Diagrammatic bosonization, aspects of criticality, and the Hohenberg-Mermin-Wagner theorem in parquet approaches

Abstract

The parquet equations present a cornerstone of some of the most important diagrammatic many-body approximations and methods currently on the market for strongly correlated materials: from non-local extensions of the dynamical mean-field theory to the functional renormalization group. The recently introduced single-boson exchange decomposition of the vertex presents an alternative set of equivalent equations in terms of screened interactions, Hedin vertices, and rest functions. This formulation has garnered much attention for several reasons: opening the door to new approximations, for avoiding vertex divergences associated with local moment formation plaguing the traditional parquet decomposition, and for its interpretative advantage in its built-in diagrammatic identification of bosons without resorting to Hubbard-Stratonovich transformations. In this work, we show how the fermionic diagrams of the particle-particle and particle-hole polarizations in the SBE formalism can be mapped to diagrammatics of a bosonic self-energy of two respective bosonic theories with pure bosonic constituents, solidifying the identification of the screened interaction with a bosonic propagator. Resorting to a spin-diagonalized basis for the bosonic fields and neglecting the coupling between singlet and triplet components is shown to recover the trace log theory known from Hubbard-Stratonovich transformations. Armed with this concrete mapping, we revisit a conjecture claiming that universal aspects of the parquet approximation coincide with those of a self-consistent large-$N$ approximation for a bosonic $O(N)$ model. We comment on the role of the self-energy and crossing symmetry in enforcing the Hohenberg-Mermin-Wagner theorem in parquet-related approaches.

Figures (18)

• Figure 1: A possible solution strategy for the Parquet equations in the SBE formalism where the vertex cycle is decoupled from the self-energy update plain_and_simple_parquet_Krien_2022.
• Figure 2: Illustration of diagrammatic bosonization of a polarization diagram. 1) An example of a diagram of the polarization $P^\mathrm{ph}(q)$. The blue rectangles mark contributions to the boson propagators $w^\mathrm{ph}$. 2) Replacing the bosonic propagator contributions with bold full propagators $w^{\mathrm{ph}}$, one obtains a collection of bosonic lines mapped together by electron propagators. In grey, we shade the remaining sub-diagrams; those irreducible with respect to splitting an arbitrary number of bosonic propagators. 3) Identifying the G-polygons with bare bosonic vertices, we arrive at a purely bosonic skeleton self-energy diagram.
• Figure 3: The 2-point, 3-point, 4-point and 5-point bare bosonic vertices in the $r=\mathrm{ph}$ bosonic theory. The Green's function arrows form a polygon with arrows meeting head-to-tail. Higher-order vertices also appear and are constructed similarly. The 2-point bare vertex, the Lindhard bubble, plays the role of a zero-order self-energy, that does not appear in the bold/skeleton diagrams. The "$s$-wave" is there to indicate that these are simple loop integrals, and so as bosonic vertices, they only connect to $s$-wave bosonic propagators.
• Figure 4: For the $r = \mathrm{pp}$ theory, the vertices are formed of Green's functions whose arrows would instead meet head-to-head and tail-to-tail. Since the theory involves only $\mathrm{pp}$ propagators $w^\mathrm{pp}$, $U(1)$ symmetry entails that only vertices with an equal number of ingoing and outgoing legs can appear.
• Figure 5: In the diagrammatic analysis for $\mathcal{V}_0^3$ and higher skeleton self-energy diagrams in the main text, we refer to by "upper lane" and "lower lane" the spatial regions marked in this figure.