Single-particle Green Function Research Articles

Exact description of quantum stochastic models as quantum resistors

We study the transport properties of generic out-of-equilibrium quantum systems connected to fermionic reservoirs. We develop a new method, based on an expansion of the current in terms of the inverse system size and out of equilibrium formulations such as the Keldysh technique and the Meir-Wingreen formula. Our method allows a simple and compact derivation of the current for a large class of systems showing diffusive/ohmic behavior. In addition, we obtain exact solutions for a large class of quantum stochastic Hamiltonians (QSHs) with time and space dependent noise, using a self consistent Born diagrammatic method in the Keldysh representation. We show that these QSHs exhibit diffusive regimes which are encoded in the Keldysh component of the single particle Green's function. The exact solution for these QSHs models confirms the validity of our system size expansion ansatz, and its efficiency in capturing the transport properties. We consider in particular three fermionic models: i) a model with local dephasing ii) the quantum simple symmetric exclusion process model iii) a model with long-range stochastic hopping. For i) and ii) we compute the full temperature and dephasing dependence of the conductance of the system, both for two- and four-points measurements. Our solution gives access to the regime of finite temperature of the reservoirs which could not be obtained by previous approaches. For iii), we unveil a novel ballistic-to-diffusive transition governed by the range and the nature (quantum or classical) of the hopping. As a by-product, our approach equally describes the mean behavior of quantum systems under continuous measurement.

Surface exceptional points in a topological Kondo insulator

Correlated materials have appeared as an arena to study non-Hermitian effects as typically exemplified by the emergence of exceptional points. We show here that topological Kondo insulators are an ideal platform for studying these phenomena due to strong correlations and surface states exhibiting a nontrivial spin texture. Using numerical simulations, we demonstrate the emergence of exceptional points in the single-particle Green's function on the surface of the material while the bulk is still insulating. We reveal how quasiparticle states with long lifetimes are created on the surface by non-Hermitian effects while the Dirac cones are smeared, which explains the surface Kondo breakdown at which heavy Dirac cones disappear from the single-particle spectrum and are replaced by light states. We further show how the non-Hermiticty changes the spin texture inherent in the surface states, which might help identify exceptional points experimentally. Besides confirming the existence of non-Hermitian effects on the surface of a topological Kondo insulator, this paper demonstrates how the eigenstates and eigenvalues of the effective non-Hermitian matrix describing the single-particle Green's function help understand the properties of correlated materials.

Local density of states and scattering rates across the many-body localization transition

Characterizing the many-body localization (MBL) transition in strongly disordered and interacting quantum systems is an important issue in the field of condensed matter physics. We study the single particle Green's functions for a disordered interacting system in one dimension using exact diagnonalization in the infinite temperature limit and provide strong evidence that single particle excitations carry signatures of delocalization to MBL transition. In the delocalized phase, the typical values of the local density of states and the scattering rate are finite while in the MBL phase, the typical values for both the quantities become vanishingly small. The probability distribution functions of the local density of states and the scattering rate are broad log-normal distributions in the delocalized phase while the distributions become very narrow and sharply peaked close to zero in the MBL phase. We also study the eigenstate Green's function for all the many-body eigenstates and demonstrate that both, the energy resolved typical scattering rate and the typical local density of states, can track the many-body mobility edges.

Diffusion and thermalization in a boundary-driven dephasing model

We study a model of noninteracting spinless fermions coupled to local dephasing and boundary drive and described within a Lindblad master equation. The model features an interplay between infinite temperature thermalization due to bulk dephasing and a nonequilibrium stationary state due to the boundary drive and dissipation. We revisit the linear and nonlinear transport properties of the model, featuring a crossover from ballistic to diffusive scaling, and compute the spectral and occupation properties encoded in the single-particle Green's functions, that we compute exactly using the Lindblad equations of motion in spite of the interacting nature of the dephasing term. We show that the distribution function in the bulk of the system becomes frequency independent and flat, consistent with infinite temperature thermalization, while near the boundaries it retains strong nonequilibrium features that reflect the continuous injection and depletion of particles due to driving and dissipation.

