Parquet Formalism in Quantum Many-Body Systems

• Parquet Formalism is a self-consistent Green’s function approach that resums infinite Feynman diagrams to treat particle–particle and particle–hole correlations equally.
• It decomposes the effective interaction into three exclusive classes—irreducible, ladder, and ring—to avoid double-counting and capture nonperturbative effects.
• Applications in nuclear structure and condensed matter demonstrate its ability to provide near-quantitative predictions compared to exact solutions.

The parquet formalism is a self-consistent Green’s function approach designed to treat strong electronic correlations in quantum many-body systems by simultaneously resumming infinite classes of Feynman diagrams in multiple scattering channels. Unlike approximate schemes restricted to selected channels (e.g., ladder or ring diagrams), the parquet method captures the nonperturbative interplay of particle–particle and particle–hole correlations on an equal footing, rendering it particularly well-suited for ab initio calculations of nuclear structure and correlated electrons in condensed matter.

1. Fundamental Principles and Diagrammatic Structure

Parquet theory is motivated by the need to go beyond single-channel resummations and to avoid the double-counting of Feynman diagrams where multiple scattering processes overlap. In the Green’s function framework, the central object is the four-point (two-particle) vertex function, denoted $\Gamma^{(4\text{--}pt)}$, which represents the effective interaction among particles. The formalism partitions all two-body diagrams into three exclusive classes based on their reducibility in different channels:

• I: Diagrams simple (irreducible) with respect to all channels.
• L: Diagrams non-simple (reducible) in the particle–particle ([12]) channel (the "ladder" channel).
• R: Diagrams non-simple (reducible) in the particle–hole ([13] or [14]) channel (the "ring" channel).

The effective interaction is then decomposed as: $\Gamma^{(4\text{--}pt)} = I + L + R\ .$ The ladder ($L$) and ring ($R$) contributions are generated by Bethe–Salpeter–like equations in which the propagation of intermediate two-particle or particle–hole pairs is resummed to all orders. The self-energy, entering the Dyson equation for the one-body Green’s function, is obtained by inserting the effective interaction into appropriate diagrams.

2. Self-Consistent Equation Set and Numerical Workflow

The coupled equations defining the parquet formalism are intrinsically nonlinear and must be solved iteratively. The key components are:

• Dyson equation for the dressed Green’s function:

$$g_{\alpha\beta}(\omega) = g^0_{\alpha\beta}(\omega) + \sum_{\gamma\delta} g^0_{\alpha\gamma}(\omega) \Sigma(\gamma,\delta; \omega) g_{\delta\beta}(\omega)$$

where $g^0$ is the free (unperturbed) propagator and $\Sigma$ is the self-energy.

• Parquet decomposition:

$$[\Gamma(\Omega)] = [I(\Omega)] + [L(\Omega)] + [R(\Omega)]$$

• Bethe–Salpeter–like equations (schematic matrix notation):

$$[L(\Omega)] = [(I+R)(\Omega)]\,[G_0^{pphh}(\Omega)]\,[I+R+L](\Omega)$$

$$[R(\Omega)] = [(I+L)(\Omega)]\,[G_0^{ph}(\Omega)]\,[I+L+R](\Omega)$$

Here, $G_0^{pphh}$ and $G_0^{ph}$ are two-particle propagators for particle–particle/hole–hole and particle–hole channels, respectively.

The frequency variables can be treated in two ways:

• Energy-independent scheme: All two-time propagators are evaluated at a fixed "starting" energy $E_{in}$.
• Energy-dependent scheme: Propagators and vertices are defined on an explicit frequency grid, allowing for full energy dependence.

A small imaginary part $i\eta$ is systematically added in energy denominators to regularize pole structures, and final results are extrapolated to $\eta \to 0$.

The self-energy is updated via

$$\Sigma(1,2;\omega) = (\text{terms involving bare V}) + \frac{1}{2} \int \frac{d\omega'}{2\pi} \sum_{\alpha\beta\gamma\delta}\! V_{1\alpha,2\beta} G_0^{pphh}(\omega+\omega') \Gamma^{(4\text{--}pt)}(\omega+\omega') g_{\gamma\alpha}(\omega')$$

and the new Green’s function is computed from the Dyson equation, providing new pole positions and residues (i.e., spectral information) for the next iteration.

3. Schematic Nuclear Model and Benchmarks

The practical testing of the parquet formalism was performed using a schematic model Hamiltonian (Bergli et al., 2010), where:

• The single-particle spectrum is a set of equally spaced levels.
• The two-body interaction consists of a pair-conserving part (pure pairing, strength $g$) and a pair-breaking term (strength $f$).

