MATERIALS SCIENCE AND CHEMISTRY
Multipoint Vertex Functions in Strongly Correlated Electron Systems
Principal Investigator:
Prof. Dr. Jan von Delft
Local Project ID:
pn25ze
HPC Platform used:
SuperMUC-NG at LRZ
Date published:
Introduction
In quantum many-body physics, correlation functions, usually abbreviated to correlators, are the quantum expectation values of operators acting on different space- time points. When a correlator involves f space-time points, it is called an f-point correlator, and when f is larger than two, it is a multipoint correlator. When the system of interest is electrons in a solid and the operators are electron creation and annihilation operators, the correlators are also called electron Green's functions. (Here "creation" means the introduction of an electron to a solid from the outside; "annihilation" is the opposite operation.) The Green's functions are important since they determine various dynamical responses and spectral properties of the solid. For example, the single- electron function, which is a two-point correlator defined by a pair of annihilation and creation operators, is probed by angle-resolved photoemission spectroscopy (ARPES) spectra. On the other hand, the two-electron function, a four-point correlator defined by two pairs of annihilation and creation operators, is relevant to the structure factor in inelastic neutron scattering as well as the optical conductivity. Four-point vertex functions are key ingredients of two-electron Green's functions and describe the effective interaction between two electrons in the presence of a many-electron background. Hence vertex functions are important in quantum field theory approaches to quantum many-body problems. Enigmatic quantum phenomena in strongly correlated electron systems, such as high-temperature superconductivity and quantum criticality, typically result from strongly interacting electrons at low temperatures. In such cases, the precise computation of the Green's and vertex functions is notoriously hard; indeed, it constitutes one of the holy grails in condensed matter physics. In this project, we have focused on computing four-point vertex functions in a class of quantum many-body systems, called quantum impurity models, as the first steps towards the holy grail. An impurity model consists of a subsystem in which electrons are interacting ("impurity") and the rest in which electrons are not interacting ("bath"). This setting of the model may sound specific, but according to the dynamical mean-field theory (DMFT) and its extensions, impurity models are useful simplifications of the lattice description of strongly correlated electrons. Once the four-point vertex function at the impurity is computed, it can be used to derive reliable estimates of Green's functions of the original lattice system by solving a set of equations from quantum field theory (see Outlook for detail).
Figure 1: Feynman diagrams of the four-point symmetric improved estimator (sIE). The four-point vertex function on the left-hand side is decomposed into the diagrams on the right-hand side under the sIE, and each diagram corresponds to a different asymptotic frequency dependence of the vertex.
Results and Methods
To compute the three- and four-point vertex functions of quantum impurity models, we use the multipoint numerical renormalization group (mpNRG) method [1,2] developed by some of us. It is a tensor network method specialized for solving impurity models; here the many- body energy eigenstates of the Hamiltonian are represented by matrix product states (MPSs) that are one-dimensional tensor network states. Our previous works [1,2] show that the mpNRG gives accurate results of the four-point vertex functions in the Matsubara formalism (MF), where the functions' arguments are imaginary frequencies. It can access much lower temperatures than the state-of-the-art quantum Monte Carlo (QMC) methods do. Of course, the imaginary frequencies are mere theoretical tools, so to compare with experimental measurements, one should be able to compute the vertex functions in the Keldysh formalism (KF), where the functions' arguments are real frequencies. To obtain reliable results of KF vertex functions, the use of an improved estimator (IE) is crucial. An IE is an analytically exact expression of the quantity of interest (e.g., vertex functions) in terms of various correlation functions. In its numerical evaluation, numerical artifacts in the correlation functions (e.g., discretization artifacts in the NRG method) are largely canceled out, leading to higher accuracy. However, the previously proposed IEs for vertex functions, including the asymmetric IE (aIE) used in Refs. [1,2], were not adequate for our purposes, especially for the real-frequency KF. For example, the aIE improves, due to the asymmetric nature of its formu- lation, the accuracy only for some components of KF vertex functions. In this project, we developed symmetric improved estimators (sIEs) [3] that yield accurate estimates of the multipoint vertex functions in KF. Our sIEs are derived by applying equations of motion recursively, leading to lower-point correlators. As a result, sIEs for a four-point vertex are given in terms of four-, three-, and two-point functions (see Fig. 1). We demonstrate the advantages of the sIE by computing the KF four-point vertex of the single-impurity Anderson model, which is a paradigmatic impurity model. Figure 2 shows how the sIE improves the quality of the vertex function by comparing the vertex functions obtained from the direct amputation, the aIE, and the sIE. The numerical artifacts are adequately eliminated only for the sIE. While the use of the sIEs for vertex functions has been found to give numerically much more accurate results than the aIEs, this gain in accuracy comes with increased computational costs. Fortunately, the evaluation of the sIEs can be parallelized over a large number of threads and nodes, which made SuperMUC-NG an ideal platform for our project. The most CPU- and memory-intensive part in calculating a four-point vertex is the computation of four-point partial spectral functions (PSFs), which are three-dimensional arrays that contain the core information of the four-point correlators appearing in the sIE. There are 120 four-point PSFs for different tuples of operators. The PSFs are independent, so they can be computed on different nodes with small communication overhead. Moreover, by slicing the PSFs, the computation can be further parallelized so that different slices are treated by different workers. Within each worker, the most of CPU time is spent on contracting tensors that represent the operators and the energy eigenstates. To obtain the sIE results shown in Fig. 2, we used roughly 1,000 nodes for 60 hours on SuperMUC-NG, which is hard to accomplish on the other HPC systems.
Figure 2: The local four-point vertex function of the single- impurity Anderson model in the Matsubara formalism (MF) and Keldysh formalism (KF). The three rows show the results obtained by direct amputation of the correlator, by using the asymmetric improved estimator (aIE), and by using the symmetric improved estimator (sIE) derived in our project. The first, second, and third columns show the MF vertex (which is purely real), the real part of the causal (c = ----) component of the KF vertex, and its imaginary part, respectively.
Ongoing Research/Outlook
Within our theoretical framework, the four-point vertex of the effective impurity model, which is the output of mpNRG and sIE calculations, will be fed into a method for solving field-theoretical equations as input. We are currently aiming at solving the parquet equations, a set of self-consistent equations that involve four-point vertex functions. A related SuperMUC-NG project, coined multiloop functional renormalization group (mfRG), is led by other members within the group of Prof. Jan von Delft. The solution of the parquet equations with the impurity vertex input amounts to the "parquet dynamical vertex approximation (DrA)", which is believed to be a powerful method for studying strongly correlated electrons in two dimensions. We plan to investigate various open prob- lems for future work within this field: strange metallicity, quantum criticality, and high-temperature superconductivity, to name a few.
References
[1] F. B. Kugler et al., Phys. Rev. X 11, 041006 (2021).
[2] S.S. B. Lee et al., Phys. Rev. X 11, 041007 (2021).
[3] J.M. Lihm et al., Phys. Rev. B 109, 125138 (2024).