Boundary condition independence of non-Hermitian Hamiltonian dynamics

Non-Hermitian skin effect, namely that the eigenvalues and eigenstates of a non-Hermitian tight-binding Hamiltonian have significant differences under open or periodic boundary conditions, is a remarkable phenomenon of non-Hermitian systems. Inspired by the presence of the non-Hermitian skin effect, we study the evolution of wave-packets in non-Hermitian systems, which can be determined using the single-particle Green's function. Surprisingly, we find that in the thermodynamical limit, the Green's function does not depend on boundary conditions, despite the presence of skin effect. We proffer a general proof for this statement in arbitrary dimension with finite hopping range, with an explicit illustration in the non-Hermitian Su-Schrieffer-Heeger model. We also explore its applications in non-interacting open quantum systems described by the master equation, where we demonstrate that the evolution of the density matrix is independent of the boundary condition.

Fast inversion, preconditioned quantum linear system solvers, fast Green's-function computation, and fast evaluation of matrix functions

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.

Phonon transport in disordered alloys: A Multiple-scattering approach

I present a formalism to calculate the configuration-averaged lattice thermal conductivity for substitutional random alloys. The method is based on multiple-scattering approach which can capture the effect of disorder-induced configuration fluctuations on single-particle and two-particle phonon Green functions in random alloys. The randomness of the system is dealt within the augmented space theorem. This is combined with a generalized Feynman diagrammatic technique to extract various useful results in the form of mathematical expressions. I show the structure of all possible scattering diagrams up to the fourth order and subsequently illustrate how to obtain Dyson's equation from a resummation of the diagrammatic series. I also study how disorder scattering affects two-particle Green functions associated with thermal response. It was shown explicitly how the disorder scattering renormalizes both the phonon propagators as well as the heat currents. I derive the relation between these renormalized heat currents and the self-energy of the propagators. I have also studied a different class of scattering diagrams which are not related to the self-energy but rather to the vertex corrections. The configuration-averaging scheme is straightforward to apply to other relevant quantities such as joint density of states, thermal diffusivity, etc., for random alloys. The developed formalism is applied to a realistic ${\mathrm{Au}}_{1\ensuremath{-}x}{\mathrm{Fe}}_{x}$ binary alloy. The effect of disorder-induced corrections (as compared to the simple virtual crystal approximation) turns out to be appreciable in this alloy. The configuration-averaged lattice thermal conductivity for ${\mathrm{Au}}_{50}{\mathrm{Fe}}_{50}$ shows a quadratic behavior in low-temperature regime ($T\ensuremath{\le}30$ K), which increases smoothly to a $T$-independent saturated value at high $T$. Simulated thermal diffusivity $D(\ensuremath{\nu})$ helps to numerically estimate the mobility $\mathrm{edge}\phantom{\rule{0.28em}{0ex}}({\ensuremath{\nu}}_{c})$ which in turn evaluates the fraction of localized states. $D(\ensuremath{\nu})$ is found to decrease smoothly (almost linearly) in the high-$\ensuremath{\nu}$ range, which when fitted to ${({\ensuremath{\nu}}_{c}\ensuremath{-}\ensuremath{\nu})}^{\ensuremath{\alpha}}$ gives the critical $\mathrm{exponent}\phantom{\rule{0.28em}{0ex}}(\ensuremath{\alpha})$ to be 1.018 for ${\mathrm{Au}}_{50}{\mathrm{Fe}}_{50}$ alloy. This agrees fairly well with the scaling and other theories of Andersen localization.

