Diagrammatic Perturbation Theory Research Articles

Phonon Thermal Transport in UO_{2} via Self-Consistent Perturbation Theory.

Computing thermal transport from first-principles in UO_{2} is complicated due to the challenges associated with Mott physics. Here, we use irreducible derivative approaches to compute the cubic and quartic phonon interactions in UO_{2} from first principles, and we perform enhanced thermal transport computations by evaluating the phonon Green's function via self-consistent diagrammatic perturbation theory. Our predicted phonon lifetimes at T=600 K agree well with our inelastic neutron scattering measurements across the entire Brillouin zone, and our thermal conductivity predictions agree well with previous measurements. Both the changes due to thermal expansion and self-consistent contributions are nontrivial at high temperatures, though the effects tend to cancel, and interband transitions yield a substantial contribution.

Extracting spinon self-energies from two-dimensional coherent spectroscopy

Two-dimensional coherent spectroscopy (2DCS) is a nonlinear spectroscopy technique capable of identifying whether apparent continua in linear response are made out of multiplets of sharp deconfined quasiparticles. This makes it a potent tool for experimental identification of fractionalized phases. Previous discussions have focused on limits where the quasiparticles in question are infinitely long lived. In this paper we discuss 2DCS in the regime where the fractionalized quasiparticles can themselves decay. We introduce a powerful path-integral-based approach, whereby the computation of nonlinear susceptibilities reduces to an efficient exercise in diagrammatic perturbation theory. We apply this method to compute the 2DCS response of the one-dimensional transverse field Ising model, in the presence of integrability-breaking perturbations. We discuss aspects of the self-energy of the fractionalized quasiparticles that may be extracted via 2DCS, such as the momentum-dependent decay rate.

Inverse Faraday effect in massive Dirac electrons

We study the inverse Faraday effect (IFE) in a Dirac Hamiltonian with random impurities using Keldysh formalism and diagrammatic perturbation theory. The mass term in the Dirac Hamiltonian is essential for IFE, where the spin magnetic moment induced by circularly polarized light is proportional to the frequency of the incident light within the THz regime. For massive Dirac electrons, the corrections due to short-range impurities on spin magnetic moment vertex exhibits mixing of the spin magnetic moment vertex and spin angular momentum vertex. The spin magnetic density response is divergently enhanced by the vertex corrections near the band edge, indicting a long-range diffusion of spin density profile in massive Dirac electrons.

Two bridge-particle-mediated RET between chiral molecules.

The problem of resonance energy transfer between a pair of chiral molecules mediated by two electrically polarizable bridging particles is solved using molecular quantum electrodynamics theory. In this framework, a single virtual photon propagates between any two-coupled entities and is responsible for the conveyance of excitation energy from the emitter to the absorber. Electric dipole and quadrupole, and magnetic dipole couplings linear in the Maxwell fields are employed for donor and acceptor, while each mediator scatters two virtual photons and responds quadratically to the electric displacement field via its electric dipole polarizability. This enables fourth-order diagrammatic perturbation theory to be used to compute the probability amplitude for the process. Individual multipole moment contributions to the Fermi golden rule rate are then extracted for oriented and isotropic systems. Discriminatory transfer rates arise when either the donor or the acceptor are electric-magnetic dipole and the other has a pure multipole moment, or when both are chiral, with mixed electric dipole-quadrupole contributions vanishing in the fluid phase. The bridge-mediated transfer rate is found to be a maximum for a collinear geometry. Moreover, a multi-level model of the mediator is necessary for energy migration. Asymptotically limiting rates for arbitrary and collinear geometries are also obtained for one center purely electric dipolar and the other purely quadrupolar, or both donor and absorber purely quadrupolar. Understanding is gained of radiationless and radiative transfer mechanisms between chiral moieties in a dielectric medium.

Field-theoretical approach to open quantum systems and the Lindblad equation

We develop a systematic field-theoretical approach to open quantum systems based on condensed-matter many-body methods. The time evolution of the reduced density matrix for the open quantum system is determined by a transmission matrix. Developing diagrammatic perturbation theory, invoking Wick's theorem in connection with a Caldeira-Leggett quantum oscillator environment in thermal equilibrium, the transmission matrix satisfies a Dyson equation characterized by an irreducible kernel. Unlike the Nakajima-Zwanzig and standard approaches, the Dyson equation is equivalent to a general non-Markovian master equation for the reduced density matrix, incorporating secular effects and independent of the initial preparation. The kernel is determined by a systematic diagrammatic expansion in powers of the interaction. We consider the Born approximation for the kernel. Applying a condensed-matter pole or, equivalently, a quasiparticle-type approximation, equivalent to the usual assumption of a timescale separation, we derive a master equation of the Markov type. Furthermore, imposing the rotating-wave approximation, we obtain a Markov master equation of the Lindblad form. To illustrate the method, we consider the standard example of a single qubit coupled to a thermal heat bath.

