHORT-RANGE … PHYS. REV. X 7, 031025 (2017) to P ¼ U=ð2a Þ. We plot this relation in Fig. 1(c) and find washed out by the comparatively large kinetic energy at asymptotic agreement with our data. half filling. As a result, we observe the entropy per site to We next determine the thermodynamic entropy per site s peak at half filling for all temperatures [Fig. 2(a)]. This is in agreement with the fact that for weak interactions the from the measured pressure at constant chemical potential: largest number of microstates is available at half filling. For strong interactions, U=t ≳ 8, and low temperatures, a Mott dP s ¼ a : ð3Þ insulator forms at half filling, μ ¼ U=2, surrounded by dT μ¼const metallic phases at higher and lower chemical potential. We observe a nonmonotonic variation of entropy versus In order to evaluate the entropy reliably, we take data sets chemical potential with a local minimum at μ − U=2 ¼ 0 nðμÞ very finely spaced in temperature increments of signaling that, at constant temperature, entropy is smaller in k ΔT ∼ 0.2t. For each data set we determine the pressure B the gapped phase and higher in the thermally connected from the recorded density profile nðμÞ and then perform a gapless phase [Figs. 2(b) and 2(c)]. By comparison of numerical derivative with respect to temperature at a fixed Figs. 2(b) and 2(c), we also show that for stronger chemical potential. In order to suppress technical noise interactions, i.e., a larger gap, this effect extends to higher entering the numerical derivative, we interpolate the data in temperatures, as expected. We attribute the deviations the temperature interval ½k T − t; k T þ t by a second- between experimental and NLCE data for the lowest B B order polynomial and calculate the slope of the fitted temperatures to the second-order polynomial fitting rou- polynomial at temperature T. A second-order polynomial tine, which we confirm by analyzing NLCE data with the is chosen as a minimal model to account for a nonconstant same routine as the experimental data and comparing to the heat capacity and at the same time to minimize the number theoretically computed entropies. of fit parameters to yield a stable fit of the data. In Fig. 2 we We now turn our attention to the comparison between the show a color map of the measured entropy per site as a thermodynamic and the local entropy which quantifies the function of both temperature and chemical potential for amount of correlations between a single lattice site and its three different interaction strengths, U=t ¼ 2.4, 8.2, 12. For environment. If one partitions a system into two subsys- the weakest interaction shown, we do not observe a Mott tems A and B, the amount of correlations between the two insulator in the density profiles since the charge gap is subsystems can be quantified by the mutual information (a) (b) (c) U/t = 2.4(2) U/t = 8.2(5) U/t = 12.0(7) s (k ) 0 log(2) log(4) 1.6 k T/t = 0.68(2) k T/t = 0.70(4) B B k T/t = 1.14(2) k T/t = 1.09(5) k T/t = 1.18(3) B B k T/t = 2.37(3) 1.2 k T/t = 2.18(5) k T/t = 2.51(3) B B 0.8 0.4 -15 -10 -5 0 -15 -10 -5 0 -15 -10 -5 0 - U/2 (t) - U/2 (t) - U/2 (t) FIG. 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 . 031025-3 s (k ) k T (t) B E. COCCHI et al. PHYS. REV. X 7, 031025 (2017) I ¼ S þ S − S , where S ¼ −k Tr½ρ logðρ Þ [Fig. 3(b)], for which the effects of interactions are A B AB X B X X denotes the entropy of the reduced density matrix ρ of generally weak, the mutual information is mostly indepen- subsystem X ¼fA; Bg, and S denotes the entropy of the dent of the interaction strength. In contrast, at half filling AB full system. In the following, we consider the subsystem A [Fig. 3(d)] we observe a larger mutual information for weak to be a single lattice site, and subsystem B to be the interactions than for strong interactions. For high temper- thermodynamic bulk excluding the single site A. The atures, the mutual information approaches zero, indicating entropy s ≡ s of a single lattice site is directly determined the absence of any correlations. 