This model reproduces essential features of nuclear correlations, including pairing (superfluidity) and core polarization. Numerical calculations show:

• For weak to moderate coupling strengths ($|g|,|f| \ll$ level spacing), parquet ground-state energies and correlation energies nearly coincide with exact diagonalization.
• As interaction strength increases or the model space grows, parquet solutions tend to "underbind" (i.e., underestimate the binding energy), but remain in qualitative agreement.
• In the pair-conserving case, the self-energy remains diagonal and spectral functions display dominant mean-field quasiparticle peaks, with fragmented spectral weight at higher energies.
• The energy-dependent formulation improves accuracy for small systems, but the proliferation of poles with increasing level number leads to a necessary increase in $\eta$, thus smearing spectral features toward the mean-field limit.

4. Implementation Details and Computational Considerations

The iterative solution of the parquet equations requires careful treatment of frequency integration and summation over many indices, often exploiting basis symmetries (e.g., angular momentum coupling). The following computational procedures are typically employed:

• Use of a finite single-particle basis (e.g., harmonic oscillator with cutoff) to make all sums tractable.
• Frequency grids for energy-dependent schemes; a fixed $E_{in}$ where appropriate for energy-independent schemes.
• Stabilization via a finite $\eta$ and extrapolation.
• Pole truncation: In spectral function computations, often only the pole nearest to the mean-field energy is retained:

$$S_h(\alpha, \omega) = \sum_{k<F} |z^k_\alpha|^2 \delta(\omega - \epsilon_k^-)$$

where $z^k_\alpha$ is a spectroscopic factor and $\epsilon_k^-$ is a removal energy.

For models with strong pair-breaking or near-degenerate levels, convergence can become delicate or unstable; in such cases, a larger $\eta$ is needed for algorithmic stability, although this may shift the results toward mean-field-like behavior.

5. Physical Insights and Comparison with Other Many-Body Approaches

The parquet formalism fundamentally extends the reach of Green’s function methods by capturing nonperturbative correlations that cannot be encompassed by ladder or ring summations alone. Specifically:

• Simultaneous inclusion of ladder (pp/hh) and ring (ph) diagrams ensures that both pairing and particle–hole polarization are treated on equal footing.
• Open-shell nuclei and states near quantum phase transitions are described more realistically, in contrast to approaches focusing on a single channel.
• Parquet calculations of ground-state and correlation energies, as well as of one-body spectral functions (e.g., spectroscopic factors), compare favorably with exact diagonalization and other ab initio approaches, including the coupled-cluster method and advanced shell model studies.

A plausible implication is that, when implemented with careful frequency treatment and starting from realistic nuclear Hamiltonians (including higher-body forces when necessary), the parquet approach may provide a computationally competitive and conceptually transparent framework for ab initio nuclear structure predictions, especially in regimes dominated by the interplay of multiple correlations.

6. Limitations and Prospective Developments

Several limitations are inherent to current parquet implementations:

• Computational cost grows rapidly with the size of the single-particle basis and the structure of the kernel in energy-dependent calculations.
• Explicit frequency dependence is difficult to manage except for the smallest systems, and schemes relying on fixed energy variables may introduce uncontrolled approximations near highly correlated or degenerate regimes.
• Instabilities may arise when off-diagonal self-energy terms (from pair-breaking interactions) induce level crossings or near-degeneracies.

Further improvements could stem from:

• Adaptive frequency grids or advanced kernel-approximation techniques (as explored in later works in the electronic context (Li et al., 2015, Li et al., 2017)).
• Systematic inclusion of three-body forces and extension toward the computation of response functions and effective interactions for odd nuclei.
• Hybrid formulations where parquet methods are embedded within auxiliary field approaches or used as the building blocks for even more advanced many-body resummations.

7. Summary Table of Core Parquet Equations and Elements

Equation/Element	Description	Key Symbol(s)
Dyson Equation	Dressed propagator from free plus self-energy	$g_{\alpha\beta}(\omega)$
Self-Energy	Diagrammatic insertion of four-point vertex	$\Sigma(\gamma,\delta; \omega)$
Parquet Decomposition	Vertex split into simple, ladder, ring contributions	$\Gamma = I + L + R$
Bethe–Salpeter (Ladder) Eqn	Ladder part constructed by convolution over int. states	$L = (I + R) \otimes G_0^{pphh} \otimes (I + R + L)$
Spectral Function (pole trunc.)	Dominant Lehmann contribution for hole removal	$S_h(\alpha, \omega)$

• "Summation of Parquet diagrams as an ab initio method in nuclear structure calculations" (Bergli et al., 2010)

The parquet formalism, through its balanced treatment of multiple correlation channels, provides a rigorously controlled diagrammatic method for nonperturbative nuclear structure calculations, with demonstrated qualitative and near-quantitative agreement to exact solutions in schematic models and promising prospects for application to realistic nuclear Hamiltonians.