D-TRILEX: Dual Triply Irreducible Local Expansion

• D-TRILEX is a diagrammatic method that extends DMFT by using a dual fermion-boson framework focused on a local three-leg vertex instead of the full four-point vertex.
• It combines nonperturbative local physics with diagrammatically reconstructed nonlocal self-energies and collective fluctuations via Hedin-like equations.
• The approach offers computational efficiency by bypassing heavy four-point vertex calculations, though its accuracy can decrease near strongly critical regimes.

Searching arXiv for D-TRILEX and related TRILEX papers to ground the article in the current literature. arXiv_search(query="D-TRILEX TRILEX dual triply irreducible local expansion", max_results=10, sort_by="relevance") to=arxiv_search code D-TRILEX, short for dual triply irreducible local expansion, is a self-consistent diagrammatic extension of dynamical mean-field theory built in a dual fermion-boson framework and organized around a local three-leg fermion-boson vertex rather than a local self-energy or a full local four-point vertex (Harkov et al., 2021, Vandelli et al., 2022). In the literature covered here, the term denotes a family of beyond-DMFT methods that start from a local or cluster reference problem, treat strong local or short-range correlations nonperturbatively, and reconstruct nonlocal electronic self-energies and collective fluctuations through Hedin-like equations involving dual fermionic and bosonic propagators (Harkov et al., 2021, Fossati et al., 8 Jul 2025). The designation should be distinguished from several adjacent usages in the TRILEX literature. One paper on triangular-lattice superconductivity uses single-site TRILEX, mostly in a simplified TRILEX approximation with $\Lambda^{\mathrm{imp},\eta}\approx 1$, and explicitly does not introduce a “D-TRILEX” formalism (Cao et al., 2017). Another paper develops a superconducting extension of TRILEX for $d$-wave pairing, but does not formally adopt “D-TRILEX” as the method name; there the phrase is at most an inferred shorthand for a $d$-wave superconducting TRILEX extension (Vucicevic et al., 2017).

1. Conceptual definition and place within beyond-DMFT methods

D-TRILEX is presented as a particularly simple yet consistent diagrammatic extension of DMFT that combines nonperturbative local physics with nonlocal charge and spin fluctuations while avoiding the explicit manipulation of the full local four-point vertex required in dual fermion, dual boson, or $D\Gamma A$-type approaches (Harkov et al., 2021, Vandelli et al., 2022). In this construction, locality is imposed not on the lattice self-energy $\Sigma(k,i\omega_n)$ itself, but on higher-order impurity objects such as the local three-leg vertex $\Lambda^\varsigma_{\nu\omega}$, local susceptibilities, and local screened bosonic propagators (Graspeuntner et al., 2022, Vandelli et al., 2022). This design allows momentum-dependent self-energies and polarizations to be reconstructed diagrammatically from a local reference system and nonlocal propagators (Graspeuntner et al., 2022, Dedov et al., 18 Nov 2025).

Within the method landscape, D-TRILEX occupies an intermediate position. Like DMFT, it begins from a self-consistent impurity model and retains a nonperturbative description of local correlations (Harkov et al., 2021, Vandelli et al., 2022). Like dual fermion and dual boson, it reformulates the problem in terms of auxiliary dual fields so that the remaining theory describes only nonlocal corrections beyond the local reference system (Harkov et al., 2021). Like TRILEX, it is organized around a three-leg fermion-boson vertex and a partially bosonized treatment of collective modes (Vandelli et al., 2022). Compared with ladder $D\Gamma A$, it avoids propagating the full local four-point vertex through ladder resummations in all frequencies and momenta, which is why the partially bosonized three-point formulation is repeatedly described as computationally cheaper (Dedov et al., 18 Nov 2025, Fossati et al., 8 Jul 2025). A further point emphasized in the dual derivation is that D-TRILEX is advertised as avoiding the Fierz-ambiguity problem that affects some channel-decomposed approaches (Fossati et al., 8 Jul 2025, Vandelli et al., 2022).

A recurring source of confusion is terminological overlap with TRILEX proper. The triangular-lattice superconductivity study on Si(111)-motivated adatom systems uses single-site TRILEX with a simplified vertex $\Lambda^{\mathrm{imp},\eta}\approx 1$, which the authors describe as “a GW+EDMFT like scheme,” but it is not a D-TRILEX paper (Cao et al., 2017). The superconducting TRILEX paper on the two-dimensional Hubbard model generalizes TRILEX to Nambu space and obtains $d$-wave superconductivity with a single-site impurity model, yet it likewise does not formally rename the method D-TRILEX (Vucicevic et al., 2017). By contrast, “D-TRILEX” is explicitly used as the central method name in the dual-space Hubbard-model analysis of van-Hove-driven ferromagnetic fluctuations and in the multiband dual formalism (Dedov et al., 18 Nov 2025, Vandelli et al., 2022).