0 A In order to gain further insight into the nature of the from the single-site reduced density matrix by s ¼ correlations between a single site and its environment, we −k p logðp Þ. Here, i ¼f↑↓; ↑; ↓; 0g labels the prob- B i i i ð1Þ abilities p for a site to be occupied with either two study the first-order correlation function G ð1Þ¼ particles, a spin-up particle, a spin-down particle, or no hc ˆ c ˆ i between neighboring lattice sites i and σ i;σ iþ1;σ particles, respectively. These probabilities are directly i þ 1. The first-order correlation function, normalized determined from the measured singles and doubles ð1Þ to the density G ð0Þ¼ n, measures the degree of density distributions as p ¼ n , p ¼ p ¼ n , and ↑↓ D ↑ ↓ S delocalization of the fermions and has not been measured p ¼ 12n − n [23]. The entropy of the entire 0 S D in optical-lattice experiments before. In the tight-binding system with L ≫ 1 sites is S ¼ Ls, where s is the AB approximation, which is well fulfilled in our optical lattice, measured thermodynamic entropy per site and, likewise, ð1Þ G ð1Þ is available through a measurement of the kinetic S ¼ðL − 1Þs. Hence, we obtain the mutual information as ð1Þ energy of site i, E ≡ −t hc ˆ c ˆ iδ ¼ −4tG ð1Þ. k;i jσ j;σ iσ ji−jj;1 I ¼ s − s, which we plot in Fig. 3 for various fillings, We deduce the kinetic energy from spatially resolved temperatures, and interaction strengths. For low temper- measurements of the total energy of the lattice gas, combin- atures, we find a mutual information greater than zero, ing the measurements of entropy, pressure, and density, and which indicates correlations between the single lattice the interaction energy given by Un : site and its environment. We observe that for low filling E ¼ðsT þ μn − Pa Þ − Un : ð4Þ k D (a) (b) In Fig. 4, we plot the magnitude of the kinetic energy for U/t = 2.4(2) Mutual 0.4 U/t = 8.2(5) the different interactions investigated. For weak interactions Information U/t = 12.0(7) the magnitude of the kinetic energy peaks at half filling, 0.2 similar to the mutual information (Fig. 3). Hence, we 0.5 identify delocalization as the mechanism to provide the correlation between a single lattice site and the environment for the lowest interaction. Conversely, for stronger inter- actions the shape of the kinetic-energy curves changes -15 -10 -5 0 123 - U/2 (t) k T (t) qualitatively as the peak shifts to a lower filling because (c) (d) B of the emerging Mott gap. At half filling the magnitude of the 0.4 0.4 kinetic energy decreases with increasing interaction strength, signaling the crossover towards a localized, Mott-insulating state. 0.2 0.2 The measured nearest-neighbor correlations provide a first ð1Þ glimpse at the single-particle density matrix G ðji−jjÞ¼ 0 0 hc ˆ c ˆ i in the lattice. With the measurements shown jσ σ iσ here, we can determine the single-particle density matrix 123 123 for the values ji − jj¼ 0, 1. Concentrating on the case of k T (t) k T (t) B B half filling (even though the discussion can be generalized ð1Þ to any filling), we find G ð0Þ¼ 1, and in the case of the FIG. 3. Mutual information between a single lattice site and its ð1Þ surrounding environment for different fillings, interactions, and metallic phase (U=t ¼ 2.4), G ð1Þ¼ 0.29ð2Þ. This drop temperatures. (a) Data used to extract the mutual information, for is in line with the expected thermal de Broglie wavelength, the exemplary case of U=t ¼ 8.2ð5Þ and k T=t ¼ 1.09ð5Þ. The which at a temperature of k T ¼ t is approximately mutual information is given by the difference between the single- λ ≃ a. For higher temperature, the spatial correlations dB site entropy s and the thermodynamic entropy per site s. Mutual drop correspondingly faster. In contrast, for a Mott insu- information for n ¼ 0.20ð2Þ (b), n ¼ 0.67ð2Þ (c), and n ¼ ð1Þ ð1Þ lator (U=t ¼ 12), the drop is from G ð0Þ¼ 1 to G ð1Þ¼ 1.00ð2Þ (d). The lines are theory predictions extracted from 0.11ð5Þ even though the temperature is even lower. This NLCE data (solid lines) and for the noninteracting Fermi gas on a indicates localization of the fermions with increasing lattice (dash-dotted lines). The color code is the same for all plots. Error bars display the statistical measurement uncertainty. interaction strength. We note, however, that even for the 031025-4 Mutual Information (k ) Entropy (k ) B B Mutual Information (k ) Mutual Information (k ) B B MEASURING ENTROPY AND SHORT-RANGE … PHYS. REV. X 7, 031025 (2017) (a) (b) (c) U/t = 2.4(2) U/t = 8.2(5) U/t = 12.0(7) |E | (t) -0.5 0 0.5 1 1.5 k T/t = 1.14(2) k T/t = 0.68(2) k T/t = 0.70(4) B B B k T/t = 2.37(3) k T/t = 2.18(5) k T/t = 2.51(3) B B B - U/2 (t) - U/2 (t) - U/2 (t) FIG. 