Diagrammatic Expansion Research Articles

Stochastic action for the entanglement of a noisy monitored two-qubit system

We study the effect of local unitary noise on the entanglement evolution of a two-qubit system subject to local monitoring and interqubit coupling by constructing a suitable stochastic path integral based on the Chantasri-Dressel-Jordan formalism. From this, we identify the optimal entanglement dynamics and deploy a diagrammatic method to find a closed-form approximation for the evolution of the average squared concurrence. We find that the optimal trajectory and diagrammatic expansion capture the oscillations of entanglement at short times. A separate numerical analysis of the long-time steady-state entanglement reveals a nonmonotonic relationship between the concurrence and noise strength. Published by the American Physical Society 2025

In-in formalism for the entropy of quantum fields in curved spacetimes

We show how to compute the purity and entanglement entropy for quantum fields in a systematic perturbative expansion. To that end, we generalize the in-in formalism to non-unitary dynamics (i.e. accounting for the presence of an environment) and to the calculation of quantum information measures, which are not observables in the usual sense. This allows us to reduce the problem to one involving standard correlation functions, and to organize their computation in a diagrammatic expansion for which we construct the corresponding Feynman rules. As an illustration, we apply the formalism to a cosmological setting inspired by the effective field theory of inflation. We find that at late times, non-linear loop corrections share the same time behavior as the linear contribution, and only yield a slight redressing of the purity. In particular, when the environment is heavy compared to the Hubble scale, the phenomenon of recoherence previously encountered is robust to the class of non-linear extensions considered. Bridging the gap between perturbative quantum field theory and open quantum systems paves the way to a better understanding of renormalization and resummation in open effective field theories. It also enables a more systematic exploration of quantum information properties in field theoretic settings.

Field theory equivalences as spans of L∞ -algebras

Semi-classically equivalent field theories are related by a quasi-isomorphism between their underlying L∞ -algebras, but such a quasi-isomorphism is not necessarily a homotopy transfer. We demonstrate that all quasi-isomorphisms can be lifted to spans of L∞ -algebras in which the quasi-isomorphic L∞ -algebras are obtained from a correspondence L∞ -algebra by a homotopy transfer. Our construction is very useful: homotopy transfer is computationally tractable, and physically, it amounts to integrating out fields in a Feynman diagram expansion. Spans of L∞ -algebras allow for a clean definition of quasi-isomorphisms of cyclic L∞ -algebras. Furthermore, they appear naturally in many contexts within physics. As examples, we first consider scalar field theory with interaction vertices blown up in different ways. We then show that (non-Abelian) T-duality can be seen as a span of L∞ -algebras, and we provide full details in the case of the principal chiral model. We also present the relevant span of L∞ -algebras for the Penrose–Ward transform in the context of self-dual Yang–Mills theory and Bogomolny monopoles.

In-in correlators and scattering amplitudes on a causal set

Causal set theory is an approach to quantum gravity in which spacetime is fundamentally discrete at the Planck scale and takes the form of an irregular Lorentzian lattice, or “causal set,” from which continuum spacetime emerges in a large-scale (low-energy) approximation. In this work, we present new developments in the framework of interacting quantum field theory on causal sets. We derive a diagrammatic expansion for in-in correlators in local scalar field theories with finite polynomial interactions. We outline how these same correlators can be computed using the double-path integral, which acts as a generating functional for the in-in correlators. We modify the in-in generating functional to obtain a generating functional for in-out correlators. We define a notion of scattering amplitudes on causal sets with noninteracting past and future regions and verify that they are given by S-matrix elements (matrix elements of the time-evolution operator). We describe how these formal developments can be implemented to compute early Universe observables under the assumption that spacetime is fundamentally discrete. Published by the American Physical Society 2024

Phase diagram and topological expansion in the complex quartic random matrix model

AbstractWe use the Riemann–Hilbert approach, together with string and Toda equations, to study the topological expansion in the quartic random matrix model. The coefficients of the topological expansion are generating functions for the numbers of 4‐valent connected graphs with j vertices on a compact Riemann surface of genus g. We explicitly evaluate these numbers for Riemann surfaces of genus 0,1,2, and 3. Also, for a Riemann surface of an arbitrary genus g, we calculate the leading term in the asymptotics of as the number of vertices tends to infinity. Using the theory of quadratic differentials, we characterize the critical contours in the complex parameter plane where phase transitions in the quartic model take place, thereby proving a result of David. These phase transitions are of the following four types: (a) one‐cut to two‐cut through the splitting of the cut at the origin, (b) two‐cut to three‐cut through the birth of a new cut at the origin, (c) one‐cut to three‐cut through the splitting of the cut at two symmetric points, and (d) one‐cut to three‐cut through the birth of two symmetric cuts.

