Dynamic correlations in electron liquids. I. General formalism.

Topological phase transitions in free fermion systems can be characterized by closing of single-particle gap and change in topological invariants. However, in the presence of electronic interactions, topological phase transitions are more complicated. In paper I of this series (arXiv:1510.07816), we have developed an efficient scheme to evaluate the topological invariants based on Green's function formalism. Here, in paper II, we demonstrate four interaction-drive topological phase transitions (TPTs) in two-dimensional (2D) interacting topological insulators (TIs) via large-scale quantum Monte Carlo (QMC) simulations, based on the scheme of evaluating topological invariants presented in paper I. Across these transitions, the defining symmetries of the TIs have been neither explicitly nor spontaneously broken. In the first two models, the topological invariants calculated from Green's function formalism succeed in characterizing interaction-driven TPTs. However, in the second two models, we find single-particle gap does not close and the topological invariants constructed from single-particle Green's function acquire no change across the TPTs. Unexpected breakdown of the Green's function formalism in constructing topological invariants is thus discovered. We thence classify the TPTs in interacting TIs into two categories: those have noninteracting correspondence can be characterized successfully by the topological invariants constructed from Green's functions, while for the others that do not have noninteracting correspondence, the Green's function formalism experiences a breakdown but more interesting and exciting phenomena, such as emergent collective critical modes at the transition, arise. Discussion on the success and breakdown of topological invariants constructed from the Green's function formalism in the context of symmetry protected topological (SPT) states is presented.

The dynamics of liquids is discussed with the help of time-dependent correlation functions. They are related to response functions by the fluctuation-dissipation theorem. This theorem enables to relate experimentally measured inelastic scattering data to the Fourier-transformed correlation and response functions. The Laplace transform of the correlation functions can be represented as a continuous fraction with suitable residual terms (memory functions). The projection formalism of Mori and Zwanzig is introduced.KeywordsCorrelation FunctionMemory FunctionRelaxation FunctionLiouville OperatorGeneralize Langevin EquationThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

A simple model of noninteracting electrons with a separable one-body potential is used to discuss the possible pole structure of single particle Green's functions for fermions on unphysical sheets in the complex frequency plane as a function of the system parameters. The poles in the exact Green's function can cross the imaginary axis, in contrast to recent claims that such a behaviour is unphysical. As the Green's function of the model has the same functional form as an approximate Green's function of coupled Luttinger liquids no definite conclusions concerning the concept of "confined coherence" can be drawn from the locations of the poles of this Green's function.

A new method for the calculation of the excitation spectrum of liquid $^{4}\mathrm{He}$ is developed based on the wave function ${\ensuremath{\chi}}_{\ensuremath{\alpha}}({x}_{1},{x}_{2})=〈0|T{\ensuremath{\Psi}}^{\ifmmode\dagger\else\textdagger\fi{}}({x}_{1})\ensuremath{\Psi}({x}_{2})|\ensuremath{\alpha}〉$ that is directly related to and completely determines the dynamic structure factor $S(\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}},\ensuremath{\omega})$ of liquid $^{4}\mathrm{He}$, and then employing this method a calculation of the sound spectrum of liquid $^{4}\mathrm{He}$ is performed taking explicitly into account the depletion effect although it is based on the McMillan's data for the ground-state momentum distribution of liquid $^{4}\mathrm{He}$. First, by employing the exponential-decay and effective-mass approximation for the single-particle Green's function, the two-time Bethe-Salpeter integral equation for ${\ensuremath{\chi}}_{\ensuremath{\alpha}}({x}_{1},{x}_{2})$ is transformed into the one-time Schr\odinger equation. This is thus an approximate Schr\odinger equation the solutions of which give $S(\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}},\ensuremath{\omega})$ of liquid $^{4}\mathrm{He}$. In the exponential-decay approximation, the numerators of the single-particle Green's function are the true momentum distribution ${N}_{ \mathrm{q}}^{\ensuremath{\rightarrow}}$ of the ground state of liquid $^{4}\mathrm{He}$, which then enters into the Schr\odinger equation in an essential way. Through this ${N}_{ \mathrm{q}}^{\ensuremath{\rightarrow}}$, both the presence of a condensate and the large depletion of particles from the zero-momentum state in $^{4}\mathrm{He}$ can be properly taken into account. And the effective mass of a helium atom is determined using the McMillan's data and the Lennard-Jones potential. The Schr\odinger equation is then solved in the limit $k\ensuremath{\rightarrow}0$ using the McMillan's data, and an excitation spectrum of $^{4}\mathrm{He}$ of the form ${[{\ensuremath{\epsilon}}_{\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}}^{\ensuremath{'}2}+\frac{2\ensuremath{\alpha}nv(\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}){\ensuremath{\epsilon}}_{\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}}^{\ensuremath{'}}}{(1+D)]}}^{\frac{1}{2}}$ in the limit $k\ensuremath{\rightarrow}0$ is obtained, which differs from the Bogoliubov form by the factor ${(1+D)}^{\ensuremath{-}1}$ and the replacement of the bare mass $m$ by the effective mass ${m}^{*}$. $D$, being proportional to $(1\ensuremath{-}\ensuremath{\alpha})n$, represents the depletion effect. The expansion of the above expression gives for the sound velocity of liquid $^{4}\mathrm{He}$ $c=190$ m/sec and verifies the positive phonon dispersion.

