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Physical review. B./Physical review. B

Linear response functions respecting Ward-Takahashi identity and fluctuation-dissipation theorem within the GW approximation

Hui Li, Zhipeng Sun, Yingze Su, Hai-Qing Lin, Huaqing Huang, Dingping Li

Published Feb 6, 2023• 5 citations
Publisher

Abstract

Fundamental equalities, such as the Ward-Takahashi identity (WTI) and the fluctuation-dissipation theorem (FDT), are important in the calculation of the response functions, which are defined as the variations of physical quantities with respect to the external sources. In this paper, the formalism of calculating the response functions according to their original definitions is presented, based on the generalized $GW$ (GGW) method which was developed for the electronic systems including spin-dependent interaction. This formalism automatically ensures the FDT, and is theorectically proved to respect the WTI. By contrast, the commonly used random phase approximation (RPA) within the GGW method violates both the WTI and the FDT, and the Bethe-Salpeter equation (BSE) satisfies the WTI but does not fulfill the FDT. The validity of this methodology is demonstrated on the two-dimensional one-band Hubbard model, and the results show that our formalism makes significant improvements over the RPA formula. Due to the similar computational cost to the BSE, our formalism is expected to be applied to realistic materials.

DOI
10.1103/physrevb.107.085106

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Fundamental equalities, such as the Ward-Takahashi identity (WTI) and the fluctuation-dissipation theorem (FDT), are important in the calculation of the response functions, which are defined as the variations of physical quantities with respect to the external sources. In this paper, the formalism of calculating the response functions according to their original definitions is presented, based on the generalized GWGWGW (GGW) method which was developed for the electronic systems including spin-dependent interaction. This formalism automatically ensures the FDT, and is theorectically proved to respect the WTI. By contrast, the commonly used random phase approximation (RPA) within the GGW method violates both the WTI and the FDT, and the Bethe-Salpeter equation (BSE) satisfies the WTI but does not fulfill the FDT. The validity of this methodology is demonstrated on the two-dimensional one-band Hubbard model, and the results show that our formalism makes significant improvements over the RPA formula. Due to the similar computational cost to the BSE, our formalism is expected to be applied to realistic materials.
Linear Response Functions Respecting Ward-Takahashi Identity and Fluctuation-Dissipation Theorem within GWGWGW Approximation
The calculation of response functions in correlated electronic systems is one of the most important problems in the condensed matter physics. To obtain a physical response function, preserving both the Ward-Takahashi identity and the fluctuation-dissipation theorem are crucial. Here we propose a self-consistent many body method within the GW framework to calculate the response functions based on the fluctuation-dissipation theorem, which also satisfies the Ward-Takahashi identity. The validity of this methodology is demonstrated on the two-dimensional one-band Hubbard model, where both the Ward-Takahashi identity and fluctuation-dissipation theorem are verified numerically. Moreover, comparing to the accurate spin susceptibility of the determinantal Monte Carlo approach, the results obtained from our method are quite satisfactory and the computational cost are greatly reduced.
A formally exact Bethe-Salpeter-like equation for the linear-response function is introduced with a kernel which depends only on the one frequency of the applied field. This is in contrast with the standard Bethe-Salpeter equation (BSE) which involves multiple-frequency integrals over the kernel and response functions. From the one-frequency kernel, known approximations are straightforwardly recovered. However, the present formalism lends itself to more powerful approximations. This is demonstrated with the exact analytical solution of the Hubbard molecule. Similarities and differences of the GW + BSE approach with the self-consistent random-phase approximation are also discussed.
A modified GWGWGW approximation to many-body systems is developed. The approximation has the same computational complexity as the traditional GWGWGW approach, but uses a different truncation scheme. This scheme neglects the high-order connected correlation functions. A covariant (preserving the Ward identities due to the charge conservation) scheme for the two-body correlators is employed, which holds the relation between the charge correlator and the charge susceptibility. The method is tested on the two-dimensional one-band Hubbard model. The results are compared with exact diagonalization, the GWGWGW approximation, the fluctuation-exchange (FLEX) theory, and determinantal Monte Carlo approach. The comparison for the (one-body) Green's function demonstrates that it is more precise in the strong-coupling regime (especially away from half filling) than the GWGWGW and FLEX approximations, which have a similar complexity. More importantly, this method indicates a Mott-Hubbard gap as the Hubbard UUU increases, whereas the GWGWGW and FLEX methods fail. In addition, the charge correlator obtained from the covariant scheme not only holds the consistency of the static charge susceptibility, but also makes a significant improvement over the random phase approximation calculations.
The fluctuation-dissipation theorem (FDT) is a central result in statistical physics, both for classical and quantum systems. It establishes a relationship between the linear response of a system under a time-dependent perturbation and time correlations of certain observables in equilibrium. Here we derive a generalization of the theorem which can be applied to any Markov quantum system and makes use of the symmetric logarithmic derivative (SLD). There are several important benefits from our approach. First, such a formulation clarifies the relation between classical and quantum versions of the equilibrium FDT. Second, and more important, it facilitates the extension of the FDT to arbitrary quantum Markovian evolutions, as given by quantum maps. Third, it brings out the full connection between the FDT and the Quantum Fisher information, the figure of merit in quantum metrology.
Using the simple (symmetric) Hubbard dimer, we analyze some important features of the GW approximation. We show that the problem of the existence of multiple quasiparticle solutions in the (perturbative) one-shot GW method and its partially self-consistent version is solved by full self-consistency. We also analyze the neutral excitation spectrum using the Bethe-Salpeter equation (BSE) formalism within the standard GW approximation and find, in particular, that 1) some neutral excitation energies become complex when the electron-electron interaction U increases, which can be traced back to the approximate nature of the GW quasiparticle energies; 2) the BSE formalism yields accurate correlation energies over a wide range of U when the trace (or plasmon) formula is employed; 3) the trace formula is sensitive to the occurrence of complex excitation energies (especially singlet), while the expression obtained from the adiabatic-connection fluctuation-dissipation theorem (ACFDT) is more stable (yet less accurate); 4) the trace formula has the correct behavior for weak (i.e., small U) interaction, unlike the ACFDT expression.
Fluctuation dissipation theorems (FDTs) connect the linear response of a physical system to a perturbation to the steady-state correlation functions. Until now, most of these theorems have been derived for finite-dimensional systems. However, many relevant physical processes are described by systems of infinite dimension in the Gaussian regime. In this work, we find a linear response theory for quantum Gaussian systems subject to time dependent Gaussian channels. In particular, we establish a FDT for the covariance matrix that connects its linear response at any time to the steady state two-time correlations. The theorem covers non-equilibrium scenarios as it does not require the steady state to be at thermal equilibrium. We further show how our results simplify the study of Gaussian systems subject to a time dependent Lindbladian master equation. Finally, we illustrate the usage of our new scheme through some examples. Due to broad generality of the Gaussian formalism, we expect our results to find an application in many physical platforms, such as opto-mechanical systems in the presence of external noise or driven quantum heat devices.