Single particle properties of the two-dimensional Hubbard model for real frequencies at weak coupling: Breakdown of the Dyson series for partial self-energy expansions

We generate the perturbative expansion of the single particle Green's function and related self-energy for a half-filled single-band Hubbard model on a square lattice. We invoke algorithmic Matsubara integration to evaluate single-particle quantities for real and Matsubara frequencies and verify results through comparison to existing data on the Matsubara axis. With low-order expansions at weak coupling we observe a number of outcomes expected at higher orders: the opening of a gap, pseudogap behavior, and Fermi-surface reconstruction. Based on low-order perturbations, we consider the phase diagram that arises from truncated expansions of the self-energy and Green's function and their relation via the Dyson equation. From Matsubara axis data, we observe insulating behavior in direct expansions of the Green's function, whereas the same order of truncation of the self-energy produces metallic behavior. This observation is supported by additional calculations for real frequencies. We attribute this difference to the order in which diagrams are implicitly summed in the Dyson series. By separating the reducible and irreducible contributions at each order we show that the reducible diagrams implicitly summed in the Dyson equation lead to incorrect physics in the half-filled Hubbard model. Our observations for this particular case lead us to question the utility of the Dyson equation for any problem that shows a disparity between reducible and irreducible contributions to the expansion of the Green's function.

Variational quantum eigensolver for dynamic correlation functions

Recent practical approaches for the use of current generation noisy quantum devices in the simulation of quantum many-body problems have been dominated by the use of a variational quantum eigensolver (VQE). These coupled quantum-classical algorithms leverage the ability to perform many repeated measurements to avoid the currently prohibitive gate depths often required for exact quantum algorithms, with the restriction of a parameterized circuit to describe the states of interest. In this work, we show how the calculation of zero-temperature dynamic correlation functions defining the linear response characteristics of quantum systems can also be recast into a modified VQE algorithm, which can be incorporated into the current variational quantum infrastructure. This allows for these important physical expectation values describing the dynamics of the system to be directly converged on the frequency axis, and they approach exactness over all frequencies as the flexibility of the parameterization increases. The frequency resolution hence does not explicitly scale with gate depth, which is approximately twice as deep as a ground state VQE. We apply the method to compute the single-particle Green's function of ab initio dihydrogen and lithium hydride molecules, and demonstrate the use of a practical active space embedding approach to extend to larger systems. While currently limited by the fidelity of two-qubit gates, whose number is increased compared to the ground state algorithm on current devices, we believe the approach shows potential for the extraction of frequency dynamics of correlated systems on near-term quantum processors.

Elementary band representations for the single-particle Green's function of interacting topological insulators

We discuss the applicability of elementary band representations (EBRs) to diagnose spatial- and time-reversal-symmetry protected topological phases in interacting insulators in terms of their single-particle Green's functions. We do so by considering an auxiliary non-interacting system $H_{\textrm{T}}(\mathbf{k}) = -G^{-1}(0, \mathbf{k})$, known as the topological Hamiltonian, whose bands can be labeled by EBRs. This labeling is stable if neither (i) the gap in the spectral function at zero frequency closes, (ii) the Green's function has a zero at zero frequency or (iii) the Green's function breaks a protecting symmetry. We demonstrate the use of EBRs applied to the Green's function on the one-dimensional Su-Schrieffer-Heeger model with Hubbard interactions, which we solve by exact diagonalization for a finite number of unit cells. Finally, the use of EBRs for the Green's function to diagnose so-called symmetry protected topological phases is discussed, but remains an open question.

Local marker for interacting topological insulators

Topological states of matter were first introduced for non-interacting fermions on an infinite uniform lattice. Since then, substantial effort has been made to generalize these concepts to more complex settings. Recently, local markers have been developed that can describe the topological state of systems without translational symmetry and well-defined gap. However, no local marker for interacting matter has been proposed yet that is capable of directly addressing an interacting system. Here we suggest such a many-body local marker based on the single-particle Green's function. Using this marker we identify topological transitions in finite lattices of a Chern insulator with Anderson disorder and Hubbard interactions. Importantly, our proposal can be straightforwardly generalised to non-equilibrium systems.

Renormalized Singles Green's Function in the T-Matrix Approximation for Accurate Quasiparticle Energy Calculation.