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. strongest interaction explored by our experiment, the atoms of constant chemical potential with a bin size of are not fully localized [24] even though the compressibility Δμ ¼ h × 100 Hz. The uncertainty of the total density n ¼ of the state is almost zero [20]. 2ðn þ n Þ is then obtained by adding the individual S D In conclusion, we measure short-range correlations in the uncertainties in quadrature. The systematic uncertainties two-dimensional Hubbard model and show that at low of the Hubbard parameters U and t are derived from the temperatures a single lattice site develops correlations with calibration uncertainties of the lattice depth along the three the surrounding environment. In particular, we find that directions and the uncertainty in the parametrization of the even for strongly interacting Mott insulators with a vanish- Feshbach resonance [20]. Statistical uncertainties of the ing compressibility the fermions are still significantly temperature T and the chemical potential μ ¼ U=2 at HF delocalized over neighboring lattice sites. We note that half filling are obtained from simultaneously fitting the the technique we present here determines the full thermo- recorded singles and doubles density profiles n ðμÞ to S;D dynamic entropy, including the entropy in the spin sector numerical simulations of the two-dimensional Hubbard without the need for spin-resolved measurements. Hence, it model [21] or the ideal (U ¼ 0) Fermi gas on a square could find use in future attempts to cool strongly correlated lattice. The systematic error of μ is negligible as the HF quantum gases by reshuffling the entropy [8,25,26]. maximum of the singles density n ðμÞ provides a model- independent calibration of the half-filling point. The We thank A. Daley and C. Kollath for discussions. The extracted temperatures are confirmed within the statistical work has been supported by DFG (SFB/TR 185), the uncertainties by fitting the low-filling regions of the density Alexander von Humboldt Stiftung, EPSRC, and ERC profile to the ideal (U ¼ 0) Fermi gas on a square lattice. (Grant No. 616082). The statistical uncertainty of the pressure PðμÞ is obtained by adding the individual statistical uncertainties APPENDIX: ERROR ANALYSIS of the total density nðμÞ in quadrature. The systematic Here, we specify the uncertainties of the measured uncertainty of the chemical potential axis is converted into quantities and Hubbard parameters and provide details an additional systematic uncertainty of the pressure only in about the error analysis of the derived thermodynamic Fig. 1(c), where the pressure at half filling is shown. quantities. The statistical uncertainty of the entropy sðμÞ is obtained The statistical uncertainties of the measured singles and from the fit errors of the second-order polynomial fit to the doubles density n are given by the standard error pressure data PðTÞ for a given μ. Systematic uncertainties of S;D the entropy originating from the chosen temperature interval resulting from averaging the recorded images over regions 031025-5 |E | (t) k T (t) B E. COCCHI et al. PHYS. REV. X 7, 031025 (2017) (b) (a) FIG. 5. Average values (a) and uncertainties (b) of the individual terms contributing to the magnitude of the kinetic energy jE j. Note that in the calculation of the error of the kinetic energy ΔE the term nΔμ cancels with the pressure uncertainty Δ P concerning the k μ uncertainty of the chemical potential. The term Δ P denotes the pressure uncertainty due to the uncertainty of the density n. 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10.1103/PhysRevX.7.031025
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PHYSICAL REVIEW X 7, 031025 (2017) 1,2 1,2 1 1 1 1 1 E. Cocchi, L. A. Miller, J. H. Drewes, C. F. Chan, D. Pertot, F. Brennecke, and M. Köhl Physikalisches Institut, University of Bonn, Wegelerstrasse 8, 53115 Bonn, Germany Cavendish Laboratory, University of Cambridge, JJ Thomson Avenue, Cambridge CB3 0HE, United Kingdom (Received 20 February 2017; revised manuscript received 1 June 2017; published 4 August 2017) 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. DOI: 10.1103/PhysRevX.7.031025 Subject Areas: Atomic and Molecular Physics, Quantum Physics, Quantum Information Quantum mechanical correlations between particles give occupation of one particle per lattice site and an energy rise to collective behavior beyond intuitive imagination. gap for the creation of particle-hole excitations of order U [1]. In contrast, for weak