Green’s function in computational chemistry: Insights and theoretical approaches

In this study, we revisit Green's function formalism in theoretical chemistry. Specifically, we investigate how it allows refining the Hartree-Fock model by introducing the self-energy's first and second-order corrections. Two distinct approaches are considered: the energy partition procedure and the Feynman diagrammatic expansion. These approaches are applied to the study of the π bond of ethylene, demonstrating that both provide better numerical results than those obtained using the Hartree-Fock model but worse than those obtained using the Configuration Interaction method. An optimization process of the energy partition procedure, capable of reproducing the results obtained by applying the Configuration Interaction method and solving the problem of the violation of particle number conservation, is also proposed.

The dilute interacting Bose gas: comparisons between Feynman diagrammatic expansion and the hierarchy equations for the imaginary time Green functions

We analyze the correspondence between two formalisms describing an interacting Bose gas, namely the standard Feynman diagrammatic expansion on the one hand, and the hierarchy equations for the imaginary-time Green functions on the other hand. We show that the Hartree–Fock approximation, as well as its first corrections at low density derived by Baym et al (2001 Eur. Phys. J. B 24 107–24), can be equivalently formulated in both formalisms. Within the hierarchy approach, the two-body correlations are expressed at lowest order in the interactions in terms of the full one-body Green function. This sheds light on the physical content of the corrections to the Hartree–Fock theory. Moreover, this representation of can be extended to higher orders, opening the way to systematic calculations of further corrections to the ideal critical temperature when the density increases.

Pairing susceptibility of the two-dimensional Hubbard model in the thermodynamic limit

We compute the diagrammatic expansion of the particle-particle susceptibility via algorithmic Matsubara integration and compute the correlated pairing susceptibility in the thermodynamic limit of the 2D Hubbard Model. We study the static susceptibility and its dependence on the pair momentum $\mathbf{q}$ for a range of temperature, interaction strength, and chemical potential. We show that $d_{x^2-y^2}$-wave pairing is expected in the model in the $U/t\to 0^+ $ limit from direct perturbation theory. From this, we identify key second and third-order diagrams that support pairing processes and note that the diagrams responsible are not a part of charge or spin susceptibility expansions. We find two key components for pairing at momenta $(0,0)$ and $(\pi,\pi)$ that can be well fit as separate bosonic modes. We extract amplitudes and correlation length scales where we find a predominantly local $(\pi,\pi)$ pairing and non-local $\mathbf{q}=(0,0)$ pairs and present the relative weights of these modes for variation in temperature, doping, and interaction strength.

Equivalence of quantum and classical third order response for weakly anharmonic coupled oscillators.

Two-dimensional (2D) infrared (IR) spectra are commonly interpreted using a quantum diagrammatic expansion that describes the changes to the density matrix of quantum systems in response to light-matter interactions. Although classical response functions (based on Newtonian dynamics) have shown promise in computational 2D IR modeling studies, a simple diagrammatic description has so far been lacking. Recently, we introduced a diagrammatic representation for the 2D IR response functions of a single, weakly anharmonic oscillator and showed that the classical and quantum 2D IR response functions for this system are identical. Here, we extend this result to systems with an arbitrary number of bilinearly coupled, weakly anharmonic oscillators. As in the single-oscillator case, quantum and classical response functions are found to be identical in the weakly anharmonic limit or, in experimental terms, when the anharmonicity is small relative to the optical linewidth. The final form of the weakly anharmonic response function is surprisingly simple and offers potential computational advantages for application to large, multi-oscillator systems.