A review of Information Field Theory for Bayesian inference of random fields

Several physical problems require Bayesian inference of spatial, or spatio-temporal phenomenon – often modeled as random fields defined on a continuous domain – from a discrete set of data points. Kriging, based on Gaussian processes, is one of the commonly used tool for such inference problems. While Gaussian joint probability distributions have known closed form solutions, several physical phenomenon exhibit non-Gaussian features which are analytically intractable. In such problems, one often approximates the underlying distribution by some known, often simpler distribution (for example, a Gaussian), and infers an assigned parametric form for its moments. More rigorous analysis involves computationally expensive methods such as Markov Chain Monte Carlo (MCMC) methods. This paper presents a review of the diagrammatic perturbation theory (following Feynman diagrams used in Physics), a particular technique developed as part of Information Field Theory, for analytically estimating moments of perturbative non-Gaussian distributions.

Quantum Coulomb glass on the Bethe lattice

We study the Coulomb glass emerging from the interplay of strong interactions and disorder in a model of spinless fermions on the Bethe lattice. In the infinite coordination number limit, strong interactions induce a metallic Coulomb glass phase with a pseudogap structure at the Fermi energy. Quantum and thermal fluctuations both melt this glass and induce a disordered quantum liquid phase. We combine self-consistent diagrammatic perturbation theory with continuous time quantum Monte-Carlo simulations to obtain the complete phase diagram of the electron glass and to characterize its dynamical properties in the quantum liquid, as well as in the replica symmetry broken glassy phase. Tunneling spectra display an Efros-Shklovskii pseudogap upon decreasing temperatures, but the density of states remains finite at the Fermi energy due to residual quantum fluctuations. Our results bear relevance to the metallic glass phase observed in Si inversion layers.

Duality between Weak and Strong Interactions in Quantum Gases.

In one-dimensional quantum gases there is a well known "duality" between hard core bosons and noninteracting fermions. However, at the field theory level, no exact duality connecting strongly interacting bosons to weakly interacting fermions is known. Here we propose a solution to this long-standing problem. Our derivation relies on regularizing the only pointlike interaction between fermions in one dimension that induces a discontinuity in the wave function proportional to its derivative. In contrast to all known regularizations our potential is weak for small interaction strengths. Crucially, this allows one to apply standard methods of diagrammatic perturbation theory to strongly interacting bosons. As a first application we compute the finite temperature spectral function of the Cheon-Shigehara model, the fermionic model dual to the celebrated Lieb-Liniger model.

Phase transitions in partial summation methods: Results from the three-dimensional Hubbard model

The accurate determination of magnetic phase transitions in electronic systems is an important task of solid state theory. While numerically exact results are readily available for model systems such as the half-filled 3D Hubbard model, the complexity of real materials requires additional approximations, such as the restriction to certain classes of diagrams in perturbation theory, that reduce the precision with which magnetic properties are described. In this work, we examine the description of magnetic properties in second order perturbation theory, GW, FLEX, and two TMatrix approximations to numerically exact CT-QMC reference data. We assess finite-size effects and compare periodic lattice simulations to cluster embedding. We find that embedding substantially improves finite size convergence. However, by analyzing different partial summation methods we find no systematic improvement in the description of magnetic properties, with most methods considered in this work predicting first-order instead of continuous transitions, leading us to the conclusion that non-perturbative methods are necessary for the accurate determination of magnetic properties and phase transitions.