2. Dual partially bosonized formalism

In the dual formulation, the lattice action is rewritten around a local impurity reference problem, after which Hubbard-Stratonovich transformations generate dual fermionic and bosonic fields (Harkov et al., 2021). The dual action contains bare dual propagators that are differences between lattice quantities dressed only by local impurity quantities and the impurity quantities themselves. In the single-band derivation this structure is written as

$$\tilde{\cal G}_{k\sigma} = \check G_{k\sigma}-g_{\nu\sigma}, \qquad \tilde{\cal W}^{\vartheta}_q = \check W^\vartheta_q - w^\vartheta_\omega,$$

with

$d$0

$d$1

(Harkov et al., 2021). In the multiband implementation, the same idea is expressed in matrix form,

$d$2

and

$d$3

(Vandelli et al., 2022).

The defining approximation of D-TRILEX is a partially bosonized representation of the local four-point impurity vertex in terms of local three-leg vertices and local screened interactions (Harkov et al., 2021, Vandelli et al., 2022). In the dual derivation this is encoded in the channel-resolved decomposition

$d$4

which is then used to approximate the exact local four-point vertex $d$5 by boson-exchange contributions in density, magnetic, and singlet channels (Harkov et al., 2021). The multiband paper gives the corresponding orbital-resolved decomposition,

$d$6

(Vandelli et al., 2022). This replacement is the precise sense in which the method is “triply irreducible”: the basic local object retained for the nonlocal expansion is the fermion-boson three-leg vertex rather than the full four-point vertex (Harkov et al., 2021, Vandelli et al., 2022).

The operational D-TRILEX equations then take the form of a self-consistent fermion-boson theory in dual space. In the square-lattice D-TRILEX analysis of ferromagnetic fluctuations, the central equations are

$d$7

$d$8

(Dedov et al., 18 Nov 2025). The dual Dyson equations are

$d$9

$d$0

(Dedov et al., 18 Nov 2025). The multiband version adds orbital and site indices to these same structures (Vandelli et al., 2022).

3. Reference systems, impurity quantities, and self-consistency

The impurity or reference problem is central because it supplies the local Green’s function $d$1, local self-energy $d$2, local susceptibilities, local polarization, and especially the local three-leg vertex $d$3 (Vandelli et al., 2022, Dedov et al., 18 Nov 2025). In the multiband formulation the reference action is written with fermionic hybridization $d$4 and optionally bosonic hybridizations $d$5, so the framework can be based on a DMFT or an EDMFT-type local problem (Vandelli et al., 2022). In practice, many of the calculations discussed in the paper use a DMFT impurity as reference (Vandelli et al., 2022).

The self-consistency loop in D-TRILEX is diagrammatic but local-input-driven. Starting from impurity quantities, one constructs the bare dual propagators, computes the dual polarization from a $d$6 bubble, obtains the dressed dual interaction, computes the dual self-energy from exchange of dressed dual bosons, updates the dressed dual Green’s function, and iterates to convergence (Vandelli et al., 2022, Dedov et al., 18 Nov 2025). The multiband paper states the convergence criterion as the relative Frobenius norm

$d$7

with convergence reached when $d$8 (Vandelli et al., 2022). Mixing is used for numerical stabilization in both fermionic and bosonic sectors (Vandelli et al., 2022).

Physical lattice quantities are reconstructed from the dual ones through exact dual-to-lattice relations. For the lattice self-energy, the single-band dual derivation gives

$d$9

while the multiband paper uses the matrix form

$D\Gamma A$0

(Harkov et al., 2021, Vandelli et al., 2022). The physical susceptibility is reconstructed from the lattice polarization through

$D\Gamma A$1

(Vandelli et al., 2022). One practical implication stressed in the multiband paper is that the divergence of the physical susceptibility coincides with the divergence of the dual renormalized interaction $D\Gamma A$2, which is why critical fluctuations can feed back strongly onto the electronic self-energy (Vandelli et al., 2022).