interactions and/or low lattice Numerous classes of many-body states whose properties fillings, the fermions delocalize into Bloch waves and occur as result of quantum correlations are known to exist, such as Bose-Einstein condensates, Mott insulators, quan- constitute a metallic state with finite charge compressibility. tum magnets, and superconductors. A general feature of a Recently, antiferromagnetic correlations in the Hubbard model have been studied in several experiments [2–8]; correlated many-body system on a lattice is a strong however, access to different correlation functions sheds correlation between a single lattice site and its surrounding light on other properties of the Hubbard model, such as the environment. These correlations induce the sensitivity and charge degree of freedom. In this work, we measure both the vulnerability of a many-body state to external perturbations mutual information between a single lattice site and its since even a very localized perturbation can destroy the environment and the single-particle density matrix between nonlocal correlations. We explore the two-dimensional Hubbard model of spin- nearest neighbors hc ˆ c ˆ i. The mutual information iþ1;σ σ i;σ 1=2 fermionic atoms in an optical lattice. The Hubbard measures the amount of correlation between different model considers the two elementary processes of tunneling subsystems of an optical lattice and has been measured between neighboring lattice sites with amplitude t and on- with bosonic atoms [9]. In contrast, the single-particle site interaction between two fermions of opposite spin with density matrix is notoriously difficult to measure. In weakly strength U. In a single-band approximation the Hubbard interacting Bose gases, a measurement has been facilitated Hamiltonian reads by interference experiments after releasing the particles from a trap [10]. However, in strongly correlated ensembles, X X interaction effects would severely challenge the interpreta- ˆ ˆ ˆ ˆ H ¼ −t c c þ U n n : ð1Þ iσ jσ i↓ i↑ tion of similar experiments. Our novel approach to meas- hi;ji;σ uring the single-particle density matrix is different: even though the correlations are of microscopic origin, they are Here, c ˆ (c ˆ ) denotes the annihilation (creation) operator iσ iσ macroscopically manifest in the thermodynamic observ- of a fermion on lattice site i in spin state σ ¼f↑; ↓g, the ables of the system. Reversing this argument, the correla- bracket h;i denotes the sum over nearest neighbors, and † tions can be determined from precise thermodynamical n ˆ ¼ c ˆ c ˆ is the number operator. A Mott insulator forms iσ iσ iσ measurements. In the particular case of the single-particle at half filling and strong repulsion, i.e., for n ¼hn ˆ iþ i↑ density matrix, the corresponding thermodynamic quantity hn i¼ 1, and U ≫ t, k T. It is characterized by an i↓ B is the kinetic energy. The measurement of kinetic energy requires knowledge of both pressure and entropy, which we determine from the density profile recorded as a function of Published by the American Physical Society under the terms of chemical potential. Previous measurements of the pressure the Creative Commons Attribution 4.0 International license. and/or entropy in cold gases have focused on continuous Further distribution of this work must maintain attribution to (i.e., nonlattice) systems for noninteracting [11] and the author(s) and the published article’s title, journal citation, and DOI. strongly interacting [12–17] Fermi gases. In a spin-polarized 2160-3308=17=7(3)=031025(7) 031025-1 Published by the American Physical Society E. COCCHI et al. PHYS. REV. X 7, 031025 (2017) (a) (b) gas in an optical lattice, the entropy has been measured site resolved in the atomic limit, i.e., disregarding fluctuations 8 8 from tunneling [18]. Our measurements extend beyond this 6 6 by providing a spatially (and thus filling-)resolved detection of the entropy without the zero-tunneling approximation. 