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arXiv:2006.09778 · Physical review. D/Physical review. D.

Spectral functions in the ϕ 4 -theory from the spectral Dyson-Schwinger equations

Jan Horak, Jan Martin Pawlowski, Nicolas Wink

Published Dec 7, 2020• 51 citations
arXivHTMLPDF

Abstract

We develop a nonperturbative functional framework for computing real-time correlation functions in strongly correlated systems. The framework is based on the spectral representation of correlation functions and dimensional regularization. Therefore, the nonperturbative spectral renormalization setup here respects all symmetries of the theories at hand. In particular, this includes space-time symmetries, as well as internal symmetries such as chiral symmetry, and gauge symmetries. Spectral renormalization can be applied within general functional approaches such as the functional renormalization group, Dyson-Schwinger equations, and two- or $n$-particle irreducible approaches. As an application, we compute the full, nonperturbative, spectral function of the scalar field in the ${\ensuremath{\phi}}^{4}$-theory in $2+1$ dimensions from spectral Dyson-Schwinger equations. We also compute the $s$-channel spectral function of the full ${\ensuremath{\phi}}^{4}$-vertex in this theory.

DOI
10.1103/physrevd.102.125016

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We develop a nonperturbative functional framework for computing real-time correlation functions in strongly correlated systems. The framework is based on the spectral representation of correlation functions and dimensional regularization. Therefore, the nonperturbative spectral renormalization setup here respects all symmetries of the theories at hand. In particular, this includes space-time symmetries, as well as internal symmetries such as chiral symmetry, and gauge symmetries. Spectral renormalization can be applied within general functional approaches such as the functional renormalization group, Dyson-Schwinger equations, and two- or nnn-particle irreducible approaches. As an application, we compute the full, nonperturbative, spectral function of the scalar field in the \ensuremathϕ4{\ensuremath{\phi}}^{4}\ensuremathϕ4-theory in 2+12+12+1 dimensions from spectral Dyson-Schwinger equations. We also compute the sss-channel spectral function of the full \ensuremathϕ4{\ensuremath{\phi}}^{4}\ensuremathϕ4-vertex in this theory.
We compute spectral functions in a scalar ϕ 4 -theory in three spacetime dimensions via the spectral functional renormalization group. This approach allows for the direct, manifestly Lorentz covariant computation of correlation functions in Minkowski spacetime, including a physical on-shell renormalization. We present numerical results for the spectral functions of the two- and four-point correlation functions for different values of the coupling parameter. These results agree very well with those obtained from another functional real-time approach, the spectral Dyson-Schwinger equation. Published by the American Physical Society 2024
We employ the functional renormalization group approach formulated on the Schwinger-Keldysh contour to calculate real-time correlation functions in scalar field theories. We provide a detailed description of the formalism, discuss suitable truncation schemes for real-time calculations, as well as the numerical procedure to self-consistently solve the flow equations for the spectral function. Subsequently, we discuss the relations to other perturbative and nonperturbative approaches to calculate spectral functions and present a detailed comparison and benchmark in d=0+1d=0+1d=0+1 dimensions.
We calculate nonperturbative self-consistent fermionic and bosonic spectral functions of ultracold Fermi gases with the spectral functional approach. This approach allows for a direct real-time computation of nonperturbative correlation functions, and in the present work we use spectral Dyson-Schwinger equations. We focus on the normal phase of the spin-balanced Fermi gas and provide numerical results for the full fermionic and bosonic spectral functions. The spectral functions are then used for the determination of the equation of state, the Tan contact, and ejection rf spectra at unitarity. These results are compared to experimental data, the self-consistent TTT-matrix approach, and lattice results. Our approach offers a wide range of applications, including the ab initio calculation of transport and spectral properties of the superfluid phase in the BCS-BEC crossover.
I review the way the many-body Green functions are used to renormalize the perturbation theory of correlated fermions. The Green functions are introduced to implement systematically dynamical corrections to the static mean-field theory. The renormalizations enter the perturbation theory via self-consistent evaluations of one-particle and two-particle Green functions. The one-particle self-consistency is discussed within the Baym-Kadanoff construction. We point to an inherent dichotomy of the Baym-Kadanoff approach in the relation between one and two-particle Green functions. They are the Schwinger-Dyson equation, directly derived from the generating Luttinger-Ward functional, and the Ward identity. The latter imposes a severe restriction on the two-particle irreducible vertex and is not guaranteed in the Baym-Kadanoff approach. No approximate theory is capable to obey both relations. I further discuss the two-particle approach with two-particle renormalizations that offers a way to reconcile the Schwinger-Dyson equation and the Ward identity. The two-particle irreducible vertex from the critical Bethe-Salpeter equation becomes a generating functional in the two-particle renormalization of the perturbation theory. The irreducible vertex is used in the Ward identity to determine the order parameter beyond the critical point and in the Bethe-Salpeter equation representing the vertex of the Schwinger-Dyson equation to determine the spectral self-energy in such a way that the thermodynamic critical behavior from both equations is qualitatively the same. Finally, I discuss a static mean-field-like approximation with a renormalized effective interaction determined self-consistently that leads to a qualitatively correct description of strong electron correlations in impurity and lattice models.
The many-body problem is usually approached from one of two perspectives: the first originates from an action and is based on Feynman diagrams, the second is centered around a Hamiltonian and deals with quantum states and operators.The connection between results obtained in either way is made through spectral (or Lehmann) representations, well known for two-point correlation functions.Here, we complete this picture by deriving generalized spectral representations for multipoint correlation functions that apply in all of the commonly used many-body frameworks: the imaginary-frequency Matsubara and the realfrequency zero-temperature and Keldysh formalisms.Our approach separates spectral from time-ordering properties and thereby elucidates the relation between the three formalisms.The spectral representations of multipoint correlation functions consist of partial spectral functions and convolution kernels.The former are formalism independent but system specific; the latter are system independent but formalism specific.Using a numerical renormalization group method described in the accompanying paper, we present numerical results for selected quantum impurity models.We focus on the four-point vertex (effective interaction) obtained for the single-impurity Anderson model and for the dynamical mean-field theory solution of the one-band Hubbard model.In the Matsubara formalism, we analyze the evolution of the vertex down to very low temperatures and describe the crossover from strongly interacting particles to weakly interacting quasiparticles.In the Keldysh formalism, we first benchmark our results at weak and infinitely strong interaction and then reveal the rich real-frequency structure of the dynamical mean-field theory vertex in the coexistence regime of a metallic and insulating solution.
We present an approximation scheme of the nonperturbative renormalization group that preserves the momentum dependence of correlation functions. This approximation scheme can be seen as a simple improvement of the local potential approximation (LPA) where the derivative terms in the effective action are promoted to arbitrary momentum-dependent functions. As in the LPA, the only field dependence comes from the effective potential, which allows us to solve the renormalization-group equations at a relatively modest numerical cost (as compared, e.g., to the Blaizot--Mend\'ez-Galain--Wschebor approximation scheme). As an application we consider the two-dimensional quantum O(NNN) model at zero temperature. We discuss not only the two-point correlation function but also higher-order correlation functions such as the scalar susceptibility (which allows for an investigation of the ``Higgs'' amplitude mode) and the conductivity. In particular, we show how, using Pad\'e approximants to perform the analytic continuation $i{\ensuremath{\omega}}_{n}\ensuremath{\rightarrow}\ensuremath{\omega}+i{0}^{+}$ of imaginary frequency correlation functions $\ensuremath{\chi}(i{\ensuremath{\omega}}_{n})$ computed numerically from the renormalization-group equations, one can obtain spectral functions in the real-frequency domain.