Fermionic Sign Structure of High-order Feynman diagrams in a Many-fermion System

In quantum mechanics, the amplitudes of fermion-fermion scattering satisfy an anti-commutation relation: permuting two incoming (or outgoing) fermions flips the amplitude’s global sign. On the one hand, the sign cancellation plays a vital role in the collective behavior of fermions (For example, it leads to the electron degeneracy pressure which prevents matter from collapsing into a single nucleusDyson and Lenard (1967); Lenard and Dyson (1968)). On the other hand, the same sign cancellation leads to the notorious sign problem in Monte Carlo simulations of many-fermion systems Loh et al. (1990); Troyer and Wiese (2005). Indeed, a Monte Carlo estimator of physical observable is a sum of various elementary scattering amplitudes. Even if each amplitude is precisely calculated, the sign-canceled signal can be easily overwhelmed by numerical noise. This paper explores the sign structure of scattering amplitudes in Feynman diagrams to better understand the fermionic sign cancellation.

In many-body physics, Feynman diagrams describe how excitations in the system scatter incoming particles. The diagram amplitude, namely the scattering amplitude, is obtained by summing all diagram topologies and integrating all internal space-time variables (where the excitations get created and annihilated). Both steps involve massive sign cancellations so that the physical scattering amplitude is merely a small fraction of the total diagram weights. Here we use an example to illustrate the non-triviality of this problem. We consider a two-dimensional homogeneous “electron gas” with Yukawa interaction (see Sec.II for the definition of the Hamiltonian). We calculate four diagrams (see Fig. 1), which are the third-order contributions to the polarization $P$ (namely, the system’s linear response function with respect to a screened external electric field perturbation). They all involve a virtual particle-hole pair but with different topologies. As shown in Fig.1, their amplitudes are functions of the external imaginary-time. Four diagrams strongly cancel each other: the summed polarization (the thick black line) is merely $0.1\%$ of individual diagrams. Later we show such cancellation is a direct consequence of the emergent charge conservation law in this diagram group—while each diagram is time-dependent, their sum turns out to be a constant of motion, reflecting the conservation of total charge in this system.

The above example leads to fundamental questions we want to address: How does the cancellation occur in diagrams? Does it originate from the summation of topologies or the integration of internal variables? Can one identify the diagram groups that feature massive sign cancellations? Concrete answers to the above questions are important at least in two aspects: i) In fermionic systems, individual-diagram-based power-counting estimation could be misleading in many cases due to the cancellations in diagrams. The sign-canceled-group-based estimation fixes this problem (See. Ref. Belitz et al. (1997) and Fig.1 for examples). ii) Numerically, the knowledge of sign-canceled diagram groups alleviates the sign problem in Monte Carlo techniques, particularly the diagrammatic Monte Carlo methods Prokof’ev and Svistunov (1998). In this approach, one calculates the physical observable in terms of diagrammatic expansions with Monte Carlo samplings. The methods have been applied to solve a series of important problems in the Hubbard model Kozik et al. (2010); Gukelberger et al. (2014); Deng et al. (2015); Wu et al. (2017); Šimkovic et al. (2017); Moutenet et al. (2018); Taheridehkordi et al. (2019); Šimkovic and Kozik (2019); Vučičević and Ferrero (2020); Kim et al. (2020), many-electron problem Van Houcke et al. (2012, 2019); Rossi et al. (2018a, b), Fermi polaron problem Prokof’ev and Svistunov (1998); Mishchenko et al. (2000); Prokof’ev and Svistunov (2008a, b); Kroiss and Pollet (2014); Mishchenko et al. (2014); Hahn et al. (2018); Mishchenko et al. (2019); Van Houcke et al. (2020) and frustrated spin systems Kulagin et al. (2013a, b); Huang et al. (2016). The conventional approaches stochastically sample individual diagrams and therefore suffer from the severe sign problem caused by the massive sign cancellation between the diagrams. Recently, a new generation of the algorithms, which sample the summed weight of groups of diagrams, has been developed  Rossi (2017); Rossi et al. (2018a); Chen and Haule (2019); Taheridehkordi et al. (2020a); Vučičević and Ferrero (2020). We will refer them to cluster DiagMC algorithms. With the proper grouping of the diagrams, those algorithms can dramatically reduce the sign problem. Indeed, a couple of cluster DiagMC algorithms are reported to be applicable on much higher-order diagrams Rossi (2017), or in problems beyond the ability of the conventional approachesChen and Haule (2019); Taheridehkordi et al. (2020a, b). Despite the impressive progress, we still lack the design principle on diagrammatic groups to minimize the sign problem.