Unexpected nontriviality of inter-band contribution and huge anisotropy of conductivity in tilted Weyl semimetals

For tilted type-I Weyl semimetals, our theoretical study indicates that the Kubo conductivity parallel to cone tilt direction increases rapidly with the Fermi energy. Hence it deviates from its Boltzmann counterpart sizably even though the electronic wavelength is so short that the quantum interference is seemingly trivial. As a result, such a system shows huge anisotropy of the conductivity. By means of the diagrammatic expansion technique, we find that the absence of backscattering and the breakdown of the time reversal symmetry in tilted Weyl semimetals are two crucial factors which bring about the nontrivial inter-band contribution to the Kubo conductivity. Our findings indicate that Boltzmann theory fails to describe reasonably the electronic transport of a tilted Weyl cone, even in the case of weak scattering and short Fermi wavelength. Though in this regime such a semiclassical transport theory is believed to be valid according to conventional arguments. This situation should be paid attention. • Via SCBA, the high order quantum corrections to conductivity are considered and it makes huge enhancement to diagonal conductivity σ z z , which is missed by the Boltzmann transport equation with the lowest order Born approximation. • The huge anisotropy of diagonal conductivity in tilted WSMs origin from the nature of the absence of backscattering and breakdown of TRS. • The interaction range of impurity potential affects the extent of anisotropy.

Effective viscosity of semi-dilute suspensions

This review is devoted to the large-scale rheology of suspensions of rigid particles in Stokes fluid. After describing recent results on the definition of the effective viscosity of such systems in the framework of homogenization theory, we turn to our new results on the asymptotic expansion of the effective viscosity in the dilute regime. This includes a new optimal proof of Einstein’s viscosity formula for the first-order expansion, as well as the continuation of this expansion to higher orders. The essential difficulty originates in the long-range nature of hydrodynamic interactions: suitable renormalizations are needed and are captured by means of diagrammatic expansions.

Learning Feynman Diagrams with Tensor Trains

We use tensor network techniques to obtain high-order perturbative diagrammatic expansions for the quantum many-body problem at very high precision. The approach is based on a tensor train parsimonious representation of the sum of all Feynman diagrams, obtained in a controlled and accurate way with the tensor cross interpolation algorithm. It yields the full time evolution of physical quantities in the presence of any arbitrary time-dependent interaction. Our benchmarks on the Anderson quantum impurity problem, within the real-time nonequilibrium Schwinger-Keldysh formalism, demonstrate that this technique supersedes diagrammatic quantum Monte Carlo by orders of magnitude in precision and speed, with convergence rates 1/N2 or faster, where N is the number of function evaluations. The method also works in parameter regimes characterized by strongly oscillatory integrals in high dimension, which suffer from a catastrophic sign problem in quantum Monte Carlo calculations. Finally, we also present two exploratory studies showing that the technique generalizes to more complex situations: a double quantum dot and a single impurity embedded in a two-dimensional lattice.11 MoreReceived 13 July 2022Revised 5 September 2022Accepted 8 September 2022DOI:https://doi.org/10.1103/PhysRevX.12.041018Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.Published by the American Physical SocietyPhysics Subject Headings (PhySH)Physical SystemsQuantum dotsTechniquesDiagrammatic methodsGreen's function methodsMachine learningQuantum Monte CarloSingle impurity modelTensor network methodsCondensed Matter, Materials & Applied Physics

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.

Unconventional superconductivity from weak coupling

We develop a joint formalism and numerical framework for analyzing the superconducting instability of metals from a weak coupling perspective. This encompasses the Kohn–Luttinger formulation of weak coupling renormalization group for superconductivity as well as the random phase approximation imposed on the diagrammatic expansion of the two-particle Green’s function. The central quantity to resolve is the effective interaction in the Cooper channel, for which we develop an optimized numerical framework. Our code is capable of treating generic multi-orbital models in two as well as three spatial dimensions and, in particular, arbitrary avenues of spin-orbit coupling.Graphic abstract

Quantum-correlated photons generated by nonlocal electron transport

Since the realization of high-quality microwave cavities coupled to quantum dots, one can envisage the possibility to investigate the coherent interaction of light and matter in semiconductor quantum devices. Here we study a parallel double quantum dot device operating as single-electron splitter interferometer, with each dot coupled to a local photon cavity. We explore how quantum correlation and entanglement between the two separated cavities are generated by the coherent transport of a single electron passing simultaneously through the two different dots. We calculate the covariance of the cavity occupations by use of a diagrammatic perturbative expansion based on Keldysh Green's functions to the fourth order in the dot-cavity interaction strength, taking into account vertex diagrams. In this way, we demonstrate the creation of entanglement by showing that the classical Cauchy-Schwarz inequality is violated if the energy levels of the two dots are almost degenerate. For large level detunings or a single dot coupled to two cavities, we show that the inequality is not violated.