Relativistic time-dependent configuration-interaction singles method

In this work, a derivation and implementation of the relativistic time-dependent configuration interaction singles (RTDCIS) method is presented. Various observables for krypton and xenon atoms obtained by RTDCIS are compared with experimental data and alternative relativistic calculations. This includes energies of occupied orbitals in the Dirac-Fock ground state, Rydberg state energies, Fano resonances and photoionization cross sections. Diagrammatic many-body perturbation theory, based on the relativistic random phase approximation, is used as a benchmark with excellent agreement between RTDCIS reported at the Tamm-Dancoff level. Results from RTDCIS are computed in the length gauge, here the negative energy states can be omitted with acceptable loss of accuracy. A complex absorbing potential, that is used to remove photoelectrons far from the ion, is implemented as a scalar potential and validated for RTDCIS. The RTDCIS methodology presented here opens for future studies of strong-field processes, such as attosecond transient absorption and high-order harmonic generation, with electron and hole spin dynamics and other relativistic effects described by first principle via the Dirac equation.

Quantum many-body states and Green's functions of nonequilibrium electron-magnon systems: Localized spin operators versus their mapping to Holstein-Primakoff bosons

The operators of localized spins within a magnetic material commute at different sites of its lattice and anticommute on the same site, so they are neither fermionic nor bosonic operators. Thus, to construct diagrammatic many-body perturbation theory, the spin operators are usually mapped to the bosonic ones with Holstein-Primakoff (HP) transformation being the most widely used in magnonics and spintronics literature. However, to make calculations tractable, the square root of operators in the HP transformation is expanded into a Taylor series truncated to some low order. This poses a question on the range of validity of truncated HP transformation when describing nonequilibrium dynamics of localized spins interacting with each other or with conduction electron spins. Here we apply exact diagonalization techniques to Hamiltonian of fermions (i.e., electrons) interacting with HP bosons vs. Hamiltonian of fermions interacting with the original localized spin operators in order to compare their many-body states and one-particle equilibrium or nonequilibrium Green functions. The Hamiltonian of fermions interacting with HP bosons gives incorrect ground state and electronic spectral function, unless large number of terms are retained in truncated HP transformation. Furthermore, tracking nonequilibrium dynamics of localized spins over longer time intervals requires progressively larger number of terms in truncated HP transformation. Finally, we show that recently proposed [M. Vogl et al., Phys. Rev. Research 2, 043243 (2020); J. K\"{o}nig et al., SciPost Phys. 10, 007 (2021)] resummed HP transformation resolves the trouble with truncated HP transformation, while allowing us to derive an exact (manifestly Hermitian) Hamiltonian consisting of finite and fixed number of boson-boson and electron-boson interacting terms.

Downfolding the Su-Schrieffer-Heeger model

Charge-density waves are responsible for symmetry-breaking displacements of atoms and concomitant changes in the electronic structure. Linear response theories, in particular density-functional perturbation theory, provide a way to study the effect of displacements on both the total energy and the electronic structure based on a single ab initio calculation. In downfolding approaches, the electronic system is reduced to a smaller number of bands, allowing for the incorporation of additional correlation and environmental effects on these bands. However, the physical contents of this downfolded model and its potential limitations are not always obvious. Here, we study the potential-energy landscape and electronic structure of the Su-Schrieffer-Heeger (SSH) model, where all relevant quantities can be evaluated analytically. We compare the exact results at arbitrary displacement with diagrammatic perturbation theory both in the full model and in a downfolded effective single-band model, which gives an instructive insight into the properties of downfolding. An exact reconstruction of the potential-energy landscape is possible in a downfolded model, which requires a dynamical electron-biphonon interaction. The dispersion of the bands upon atomic displacement is also found correctly, where the downfolded model by construction only captures spectral weight in the target space. In the SSH model, the electron-phonon coupling mechanism involves exclusively hybridization between the low- and high-energy bands and this limits the computational efficiency gain of downfolded models.

QCD beyond diagrams

Quantum chromodynamics (QCD), the theory of the strong interactions, involves quarks interacting with non-Abelian gluon fields. This theory has many features that are difficult to impossible to see in conventional diagrammatic perturbation theory. This includes quark confinement, mass generation and chiral symmetry breaking. This paper is a colloquium level overview of the framework for understanding how these effects come about.

