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arXiv:1910.09369 · Physical review. D/Physical review. D.

Flowing with the temporal renormalization group

Lukas Corell, Anton K. Cyrol, Markus Heller, Jan Martin Pawlowski

Published Jul 6, 2021• 24 citations
arXivHTMLPDF

Abstract

We discuss the far-from-equilibrium evolution of ${\ensuremath{\phi}}^{3}$ theory in $1+1$ dimensions with the temporal functional renormalization group. In particular, we show that this manifestly causal approach leads to novel one-loop exact equations for fully dressed correlation functions. Within this setup, we numerically compute the dynamical propagator. Its behavior suggests self-similarity far from equilibrium in a restricted momentum regime. We discuss the scaling exponents for our solution, as well as the numerical satisfaction of energy and particle number conservation. We also derive a simple exact representation of the expectation value of the energy-momentum tensor solely in terms of the propagator.

DOI
10.1103/physrevd.104.025005

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We discuss the far-from-equilibrium evolution of \ensuremathϕ3{\ensuremath{\phi}}^{3}\ensuremathϕ3 theory in 1+11+11+1 dimensions with the temporal functional renormalization group. In particular, we show that this manifestly causal approach leads to novel one-loop exact equations for fully dressed correlation functions. Within this setup, we numerically compute the dynamical propagator. Its behavior suggests self-similarity far from equilibrium in a restricted momentum regime. We discuss the scaling exponents for our solution, as well as the numerical satisfaction of energy and particle number conservation. We also derive a simple exact representation of the expectation value of the energy-momentum tensor solely in terms of the propagator.
In this work, we substantially advance the formalism of the temporal functional renormalisation group which constitutes a non-perturbative framework for computing the dynamics of correlation functions in quantum field theories. To that end we carefully revisit the derivation of the temporal flow equation, paying particular attention to properties arising from a causal temporal regulator. We use the manifest causality of the formalism to integrate the general temporal flow analytically. The result are novel one-loop exact equations for fully dressed correlation functions. Further leveraging causality, we derive the complete Dyson-Schwinger hierarchy and the s-channel effective vertex in terms of specific truncations of the temporal flow. We solve the problem of renormalising the general causal temporal flow. We demonstrate that certain types of causal integral equations can be solved by an explicit numerical method. We numerically solve the integrated flow in a truncation involving the propagator of the phi3\\phi^3phi3-theory in 1+11+11+1 dimensions. Our results indicate the emergence of universal dynamics. Due to the high degree of flexibility of approximation schemes of the temporal flow, energy conservation in generic truncations is not guaranteed automatically but becomes a non-trivial feature instead. We explore energy-conserving truncations by deriving the causal temporal flow of the energy-momentum tensor, which we integrate analytically.
We develop a functional renormalization group approach to obtain the time evolution of the momentum distribution function of interacting bosons out of equilibrium. Using an external out-scattering rate as flow parameter, we derive formally exact renormalization group flow equations for the nonequilibrium self-energies in the Keldysh basis. A simple perturbative truncation of these flow equations leads to an approximate solution of the quantum Boltzmann equation, which does not suffer from secular terms and gives accurate results even for long times. We demonstrate this explicitly within a simple exactly solvable toy model describing a quartic oscillator with off-diagonal pairing terms.
We give a self-contained derivation of the differential equations for Wilson's renormalization group for the one-particle irreducible Green functions in fermionic systems. The application of this equation to the (t,t′)-Hubbard model appears in Ref. [HSFR]. Here we focus on theoretical aspects. After deriving the equations, we discuss the restrictions imposed by symmetries on the effective action. We discuss scaling properties due to the geometry of the Fermi surface and give precise criteria to determine when they justify the use of one-loop flows. We also discuss the relationship of this approach to other RG treatments, as well as aspects of the practical treatment of truncated equations, such as the projection to the Fermi surface and the calculation of susceptibilities.
The experimental progress in cold atomic gases and mesoscopic systems has allowed unprecedented access to low dimensional correlated quantum systems far from equilibrium. This has revealed a variety of novel phenomena such as many-body localization, time crystals and dynamical phase transitions. The theoretical description of such systems has remained a challenge: The numerically exact solution of generic, correlated quantum systems requires computational effort that scales exponentially in the number of fermions, rendering larger systems and therefore collective behavior inaccessible. This necessitates approximations. In low dimensions, infrared divergences prohibit the use of a purely perturbative approach. Renormalization group based techniques, where energy scales are treated successively remedy some of these shortcomings. We therefore propose the use of the functional renormalization group to access correlation functions far from equilibrium in large, interacting systems. Such techniques were previously successfully applied in the investigation of quantum impurity problems and quenches in tight-binding chains and are widely used in the study of equilibrium phase diagrams. After a presentation of the Keldysh formalism, that constitutes a natural language out of equilibrium, we present perturbation theory and the functional renormalization group approach in a diagrammatic language. We then develop, optimize and apply flexible fRG techniques to four different problems in one-dimensional tight-binding chains with short-ranged two-particle interaction. First, we discuss the emergence of anomalous transport in the presence of a quasiperiodic potential by analyzing linear response and a far-from-equilibrium quench protocol. We then develop a variant of the fRG to access pure excited eigenstates. We use this algorithm to investigate many-body states in large tight-binding chains and show that generic excitations appear thermal while some states show unique spectral properties. To study the influence of inelastic scattering on the transport properties in such a chain we then go beyond an effective single-particle picture; to that end, we present an algorithm, that includes scattering in long chains. Finally, we discuss infinite chains coupled to a substrate while driven out of equilibrium by an electric field. Exploiting the inherent symmetries, we develop efficient algorithms and apply them to a simple tight-binding chain with nearest-neighbor interaction to explore the interplay of spontaneous symmetry breaking and driving. We demonstrate, that ordering is suppressed and provide results that indicate unconventional transport in the driven system.
We include spontaneous symmetry breaking in the functional renormalization group equations for the irreducible vertices of Ginzburg–Landau theories by augmenting these equations by a flow equation for the order parameter, which is determined from the requirement that at each renormalization group (RG) step the vertex with one external leg vanishes identically. Using this strategy, we propose a simple truncation of the coupled RG flow equations for the vertices in the broken symmetry phase of the Ising universality class in D dimensions. Our truncation yields the full momentum dependence of the self-energy Σ( k ) and interpolates between lowest-order perturbation theory at large momenta k and the critical scaling regime for small k . Close to the critical point, our method yields the self-energy in the scaling form Σ( k ) = k c 2 σ − (| k |ξ,| k |/ k c ), where ξ is the order parameter correlation length, k c is the Ginzburg scale, and σ − ( x , y ) is a dimensionless two-parameter scaling function for the broken symmetry phase which we calculate explicitly within our truncation.