4 4 In our experiment, we prepare a spin-balanced quantum 2 2 degenerate mixture of the two lowest hyperfine states jF ¼ 9=2;m ¼ −9=2i and jF ¼ 9=2;m ¼ −7=2i of fer- F F 0 0 mionic Katoms [19,20]. We load the quantum gas into an -20 -16 -12 -8 -4 0 4 0 0.5 1 1.5 2 - U/2 (t) anisotropic, three-dimensional optical lattice in which tun- (c) neling is suppressed along the vertical direction. Hence, the 8 k T/t U/t dynamics is restricted to two-dimensional planes within -) 0.2(3) 1.35(4 which we choose a lattice depth of 5.2ð1ÞE ≤ V ≤ 6 rec xy 2 2 2 2.4(2) 0.78(3) 6.6ð1ÞE , where E ¼ ℏ π =ð2ma Þ denotes the recoil rec rec 8.2(5) 0.63(2) energy, a ¼ 532 nm is the lattice period, and m is the atomic 12.0(7) 0.70(4) mass. The Hubbard interaction parameter U is controlled by 2 utilizing a Feshbach resonance near 202 G, which provides us 19.5(1.3) 1.41(5) with access to the parameter range from weak to strong 048 12 interactions, 0 ≲ U=t ≲ 20. The temperature of the gas is U/t adjusted by heating due to a hold time in the optical lattice potential or periodic modulation of the trapping potential FIG. 1. Pressure as a function of interaction strength and temperature. (a) Pressure versus chemical potential for different followed by a thermalization time. Thereby, we prepare interaction strengths and temperatures. The uncertainties of the equilibrium systems with well-defined parameters t, U, pressure and the chemical potential are smaller or equal to the and k T. By combining radio-frequency spectroscopy and marker size. (b) Pressure versus filling for the same interactions absorption imaging, we simultaneously detect the in and the same temperatures as in (a). Horizontal error bars display situ density distributions of singly occupied lattice sites the standard error obtained from averaging the density data over (“singles”), n ¼hn ˆ − n ˆ n ˆ i, and doubly occupied lattice S i↑ i↑ i↓ regions of constant chemical potential. The solid lines in (a) and sites (“doubles”), n ¼hn ˆ n ˆ i, in a single two-dimensional D i↑ i↓ (b) are the predictions from NLCE data [21] with the exception of layer of the optical lattice. Our technique gives direct access to the purple solid line, which represents the ideal Fermi gas on a lattice; the black, dashed line in (b) is the T ¼ 0 prediction of the the density distribution nðμÞ as a function of the chemical free Fermi gas using the effective mass at the bottom of the lowest potential μ. We perform thermometry by fitting the measured band. (c) Pressure at half filling n ¼ 1 versus interaction strength. density profile nðμÞ with numerical linked cluster expansion The horizontal error bars display the systematic uncertainties of (NLCE) calculations of the two-dimensional Hubbard model U=t, the vertical error bars indicate the uncertainty of determining [21] and the ideal (U ¼ 0) Fermi gas on a square lattice. half filling from the density profiles nðμÞ. The dash-dotted line is To access the thermodynamics of the Hubbard model, we the infinite-U and zero-temperature prediction P ¼ U=ð2a Þ. first determine the pressure from the measured density profile nðμÞ [22] by employing the Gibbs-Duhem relation determined pressures are nearly independent of interaction SdT − AdP þ Ndμ ¼ 0, where S denotes the entropy, A strength and agree well with the theoretical prediction of the area, P the pressure, and N the total particle number. the free Fermi gas. We attribute this behavior to the Expressing all extensive quantities in units per lattice site, suppression of interaction effects at low filling and pressure and density are related to each other in thermal the nearly harmonic dispersion relation at the bottom of equilibrium and at constant temperature by the band. However, for n ≳ 0.5, we observe deviations from 1 the free Fermi gas behavior. For weak interactions, 0 0 Pðμ;TÞ¼ nðμ ;TÞdμ : ð2Þ U=t ≲ 3, the pressure is smaller than that of the free −∞ Fermi gas since for n ≳ 0.5 the particles experience the In order to limit the accumulation of technical noise in the nonharmonic dispersion, which affects the pressure versus density relation. For strong interactions, U=t ≳ 8, the numerical integration of the experimental data, we choose a pressure increases over that of the free Fermi gas and, in lower bound of the integration region μ corresponding to min particular, develops a near-vertical slope at half filling when an average lattice site occupation of nðμ Þ¼ 0.01. The min the lattice gas enters into a Mott insulator. This behavior is resulting systematic uncertainty of the pressure is compa- associated with the opening of the charge gap of the Mott rable to or below the statistical uncertainty of our data. In insulator, and one can understand the pressure at half filling Fig. 1, we show the measured pressure as a function of the in the limit of zero temperature and infinite interactions by chemical potential μ [Fig. 1(a)] and