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arXiv:2309.10496

Note on general functional flows in equilibrium systems

Kiyoharu Kawana

Published Sep 19, 2023• 1 citation
arXivHTMLPDF

Abstract

We study the response of generating functionals to a variation of parameters (couplings) in equilibrium systems i.e. in quantum field theory (QFT) and equilibrium statistical mechanics. These parameters can be either physical ones such as coupling constants or artificial ones which are intentionally introduced such as the renormalization scale in field theories. We first derive general functional flow equations for the generating functional (grand-canonical potential) $W[J]$ of the connected diagrams. Then, we obtain functional flow equations for the one-particle irreducible ($1$PI) vertex functional (canonical potential) $Γ[ϕ]$ by performing the Legendre transformation. By taking the functional derivatives of the flow equations, we can obtain an infinite hierarchical equations for the $1$PI vertices. We also point out that a Callan-Symanzik type equation holds among the vertices when partition function is invariant under some changes of the parameters. After discussing general aspects of parameter response, we apply our formalism to several examples and reproduce the well-known functional flow equations. Our response theory provides us a systematic and general way to obtain various functional flow equations in equilibrium systems.

DOI
10.48550/arxiv.2309.10496

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Numerical fluid dynamics for FRG flow equations: Zero-dimensional QFTs as numerical test cases - Part I: The O(N)O(N)O(N) model
The functional renormalization group (FRG) approach is a powerful tool for studies of a large variety of systems, ranging from statistical physics over the theory of the strong interaction to gravity. The practical application of this approach relies on the derivation of so-called flow equations, which describe the change of the quantum effective action under the variation of a coarse-graining parameter. In the present work, we discuss in detail a novel approach to solve such flow equations. This approach relies on the fact that RG equations can be rewritten such that they exhibit similarities with the conservation laws of fluid dynamics. This observation can be exploited in different ways. First of all, we show that this allows to employ powerful numerical techniques developed in the context of fluid dynamics to solve RG equations. In particular, it allows to reliably treat the emergence of non-analytic behavior in the RG flow of the effective action as it is expected to occur in studies of, e.g., spontaneous symmetry breaking. Second, the analogy between RG equations and fluid dynamics offers the opportunity to gain novel insights into RG flows and their interpretation in general, including the irreversibility of RG flows. We work out this connection in practice by applying it to zero-dimensional quantum-field theoretical models. The generalization to higher-dimensional models is also discussed. Our findings are expected to help improving future FRG studies of quantum field theories in higher dimensions both on a qualitative and quantitative level.
The topic of this thesis is the functional renormalization group. We discuss some approximations schemes. Thereafter we apply these approximations to study different fields of condensed matter physics. Generally we have to evaluate an infinite set of vertex functions describing the scattering of particles. These vertex functions get renormalized away from their bare values governed by an infinite hierarchy of flow equations. We cannot expect to actually solve these equations but have to apply a couple of approximations. The aim is to somehow separate relevant contributions from irrelevant ones. One possible scheme opens up if we rescale fields and vertices. Here "relevance" is used in a quantitative way to describe the scaling behaviour of vertices close to a fixed point of the RG. One disadvantage of describing the system in terms of infinitely many vertices is that the majority of these vertices we have to evaluate are not of interest to us. In most cases we are just looking for the self-energy or the two-particle effective interaction. However there might be contributions to the flow of these vertices that are generated by irrelevant vertices. We generally assume that we can express irrelevant vertices in terms of the relevant and marginal ones. Then in turn it should be possible to write the contributions of these irrelevant vertices to the flow of relevant and marginal ones in terms of relevant and marginal vertices as well. We show how this can be achieved by what we term the adiabatic approximation. We now consider weakly interacting bosons at the critical point of Bose-Einstein condensation. As the transition takes place at a finite temperature this temperature defines an effective ultraviolet cut-off. For the investigation of physical properties that depend on momenta smaller than this cut-off it is therefore sufficient to describe the system by a classical field theory. Our central topic here is the self-energy of the bosons and we are able to evaluate it with the full momentum dependence. For small momenta it approaches a scaling form and as the momentum is gradually increased we observe a crossover to the perturbative regime. As a test for the reliability of our expression for the selfenergy we investigate the interaction induced shift of the critical. Our results compare quite satisfactory to the best available estimates for this shift. For the anomalous dimension our approach predicts the correct order of magnitude however with a considerable error. As an improvement we include more vertices into our calculations. Here we observe that our fixed point estimates indeed approach the best known results but this convergence is quite weak. We turn toward systems of interacting fermions. The formulation of the functional renormalization group implicitly