We combine the renormalized singles (RS) Green's function with the T-matrix approximation for the single-particle Green's function to compute quasiparticle energies for valence and core states of molecular systems. The GRST0 method uses the RS Green's function that incorporates singles contributions as the initial Green's function. The GRSTRS method further calculates the generalized effective interaction with the RS Green's function by using RS eigenvalues in the T-matrix calculation through the particle-particle random phase approximation. The GRSTRS method provides significant improvements over one-shot methods G0T0 and G0W0 as demonstrated in calculations for GW100 and CORE65 test sets. It also systematically eliminates the dependence of G0T0 on the choice of density functional approximations. For valence states, the GRSTRS method provides excellent accuracy, which is better than that of G0T0 and G0W0. For core states, the GRSTRS method identifies correct peaks in the spectral function and significantly outperforms G0T0 on core-level binding energies (CLBEs) and relative CLBEs.

Impurities in systems of noninteracting trapped fermions

We study the properties of spin-less non-interacting fermions trapped in a confining potential in one dimension but in the presence of one or more impurities which are modelled by delta function potentials. We use a method based on the single particle Green's function. For a single impurity placed in the bulk, we compute the density of the Fermi gas near the impurity. Our results, in addition to recovering the Friedel oscillations at large distance from the impurity, allow the exact computation of the density at short distances. We also show how the density of the Fermi gas is modified when the impurity is placed near the edge of the trap in the region where the unperturbed system is described by the Airy gas. Our method also allows us to compute the effective potential felt by the impurity both in the bulk and at the edge. In the bulk this effective potential is shown to be a universal function only of the local Fermi wave vector, or equivalently of the local fermion density. When the impurity is placed near the edge of the Fermi gas, the effective potential can be expressed in terms of Airy functions. For an attractive impurity placed far outside the support of the fermion density, we show that an interesting transition occurs where a single fermion is pulled out of the Fermi sea and forms a bound state with the impurity. This is a quantum analogue of the well-known Baik-Ben Arous-Péché (BBP) transition, known in the theory of spiked random matrices. The density at the location of the impurity plays the role of an order parameter. We also consider the case of two impurities in the bulk and compute exactly the effective force between them mediated by the background Fermi gas.

Motion of an impurity in a two-leg ladder

We study the motion of an impurity in a two-leg ladder interacting with two fermionic baths along each leg, a simple model bridging cold atom quantum simulators with an idealised description of the basic transport processes in a layered heterostructure. Using the linked-cluster expansion we obtain exact analytical results for the single-particle Green's function and find that the long-time behaviour is dominated by an intrinsic orthogonality catastrophe associated to the motion of the impurity in each one-dimensional chain. We explore both the case of two identical legs as well as the case where the legs are characterised by different interaction strengths: in the latter case we observe a subleading correction which can be relevant for intermediate-time transport at an interface between different materials. In all the cases we do not find significant differences between the intra- and inter-leg Green's functions in the long-time limit.

Dual boson diagrammatic Monte Carlo approach applied to the extended Hubbard model

In this work we introduce the Dual Boson Diagrammatic Monte Carlo technique for strongly interacting electronic systems. This method combines the strength of dynamical mean-filed theory for non-perturbative description of local correlations with the systematic account of non-local corrections in the Dual Boson theory by the diagrammatic Monte Carlo approach. It allows us to get a numerically exact solution of the dual boson theory at the two-particle local vertex level for the extended Hubbard model. We show that it can be efficiently applied to description of single particle observables in a wide range of interaction strengths. We compare our exact results for the self-energy with the ladder Dual Boson approach and determine a physical regime, where description of collective electronic effects requires more accurate consideration beyond the ladder approximation. Additionally, we find that the order-by-order analysis of the perturbative diagrammatic series for the single-particle Green's function allows to estimate the transition point to the charge density wave phase.

