The Variational Discrete Action Theory (VDAT) of Cheng and Marianetti (Phys. Rev. Lett. 126, 206402 (2021)) is a variational approach to quantum many-body ground states built on the sequential product density matrix (SPD), an ansatz with an integer level N\mathcal{N} that monotonically approaches the exact solution, evaluated exactly in d=∞d=\infty via the self-consistent canonical discrete action approximation (SCDA), the integer-time analogue of dynamical mean-field theory (DMFT). The founding papers posed several unresolved directions: scalable (diagrammatic or quantum Monte Carlo) evaluation of the discrete action, VDAT with unitary projectors, multi-orbital extensions, the parameterization of energy functionals for the off-shell effective energy theory, and a practical DFT+VDAT workflow for first-principles correlated materials. Later literature by the authors substantially resolved the multi-orbital d=∞d=\infty extension (PRB 106, 205129 (2022); PRB 108, 035127 (2023); PRB 113, 085137 (2026)), and SCDA has been applied pragmatically to a two-dimensional moiré Hubbard model (PNAS 121, e2311486121 (2024)), but the remaining directions are open: no discrete-action diagrammatic/QMC method, no developed unitary-SPD formalism, and no implemented DFT+VDAT workflow exist at present, and the energy-functional parameterization has advanced only to a recent preprint. The surviving open core is the completion of VDAT as a scalable and practically applicable method for strongly correlated electrons beyond the exactly treatable d=∞d=\infty density-density class.

Background

The Hubbard model — electrons hopping between lattice sites with a local repulsion UU — is the paradigmatic model of strongly correlated electrons, and much of condensed-matter theory is devoted to solving it and its multi-orbital generalizations. Classic variational wave functions include the Gutzwiller wave function, which suppresses energetically costly double occupancy by a site-local projector, and the Baeriswyl wave function, which instead applies a projector built from the kinetic-energy operator. Dynamical mean-field theory (DMFT) maps the lattice model onto a self-consistent Anderson impurity model (AIM, a correlated site hybridized with a non-interacting bath) and becomes exact in the limit of infinite spatial dimension d=∞d=\infty; combined with density-functional theory (DFT) it yields DFT+DMFT, the current standard for first-principles correlated materials, at a substantial computational cost dominated by the impurity solver.

VDAT, introduced by Z. Cheng and C. A. Marianetti (Phys. Rev. Lett. 126, 206402 (2021)) and elaborated in the companion paper (Phys. Rev. B 103, 195138 (2021)), unifies these threads. Its ansatz is the sequential product density matrix (SPD): an alternating product of exponentials of the non-interacting and interacting parts of the Hamiltonian, exp⁡(γ1H^0)exp⁡(g1V^)⋯exp⁡(γNH^0)exp⁡(gNV^)\exp(\gamma_1 \hat H_0)\exp(g_1 \hat V)\cdots\exp(\gamma_{\mathcal{N}} \hat H_0)\exp(g_{\mathcal{N}} \hat V) applied to a Slater determinant, a variational analogue of a Trotter–Suzuki decomposition. The integer N\mathcal{N} controls the variational power: N=1\mathcal{N}=1 recovers Hartree–Fock, N=2\mathcal{N}=2 the Gutzwiller approximation (GA), N=3\mathcal{N}=3 the Gutzwiller–Baeriswyl wave function, and N→∞\mathcal{N}\to\infty the exact ground state. The key technical innovation is the discrete action theory (DAT): a Green's-function formalism on a discrete 'integer' time index τ=1,…,N\tau=1,\dots,\mathcal{N}, with its own Wick theorem, path integral, generating functional, Dyson equation, and Bethe–Salpeter equation, which evaluates expectation values under an SPD. For a lattice SPD (class SPD-d), the self-consistent canonical discrete action approximation (SCDA) assumes a local integer-time self-energy, Σij(g)=δijΣloc(gloc)\Sigma_{ij}(g)=\delta_{ij}\Sigma_{\mathrm{loc}}(g_{\mathrm{loc}}), exactly parallel to DMFT's local self-energy, and is proven to evaluate the SPD exactly in d=∞d=\infty. Because the evaluated energy is a rigorous variational upper bound, VDAT at N=3\mathcal{N}=3 reproduces Mott physics in the AIM and the d=∞d=\infty Hubbard model at a cost comparable to the GA — a 'minimal yet precise' description positioned between GA and DMFT.