Negative thermal expansion property in Nb14W3O44

Nb14W3O44, a promising electrode material of lithium-ion batteries/capacitors, was prepared using a solid-state method. Its geometrical structure, phase diagram, and thermal expansion properties were all thoroughly investigated. The material has a tetragonal structure and an intriguing linear negative thermal expansion (NTE) property with an impressive coefficient of thermal expansion (−4.27 × 10−6 K−1, 300–433 K). The NTE phenomenon can be attributed to the vibrations of the bridge-site atoms, which caused the coupling rotation of the rigid unit models. The lattice distortion, and pores could also affect the thermal expansion performance in the bulk materials.

Fast principal minor algorithms for diagrammatic Monte Carlo

The computation of determinants plays a central role in diagrammatic Monte Carlo algorithms for strongly correlated systems. The evaluation of large numbers of determinants can often be the limiting computational factor determining the number of attainable diagrammatic expansion orders. In this work we build on the algorithm presented in [K. Griffin and M. J. Tsatsomeros, Principal minors, part i: A method for computing all the principal minors of a matrix, Linear Algebra Appl. 419, 107 (2006)] which computes all principal minors of a matrix in $O({2}^{n})$ operations. We present multiple generalizations of the algorithm to the efficient evaluation of certain subsets of all principal minors with immediate applications to connected determinant diagrammatic Monte Carlo within the normal and symmetry-broken phases as well as continuous-time quantum Monte Carlo. Additionally, we improve the asymptotic scaling of diagrammatic Monte Carlo formulated in real time to $O({2}^{n})$ and report speedups of up to a factor 25 at computationally realistic expansion orders. We further show that all permanent principal minors, corresponding to sums of bosonic Feynman diagrams, can be computed in $O({3}^{n})$, thus encouraging the investigation of bosonic and mixed systems by means of diagrammatic Monte Carlo.

Cosmological Scattering Equations.

We propose a world sheet formula for tree-level correlation functions describing a scalar field with arbitrary mass and quartic self-interaction in de Sitter space, which is a simple model for inflationary cosmology. The correlation functions are located on the future boundary of the spacetime and are Fourier-transformed to momentum space. Our formula is supported on mass-deformed scattering equations involving conformal generators in momentum space and reduces to the CHY formula for ϕ^{4} amplitudes in the flat space limit. Using the global residue theorem, we verify that it reproduces the Witten diagram expansion at four and six points, and sketch the extension to n points.

Optical dressing of the electronic response of two-dimensional semiconductors in quantum and classical descriptions of cavity electrodynamics

We study quantum effects of the vacuum light-matter interaction in materials embedded in optical cavities. We focus on the electronic response of a two-dimensional semiconductor placed inside a planar cavity. By using a diagrammatic expansion of the electron-photon interaction, we describe signatures of light-matter hybridization characterized by large asymmetric shifts of the spectral weight at resonant frequencies. We follow the evolution of the light dressing from the cavity to the free-space limit. In the cavity limit, light-matter hybridization results in a modification of the optical gap with sizable spectral weight appearing below the bare gap edge. In the limit of large cavities, we find a residual redistribution of spectral weight which becomes independent of the distance between the two mirrors. We show that the photon dressing of the electronic response can be fully explained by using a classical description of light. The classical description is found to hold up to a strong coupling regime of the light-matter interaction highlighted by the large modification of the photon spectra with respect to the empty cavity. We show that, despite the strong coupling, quantum corrections are negligibly small and weakly dependent on the cavity confinement. As a consequence, in contrast to the optical gap, the single-particle electronic band gap is not sensibly modified by strong coupling. Our results show that quantum corrections are dominated by off-resonant photon modes at high energy. As such, cavity confinement can hardly be seen as a knob to control the quantum effects of the light-matter interaction in vacuum.

Diagrammatic strong coupling expansion of a U(1) lattice model in the Fourier basis

The transfer-matrix of the U(1) lattice gauge theory is investigated in the field Fourier space, the basis of which consists of the quantized currents on lattice links. Based on a lattice version of the current conservation, the transfer-matrix elements are shown to be nonzero only between current states that differ in circulating currents inside plaquettes. In the strong coupling limit, a series expansion is developed for the elements of the transfer matrix, to which a diagrammatic representation based on the occurrence of virtual link and loop currents can be associated. With $g$ as the coupling, the weight of each virtual current in the expansion is $1/{g}^{2}$, by which at any given order the relevant diagrams are determined. Either by interpretation or through their role in fixing the relevant terms, the diagrams are reminiscent of the Feynman ones of the perturbative small coupling expansions.