Abstract
Precise understanding of strongly interacting fermions, from electrons in modern materials to nuclear matter, presents a major goal in modern physics. However, the theoretical description of interacting Fermi systems is usually plagued by the intricate quantum statistics at play. Here we present a crossvalidation between a new theoretical approach, bold diagrammatic Monte Carlo^{1,2,3}, and precision experiments on ultracold atoms. Specifically, we compute and measure, with unprecedented precision, the normalstate equation of state of the unitary gas, a prototypical example of a strongly correlated fermionic system^{4,5,6}. Excellent agreement demonstrates that a series of Feynman diagrams can be controllably resummed in a nonperturbative regime using bold diagrammatic Monte Carlo.
Main
In his seminal 1981 lecture^{7}, Richard Feynman argued that an arbitrary quantum system cannot be efficiently simulated with a classical universal computer, because generally, quantum statistics can only be imitated with a classical theory if probabilities are replaced with negative (or complex) weighting factors. For the majority of manyparticle models this indeed leads to the socalled sign problem, which has remained an insurmountable obstacle. According to Feynman, the only way out is to employ computers made out of quantummechanical elements^{7}. The recent experimental breakthroughs in cooling, probing and controlling strongly interacting quantum gases prompted a challenging effort to use this new form of quantum matter to realize Feynman’s emulators of fundamental microscopic models^{7,8}. Somewhat ironically, Feynman’s arguments, which led him to the idea of emulators, may be defied by a theoretical method that he himself devised, namely Feynman diagrams. This technique organizes the calculation of a given physical quantity as a series of diagrams representing all the possible ways particles can propagate and interact (for example, ref. 9). For the manybody problem, this diagrammatic expansion is commonly used either in perturbative regimes or within uncontrolled approximations. However, the introduction of diagrammatic Monte Carlo recently allowed one to go well beyond the first few diagrams, and even reach convergence of the series in a moderately correlated regime^{1,10}.
In this Letter we show that for a strongly correlated system and down to a phase transition, the diagrammatic series can still be given a mathematical meaning and leads to controllable results within bold diagrammatic Monte Carlo (BDMC). This approach, proposed in refs 1, 2, 3, is first implemented here for the manybody problem. We focus on the unitary gas, that is, spin1/2 fermions with zerorange interactions at infinite scattering length^{4,5,6}. This system offers the unique possibility to stringently test our theory against a quantum emulator realized here with trapped ultracold ^{6}Li atoms at a broad Feshbach resonance^{4,5,6}. This experimental validation is indispensable for our theory, based on resummation of a possibly divergent series: although the physical answer is shown to be independent of the applied resummation technique—suggesting that the procedure is adequate—its mathematical validity remains to be proven. In essence, nature provides the ‘proof’. This presents the first—although longanticipated—compelling example of how ultracold atoms can guide new microscopic theories for strongly interacting quantum matter.
At unitarity, the disappearance of an interactionimposed length scale leads to scale invariance. This property renders the model relevant for other physical systems such as neutron matter. It also makes the balanced (that is, spinunpolarized) unitary gas ideally suited for the experimental highprecision determination of the equation of state (EOS) described below. Finally, it implies the absence of a small parameter, making the problem notoriously difficult to solve.
In traditional Monte Carlo approaches, which simulate a finite piece of matter, the sign problem causes an exponential increase of the computing time with system size and inverse temperature. In contrast, BDMC simulates a mathematical answer in the thermodynamic limit. This radically changes the role of the fermionic sign. Diagrammatic contributions are signalternating with order, topology and values of internal variables. Because the number of graphs grows factorially with diagram order, a nearcancellation between these contributions is actually necessary for the series to be resummable by techniques requiring a finite radius of convergence. We find that this ‘sign blessing’ indeed takes place.
In essence, BDMC solves the full quantum manybody problem by stochastically summing all the skeleton diagrams for irreducible singleparticle selfenergy Σ and pair selfenergy Π, expressed in terms of bold (that is, fully dressed) singleparticle and pair propagators G and Γ which are determined selfconsistently (Fig. 1). The density EOS (that is, the relation between total density n, chemical potential μ and temperature T) is given by G at zero distance and imaginary time, n(μ,T)=2 G(r=0,τ=0^{−}). The thermodynamic limit can be taken analytically. The sum of ladder diagrams built on the bare singleparticle propagator defines a partially dressed pair propagator Γ^{0}. As Γ^{0} is well defined for the zerorange continuousspace interaction, the zerorange limit can also be taken analytically. This is in sharp contrast with other numerical methods^{11,12,13}, where taking the thermodynamic and zerorange limits is computationally very expensive. BDMC performs a random walk in the space of irreducible diagrams using local updates. The simulation is run in a selfconsistent cycle (along the lines of ref. 2) until convergence is reached. Full details will be presented elsewhere. In essence, our approach upgrades the standard manybody theories based on one lowestorder diagram (for example, refs 14, 15) to millions of graphs.
