Abstract

We develop a non-perturbative 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 regularisation. Therefore, the non-perturbative spectral renormalisation 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 renormalisation can be applied within general functional approaches such as the functional renormalisation group, Dyson-Schwinger equations, and two- or $n$-particle irreducible approaches. As an application we compute the full, non-perturbative, spectral function of the scalar field in the $\phi^4$-theory in $2+1$ dimensions from spectral Dyson-Schwinger equations. We also compute the $s$-channel spectral function of the full $\phi^4$-vertex in this theory.

Highlights

  • The study of the dynamics and resonance structure of strongly correlated systems requires the knowledge of realtime correlation functions

  • In this work. we develop a novel approach for the direct computation of real-time (Minkowski) observables that is based on spectral representations of correlation functions

  • We developed a spectral functional approach for the direct nonperturbative computation of real-time correlation functions

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Summary

INTRODUCTION

The study of the dynamics and resonance structure of strongly correlated systems requires the knowledge of realtime (timelike) correlation functions. Spectral functions encode the full, nonperturbative, information of the respective degrees of freedom and open the door to additional real-time quantities such as transport coefficients They are a useful tool when discussing resonances and bound states, since they give direct access to the spectrum of excitations in a given theory. We develop a novel approach for the direct computation of real-time (Minkowski) observables that is based on spectral representations of correlation functions. We apply our novel approach to the scalar φ4-theory in d 1⁄4 2 þ 1 dimensions This theory is a simple strongly correlated system and serves as a good benchmark for new techniques before applying them more involved theories such as non-Abelian ones.

SPECTRAL RENORMALIZATION
Dyson-Schwinger equations
Källen-Lehmann spectral representation
Ziδðλ i
Spectral dimensional renormalization
Spectral BPHZ-renormalization
Nonperturbative spectral renormalization
Momentum integration and spectral renormalization
Analytic continuation
Spectral integration and iteration
Compute from propagator
RESULTS
DSE with classical four-point vertex
Fully nonperturbative DSE
Bubble-resummed s-channel four-point function
Tadpole contribution to the sunset topology
Vertex approximation in the skeleton expansion
Spectral representation for the four-point function
Results for the coupled system of propagator and vertices
Self-consistent skeleton expansion
Vertex-approximation in the self-consistent skeleton expansion
Results
Low-lying bound state close to phase transition
CONCLUSION

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