The founding papers explicitly listed what VDAT could not yet do. First, exact SPD evaluation relies on summing integer-time Wick (Taylor) expansions of the interacting projectors whose term count grows explosively with N\mathcal{N} and system size, so a diagrammatic or auxiliary-field quantum Monte Carlo (AFQMC, a stochastic sampling of the Trotter time path) adaptation of the discrete-action formalism was posed as an open computational challenge. Second, the SPDs used are projective — Hermitian, real, semidefinite — while a unitary SPD, in which the projectors are unitary operators, would connect VDAT to unitary coupled cluster (UCC, a variational truncated Trotter ansatz used in quantum chemistry) and to variational quantum algorithms on quantum computers; Foundations defined this class but treated only pedagogical two-site examples. Third, SCDA exactness is proven only in d=∞d=\infty; the Foundations paper noted that the SCDA lattice SPD 'can also be applied as an approximation in finite dimensions, improving upon well characterized theories such as the Gutzwiller approximation', but no controlled finite-dimensional theory was developed. Fourth, the authors proposed DFT+VDAT, in the same spirit as DFT+U and DFT+Gutzwiller, as a potentially efficient first-principles rival to DFT+DMFT — but no such workflow existed. Fifth, VDAT was proposed as a parameterization of energy functionals for the authors' off-shell effective energy theory (Phys. Rev. B 101, 081105 (2020)), a formally exact construction based on an N=2\mathcal{N}=2 SPD.

Since 2021 the authors' group has advanced the d=∞d=\infty machinery considerably — decoupled minimization for multi-orbital models (PRB 106, 205129 (2022)), a closed-form gauge-constrained algorithm at N=3\mathcal{N}=3 for general density-density interactions up to eight orbitals (PRB 108, 035127 (2023)), and a mathematically equivalent 'qubit parametrization' reformulating the variational parameters as the pure state of an auxiliary two-level system per spin-orbital (PRB 113, 085137 (2026)) — and an external collaboration has applied SCDA at N=3\mathcal{N}=3 to a two-dimensional moiré Hubbard model with encouraging accuracy (PNAS 121, e2311486121 (2024)). None of this, however, delivers stochastic or diagrammatic evaluation at large N\mathcal{N}, a unitary-SPD theory, a realized energy-functional parameterization, or a DFT-integrated materials workflow.

Problem Statement

For the Variational Discrete Action Theory (VDAT) of Cheng and Marianetti, whose sequential product density matrix (SPD) ansatz — an alternating product of exponentials of the non-interacting and interacting parts of the Hamiltonian at integer level N\mathcal{N} — is exactly evaluable within the self-consistent canonical discrete action approximation (SCDA, the integer-time analogue of DMFT with a local integer-time self-energy) only in d=∞d=\infty for density-density multi-orbital Hubbard models (the multi-orbital extension posed in the source is thereby substantially complete), develop the method's remaining open development directions, posed as possible future directions in the concluding sections of the founding papers, which would make VDAT a scalable and practically applicable theory for strongly correlated electron systems:

  1. Scalable evaluation of the discrete action. Replace the direct integer-time Wick/Taylor expansion of the interacting projectors — whose term count grows explosively with N\mathcal{N} and system size — by an efficient general evaluation scheme applicable to general SPDs at large N\mathcal{N}, such as a diagrammatic resummation or an auxiliary-field quantum Monte Carlo adaptation of the integer-time (compound-space) path integral.

  2. VDAT with unitary projectors. Develop VDAT for SPDs whose projectors are unitary (complex) rather than Hermitian real projectors — rebuilding the evaluation theory, which currently relies on semidefinite projective density matrices, for this class — so that VDAT connects quantitatively to unitary coupled cluster and to variational quantum-computing ansätze.

  3. A practical DFT+VDAT workflow. Realize the proposed combination of VDAT with density-functional theory, in the same spirit as DFT+U and DFT+Gutzwiller, so that DFT+VDAT becomes an efficient first-principles approach to strongly correlated materials which can rival DFT+DMFT.

