Abstract
Two of the most influential ideas developed by Richard Feynman are the Feynman diagram technique and his variational approach. Here we show that combining both, and introducing a diagrammatic quantum Monte Carlo method, results in a powerful and accurate solver to the generic solid state problem, in which a macroscopic number of electrons interact by the long range Coulomb repulsion. We apply it to the quintessential problem of solid state, the uniform electron gas, which is at the heart of the density functional theory success in describing real materials, yet it has not been adequately solved for over 90 years. Our method allows us to calculate numerically exact momentum and frequency resolved spin and charge response functions. This method can be applied to a number of moderately interacting electron systems, including models of realistic metallic and semiconducting solids.
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Introduction
The success of the Feynman’s diagram technique^{1} rests on two pillars, the quality of the chosen starting point and one’s ability to compute the contributions of highenough order, so that the sum ultimately can be extrapolated to the infinite order.
Here we address the former by introducing the variationally optimized starting point, as discussed below, and we solve the latter by developing a powerful Monte Carlo method which can sum factorially large number of diagrams while massively reducing the fermionic sign problem by organizing Feynman diagrams into signblessed groups. The resulting Variational Diagramatic Monte Carlo (VDMC) method is a generic manybody solver, which is here tested on the classic solid state problem. We compute the spin and the charge response functions, which are directly accessed by the experiment, but remain challenging to compute by other methods. The accuracy of the calculated response functions is sufficiently high, so as to uncover previously missed fine structure in these responses.
Results
The optimized starting point Lagrangian
In the Feynman diagrammatic approach, one splits the Lagrangian of a system, L, into a solvable part L_{0} and the interaction ΔL = L − L_{0}. The effect of the interaction is included with a power expansion in ΔL, constructed using the Feynman diagrams. Such diagrammatic series achieves the most rapid convergence when the leading term L_{0} captures the emergent collective behavior of the system, which can be very different from the noninteracting problem^{2}. In the metallic state, which is of special interest in this paper, the lowtemperature physics is described by the emergent quasiparticles interacting with a screened Coulomb interaction. We build an effective Feynman diagrammatic approach by explicitly encoding such physics in L_{0}. We screen the interaction in L_{0} with a screening parameter λ, rendering the Coulomb interaction shortranged (\(V(r) \propto {\mathrm{{exp}}}(  r\sqrt \lambda )/r\)). Correspondingly, a λ counterterm must be added to ΔL to capture the nonlocal effects of the Coulomb interaction with highorder diagrams (see the Methods section). Similarly, we introduce an electron potential v_{k} which properly renormalizes the electron dispersion and also fixes the Fermi surface of L_{0} to the exact physical volume, which is enforced by the Luttinger’s theorem^{3} (see the Methods section). In our simulations, such choice shows the most rapid and uniform convergence of the response functions for both small and large momenta.
Motivated by Feynman’s variational approach^{4}, we take the screening parameter λ as variational parameter which should be optimized to accelerate the rate of convergence. It was shown in the development of optimized perturbation theory^{5} and variational perturbation theory^{6,7} that the best choice of a variational parameter is the value at which the targeted observable is least sensitive to the change of the parameter. This technique is called the principle of minimal sensitivity (PMS). In refs. ^{7,8,9,10}, it was shown that the perturbative expansion optimized with the PMS can succeed even when interaction is strong, and regular perturbation theory fails. In this work, we optimize the screening parameter λ with PMS and observe a vast improvement to the convergence of the targeted response functions with expansion order.