A methodological subtlety appears in the choice of reference problem. In the self-consistent D-TRILEX analysis of the Hubbard model, updating the impurity hybridization to satisfy the standard dual condition $D\Gamma A$3 can substantially improve results when DMFT is not the best local starting point (Harkov et al., 2021). In the cluster extension, a further issue is translational symmetry breaking inherited from CDMFT-like reference problems; there the authors argue that the reference cluster itself is the main source of periodization ambiguity and propose a rotated-basis cluster reference with diagonal hybridization to improve the starting point for the dual expansion (Fossati et al., 8 Jul 2025).

4. Single-site, multiband, and cluster realizations

The literature represented here includes several realizations of the same core idea. The multiband paper generalizes D-TRILEX to systems with multiple orbitals and several atoms in the unit cell, with local Coulomb tensors, channel-dependent long-ranged interactions, and matrix-valued Green’s functions, self-energies, susceptibilities, polarizations, and three-leg vertices (Vandelli et al., 2022). The combined orbital-site index $D\Gamma A$4 and the channel-resolved interaction tensors $D\Gamma A$5, $D\Gamma A$6, $D\Gamma A$7, and $D\Gamma A$8 define the formal architecture that makes the method applicable to extended Hubbard, Hubbard-Kanamori, and bilayer models (Vandelli et al., 2022). This implementation is explicitly able to account for frequency- and channel-dependent long-ranged electronic interactions (Vandelli et al., 2022).

The cluster-diagrammatic extension replaces the single-site reference problem by a cluster reference system so that short-range correlations are treated exactly within the cluster and long-range collective fluctuations are added diagrammatically beyond it (Fossati et al., 8 Jul 2025). In the benchmark one-dimensional nano-ring study, the authors use a dimer cluster and transform to a bonding-antibonding basis that diagonalizes the local part of the single-particle Hamiltonian,

$D\Gamma A$9

thereby allowing a diagonal hybridization function in the impurity problem and generating the off-diagonal self-energy diagrammatically (Fossati et al., 8 Jul 2025). This rotated-basis construction is justified because the local average of the dispersion provides the leading contribution to the rotated hybridization (Fossati et al., 8 Jul 2025). The cluster formalism retains the same dual structure, with

$\Sigma(k,i\omega_n)$0

and a Hedin-like dual polarization

$\Sigma(k,i\omega_n)$1

(Fossati et al., 8 Jul 2025).

The cluster study is also relevant because it sharpens what D-TRILEX is not. It is not merely cluster DMFT with periodization, because inter-cluster self-energy is generated diagrammatically rather than set to zero (Fossati et al., 8 Jul 2025). It is likewise not a parquet method, because it still avoids explicit four-point-vertex parquet machinery and Bethe-Salpeter equations in frequency space (Fossati et al., 8 Jul 2025). The authors therefore describe it as a hybrid between cluster DMFT and a dual boson/TRILEX-type Hedin-like extension (Fossati et al., 8 Jul 2025).

5. Benchmarking, accuracy, and physical regimes

A central contribution of the dual-theory analysis of collective fluctuations is to clarify why D-TRILEX can work well despite discarding large parts of the full local four-point vertex (Harkov et al., 2021). By comparing ladder dual fermion, exact dual diagrammatic Monte Carlo, diagrammatic Monte Carlo for the partially bosonized dual theory, and D-TRILEX, the paper concludes that the components of the local four-point vertex not represented by the partially bosonized approximation have only a minor effect on the electronic self-energy in a broad range of parameters (Harkov et al., 2021). At $\Sigma(k,i\omega_n)$2 for the half-filled square-lattice Hubbard model, the normalized deviation from the dual DiagMC reference is only $\Sigma(k,i\omega_n)$3 at $\Sigma(k,i\omega_n)$4 and $\Sigma(k,i\omega_n)$5 at $\Sigma(k,i\omega_n)$6 for D-TRILEX, while the largest discrepancy appears around $\Sigma(k,i\omega_n)$7, where $\Sigma(k,i\omega_n)$8 and antiferromagnetic fluctuations are strongest (Harkov et al., 2021). The interpretation offered is that in regimes where ladder dual fermion is accurate, the self-energy is dominated by longitudinal bosonic modes, whereas transverse particle-hole and particle-particle contributions largely cancel (Harkov et al., 2021).