Constraining the nonanalytic terms in the isospin-asymmetry expansion of the nuclear equation of state

We examine the properties of the isospin-asymmetry expansion of the nuclear equation of state from chiral two- and three-body forces. We focus on extracting the high-order symmetry energy coefficients that consist of both normal terms (occurring with even powers of the isospin asymmetry) as well as terms involving the logarithm of the isospin asymmetry that are formally nonanalytic around the expansion point of isospin-symmetric nuclear matter. These coefficients are extracted from numerically precise perturbation theory calculations of the equation of state coupled with a new set of finite difference formulas that achieve stability by explicitly removing the effects of higher-order terms in the expansion. We consider contributions to the symmetry energy coefficients from both two- and three-body interactions. It is found that the coefficients of the logarithmic terms are generically larger in magnitude than those of the normal terms from second-order perturbation theory diagrams, but overall the normal terms give larger contributions to the ground state energy. The high-order isospin-asymmetry terms are especially relevant at large densities where they affect the proton fraction in beta-equilibrium matter.

Topological transport of mobile impurities

We study the Hall response of topologically-trivial mobile impurities (Fermi polarons) interacting weakly with majority fermions forming a Chern-insulator background. This setting involves a rich interplay between the genuine many-body character of the polaron problem and the topological nature of the surrounding cloud. When the majority fermions are accelerated by an external field, a transverse impurity current can be induced. To quantify this polaronic Hall effect, we compute the drag transconductivity, employing controlled diagrammatic perturbation theory in the impurity-fermion interaction. We show that the impurity Hall drag is not simply proportional to the Chern number characterizing the topological transport of the insulator on its own - it also depends continuously on particle-hole breaking terms, to which the Chern number is insensitive. However, when the insulator is tuned across a topological phase transition, a sharp jump of the impurity Hall drag results, for which we derive an analytical expression. We describe how the Hall drag and jump can be extracted from a circular dichroic measurement of impurity excitation rates, particularly suited for ultracold gas experiments.

Moiré superlattice on the surface of a topological insulator

Twisting van der Waals heterostructures to induce correlated many-body states provides a novel tuning mechanism in solid-state physics. In this work, we theoretically investigate the fate of the surface Dirac cone of a three-dimensional topological insulator subject to a superlattice potential. Using a combination of diagrammatic perturbation theory, lattice model simulations, and ab initio calculations we elucidate the unique aspects of twisting a single Dirac cone with an induced moir\'e potential and the role of the bulk topology on the reconstructed surface band structure. We report a dramatic renormalization of the surface Dirac cone velocity as well as demonstrate a topological obstruction to the formation of isolated minibands. Due to the topological nature of the bulk, surface band gaps cannot open; instead, additional satellite Dirac cones emerge, which can be highly anisotropic and made quite flat. We discuss the implications of our findings for future experiments.

Resonance energy transfer mediated by a chiral molecule.

The problem of resonant energy transfer (RET) between an electric dipole donor, D, and an electric dipole acceptor, A, mediated by a passive, chiral third-body, T, is considered within the framework of molecular quantum electrodynamics theory. To account for the optical activity of the mediator, magnetic dipole and electric quadrupole coupling terms are included in addition to the leading electric dipole interaction term. Fourth-order diagrammatic time-dependent perturbation theory is used to obtain the matrix element. It is found that the Fermi golden rule rate depends on pure multipole moment polarizabilities and susceptibilities of T, as well as on various mixed electric and magnetic multipole moment response functions. The handedness of T manifests through mixed electric-magnetic dipole and mixed electric dipole-quadrupole polarizabilities, which affect the rate and, respectively, require the use of fourth-rank and sixth-rank Cartesian tensor averages over T, yielding non-vanishing isotropic rate formulae applicable to a chiral fluid medium. Terms of a similar order of magnitude proportional to the product of electric dipole polarizability and either magnetic dipole susceptibility or electric quadrupole polarizability of T are also computed for oriented and freely tumbling molecules. Migration rates dependent upon the product of the pure electric dipole or magnetic dipole polarizability with the mixed electric-magnetic or electric dipole-quadrupole analogs, which require fourth- and fifth-rank Cartesian tensor averaging, vanish for randomly oriented systems. Asymptotically limiting rate expressions are also evaluated. Insight is gained into RET occurring in complex media.

Discriminatory Resonance Energy Transfer Mediated by a Third Molecule.

Relay of resonant excitation energy between two chiral molecules by an inert third particle is studied using molecular quantum electrodynamics theory. A single virtual photon propagates between each interacting pair. Fourth-order diagrammatic time-dependent perturbation theory is employed to compute the matrix element. Rate terms dependent upon the chirality of the donor and acceptor species are extracted using the Fermi golden rule. Interestingly, the mediated rate is discriminatory. For freely tumbling particles it exhibits an inverse-square dependence on each interparticle separation distance, indicating a purely radiative exchange mechanism. Furthermore, the isotropic rate is found to be a maximum for a collinear geometry and vanishes when the angle between the donor, mediator, and acceptor is 90°. The indirect rate is compared with direct transfer between two chiral molecules. Insight is gained into discriminatory migration of energy in a dielectric medium.