The signblessed Monte Carlo algorithm
While our setup of the expansion (with the static screening and the physical Fermi surface) is not entirely new^{11,12,13,14}, its evaluation to highenough order until ultimate convergence has not to our knowledge been achieved before in any realistic model containing the longrange Coulomb interactions that are relevant for realistic solids. Our solution employs a recently developed diagrammatic Monte Carlo method^{15,16,17,18,19,20,21,22}, which is here optimized to take a maximal advantage of the sign blessing in fermionic systems^{16}. Namely, by carefully arranging and grouping the Feynman diagrams, it is possible to ensure a massive sign cancellation for different diagrams in the same group, before the MC sampling is performed^{21,23}. The conventional diagrammatic Monte Carlo algorithms^{15,16,17,18,19,20}, which were sampling the diagrams one by one, are highly inefficient here.
We evaluate diagrams in the momentum and imaginarytime representation, and for each configuration of random momenta (k_{0}, k_{1}, k_{2}, ⋯, k_{N}) and times (τ_{1}, τ_{2}, ⋯, τ_{2N}) generated by the Markov chain, we sum the contribution of all diagrams at a given order N, which have the same number of momenta and time variables^{23}. For example, when computing the polarization at order N = 6, the sector without counterterms contains 14,593 Feynman diagrams (see Fig. 1). These are regrouped into a much smaller number of signblessed groups to boost the efficiency of the MC sampling. For example, motivated by the crossing symmetry, at the lowest order in the crossing exchange, we get from standard Feyman diagrams to socalled Hugenholtz diagrams^{24} where the direct and exchange interactions are combined into an antisymmetrized four point vertex (see Fig. 1, green box). That exchange operation keeps the diagram exactly the same, except for a change of the overall sign and a change of momentum on a single interaction line, hence the pairs of such diagrams largely cancel. After this operation, there are only 877 Hugenholtz diagrams at order 6. To further reduce the number of diagrams, we then combine the polarization diagrams that can be derived from the same freeenergy diagram by attaching two external vertices to propagators. Mathematically, adding external vertices to a freeenergy diagram corresponds to taking its functional derivative with respect to the inverse propagator. Therefore, the above step groups the polarization diagrams into a conserving group in the Baym–Kadanoff sense^{25}, and the sign cancellation is guaranteed by the Ward identities (see Supplementary Note 1). For example, at order N = 6 there are only 41 such freeenergy groups (see Fig. 1). We thus managed to reduce the complexity from 14,593 individual diagrams to 41 groups. The diagrams in the same group are very similar, and hence can share the identical momentum/time variables (except the external vertices). This not only ensures the massive sign cancellation between different diagrams but also reduces the cost of computing the total weight of Feynman diagrams in Monte Carlo updates.
Finally, beyond variationally optimizing the zeroth order term (L_{0}) we can also look for improvement of the highorder vertex functions. One of our choices is to sum up all ladder diagrams dressing the vertices (see Supplementary Fig. 3). We call this scheme the Vertex Corrected Constant Fermi Surface (VCCFS). The original diagrammatic expansion is here called Constant Fermi Surface (CFS) scheme. The name originates in the above described principle that electron potential v_{k} is determined in such a way that L and L_{0} share the same physical Fermi surface volume.
Spin susceptibility of the uniform electron gas
All results in this work are obtained at temperature T = 0.04E_{F}, which is much lower than any other scale in UEG problem^{26}; hence, results are the zero temperature equivalent. We want to point out that finite temperature calculations are very hard in the Diffusion Monte Carlo (DMC)^{27}, while our method is very well suited for finite temperature calculations, and converges even faster with the increasing order. While wave function properties, such as energy and pair distribution function, are very precisely computed by DMC^{27}, and some of them are also are amenable to approximations such as GW^{28,29}, the response functions are more challenging to evaluate with the existing techniques. The strength of our approach is that it can be used to compute both the static and the dynamic, the single and the multiparticle correlation functions.