as a function of n [Fig. 1(b)]. We find that, for low filling, experimentally considering the internal energy E ¼hHi¼ 0, which leads 031025-2 -2 -2 Pressure P (ta ) Pressure P (ta ) -2 Pressure P (t a ) MEASURING ENTROPY AND SHORT-RANGE … PHYS. REV. X 7, 031025 (2017) to P ¼ U=ð2a Þ. We plot this relation in Fig. 1(c) and find washed out by the comparatively large kinetic energy at asymptotic agreement with our data. half filling. As a result, we observe the entropy per site to We next determine the thermodynamic entropy per site s peak at half filling for all temperatures [Fig. 2(a)]. This is in agreement with the fact that for weak interactions the from the measured pressure at constant chemical potential: largest number of microstates is available at half filling. For strong interactions, U=t ≳ 8, and low temperatures, a Mott dP s ¼ a : ð3Þ insulator forms at half filling, μ ¼ U=2, surrounded by dT μ¼const metallic phases at higher and lower chemical potential. We observe a nonmonotonic variation of entropy versus In order to evaluate the entropy reliably, we take data sets chemical potential with a local minimum at μ − U=2 ¼ 0 nðμÞ very finely spaced in temperature increments of signaling that, at constant temperature, entropy is smaller in k ΔT ∼ 0.2t. For each data set we determine the pressure B the gapped phase and higher in the thermally connected from the recorded density profile nðμÞ and then perform a gapless phase [Figs. 2(b) and 2(c)]. By comparison of numerical derivative with respect to temperature at a fixed Figs. 2(b) and 2(c), we also show that for stronger chemical potential. In order to suppress technical noise interactions, i.e., a larger gap, this effect extends to higher entering the numerical derivative, we interpolate the data in temperatures, as expected. We attribute the deviations the temperature interval ½k T − t; k T þ t by a second- between experimental and NLCE data for the lowest B B order polynomial and calculate the slope of the fitted temperatures to the second-order polynomial fitting rou- polynomial at temperature T. A second-order polynomial tine, which we confirm by analyzing NLCE data with the is chosen as a minimal model to account for a nonconstant same routine as the experimental data and comparing to the heat capacity and at the same time to minimize the number theoretically computed entropies. of fit parameters to yield a stable fit of the data. In Fig. 2 we We now turn our attention to the comparison between the show a color map of the measured entropy per site as a thermodynamic and the local entropy which quantifies the function of both temperature and chemical potential for amount of correlations between a single lattice site and its three different interaction strengths, U=t ¼ 2.4, 8.2, 12. For environment. If one partitions a system into two subsys- the weakest interaction shown, we do not observe a Mott tems A and B, the amount of correlations between the two insulator in the density profiles since the charge gap is subsystems can be quantified by the mutual information (a) (b) (c) U/t = 2.4(2) U/t = 8.2(5) U/t = 12.0(7) s (k ) 0 log(2) log(4) 1.6 k T/t = 0.68(2) k T/t = 0.70(4) B B k T/t = 1.14(2) k T/t = 1.09(5) k T/t = 1.18(3) B B k T/t = 2.37(3) 1.2 k T/t = 2.18(5) k T/t = 2.51(3) B B 0.8 0.4 -15 -10 -5 0 -15 -10 -5 0 -15 -10 -5 0 - U/2 (t) - U/2 (t) - U/2 (t) FIG. 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 . 031025-3 s (k ) k T (t) B E. COCCHI et al. PHYS. REV. X 7, 031025 (2017) I ¼ S þ S − S , where S ¼ −k Tr½ρ logðρ Þ [Fig. 3(b)], for which the effects of interactions are A B AB X B X X denotes the entropy of the reduced density matrix ρ of generally weak, the mutual information is mostly indepen- subsystem X ¼fA; Bg, and S denotes the entropy of the dent of the interaction strength. In contrast, at half filling AB full system. In the following, we consider the subsystem A [Fig. 3(d)] we observe a larger mutual information for weak to be a single lattice site, and subsystem B to be the interactions than for strong interactions. For high temper- thermodynamic bulk excluding the single site A. The atures, the mutual information approaches zero, indicating entropy s ≡ s of a single lattice site is directly determined the absence of any correlations. 