In the quantum degenerate regime, we do not observe convergence of the diagrammatic series for Σ and Π evaluated up to order 9. Here, order N means Σdiagrams with N vertices (that is, N Γlines) and Πdiagrams with N−1 vertices. To extract the infiniteorder result, we apply the following Abelian resummation methods^{16}. The contribution of all diagrams of order N is multiplied by {\text{e}}^{\epsilon {\lambda}_{N1}}, where λ_{n} depends on the resummation method: (1) λ_{n}=n logn (with λ_{0}=0) for Lindelöf^{16}, (2) λ_{n}=(n−1)log(n−1) (with λ_{0}=λ_{1}=0) for ‘shifted Lindelöf’, or (3) λ_{n}=n^{2} for Gaussian^{17}. A full simulation is performed for each ε, and the final result is obtained by extrapolating to ε=0 (Fig. 2).
This protocol relies on the following crucial mathematical assumptions: (1) the Nth order contribution of the diagrammatic expansion for Σ (for fixed external variables) is the Nth coefficient of the Taylor series at z=0 of a function g(z) which has a nonzero convergence radius, (2) the analytic continuation g(1), performed by the above resummation methods^{16,17}, is the physically correct value of Σ. The same assumptions should hold for Π.
Proving these assumptions is an open mathematical challenge. Note that Dyson’s collapse argument^{18} is not applicable to immediately disprove the assumption (1) of a nonzero convergence radius: indeed, unlike QED, our skeleton series is not an expansion in powers of a coupling constant whose sign change would lead to an instability. The first important evidence for the validity of our mathematical assumptions is that the three different resummation methods yield consistent results. For an independent test, we turn to experiments.
The present experiment furnishes highprecision data for the density n as a function of the local value V of the trapping potential (Fig. 3 and Methods). We start the process by obtaining the EOS at high temperatures in the nondegenerate wings of the atom cloud, where the virial expansion is applicable. Once the temperature and the chemical potential have been determined from fits to the wings of the cloud, the data closer to the cloud centre provides a new prediction of the EOS. The process is iterated to access lower temperatures.
Scale invariance allows one to write the density EOS as n(μ,T)λ^{3}=f(β μ), with the thermal de Broglie wavelength, β=1/(k_{B}T)the inverse temperature and f a universal function. A convenient normalization of the data is provided by the EOS of a noninteracting Fermi gas, n_{0}λ^{3}=f_{0}(β μ). In Fig. 4a, we thus report the ratio n(μ,T)/n_{0}(μ,T)=f(β μ)/f_{0}(β μ), bringing out the difference between the ideal and the strongly interacting Fermi gas. The Gibbs–Duhem relation allows us to also calculate the pressure at a given chemical potential, , where . We normalize it by the pressure of the ideal Fermi gas and show F(β μ)/F_{0}(β μ) (Fig. 4b). The agreement between BDMC and experiment is excellent. The comparison is sufficiently sensitive to validate the procedure of resumming and extrapolating (Fig. 2). The result was checked to be independent of the maximal sampled diagram order N_{max}∈{7;8;9} within the error bars shown in Fig. 2 for each ε. The BDMC final error bar in Fig. 4 is the sum of the conservatively estimated systematic errors from the uncertainty of the ε→0 extrapolation and from the dependence on numerical grids and cutoffs, the latter being reduced by analytically treating highmomentum shorttime singular parts. The systematic error in the experiment is determined to be about 1% by the independent determination of the EOS of the noninteracting Fermi gas. The experimental error bars of Fig. 4 also include the statistical error, which is <0.5%, thanks to the scale invariance of the balanced unitary gas: irrespective of shottoshot fluctuations of atom number and temperature, all experimental profiles contribute to the same scaled EOSfunction f. The dominant uncertainty on the experimental EOS stems from the uncertainty in the position of the ^{6}Li Feshbach resonance, known to be at 834.15±1.5 G from spectroscopic measurements^{19}. The change in energy, pressure and density with respect to the interaction strength is controlled by the socalled contact^{20} that is obtained from Γ in the BDMC calculation. This allows us to define the uncertainty margins above and below the experimental data (Fig. 4) that give the prediction for the unitary EOS if the true Feshbach resonance lies 1.5 G below or above 834.15 G, respectively.
Our results clearly differ from previous theoretical and experimental results. Deviations from the theory based on the firstorder Feynman diagrams^{15,21} are expected, and rather remarkably moderate. Differences with lattice Monte Carlo data^{11,13} may seem more surprising, as in the particular case of the balanced system these algorithms are free of the sign problem, allowing one in principle to approach the balanced unitary gas model in an unbiased way. However, eliminating systematic errors from latticediscretization and finite volume requires extrapolations which are either not done^{11} or difficult to control^{12,13}. The ENS experimental pressure EOS (ref. 22) lies systematically below ours, slightly outside the reported error bar. The experimental results from Tokyo^{23} do not agree with the virial expansion at high temperature. The BDMC results agree well with the present experimental data all the way down to the critical temperature for superfluidity (Fig. 4). On approaching (β μ)_{c}, we observe the growth of the correlation length in the BDMC pair correlation function Γ. A protocol for extracting the critical temperature itself from the BDMC simulation will be presented elsewhere.