This picture also identifies the method’s main limitation. In the weak-coupling Slater regime close to antiferromagnetic instability, the neglected transverse and anharmonic fluctuation contributions become important, and D-TRILEX can miss pseudogap formation or place it at too low a temperature (Harkov et al., 2021). At $\Sigma(k,i\omega_n)$9, the estimated antinodal pseudogap onset temperature is $\Lambda^\varsigma_{\nu\omega}$0 in exact DiagMC, $\Lambda^\varsigma_{\nu\omega}$1 in ladder dual fermion, and only $\Lambda^\varsigma_{\nu\omega}$2 in D-TRILEX (Harkov et al., 2021). At $\Lambda^\varsigma_{\nu\omega}$3, however, the agreement improves and the self-consistent version captures the nodal/antinodal differentiation much more accurately (Harkov et al., 2021). This suggests that the method becomes more reliable when magnetism is more local and harmonic rather than weak-coupling and strongly itinerant (Harkov et al., 2021).

The multiband benchmarks reinforce this pattern. For the Hubbard-Kanamori dimer, D-TRILEX nearly coincides with exact diagonalization for the Green’s function and yields an average energy error of about $\Lambda^\varsigma_{\nu\omega}$4 at $\Lambda^\varsigma_{\nu\omega}$5, decreasing with $\Lambda^\varsigma_{\nu\omega}$6 (Vandelli et al., 2022). At half filling and moderate interaction, where DMFT fails because the self-energy develops strong nonlocal real parts, D-TRILEX remains close to exact diagonalization (Vandelli et al., 2022). In the extended Hubbard model on the square lattice, comparison with DiagMC@DB shows almost perfect agreement at $\Lambda^\varsigma_{\nu\omega}$7, good agreement at $\Lambda^\varsigma_{\nu\omega}$8, and a clear deterioration at $\Lambda^\varsigma_{\nu\omega}$9, especially in $D\Gamma A$0, again consistent with the breakdown of ladder-like approximations in strongly fluctuating regimes (Vandelli et al., 2022).

The cluster nano-ring benchmark further illustrates the method’s selectivity. At the Fermi momentum $D\Gamma A$1 in metallic $D\Gamma A$2 and $D\Gamma A$3 rings, exact QMC shows insulating low-frequency behavior, single-site and cluster DMFT remain too metallic, parquet $D\Gamma A$4 performs surprisingly poorly for $D\Gamma A$5, and D-TRILEX gives the best low-frequency self-energy among the approximate methods (Fossati et al., 8 Jul 2025). Away from the Fermi energy at $D\Gamma A$6, by contrast, parquet $D\Gamma A$7 is more accurate, and D-TRILEX can even show noncausal low-frequency behavior for the smallest rings, which the authors attribute to the omission of particle-particle correlations and other missing diagram classes (Fossati et al., 8 Jul 2025). This suggests that D-TRILEX is especially adapted to low-energy particle-hole fluctuation physics near the Fermi surface, but less complete away from that regime (Fossati et al., 8 Jul 2025).

6. Applications and physical phenomena

The method has been applied to a broad range of correlated-electron problems. In pyrochlore iridates, TRILEX rather than D-TRILEX is used, but the study is still instructive because it demonstrates the core philosophy of reconstructing nonlocal self-energy effects from a local three-leg vertex (Graspeuntner et al., 2022). For $D\Gamma A$8, single-site DMFT gives a direct transition from a paramagnetic metal to an all-in/all-out antiferromagnetic insulator, whereas TRILEX produces strong momentum-dependent self-energy structure near the transition and yields evidence for a Weyl semimetal or at least a Weyl metal phase (Graspeuntner et al., 2022). The critical interaction is estimated around $D\Gamma A$9 eV in TRILEX versus around $\Lambda^{\mathrm{imp},\eta}\approx 1$0 eV in single-site DMFT, and the momentum variation of the self-energy becomes two orders of magnitude larger between $\Lambda^{\mathrm{imp},\eta}\approx 1$1 eV and $\Lambda^{\mathrm{imp},\eta}\approx 1$2 eV (Graspeuntner et al., 2022). The implication is that local-vertex diagrammatic extensions can qualitatively alter topological low-energy physics (Graspeuntner et al., 2022).