In Figs. 2 and 3 we show the momentumdependent (Pauli) spin susceptibility at zero frequency, which has never been precisely calculated before to our knowledge even though its overall shape is crucial for the design of appropriate exchangecorrelation functionals of the density functional (DFT) to predict magnetic order in real materials. In panels (a) and (b) we show how the convergence properties of the susceptibility χ_{s} depends on the screening parameter λ in the theory. Note that the static screening in L_{0} is always compensated by the counterterm in ΔL, so that for any value of λ the UEG model is recovered at infinite order limit. The observable χ_{s}(q = 0) develops a broad plateau as a function of λ (Fig. 2a, b) at the point \(\lambda _N^ \ast\), which is slightly increasing with the increasing order. This shows that if expansion is carried out to highenough order, the physics becomes more and more local and allows one to use very shortrange form of the interaction, which greatly improves the efficiency of the method. We note that this property will be very beneficial in the realistic material applications, where the converged result is extremely difficult to obtain due to the longrange nature of the bare Coulomb interaction. Figure 2c shows the value of χ_{s}(q = 0) at the optimized \(\lambda _N^ \ast\) versus perturbation orders. When the PMS is used, such that the variational parameter λ is optimized order by order, the convergence is very rapid, even when the bare interaction is strong. The value χ_{s}(q = 0) at the optimized \(\lambda _N^ \ast\) is monotonically increasing with the increasing order in the CFS scheme, and beyond the second order is oscillating around the converged value in VCCFS scheme. Both schemes converge towards the same value, and the systematic error bar at a given truncation order can be estimated from comparison between the two methods, allowing one to extract very precise value of χ_{s}(q = 0) even at a moderate expansion order (see Fig. 2c, Table 1).
Figure 2d shows the momentum dependence of spin susceptibility χ_{s}(q) at λ^{*}/E_{F} = 0.75, optimized at the highest order (N = 6) and its comparison to the noninteracting (RPA) result, which underestimates χ_{s} up to 57%.
In Fig. 3a we show the same spin susceptibility as in Fig. 2d, but for other values of density parameter r_{s} = 1–4 (here density \(n = 3/(4\pi r_{\mathrm{s}}^{\mathrm{3}})\)). Both VCCFS and CFS schemes agree with each other within the statistical error bar at order N = 6 for all r_{s} ≤ 4. We note that this spin susceptibility plays a central role in construction of the DFT exchangecorrelation kernel for magnetically ordered systems. Finally, Fig. 3b displays the static localfield correction, which measures the deviation from the noninteracting electron gas (χ_{RPA}), \(G(q) \equiv \frac{{q^2}}{{8\pi }}(\chi _{{\mathrm{RPA}}}^{  1}(q,\omega = 0)  \chi ^{  1}(q,\omega = 0))\). It is a very sensitive measure of electron correlations. It has been suggested in the literature that the possible peak near k ~ 2k_{F} is of great importance for understanding the quasiparticle properties^{30}. Within the local density approximation, the function G(q) is approximated by the quadratic parabola depicted in Fig. 3b^{31}, which is an excellent approximation at small q ≤ k_{F}, but its deviation from the quadratic function is very pronounced near 2k_{F}. Note that within RPA G(q) vanishes, as RPA does not take into account the exchangecorrelation kernel. We note that our calculation clearly shows that in the exact solution, the localfield correction displays nontrivial maximum just above 2k_{F}, which has not been established before.
Charge response function
Figure 4 shows the dielectric function \(\epsilon (q)\) for densities r_{s} = 1 to r_{s} = 4, and its comparison to RPA and DMC^{31,32} results. We show several orders (N = 2–5) using VCCFS scheme, and also the extrapolated result to N = ∞ using standard secondorder Richardson extrapolation. The DMC data are in agreement with our prediction, but notice that DMC allows one to calculate only a set of discrete points, while the developed VDMC method gives a smooth and very accurate continuous curve, which allows one to resolve the fine structure. For example, we notice that there is a clear kink of \(1/\epsilon\) curve near 2k_{F}. This feature has been proposed in some theories (e.g. ref. ^{33}), but the previous DMC results in refs. ^{31,32} were not precise enough to confirm or disprove it.