We are not aware of any system of strongly correlated fermions in nature where experimental and unbiased theoretical results were compared at the same level of accuracy. Even for bosons, the only analogue is liquid ^{4}He. This promotes the unitary gas to the major testing ground for unbiased theoretical treatments. The present BDMC implementation should remain applicable at finite polarization and/or finite scattering length, opening the way to rich physics which was already addressed by cold atoms experiments^{6,24,25,26,27}. We also plan to extend BDMC to superfluid phases by introducing anomalous propagators. Moreover, as the method is generic, we expect numerous other important applications to longstanding problems across many fields.
Note added in proof: After a preprint of this work became available, new auxiliaryfield quantum Monte Carlo data were presented^{28}, with undetermined systematic errors whose evaluation in future work is called for by the authors of ref. 28.
Methods
The experimental setup has been described previously^{24}. In short, ultracold fermionic ^{6}Li is brought to degeneracy by sympathetic cooling with ^{23}Na. A twostate mixture of the two lowest hyperfine states of ^{6}Li is further cooled in a hybrid magnetic and optical trap at the broad Feshbach resonance at 834 G. We employ highresolution in situ absorption imaging to obtain the column density of the gas, that is converted into the full 3D density using the inverse Abel transform^{29}. Equidensity lines provide equipotential lines that are precisely calibrated using the known axial, harmonic potential (axial frequency ν_{z}=22.83±0.05 Hz). Equipotential averaging yields lownoise profiles of density n versus potential V. Density is absolutely calibrated by imaging a highly degenerate, highly imbalanced Fermi mixture, and fitting the majority density profile to the ideal Fermi gas EOS (ref. 24). In contrast to previous studies^{22,23}, our analysis does not rely on the assumption of harmonic trapping.
Thermometry is performed by fitting the density profile to the EOS constructed thus far, restricting the fit to the portion of the density profile where the EOS is valid. In the hightemperature regime, the EOS is given by the virial expansion
where the virial coefficients are (ref. 30), and b_{3}=−0.29095295 (ref. 31). Fitting a hightemperature cloud to the virial expansion gives the temperature T and the chemical potential μ of the cloud, and the EOS n λ^{3}=f(β μ) can be constructed. We have used equation (1) for β μ<(β μ)_{m a x}=−1.25 and we checked that our EOS did not change within statistical noise if we instead used (β μ)_{max}=−0.85. Once a new patch of EOS has been produced, it can then in turn be used to fit colder clouds. Iteration of this method allows us to construct the EOS to arbitrarily low temperature. A total of ∼ 1,000 profiles were used, with 10–100 profiles (50 on average) contributing at any given β μ.
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Acknowledgements
We thank R. Haussmann for providing propagator data from refs 15, 21 for comparison, and the authors of refs 11, 13, 22, 23 for sending us their data. This collaboration was supported by a grant from the Army Research Office with funding from the Defense Advanced Research Projects Agency (DARPA) Optical Lattice Emulator program. Theorists acknowledge the financial support of the Research Foundation Flanders (FWO) (K.V.H.), National Science Foundation (NSF) grant PHY1005543 (University of Massachusetts group), Swiss National Science Foundation (SNF) Fellowship for Advanced Researchers (E.K.), and l’Institut Francilien de Recherche sur les Atomes Froids (IFRAF) (F.W.). Simulations ran on the clusters CM at UMass and Brutus at ETH. The MIT work was supported by the NSF, AFOSRMURI, AROMURI, Office of Naval Research (ONR), DARPA Young Faculty Award, an AFOSR Presidential Early Career Award for Scientists and Engineers (PECASE), the David and Lucile Packard Foundation, and the Alfred P. Sloan Foundation.
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K.V.H. (theory) and M.J.H.K. (experiment) contributed equally to this work. K.V.H., F.W., E.K., N.P. and B.S. developed the BDMC approach for unitary fermions; the computer code was written by K.V.H. assisted by F.W.; simulation data were produced by F.W., E.K. and K.V.H.; M.J.H.K., A.T.S., L.W.C., A.S. and M.W.Z. all contributed to the experimental work and the analysis. All authors participated in the manuscript preparation.
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Van Houcke, K., Werner, F., Kozik, E. et al. Feynman diagrams versus Fermigas Feynman emulator. Nature Phys 8, 366–370 (2012). https://doi.org/10.1038/nphys2273
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DOI: https://doi.org/10.1038/nphys2273
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