A fully consistent relativistic continuum random phase approximation (RCRPA) is constructed, where the contribution of the continuum spectrum to nuclear excitations is treated exactly by the single-particle Green's function technique. The full consistency of the calculations is achieved that the same effective Lagrangian is adopted for the ground state and the excited states. The negative energy states in the Dirac sea are also included in the single-particle Green's function in the no-sea approximation. The currents from the vector meson and photon exchanges and the Coulomb interaction in RCRPA are treated exactly. The spin-orbit interaction is included naturally in the relativistic frame. Numerical results of the RCRPA are checked with the constrained relativistic mean-field theory. We study the effects of the inconsistency, particularly the currents and Coulomb interaction in various collective multipole excitations.

Theory of a weakly non-ideal Bose gas in the canonical ensemble is developed without assumption of the C-number representation of the creation and annihilation operators with zero momentum. Instead of this assumption, we use the assumption on the C-number nature of the density operator N0 = a†0a0 with zero momentum. It is shown that the pole of the “density-density” Green function (DDGF), as well as the pole of the single-particle Green function (SPGF), exactly coincide with the Bogoliubov phonon-roton spectrum of excitations for both assumptions. This spectrum, as is known confirmed by many neutron and x-ray experimental measurements of the dynamic structure factor in He II, is straightly related to the DDGF. At the same time, we show that in the other case under consideration, when neither N0 nor a†0 and a0 are C-numbers, a gap can exist in SPGF. This gap in SPGF excitations is straightly related to the density of particles in the “condensate”. Therefore, the spectra of excitations for the DDGF and SPGF in the last case under consideration are different, in contrast to the Bogoliubov-type theory where these spectra are identical.

The exact reduced density-matrix functional is derived from the Luttinger-Ward functional of the single-particle Green's function. Thereby, a formal link is provided between diagrammatic many-body approaches using Green's functions on the one hand and theories based on many-body wave functions on the other. This link can be used to explicitly construct approximations for the density-matrix functional that are equivalent to standard diagrammatic re-summation techniques and to non-perturbative dynamical mean-field theory in particular. Contrary to functionals of the Green's-function, the exact density-matrix functional is convex and thus provides a true minimum principle which facilitates the calculation of the grand potential and derived equilibrium properties. The benefits of the proposed Green's-function-based density-matrix functional theory for geometrical structure optimization of strongly correlated materials are discussed.

We calculate the static and dynamic single-particle correlation functions in one-dimensional (1D) trapped Bose gases and discuss experimental measurements that can directly probe such correlation functions. Using a quantized hydrodynamic theory for the low energy excitations, we calculate both the static and dynamic single-particle correlation functions for a 1D Bose gas that is a phase-fluctuating quasi-condensate. For the static (equal-time) correlation function, our approximations and results are equivalent to those of Petrov, Shlyapnikov and Walraven. The Fourier transform of the static single-particle correlation function gives the momentum distribution, which can be measured using Doppler-sensitive Bragg scattering experiments on a highly elongated Bose gas. We show how a two-photon Raman out-coupling experiment can measure the characteristic features of the dynamic or time-dependent single-particle correlation function of a 1D Bose quasi-condensate.