In this paper, we investigate two universal mechanisms of the sign cancellation, one of which is imposed by the crossing symmetry of the two-body scattering. The other is imposed by the global $U(1)$ symmetry or the charge conservation law, as previously discussed. We show that the symmetries constrain the sign-canceled diagram groups’ structure and fix the internal variables’ arrangements. We numerically test the idea by calculating the diagrams up to the sixth order for a representative many-fermion system—the uniform Fermi gas with Yukawa interaction. We observe massive sign cancellations in the high-order diagrams and find a significant portion of the overall cancellation originates from the summation of topologies. We then utilize the grouping scheme to formulate a highly efficient cluster DiagMC algorithm. Compared with the conventional approaches, our new algorithm improves the efficiency by $\sim 10^{5}$ at the sixth order.

We have made three contributions in this paper: i) We prove that in the imaginary-time and momentum representation, the charge conservation law is a topological property of Feynman diagrams. We then clarify its essential role in the diagrammatic sign structure. ii) We develop the protocol to quantify the sign cancellations from different mechanisms in high-order diagrams. It enables a systematic investigation of the sign-cancellation effects caused by the two symmetries. iii) A more efficient cluster DiagMC algorithm is proposed. Thanks to the symmetry analysis, we can optimize the grouping scheme systematically. The sign problem considerably is significantly alleviated as compared to the algorithm proposed in Ref.33.

We now sketch the content of the paper and outline how it is organized. In Sec. II, we describe a two-dimensional uniform Fermi gas with Yukawa potential, which is used to quantitatively investigate Feynman diagrams’ sign structure. In Sec. III, we mathematically formulate the concepts of overall and topological sign cancellation. We also clarify the relation between the topological sign cancellation and the efficiency of a typical cluster DiagMC algorithm. Sec. IV and V are dedicated to crossing symmetry and global $U(1)$ symmetry induced sign cancellations. The topological sign cancellation and efficiency of the cluster DiagMC are quantitatively investigated. Finally, we outline our conclusions and discuss the outlook for future development and applications in Sec. VI.

To study the sign structure of Feynman diagrams at a quantitative level, we consider a representative model, which is a 2D uniform spin-polarized interacting Fermi gas described by the Hamiltonian,

where $\hat{c}_{\bm{k}}^{\dagger}(\hat{c}_{\bm{k}})$ is the creation (annihilation) operator of a fermion with momentum $\bm{k}$, $\epsilon_{k}=k^{2}/(2m)-\mu$ is the kinetic energy shifted with the chemical potential $\mu$. $V(q)=4\pi e^{2}/(q^{2}+\lambda^{2})$ is the Yukawa interaction between fermions with $e$ the electric charge and $\lambda^{-1}$ the static screening length. The fermions in this model can be understood as the electrons with a renormalized Coulomb interactionChen and Haule (2019). We adapt the natural units $\hbar=c=\varepsilon_{0}=k_{B}=1$. With a given screening length $\lambda^{-1}$ and temperature $T$, the physics of the system is controlled by the characteristic length scale, namely the Wigner–Seitz radius $r_{s}=(1/\pi n)^{1/2}$ with $n$ the density of the fermions.

Hereby we focus on the physical observable polarization $P$, which is the linear response function of the system with respect to a screened external electric field. It is closely related to the dielectric function, Pauli spin susceptibility and charge compressibility of the system Giuliani and Vignale (2005). More specifically, one couples the system to a small external electric field $\delta E$,

where $\hat{n}$ is the charge density operator. We assume the external electric field doesn’t change with the time since we are mainly interested in the static polarization. The generating functional for the connected correlation function is the grand potential of the system $\Phi[\delta E]=\ln\operatorname{Tr}\exp\{-\beta\hat{H}[\delta E]\}$. In particular, the two particle correlation function (charge susceptibility) $\chi$ is given by,

where the indexes ${\bm{x}}_{1}$ and ${\bm{x}}_{2}$ are the spatial coordinates of the external perturbation. The polarization is the charge response function with respect to the screened external electric field, rather than the original field $\delta E$. It can be derived from the charge susceptibility,

When the fermionic interaction $V(\bm{q})$ is relatively weak, the polarization can be effectively calculated with the Feynman diagrammatic technique. One first performs a double power expansion of the generating functional $\Phi[\delta E]$ in both $V(\bm{q})$ and $\delta E$, then derives the power series of the charge susceptibility using Eq.(3). Each term in the power series can be vividly represented by a Feynman diagram. They are connected diagrams composed of bare propagator lines and bare interaction lines. All of them have two external vertices connecting two external perturbations. In the end, the polarization diagrams can be derived from the charge susceptibility diagrams using Eq.(4). Diagrammatically, it means one should exclude all reducible diagrams that would fall into two pieces if one interaction line were cut off. We show some of the first two orders of polarization diagrams in Fig. 2. We use the momentum and imaginary-time representation so that all diagram weights are real-valued.