The 2025 D-TRILEX study of the square-lattice Hubbard model with $\Lambda^{\mathrm{imp},\eta}\approx 1$3 applies the fully self-consistent dual formalism to the regime $\Lambda^{\mathrm{imp},\eta}\approx 1$4, $\Lambda^{\mathrm{imp},\eta}\approx 1$5, $\Lambda^{\mathrm{imp},\eta}\approx 1$6, where van-Hove-enhanced ferromagnetic fluctuations dominate (Dedov et al., 18 Nov 2025). There the central finding is a low-temperature split spectral structure with only weak momentum dependence, but only one split branch crosses the Fermi level, so the Fermi surface itself remains unsplit while its area increases (Dedov et al., 18 Nov 2025). The state is interpreted as non-Fermi-liquid because the quasiparticle damping at the van Hove point $\Lambda^{\mathrm{imp},\eta}\approx 1$7 is about twice the DMFT value and remains nearly temperature independent or even increases slightly on cooling, while the first-Matsubara-frequency-rule test is completely violated at $\Lambda^{\mathrm{imp},\eta}\approx 1$8 and also violated at nodal and antinodal points (Dedov et al., 18 Nov 2025). The paper emphasizes that both self-consistent nonlocal self-energy feedback and proper treatment of the impurity three-leg vertex are necessary to obtain this behavior (Dedov et al., 18 Nov 2025).

The multiband D-TRILEX paper adds further model applications. In the two-orbital Hubbard-Kanamori model, D-TRILEX produces stronger correlation effects than DMFT: at $\Lambda^{\mathrm{imp},\eta}\approx 1$9 it reduces the quasiparticle peak in the wider band and opens a pseudogap in the narrower one, and at $d$0 it destroys the quasiparticle peak in both orbitals while DMFT still keeps one in the wider band (Vandelli et al., 2022). In the bilayer Hubbard model it captures the crossover from intralayer antiferromagnetic pseudogap behavior at small $d$1 to a dimer- or band-insulating regime with a four-peak density of states structure at $d$2 (Vandelli et al., 2022). In the extended Hubbard model with nearest-neighbor $d$3, it shows that increasing $d$4 strongly enhances the charge susceptibility at $d$5 while only slightly reducing the spin susceptibility, which is interpreted as screening of the magnetic channel by charge fluctuations (Vandelli et al., 2022).

It is important not to collapse these D-TRILEX results into the broader TRILEX literature without distinction. The triangular-lattice superconductivity paper studies a single-band triangular lattice with realistic $d$6 Coulomb tail and finds a dome-shaped chiral $d$7-wave superconducting phase in hole-doped systems, with both charge and spin fluctuations contributing cumulatively because the two $d$8-wave harmonics are symmetry-degenerate on the triangular lattice (Cao et al., 2017). However, the method actually used there is single-site TRILEX, mostly in a simplified vertex-unity approximation, not D-TRILEX (Cao et al., 2017). Likewise, the superconducting extension of TRILEX for the square-lattice Hubbard model yields a $d$9-wave dome at strong coupling and around $$\tilde{\cal G}_{k\sigma} = \check G_{k\sigma}-g_{\nu\sigma}, \qquad \tilde{\cal W}^{\vartheta}_q = \check W^\vartheta_q - w^\vartheta_\omega,$$0 doping for $$\tilde{\cal G}_{k\sigma} = \check G_{k\sigma}-g_{\nu\sigma}, \qquad \tilde{\cal W}^{\vartheta}_q = \check W^\vartheta_q - w^\vartheta_\omega,$$1, $$\tilde{\cal G}_{k\sigma} = \check G_{k\sigma}-g_{\nu\sigma}, \qquad \tilde{\cal W}^{\vartheta}_q = \check W^\vartheta_q - w^\vartheta_\omega,$$2, and $$\tilde{\cal G}_{k\sigma} = \check G_{k\sigma}-g_{\nu\sigma}, \qquad \tilde{\cal W}^{\vartheta}_q = \check W^\vartheta_q - w^\vartheta_\omega,$$3, with $$\tilde{\cal G}_{k\sigma} = \check G_{k\sigma}-g_{\nu\sigma}, \qquad \tilde{\cal W}^{\vartheta}_q = \check W^\vartheta_q - w^\vartheta_\omega,$$4 inferred from the eigenvalue condition $$\tilde{\cal G}_{k\sigma} = \check G_{k\sigma}-g_{\nu\sigma}, \qquad \tilde{\cal W}^{\vartheta}_q = \check W^\vartheta_q - w^\vartheta_\omega,$$5, but this is still explicitly formulated as superconducting TRILEX rather than D-TRILEX (Vucicevic et al., 2017).