  4. Parameterization of energy functionals. Develop the proposed use of VDAT as a key tool for parameterizing energy functionals in the context of the authors' off-shell effective energy theory (Phys. Rev. B 101, 081105 (2020)) — viewed in the source as a formally exact construction based on an N=2\mathcal{N}=2 SPD — beyond the preprint-stage progress noted below.

Each of these directions was posed in the concluding sections of the founding papers: the PRL conclusion describes 'many possible directions for future development, given the obvious parallels between the discrete action theory and traditional many-body Green's function theory', and the Foundations conclusion (Section VIII) lists 'many near term and far term directions to consider for VDAT'. None of the four is discharged at present.

These four directions stand independently rather than jointly: advancing any one of them with a concrete, self-contained result discharges the task, and full resolution of the remaining directions is not required. The required deliverable is such a worked result — a derivation, method, or proof-of-principle that makes demonstrable progress on the chosen direction — rather than a research plan or proposal, and the depth required is that demonstrable progress rather than completion of the corresponding multi-year program.

References

  1. Z. Cheng and C. A. Marianetti, 'Variational Discrete Action Theory', Phys. Rev. Lett. 126, 206402 (2021). doi:10.1103/PhysRevLett.126.206402; arXiv:2011.14510; https://arxiv.org/abs/2011.14510
  2. Z. Cheng and C. A. Marianetti, 'Foundations of variational discrete action theory', Phys. Rev. B 103, 195138 (2021). doi:10.1103/PhysRevB.103.195138; arXiv:2011.14509; https://arxiv.org/abs/2011.14509
  3. Z. Cheng and C. A. Marianetti, 'Precise ground state of multiorbital Mott systems via the variational discrete action theory', Phys. Rev. B 106, 205129 (2022). doi:10.1103/PhysRevB.106.205129; arXiv:2206.08433; https://arxiv.org/abs/2206.08433
  4. Z. Cheng and C. A. Marianetti, 'Gauge constrained algorithm of variational discrete action theory at N=3\mathcal{N}=3 for the multiorbital Hubbard model', Phys. Rev. B 108, 035127 (2023). doi:10.1103/PhysRevB.108.035127; arXiv:2304.14616; https://arxiv.org/abs/2304.14616
  5. Z. Cheng and C. A. Marianetti, 'Qubit parametrization of the variational discrete action theory for the multiorbital Hubbard model', Phys. Rev. B 113, 085137 (2026). doi:10.1103/g2t7-cy32; arXiv:2508.20111; https://arxiv.org/abs/2508.20111
  6. Z. Cheng and C. A. Marianetti, 'Mott transition from the non-analyticity of the one-body reduced density-matrix functional', arXiv preprint (2025). arXiv:2508.14110; https://arxiv.org/abs/2508.14110
  7. P. Tscheppe, J. Zang, M. Klett, S. Karakuzu, A. Celarier, Z. Cheng, C. A. Marianetti, T. A. Maier, M. Ferrero, A. J. Millis, and T. Schäfer, 'Magnetism and metallicity in moiré transition metal dichalcogenides', Proc. Natl. Acad. Sci. USA 121, e2311486121 (2024). doi:10.1073/pnas.2311486121; arXiv:2303.13578; https://arxiv.org/abs/2303.13578
  8. Z. Cheng and C. A. Marianetti, 'Off-shell effective energy theory: A unified treatment of the Hubbard model from d=1d=1 to d=∞d=\infty', Phys. Rev. B 101, 081105 (2020). doi:10.1103/PhysRevB.101.081105; arXiv:1911.03102; https://arxiv.org/abs/1911.03102
  9. J. Zhang and Z. Cheng, 'Density-matrix mean-field theory', SciPost Phys. 17, 062 (2024). doi:10.21468/SciPostPhys.17.2.062; https://scipost.org/10.21468/SciPostPhys.17.2.062
  10. K. Li, Y.-B. Zhang, and H. C. Po, 'Constructive fermionic matrix product states for projected Fermi sea', Phys. Rev. Res. 7, 013182 (2025). doi:10.1103/PhysRevResearch.7.013182; https://doi.org/10.1103/PhysRevResearch.7.013182