To further improve the convergence of the diagrammatic series, we replace each bare Green’s function line in the diagrams with the renormalized Green’s function which resumes the Hatree-Fock self-energy,

where $\epsilon_{\bm{k}}={\bm{k}}^{2}/{2m}-\mu-\epsilon_{HF}(\bm{k})$ with $\epsilon_{HF}(\bm{k})$ the Hatree-Fock self-energy which doesn’t depend on the imaginary-time. To avoid double-counting in the diagrammatic series, all diagrams which contain at least one Hatree-Fock sub-diagram are removed.

In our numerical calculation, we fine-tune the chemical potential $\mu$ so that the Fermi momentum of the Green’s function Eq. 5 matches $k_{F}=(4/S)^{1/2}/r_{s}$, where the spin factor $S=1$ and the radius $r_{s}=\sqrt{2}$. As one includes higher-order quantum corrections, we expect the Fermi momentum, as well as the Wigner–Seitz radius, get renormalized. Therefore, both $k_{F}$ and $r_{s}$ should be slightly different from the actual physical values. It does not affect the claims in this paper. At the temperature $T=0.2E_{F}$, we calculate static polarization diagrams $P^{(n)}(q=0,\omega=0)$ up to order $n=6$. The result is shown in Fig. 3. With the above parameters, we observe exponential convergence with the truncated diagram order.

This section quantifies the concepts of overall and topological sign cancellation in Feynman diagrams. We also clarify the connection between topological sign cancellation and cluster diagrammatic Monte Carlo algorithms’ efficiency.

We consider a diagrammatic series for a positively defined physical quantity $F$ (for example, the polarization Eq.(4)),

where $f(n,\xi_{n},\bm{x})$ is the weight of an $n$-th order diagram with a topological index $\xi_{n}$ and a set of internal variables $\bm{x}$ (in this paper, they are the internal momenta and imaginary times). Due to the sign cancellation between diagrams, we expect $F^{n}$ to be much smaller than the weight function,

where sums the absolute weight of the diagrams.

We numerically calculate the ratio $F^{n}/F^{n}_{\rm abs}$, where $F^{n}$ is the power series of the polarization in Fig.4. As shown in the red curve in Fig.4, we find this ratio roughly exponentially decays to zero with increasing orders, indicating massive sign cancellations. As a result, while the absolute weight function $F^{n}_{\rm abs}$ factorially diverge (the number of Feynman diagrams diverge as $2^{n}n!$), the sign-canceled net contribution to the polarization exponentially converges. In literature, this effect is referred as the sign blessing  Prokof’ev and Svistunov (2008a). In what follows, we will define the overall sign cancellation as the ratio $F^{n}/F^{n}_{\rm abs}$.

We expect a significant amount of the overall sign cancellation can be traced back to the sign cancellation between different diagram topologies before the internal variables are integrated out. To measure the topological sign cancellation, one should classify the diagrams into sign-canceled topological clusters $C_{n}$, then calculate the absolute weight of the clusters rather than individual diagrams,

This weight function satisfies an exact inequality,

Of course, the magnitude of $F_{\rm clu}^{n}$ strongly depends on the choice of the topological cluster $C_{n}$ and the arrangement of internal variables. Our overall goal is to find the optimized topological clusters to make $F_{\rm clu}^{n}$ as small as possible. We calculate the ratio $F_{\rm clu}^{n}/F^{n}_{\rm abs}$ for the polarization using the clustering scheme discussed in the following section. As shown in the blue curve in Fig.4, we find the ratio $F_{\rm clu}^{n}/F^{n}_{\rm abs}\ll 1$, indicating a significant amount of the overall sign cancellation indeed has a topological origin.