7. Limitations, controversies, and outlook

The limitations of D-TRILEX are stated with unusual clarity across the cited literature. First, the method is approximate because it replaces the exact local four-point vertex by a partially bosonized decomposition and then retains only a restricted set of diagrams, typically longitudinal particle-hole charge and spin fluctuations (Harkov et al., 2021, Vandelli et al., 2022). This is the price paid for the large reduction in computational complexity relative to dual boson, dual fermion, or parquet $$\tilde{\cal G}_{k\sigma} = \check G_{k\sigma}-g_{\nu\sigma}, \qquad \tilde{\cal W}^{\vartheta}_q = \check W^\vartheta_q - w^\vartheta_\omega,$$6 (Vandelli et al., 2022, Fossati et al., 8 Jul 2025). A practical implication is that the method can overestimate crossover temperatures $$\tilde{\cal G}_{k\sigma} = \check G_{k\sigma}-g_{\nu\sigma}, \qquad \tilde{\cal W}^{\vartheta}_q = \check W^\vartheta_q - w^\vartheta_\omega,$$7 relative to DMFT because the full four-point vertex is approximated by three-point vertices connected through bosonic fluctuations (Dedov et al., 18 Nov 2025).

Second, accuracy degrades near strong criticality or in regimes dominated by strongly nonlinear fluctuations. This is seen in the weak-coupling antiferromagnetic pseudogap regime of the Hubbard model, in the deterioration relative to DiagMC@DB at $$\tilde{\cal G}_{k\sigma} = \check G_{k\sigma}-g_{\nu\sigma}, \qquad \tilde{\cal W}^{\vartheta}_q = \check W^\vartheta_q - w^\vartheta_\omega,$$8 in the extended Hubbard model, and in the noncausal behavior at $$\tilde{\cal G}_{k\sigma} = \check G_{k\sigma}-g_{\nu\sigma}, \qquad \tilde{\cal W}^{\vartheta}_q = \check W^\vartheta_q - w^\vartheta_\omega,$$9 for small nano-rings where omitted particle-particle or more complex diagrams are likely important (Harkov et al., 2021, Vandelli et al., 2022, Fossati et al., 8 Jul 2025). Third, the quality of the chosen reference problem matters. Self-consistent updating of the impurity hybridization can improve results when DMFT is a poor local starting point, but in cluster implementations the translational-symmetry breaking built into the cluster reference remains a bottleneck unless a much more expensive outer self-consistency loop is added (Harkov et al., 2021, Fossati et al., 8 Jul 2025).

Fourth, the bosonic sector can become numerically fragile near phase boundaries. In the pyrochlore iridate TRILEX study, fully self-consistent calculations could only be converged up to $d$00 eV because Monte Carlo slowing down and noisy bosonic propagators prevented stable Fourier transforms closer to the transition, forcing the physically interesting near-critical regime to be explored by one-shot TRILEX instead (Graspeuntner et al., 2022). The D-TRILEX ferromagnetic-fluctuation study does not report the same breakdown, but it likewise relies on careful impurity-vertex measurement and emphasizes that proper treatment of the local three-leg vertex is essential (Dedov et al., 18 Nov 2025).

A final source of confusion is the breadth of the TRILEX family itself. Single-site TRILEX, cluster TRILEX, TRILEX $d$01, GW+EDMFT-like simplified TRILEX, and dual D-TRILEX are distinct although related constructions (Cao et al., 2017, Richter et al., 2021, Vandelli et al., 2022, Fossati et al., 8 Jul 2025). The honeycomb-lattice spin-orbit-coupling paper, for example, uses a hermiticity-preserving “TRILEX $d$02” variant and explicitly notes that it is related to dual-boson and D-TRILEX approaches, but what is actually implemented is not D-TRILEX itself (Richter et al., 2021). This suggests that “D-TRILEX” is best reserved for methods explicitly formulated in dual space and built as a dual triply irreducible local expansion, not as a generic label for all three-leg-vertex extensions of DMFT (Vandelli et al., 2022, Harkov et al., 2021).

The outlook in the literature is therefore twofold. On the one hand, D-TRILEX is presented as a particularly attractive compromise between accuracy and feasibility for multiorbital materials and low-dimensional systems where local strong-correlation physics and nonlocal collective fluctuations must be combined (Vandelli et al., 2022, Fossati et al., 8 Jul 2025). On the other hand, the cited papers explicitly point toward extensions that include better reference systems, more complete diagrammatics, particle-particle channels, superconducting symmetry breaking, and outer self-consistency loops (Vandelli et al., 2022, Fossati et al., 8 Jul 2025). A plausible implication is that D-TRILEX serves less as an endpoint than as an efficient organizing principle: localize the right higher-order vertex object, retain self-consistent fermion-boson feedback, and generate nonlocal electronic structure diagrammatically without the full cost of four-point-vertex many-body methods (Harkov et al., 2021, Vandelli et al., 2022).