requires knowledge of the true Fermi surface of the full interacting system. In general however we can just calculate it a-posteriori from the self-energy. The requirement to flow into a fixed point can be translated into a fine-tuning of the frequency/momentum independent part r_0 of the rescaled 2-point function. We show how this bare value is related to the momentum dependent effective interaction along the complete trajectory of the RG. On the other hand r_0 expresses the difference between the bare and the true Fermi surface. Putting both equations together results into an exact selfconsistency equation for the Fermi surface. We apply our self-consistency equation above to tackle the problem of finding the true Fermi surface of interacting fermions in low dimensions. The most simple non-trivial model with an inhomogeneous Fermi surface is a system of two coupled metallic chains. The process of interband backward scattering leads to a smoothing of the Fermi surface. Of special interest is if the Fermi momenta of the two bands collapse into just one value. We propose the term confinement transition for this behaviour. We bosonize the interband backward scattering by means of a Hubbard-Stratonovich transformation and treat our system as a single channel problem. This bosonization together with the adiabatic approximation allows us to investigate the system even at strong coupling. Within a simple one-loop treatment our method predicts a confinement transition at strong coupling. However taken vertex renormalizations into account we observe that this confinement is destroyed by fluctuations beyond one-loop. Actually we observe how the confined phase can be stabilized by the inclusion of interband umklapp scattering. Thereafter we consider the physically more relevant case of a two-dimensional system of infinitely many coupled metallic chains. Here the Fermi surface consists of two disconnected weakly curved sheets. We are able to repeat the calculations we have performed for our toy model. Within a self-consistent 2-loop calculation indeed signs for a confinement transition at finite coupling strength emerge.
The functional renormalization group (FRG) approach is a powerful tool for studies of a large variety of systems, ranging from statistical physics over the theory of the strong interaction to gravity. The practical application of this approach relies on the derivation of so-called flow equations, which describe the change of the quantum effective action under the variation of a coarse-graining parameter. In the present work, we discuss in detail a novel approach to solve such flow equations. This approach relies on the fact that RG equations can be rewritten such that they exhibit similarities with the conservation laws of fluid dynamics. This observation can be exploited in different ways. First of all, we show that this allows to employ powerful numerical techniques developed in the context of fluid dynamics to solve RG equations. In particular, it allows us to reliably treat the emergence of nonanalytic behavior in the RG flow of the effective action as it is expected to occur in studies of, e.g., spontaneous symmetry breaking. Second, the analogy between RG equations and fluid dynamics offers the opportunity to gain novel insights into RG flows and their interpretation in general, including the irreversibility of RG flows. We work out this connection in practice by applying it to zero-dimensional quantum-field theoretical models. The generalization to higher-dimensional models is also discussed. Our findings are expected to help improving future FRG studies of quantum field theories in higher dimensions both on a qualitative and quantitative level.
The Statistical Mechanics of the N-particle system is derived from Quantum Random Field Theory. An asymptotically measure preserving, flow operator has been obtained from the QFT evolution operator. A basic assumption is that the field Lagrangian is a generalized, infinitely divisible random field, and the interaction Hamiltonian varies in time in a step-wise manner at the transition lattice space-time points. Macroscopic boundary conditions are imposed on the wave functions. The density matrix, the partition function, the Bose-Einstein, the Fermi-Dirac distributions and the thermodynamic temperature follow. The equilibrium or non-equilibrium temperature have been obtained as functionals of the interaction Hamiltonian. The chemical potential has also been derived from the theory.
We consider formulations of the functional renormalization-group (fRG) flow for correlated electronic systems with the dynamical mean-field theory as a starting point. We classify the corresponding renormalization-group schemes into those neglecting one-particle irreducible six-point vertices (with respect to the local Green’s functions) and neglecting one-particle reducible six-point vertices. The former class is represented by the recently introduced DMF 2 RG approach [31], but also by the scale-dependent generalization of the one-particle irreducible representation (with respect to local Green’s functions, 1PI-LGF) of the generating functional [20]. The second class is represented by the fRG flow within the dual fermion approach [16, 32]. We compare formulations of the fRG approach in each of these cases and suggest their further application to study 2D systems within the Hubbard model.
Abstract By exploiting the convexity of the two-particle-irreducible effective action, we describe a procedure for extracting n-point vertex functions. This procedure is developed within the context of a zero-dimensional ‘quantum field theory’ and subsequently extended to higher dimensions. These results extend the practicability and utility of a recent, alternative approach to the functional renormalization group programme (see Alexander et al 2021 Phys. Rev. D 104 069906; Millington and Saffin 2021 J. Phys. A: Math. Theor. 54 465401), and clarify the relationship between the flow equations for coupling parameters and vertices.