Now we discuss the relation between the above weight functions and the efficiency of DiagMC algorithms. In conventional diagrammatic Monte Carlo approaches Prokof’ev and Svistunov (1998); Prokof’ev and Svistunov (2008a); Van Houcke et al. (2010); Kozik et al. (2010); Van Houcke et al. (2012); Deng et al. (2015), the weight function Eq.(7) is treated as the weight of the configuration space spanned by the toplogies, the internal variables as well as the order of diagrams. It defines a probability density, $\rho(n,\xi_{n},\bm{x})=\left|f(n,\xi_{n},\bm{x})\right|/F^{n}_{\rm abs}$, which can be efficiently sampled with a Metropolis algorithm. Supplemented with a normalization scheme, one can use the Monte Carlo estimator $f(n,\xi_{n},\bm{x})/\left|f(n,\xi_{n},\bm{x})\right|$ to calculate diagram weight $F^{n}$.

In contrast to the conventional algorithm, the recent cluster algorithms Rossi (2017); Rossi et al. (2018a); Chen and Haule (2019); Taheridehkordi et al. (2020a) propose to use the weight function Eq.(8) instead of Eq.(7). It is reported that the sign problem is greatly alleviated. We derive the following relation to explain such observation,

The detailed derivation is presented in Appendix References. Here $M$ is the number of Monte Carlo samples, $\epsilon_{\rm MC}$ is the statistical absolute error in a typical set up of DiagMC algorithm, and $Z$ is the weight of the configuration space. To achieve the desired relative accuracy, the number of the samples should be proportional to the squared ratio $\sim(Z/F^{n})^{2}$. Therefore, the smaller $Z$, the less sign problem. Indeed, the conventional DiagMC choose $Z=F^{n}_{\rm abs}$, while the cluster algorithms choose $Z=F^{n}_{\rm clus}$. we expect the latter improves the efficiency by a factor,

To achieve better efficiency, one should minimize the weight function $F^{n}_{\rm clu}$. Of course, this is quite challenging in general. In fact, we find that a randomly picked arrangement of internal variables can hardly improve the sign problem, namely $F^{n}\ll F^{n}_{\rm clu}\approx F^{n}_{\rm abs}$. To make $F^{n}_{\rm clu}$ small, one has to carefully group the topology of diagrams and arrange the internal variables, so that the diagrams with similar topologies maximally cancel with each other before the integration of internal variables $\mathbf{x}$. Thus, we are motivated to explore the underlying mechanism of the topological sign cancellation in the next two sections.

In this section, we explore how different symmetries lead to the topological sign cancellation between Feynman diagrams. We focus on two most common symmetries: the crossing symmetry in the two-fermion scattering amplitudes, and the $U(1)$ global gauge symmetry associated with the charge conservation.

In conclusion, we reveal the fermionic sign structure of high-order Feynman diagrams constrained by the crossing symmetry and the charge conservation law, which are two universal features of many-fermion systems. As a by-product, we prove that the charge conservation law is a topological property of Feynman diagrams in the imaginary-time/momentum representation. With modern numerical techniques, we investigate the significance of sign cancellations from the crossing symmetry and the conservation law.

The diagrammatic sign structure knowledge is used to optimize the internal-variable arrangements and construct the sign-canceled diagram groups. We then propose a cluster Diagrammatic Monte Carlo algorithm, which samples the sign-canceled diagram groups instead of individual diagrams. Compared to the conventional algorithm, we achieve an efficiency boost up to a factor $\sim 10^{5}$ at the diagram order $6$.

Besides the numerical application, identifying the sign-canceled groups provides important hints in constructing the relevant perturbative effective field theory of many-fermion systems. The diagrams in Fig.1, which forms a sign-canceled group, provides a particularly relevant example. The massive sign cancellation between those diagrams results in an emergent “unreasonably” small number, invalidating the conventional power-counting argument. It indicates that the relevance of perturbation in fermionic systems is better discussed in the context of sign-canceled groups rather than individual diagrams.

In the end, we summarize open questions for future work. There are questions associated with the charge conservation law in the topological structure of diagrams. Are there similar properties for other conservation laws (e.g., energy, momentum, and angular momentum)? Does the type of internal variables (space versus momentum, time versus frequency) matter? One probably needs to formulate a stronger version of the celebrated Baym-Kadanoff theory to address the above questions. There are also technical problems. For example, in Sec.IV.2, we point out that the momentum symmetrization operation $k\rightarrow k_{F}-k$ leads to a significant sign cancellation as indicated by Fig. 8. However, it is not clear how to implement it in the Monte Carlo algorithm to alleviate the sign problem.