Improved Field Theoretical Approach to Noninteracting Brownian Particles in a Quenched Random Potential

We construct a dynamical field theory for noninteracting Brownian particles in the presence of a quenched Gaussian random potential. The main variable for the field theory is the density fluctuation which measures the difference between the local density and its average value. The average density is spatially inhomogeneous for given realization of the random potential. It becomes uniform only after averaged over the disorder configurations. We develop the diagrammatic perturbation theory for the density correlation function and calculate the zero-frequency component of the response function exactly by summing all the diagrams contributing to it. From this exact result and the fluctuation dissipation relation, which holds in an equilibrium dynamics, we find that the connected density correlation function always decays to zero in the long-time limit for all values of disorder strength implying that the system always remains ergodic. This nonperturbative calculation relies on the simple diagrammatic structure of the present field theoretical scheme. We compare in detail our diagrammatic perturbation theory with the one used in a recent paper [B.\ Kim, M.\ Fuchs and V.\ Krakoviack, J.\ Stat.\ Mech.\ (2020) 023301], which uses the density fluctuation around the uniform average, and discuss the difference in the diagrammatic structures of the two formulations.

ADG: Automated generation and evaluation of many-body diagrams II. Particle-number projected Bogoliubov many-body perturbation theory

We describe the second version (v2.0.0) of the code ADG that automatically (1) generates all valid off-diagonal Bogoliubov many-body perturbation theory diagrams at play in particle-number projected Bogoliubov many-body perturbation theory (PNP-BMBPT) and (2) evaluates their algebraic expression to be implemented for numerical applications. This is achieved at any perturbative order p for a Hamiltonian containing both two-body (four-legs) and three-body (six-legs) interactions (vertices). All valid off-diagonal BMBPT diagrams of order p are systematically generated from the set of diagonal, i.e.,unprojected, BMBPT diagrams. The production of the latter were described at length in Arthuis et al. (2019) dealing with the first version of ADG. The automated evaluation of off-diagonal BMBPT diagrams relies both on the application of algebraic Feynman’s rules and on the identification of a powerful diagrammatic rule providing the result of the remaining p-tuple time integral. The new diagrammatic rule generalizes the one already identified in Arthuis et al. (2019) to evaluate diagonal BMBPT diagrams independently of their perturbative order and topology. The code ADG is written in Python3 and uses the graph manipulation package NetworkX. The code is kept flexible enough to be further expanded throughout the years to tackle the diagrammatics at play in various many-body formalisms that already exist or are yet to be formulated. Program summaryProgram Title:ADGCPC Library link to program files:https://doi.org/10.17632/6h4xrydwfb.2Licensing provisions: GPLv3Programming language: Python3Journal reference of previous version: P. Arthuis, T. Duguet, A. Tichai, R.-D. Lasseri and J.-P. Ebran, ”ADG: Automated generation and evaluation of many-body diagrams I. Bogoliubov many-body perturbation theory”, Computer Physics Communications 240 (2019), pp. 202-227.Does the new version supersede the previous version?: Yes.Reasons for the new version: Incorporation of a new formalism into the program.Summary of revisions: Addition of off-diagonal BMBPT to the formalisms for which diagrams can be generated, fix of a wrong symmetry factor, move of the codebase from Python2 to Python3 while maintaining support for Python2, various optimizations to reduce the time and memory necessary to the program.Nature of problem: As formal and numerical developments in many-body-perturbation-theory-based ab initio methods make higher orders reachable, manually producing and evaluating all the diagrams becomes rapidly untractable as both their number and complexity grow quickly, making it prone to mistakes and oversights.Solution method: Diagonal BMBPT diagrams are encoded as square matrices known as oriented adjacency matrices in graph theory, and then turned into graph objects using the NetworkX package. Off-diagonal BMBPT diagrams can then be generated from those graphs. Checks on the diagrams and evaluation of their time-integrated expression are eventually done on a purely diagrammatic basis. HF-MBPT diagrams are produced and evaluated as well using the same principle.