We present a hardware-reconfigurable ansatz on Nq-qubits for the variational preparation of many-body states of the Anderson impurity model (AIM) with Nimp+Nbath=Nq/2 sites, which conserves total charge and spin z component within each variational search subspace. The many-body ground state of the AIM is determined as the minimum over all minima of O(Nq2) distinct charge-spin sectors. Hamiltonian expectation values are shown to require ω(Nq)<Nmeas.≤O(NimpNbath) symmetry-preserving, parallelizable measurement circuits, each amenable to postselection. To obtain the one-particle impurity Green's function we show how initial Krylov vectors can be computed via midcircuit measurement and how Lanczos iterations can be computed using the symmetry-preserving ansatz. For a single-impurity Anderson model with a number of bath sites increasing from one to seven, we show using numerical emulation that the ease of variational ground-state preparation is suggestive of linear scaling in circuit depth and subquartic scaling in optimizer complexity. We therefore expect that, combined with time-dependent methods for Green's function computation, our ansatz provides a useful tool to account for electronic correlations on early fault-tolerant processors. Finally, with a view towards computing real materials properties of interest like magnetic susceptibilities and electron-hole propagators, we provide a straightforward method to compute many-body, time-dependent correlation functions using a combination of time evolution, midcircuit measurement-conditioned operations, and the Hadamard test. Published by the American Physical Society 2025

We present a scalable single-particle framework to treat electronic correlation in molecules and materials motivated by Green's function theory. We derive a size-extensive Brillouin-Wigner perturbation theory from the single-particle Green's function by introducing the Goldstone self-energy. This new ground state correlation energy, referred to as Quasi-Particle MP2 theory (QPMP2), avoids the characteristic divergences present in both second-order Møller-Plesset perturbation theory and Coupled Cluster Singles and Doubles within the strongly correlated regime. We show that the exact ground state energy and properties of the Hubbard dimer are reproduced by QPMP2 and demonstrate the advantages of the approach for larger Hubbard models where the metal-to-insulator transition is qualitatively reproduced, contrasting with the complete failure of traditional methods. We apply this formalism to characteristic strongly correlated molecular systems and show that QPMP2 provides an efficient, size-consistent regularization of MP2.

Preconditioning is the most widely used and effective way for treating ill-conditioned linear systems in the context of classical iterative linear system solvers. We introduce a quantum primitive called fast inversion, which can be used as a preconditioner for solving quantum linear systems. The key idea of fast inversion is to directly block-encode a matrix inverse through a quantum circuit implementing the inversion of eigenvalues via classical arithmetics. We demonstrate the application of preconditioned linear system solvers for computing single-particle Green's functions of quantum many-body systems, which are widely used in quantum physics, chemistry, and materials science. We analyze the complexities in three scenarios: the Hubbard model, the quantum many-body Hamiltonian in the planewave-dual basis, and the Schwinger model. We also provide a method for performing Green's function calculation in second quantization within a fixed particle manifold and note that this approach may be valuable for simulation more broadly. Besides solving linear systems, fast inversion also allows us to develop fast algorithms for computing matrix functions, such as the efficient preparation of Gibbs states. We introduce two efficient approaches for such a task, based on the contour integral formulation and the inverse transform respectively.

We investigate the electron correlation in few‐electron closed‐shell atomic systems and similarly in few‐electron quantum dots under weak confinement. As usual we start with restricted Hartree–Fock (HF) calculations and add electron correlation in steps in a series of approximations based on the single particle Green's function approach: (i) second‐order Green function (GF); (ii) 2ph‐Tamm‐Dancoff approximation (TDA); and (iii) an extended version thereof which introduces ground‐state correlation into the TDA. Our studies exhibit similarities and differences between weakly confined quantum dots and standard atomic systems. The calculations support the application of HF, GF, and TDA techniques in the modeling of three‐dimensional quantum dot systems. The observed differences emphasize the significance of confinement and electronic features unique to quantum dots, such as the increased binding of electrons with higher angular momentum and thus—compared to atomic systems—modified shell‐filling sequences. © 2008 Wiley Periodicals, Inc. Int J Quantum Chem, 2008

An approximate solution of a Dyson equation satisfied by a single particle Green's function in many-impurity problems in solids is obtained for an extended impurity perturbation potential. We assume that there is no overlapping of any two impurity potentials and that the range of the impurity potential is arbitrary. The coordinate representation of the Dyson equation is solved by transforming an iterative solution of the equation into a form with no repetitive summation indices appearing in any term of given order. By neglecting fluctuations of the impurity distribution, a set of equations are derived to determine the Green's function and the self-energy in a self-consistent manner. A group-theoretic consideration of the self-energy and some sum rules are given.

A general formalism for the calculation of free energies of inhomogeneous dielectric and electrolytic systems

We introduce a formalism for calculating dynamic response functions using experimental single-particle Green’s functions. As an illustration of this procedure, we estimate the dynamic spin-response of the cuprate superconductor Bi_2Sr_2CaCu_2O_8+δ. We find good agreement with superconducting state neutron data, in particular, the (π,π) resonance with its unusual “hourglass” shaped dispersion. We anticipate that our formalism will also be useful in interpreting results from other spectroscopies, such as optical and Raman responses.