Bosonization of the Q=0 continuum of Dirac fermions

We develop a bosonization formalism that captures non-perturbatively the interaction effects on the $\mathbf{Q}=0$ continuum of excitations of nodal fermions above one dimension. Our approach is a natural extension of the classic bosonization scheme for higher dimensional Fermi surfaces to include the $\mathbf{Q}=0$ neutral excitations that would be absent in a single-band system. The problem is reduced to solving a boson bilinear Hamiltonian. We establish a rigorous microscopic footing for this approach by showing that the solution of such boson bilinear Hamiltonian is exactly equivalent to performing the infinite sum of Feynman diagrams associated with the Kadanoff-Baym particle-hole propagator that arises from the self-consistent Hartree-Fock approximation to the single particle Green's function. We apply this machinery to compute the interaction corrections to the optical conductivity of 2D Dirac Fermions with Coulomb interactions reproducing the results of perturbative renormalization group at weak coupling and extending them to the strong coupling regime.

On the Self-Consistency Conditions in the Statistical Thermodynamics of the Coulomb System

The thermodynamic potential for a homogeneous and isotropic equilibrium nonrelativistic system of electrons and nuclei interacting with each other according to the Coulomb law is considered. Using the virial theorem and exact relations for the Green functions, it is shown that the thermodynamic functions of the Coulomb system are uniquely determined by the single-particle Green functions of electrons and nuclei outside the framework of perturbation theory. On this basis, self-consistency conditions are formulated for the thermodynamic functions that determine the applicability of the approximations used for single-particle Green functions.

Minimal effective Gibbs ansatz: A simple protocol for extracting an accurate thermal representation for quantum simulation

Quantum Gibbs state sampling algorithms generally suffer from either scaling exponentially with system size or requiring specific knowledge of spectral properties \textit{a priori}. Also, these algorithms require a large overhead of bath or scratch/ancilla qubits. We propose a method, termed the minimal effective Gibbs ansatz (MEGA), which uses a quantum computer to determine a minimal ensemble of pure states that accurately reproduce thermal averages of typical observables. This technique employs properties of correlation functions that can be split into a lesser and greater part; here, we primarily focus on single-particle Green's functions. When properly measured, these correlation functions provide a simple test to indicate how close a given pure state or ensemble of pure states are to providing accurate thermal expectation values. Further, we show that when properties such as the eigenstate thermalization hypothesis hold, this approach leads to accurate results with a sparse ensemble of pure states; sometimes only one suffices. We illustrate the ansatz using exact diagonalization simulations on small clusters for the Fermi-Hubbard and Hubbard-like models. Even if MEGA becomes as computationally complex as other Gibbs state samplers, it still gains an advantage due to its ease of implementation without any \textit{a priori} information about the Hamiltonian and in the efficient allocation of available qubits by eliminating bath qubits and using a minimal number of ancilla.

Combined spontaneous symmetry-breaking and symmetry-protected topological order from cluster charge interactions

The study of symmetry-protected topological states in presence of electron correlations has recently aroused great interest as rich and exotic phenomena can emerge. Here, we report a concrete example by employing large-scale unbiased quantum Monte Carlo study of the Kane-Mele model with cluster charge interactions. The ground-state phase diagram for the model at half filling is established. Our simulation identifies the coexistence of a symmetry-protected topological order with a symmetry-breaking Kekul$\acute{e}$ valence bond order and shows that the spontaneous symmetry-breaking is accompanied by an interaction-driven topological phase transition (TPT). This TPT features appearance of zeros of single-particle Green's function and gap closing in spin channel rather than single-particle excitation spectrum, and thus has no mean-field correspondence.

Phenomenological single-particle Green's function for the pseudogap and superconducting phases of high- Tc cuprates

We present a phenomenological Green's function to characterize the superconducting and pseudogap phases of the cuprates based on a microscopic theory of doped Mott insulators. In this framework, the "Fermi arc" and "kink" phenomena observed by angle-resolved photoemission spectroscopy (ARPES) experiments in the pseudogap phase can be systematically explained as a function of doping, which is further connected to the two-gap feature in the superconducting phase with dichotomy between the nodal and antinodal physics. We demonstrate that a phase-string-induced fractionalization plays the key role in giving rise to such a peculiar Green's function with a unique two-component structure.