Finally, in Table 1 we give our best estimates for the static spin and charge response with estimation of the error bar. Within our method the spin response shows faster convergence with increasing order; hence, it allows us to compute the spin response more precisely than the charge response. Therefore, our values for χ_{s}/N_{F} are more precise than currently available literature (compare columns one and two). Note that the previous estimate for the spin susceptibility relied on an uncontrolled ansatz for the spin dependence of the susceptibility, hence large uncertainty.
Contrary to the spin response, or finite momentum charge response, the static uniform charge response P(q = 0, ω = 0) can be obtained from the ground state energy of the system, without explicitly introducing a modulated external potential, and hence it can be extracted very precisely from the existing DMC calculations. We compare it with our results, and find excellent agreement. We note that static P(q = 0, ω = 0) at r_{s} = 4 convergences very slowly in our method, due to proximity to the well known charge instability at r_{s} ≈ 5.2; hence, we can not reliably extrapolate its value to infinite order at r_{s} ≥ 4.
Discussion
The prospects of combining the Variational diagrammatic Monte Carlo with DFT to obtain theoretically controlled results in real solids are particularly exciting, as the DFT potential is semilocal and can be added to v_{k}, so that it will play a role of a counterterm in the expansion. The complexity would be modest, because no expensive selfconsistency is required, and because the interaction is statically screened, hence the scaling of this method should be similar to the complexity of screened hybrids^{34} rather than the selfconsistent GW approximation^{35}.
Methods
Model definition
The UEG model describes electrons in a solid where the positive charges, which are the atomic nuclei, are assumed to be uniformly distributed in space. The electrons interact with the other charges through a longrange Coulomb interaction. The secondquantized Hamiltonian is
where \(\hat \psi\)/\(\hat \psi ^\dagger\) are the annihilation/creation operator of an electron, μ is the chemical potential controlling the density of the electron in the system. We measure the energy in units of Rydbergs, and the wave number k, q in units of inverse Bohr radius.
Lagrangian with the counterterms
In the path integral representation, using the standard Hubbard–Stratonovich transformation, the Lagrangian of the uniform electron gas can be cast into the form in which the Coulomb interaction is mediated by an auxiliary bosonic field ϕ_{q}. Motivated by the well known fact that the longrange Coulomb interaction is screened in the solid, and that the effective potential of emerging quasiparticles differs from the bare potential, we introduce the screening parameter λ_{q} and an electron potential v_{k} into L_{0}, which then takes the form
and represents well the lowenergy degrees of freedom in the problem when parameters λ_{q} and v_{k} are properly optimized. To compensate for this choice of L_{0}, we have to add the following interaction:
so that, when the number ξ is set to unity, L = L_{0} + ΔL(ξ) is exactly the UEG Lagrangian. The density ρ is \(\rho _{\mathbf{q}} = \mathop {\sum}\nolimits_{{\mathbf{k}}\sigma } {\psi _{{\mathbf{k}}\sigma }^\dagger } \psi _{{\mathbf{k}} + {\mathbf{q}}\sigma }\). Note that the first two terms in ΔL are the counterterms^{14} which exactly cancel the two terms we added to L_{0} above. We use the number ξ to track the order of the Feynman diagrams, so that order N contribution sums up all diagrams carrying the factor ξ^{N}. We set ξ = 1 at the end of the calculation. Note also that this arrangement bears similarity with the well established methods, such as G0W0^{35}, which computes the selfenergy at the lowest order (ξ^{1}) and sets v_{k} to the DFT Kohn–Sham potential, and λ_{q} to the bubble diagram (λ_{q} = g^{0}g^{0} with \(g_{\mathbf{k}}^{0  1} = (i\omega + \mu  \frac{{{\mathbf{k}}^2}}{{2m}}  v_{\mathbf{k}})\)). The socalled skeleton Feynman diagram technique is recovered when v_{k} and λ_{q} are equated with the selfconsistently determined selfenergy and polarization. However, note that such diagram expansion can be dangerous, as it can lead to false convergence to the wrong solution^{36}.