0 A In order to gain further insight into the nature of the from the single-site reduced density matrix by s ¼ correlations between a single site and its environment, we −k p logðp Þ. Here, i ¼f↑↓; ↑; ↓; 0g labels the prob- B i i i ð1Þ abilities p for a site to be occupied with either two study the first-order correlation function G ð1Þ¼ particles, a spin-up particle, a spin-down particle, or no hc ˆ c ˆ i between neighboring lattice sites i and σ i;σ iþ1;σ particles, respectively. These probabilities are directly i þ 1. The first-order correlation function, normalized determined from the measured singles and doubles ð1Þ to the density G ð0Þ¼ n, measures the degree of density distributions as p ¼ n , p ¼ p ¼ n , and ↑↓ D ↑ ↓ S delocalization of the fermions and has not been measured p ¼ 12n − n [23]. The entropy of the entire 0 S D in optical-lattice experiments before. In the tight-binding system with L ≫ 1 sites is S ¼ Ls, where s is the AB approximation, which is well fulfilled in our optical lattice, measured thermodynamic entropy per site and, likewise, ð1Þ G ð1Þ is available through a measurement of the kinetic S ¼ðL − 1Þs. Hence, we obtain the mutual information as ð1Þ energy of site i, E ≡ −t hc ˆ c ˆ iδ ¼ −4tG ð1Þ. k;i jσ j;σ iσ ji−jj;1 I ¼ s − s, which we plot in Fig. 3 for various fillings, We deduce the kinetic energy from spatially resolved temperatures, and interaction strengths. For low temper- measurements of the total energy of the lattice gas, combin- atures, we find a mutual information greater than zero, ing the measurements of entropy, pressure, and density, and which indicates correlations between the single lattice the interaction energy given by Un : site and its environment. We observe that for low filling E ¼ðsT þ μn − Pa Þ − Un : ð4Þ k D (a) (b) In Fig. 4, we plot the magnitude of the kinetic energy for U/t = 2.4(2) Mutual 0.4 U/t = 8.2(5) the different interactions investigated. For weak interactions Information U/t = 12.0(7) the magnitude of the kinetic energy peaks at half filling, 0.2 similar to the mutual information (Fig. 3). Hence, we 0.5 identify delocalization as the mechanism to provide the correlation between a single lattice site and the environment for the lowest interaction. Conversely, for stronger inter- actions the shape of the kinetic-energy curves changes -15 -10 -5 0 123 - U/2 (t) k T (t) qualitatively as the peak shifts to a lower filling because (c) (d) B of the emerging Mott gap. At half filling the magnitude of the 0.4 0.4 kinetic energy decreases with increasing interaction strength, signaling the crossover towards a localized, Mott-insulating state. 0.2 0.2 The measured nearest-neighbor correlations provide a first ð1Þ glimpse at the single-particle density matrix G ðji−jjÞ¼ 0 0 hc ˆ c ˆ i in the lattice. With the measurements shown jσ σ iσ here, we can determine the single-particle density matrix 123 123 for the values ji − jj¼ 0, 1. Concentrating on the case of k T (t) k T (t) B B half filling (even though the discussion can be generalized ð1Þ to any filling), we find G ð0Þ¼ 1, and in the case of the FIG. 3. Mutual information between a single lattice site and its ð1Þ surrounding environment for different fillings, interactions, and metallic phase (U=t ¼ 2.4), G ð1Þ¼ 0.29ð2Þ. This drop temperatures. (a) Data used to extract the mutual information, for is in line with the expected thermal de Broglie wavelength, the exemplary case of U=t ¼ 8.2ð5Þ and k T=t ¼ 1.09ð5Þ. The which at a temperature of k T ¼ t is approximately mutual information is given by the difference between the single- λ ≃ a. For higher temperature, the spatial correlations dB site entropy s and the thermodynamic entropy per site s. Mutual drop correspondingly faster. In contrast, for a Mott insu- information for n ¼ 0.20ð2Þ (b), n ¼ 0.67ð2Þ (c), and n ¼ ð1Þ ð1Þ lator (U=t ¼ 12), the drop is from G ð0Þ¼ 1 to G ð1Þ¼ 1.00ð2Þ (d). The lines are theory predictions extracted from 0.11ð5Þ even though the temperature is even lower. This NLCE data (solid lines) and for the noninteracting Fermi gas on a indicates localization of the fermions with increasing lattice (dash-dotted lines). The color code is the same for all plots. Error bars display the statistical measurement uncertainty. interaction strength. We note, however, that even for the 031025-4 Mutual Information (k ) Entropy (k ) B B Mutual Information (k ) Mutual Information (k ) B B MEASURING ENTROPY AND SHORT-RANGE … PHYS. REV. X 7, 031025 (2017) (a) (b) (c) U/t = 2.4(2) U/t = 8.2(5) U/t = 12.0(7) |E | (t) -0.5 0 0.5 1 1.5 k T/t = 1.14(2) k T/t = 0.68(2) k T/t = 0.70(4) B B B k T/t = 2.37(3) k T/t = 2.18(5) k T/t = 2.51(3) B B B - U/2 (t) - U/2 (t) - U/2 (t) FIG. 