Variational optimization
In optimizing the screening parameter λ_{q} by the principle of minimal sensitivity, we found it is sufficient to take a constant λ_{q} = λ. Furthermore, we found that the uniform convergence for all momenta is best achieved when the electron potential v_{k} preserves the Fermi surface volume of \(g_{\mathbf{k}}^0\); therefore, we expand \(v_{\mathbf{k}} = \xi {\kern 1pt} (\Sigma _{\mathbf{k}}^{\mathrm{x}}  \Sigma _{{\mathrm{k}}_{\mathrm{F}}}^{\mathrm{x}}) + \xi ^2{\kern 1pt} s_2 + \xi ^3{\kern 1pt} s_3 \cdots\), and we determine s_{N} so that all contributions at order N do not alter the physical volume of the Fermi surface. In other words, we ensure the density, which can be calculated with the identity n = −P_{q}(τ = 0) where \({\mathbf{q}} \gg k_{\mathrm{F}}\), remains fixed order by order. Since the exchange (\(\Sigma _{\mathbf{k}}^{\mathrm{x}}\)) is static, and is typically large, we accomodate it at the first order into the effective potential, so that at the first order we recover the screened Hartree–Fock approximation, i.e., interaction screened to \(\sim \hskip 2pt {\mathrm{{exp}}}(  r\sqrt \lambda )/r\) and optimized λ.
We also introduce a vertex correction scheme (VCCFS) to further improve the convergence of the series. In practice, within the VCCFS scheme, we precompute the threepoint ladder vertex, and attach it to both sides of a polarization Feynman diagram, and at the same time, we eliminate all laddertype diagrams from the sampling, to avoid double counting of diagrams (see Supplementary Fig. 4).
Comparison to existing approaches
Finally, we discuss the advantages and limitations of the proposed method. The current variational approach is very effective at weak to intermediate correlation strength (spin/charge response up to r_{s} ≈ 4), but to extend it to the regime with stronger correlations, one would need to introduce more sophisticated counterterms, such as the three and the four point vertex renormalization, to capture the emergent charge instability around r_{s} ≈ 5.2. Beyond the variational approach, we also want to point out that our developed Monte Carlo algorithm is a very generic Feynman diagram calculator for manyelectron systems with longrange Coulomb repulsion, and is more efficient and simpler that the existing conventional diagrammatic Monte Carlo of refs. ^{15,16,17,18,19,20}. For example, VDMC requires only three updates, while the conventional approach needs about dozen updates. More importantly, this algorithm utilizes the signblessed grouping techinque to dramatically improve the sampling efficiency. Comparing to the recently proposed Determinant Diagrammatic Monte Carlo algorithm^{21}, our method is more generic in the sense that the algorithm can directly work in any representation (momentum/frequency, space/time) and can handle any vertex renormalization without sacrificing the efficiency.
Data availability
All analytical data not given in the Supplementary Information are available on request.
Code availability
The code is available at https://github.com/haulek/VDMC with https://doi.org/10.5281/zenodo.3309567, and https://github.com/kunyuan/FeynCalc with https://doi.org/10.5281/zenodo.3308233.
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Acknowledgements
We thank G. Kotliar, N. Prokof’ev, B. Svistunov, and Y. Deng for stimulating discussion. This work is supported by the Simons Collaboration on the Many Electron Problem and NSF DMR1709229.
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Both K.C. and K.H. developed the MC code, created the theoretical formalism, carried out the calculation, and analyzed the results and wrote the paper. K.H. supervised the project.
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Chen, K., Haule, K. A combined variational and diagrammatic quantum Monte Carlo approach to the manyelectron problem. Nat Commun 10, 3725 (2019). https://doi.org/10.1038/s41467019117086
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DOI: https://doi.org/10.1038/s41467019117086
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