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. strongest interaction explored by our experiment, the atoms of constant chemical potential with a bin size of are not fully localized [24] even though the compressibility Δμ ¼ h × 100 Hz. The uncertainty of the total density n ¼ of the state is almost zero [20]. 2ðn þ n Þ is then obtained by adding the individual S D In conclusion, we measure short-range correlations in the uncertainties in quadrature. The systematic uncertainties two-dimensional Hubbard model and show that at low of the Hubbard parameters U and t are derived from the temperatures a single lattice site develops correlations with calibration uncertainties of the lattice depth along the three the surrounding environment. In particular, we find that directions and the uncertainty in the parametrization of the even for strongly interacting Mott insulators with a vanish- Feshbach resonance [20]. Statistical uncertainties of the ing compressibility the fermions are still significantly temperature T and the chemical potential μ ¼ U=2 at HF delocalized over neighboring lattice sites. We note that half filling are obtained from simultaneously fitting the the technique we present here determines the full thermo- recorded singles and doubles density profiles n ðμÞ to S;D dynamic entropy, including the entropy in the spin sector numerical simulations of the two-dimensional Hubbard without the need for spin-resolved measurements. Hence, it model [21] or the ideal (U ¼ 0) Fermi gas on a square could find use in future attempts to cool strongly correlated lattice. The systematic error of μ is negligible as the HF quantum gases by reshuffling the entropy [8,25,26]. maximum of the singles density n ðμÞ provides a model- independent calibration of the half-filling point. The We thank A. Daley and C. Kollath for discussions. The extracted temperatures are confirmed within the statistical work has been supported by DFG (SFB/TR 185), the uncertainties by fitting the low-filling regions of the density Alexander von Humboldt Stiftung, EPSRC, and ERC profile to the ideal (U ¼ 0) Fermi gas on a square lattice. (Grant No. 616082). The statistical uncertainty of the pressure PðμÞ is obtained by adding the individual statistical uncertainties APPENDIX: ERROR ANALYSIS of the total density nðμÞ in quadrature. The systematic Here, we specify the uncertainties of the measured uncertainty of the chemical potential axis is converted into quantities and Hubbard parameters and provide details an additional systematic uncertainty of the pressure only in about the error analysis of the derived thermodynamic Fig. 1(c), where the pressure at half filling is shown. quantities. The statistical uncertainty of the entropy sðμÞ is obtained The statistical uncertainties of the measured singles and from the fit errors of the second-order polynomial fit to the doubles density n are given by the standard error pressure data PðTÞ for a given μ. Systematic uncertainties of S;D the entropy originating from the chosen temperature interval resulting from averaging the recorded images over regions 031025-5 |E | (t) k T (t) B E. COCCHI et al. PHYS. REV. X 7, 031025 (2017) (b) (a) FIG. 5. Average values (a) and uncertainties (b) of the individual terms contributing to the magnitude of the kinetic energy jE j. Note that in the calculation of the error of the kinetic energy ΔE the term nΔμ cancels with the pressure uncertainty Δ P concerning the k μ uncertainty of the chemical potential. The term Δ 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]. [4] M. F. Parsons, A. Mazurenko, C. S. Chiu, G. Ji, D. Greif, and the order of the polynomial fit function have been and M. Greiner, Site-Resolved Measurement of the estimated by applying the same routine to numerical density Spin-Correlation Function in the Fermi-Hubbard Model, data of the two-dimensional Hubbard model [21].In the Science 353, 1253 (2016). explored temperature regime, we obtain a maximum deviation [5] M. Boll, T. A. Hilker, G. Salomon, A. Omran, J. Nespolo, L. between the theoretical value of the entropy and the value Pollet, I. Bloch, and C. Gross, Spin- and Density-Resolved obtained with our routine of 0.1k . 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Published: Jul 1, 2017

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