Dynamical Hubbard Functional Overview

• Dynamical Hubbard functional is a framework that encodes Hubbard correlations using time-, frequency-, or coordinate-dependent functionals rather than static on-site parameters.
• It underpins diverse methodologies including nonequilibrium variational self-energy approaches, Green’s-function spectral functionals, DFT/TDDFT-based methods, and lattice structural coupling formulations.
• The approach delivers practical insights into dynamical Mott transitions, frequency-dependent screening, and the impact of structural modes on effective electron interactions.

Searching arXiv for papers on “dynamical Hubbard functional” and related formulations. “Dynamical Hubbard Functional” is not a single formalism but a family of closely related constructions in which Hubbard correlations are encoded in an explicitly time-, frequency-, or coordinate-dependent functional rather than in a purely static on-site parameter. In the literature, the term is used for a nonequilibrium grand-potential functional of the self-energy on the Keldysh–Matsubara contour, a Klein or Luttinger–Ward-type Green’s-function functional with a local screened interaction $U(\omega)$, a density-functional exchange–correlation potential derived from DMFT, a Hubbard augmentation kernel in TDDFPT, and a structural interaction law $U[Q]$ obtained from cRPA and embedded in DFT+DMFT (Hofmann et al., 2016, Chiarotti et al., 2023, Karlsson et al., 2010, Binci et al., 2024, Notter et al., 30 Oct 2025). The common element is the replacement of a static Hubbard correction by a dynamical object whose stationarity, functional derivative, or parametric dependence generates equations of motion, self-energies, response kernels, or lattice-controlled effective interactions.

1. Conceptual scope and principal meanings

The term covers several technically distinct uses. In nonequilibrium self-energy functional theory, the functional variable is the contour-ordered self-energy $\Sigma$, and the central object is a grand-potential functional $\Omega[\Sigma]$. In Green’s-function approaches for correlated solids, the functional variable is the interacting one-particle Green’s function $G(\omega)$, with a local dynamical exchange–correlation contribution $\Phi_{\text{dynH}}[\mathbf{G}]$. In lattice DFT and TDDFT, the “dynamical Hubbard functional” is an exchange–correlation potential $v_{xc}(n)$ derived from a dynamical many-body reference problem, typically DMFT. In TDDFPT, the relevant object is instead the linear variation of a Hubbard energy functional, which enters the dynamical susceptibility through a kernel $f_U$. In lattice-coupled correlated materials, one also encounters a coordinate-dependent interaction $U[Q]$, where a structural amplitude controls screened Coulomb interactions and Hubbard bands (Hofmann et al., 2016, Chiarotti et al., 2023, Karlsson et al., 2010, Binci et al., 2024, Notter et al., 30 Oct 2025).

Formulation	Central variable	Representative papers
Nonequilibrium SFT	$\Omega[\Sigma]$	(Hofmann et al., 2016, Hofmann et al., 2015)
Spectral Green’s-function dynH	$U[Q]$0, $U[Q]$1	(Chiarotti et al., 2023, Chiarotti et al., 25 Aug 2025, Caserta et al., 13 Mar 2025)
DMFT-based lattice DFT/TDDFT	$U[Q]$2, $U[Q]$3	(Karlsson et al., 2010, Boström et al., 2019)
TDDFPT Hubbard response	$U[Q]$4, $U[Q]$5, $U[Q]$6	(Binci et al., 2024)
Structural interaction functional	$U[Q]$7, $U[Q]$8	(Notter et al., 30 Oct 2025)

A plausible implication is that the phrase is best understood as a category label for dynamical functionalizations of Hubbard physics rather than as the name of a unique formal theory. That plurality is explicit across the arXiv literature.

2. Nonequilibrium self-energy functionals and real-time Mott physics

In the nonequilibrium SFT formulation of the Fermi–Hubbard model, the basic real-time object is a contour-ordered Green’s function on the Keldysh–Matsubara contour $U[Q]$9,

$\Sigma$0

with Dyson equation

$\Sigma$1

The dynamical Hubbard functional is the grand-potential functional

$\Sigma$2

evaluated on trial self-energies $\Sigma$3 generated by a solvable reference system. Stationarity with respect to the time-dependent one-body parameters $\Sigma$4 of the reference system,

$\Sigma$5

defines the optimal approximation. In the two-site dynamical impurity approximation, the only variational parameter is the hybridization $\Sigma$6 between a correlated impurity and a single auxiliary bath site (Hofmann et al., 2016).

For the half-filled Hubbard Hamiltonian

$\Sigma$7

the formalism was applied to sudden quenches and finite-time cosine ramps of $\Sigma$8, with $\Sigma$9, $\Omega[\Sigma]$0, a one-dimensional chain of $\Omega[\Sigma]$1 sites, and time windows up to $\Omega[\Sigma]$2. The optimized $\Omega[\Sigma]$3, the double occupancy $\Omega[\Sigma]$4, the momentum distribution $\Omega[\Sigma]$5, and $\Omega[\Sigma]$6 diagnose the dynamical Mott transition. Two regimes emerge: weak coupling with prethermalization and a persistent Fermi-surface jump, and strong coupling with collapse-and-revival oscillations at frequency $\Omega[\Sigma]$7. These are separated by a sharp critical interaction $\Omega[\Sigma]$8, at which the low-energy bath decouples dynamically,

$\Omega[\Sigma]$9

For a sudden quench, $G(\omega)$0 in the 1D-chain units used in the paper, corresponding to $G(\omega)$1 after rescaling to DOS variance; for a ramp with $G(\omega)$2, $G(\omega)$3, and $G(\omega)$4 increases monotonically toward the equilibrium zero-temperature upper spinodal $G(\omega)$5 (Hofmann et al., 2016).

The nonequilibrium variational-cluster approach provides the methodological precursor for stable real-time propagation within this framework. There, direct solution of the Euler equation was found to be numerically unstable, and the practical condition was replaced by

$G(\omega)$6

which preserves causal time stepping because updates at time $G(\omega)$7 do not affect earlier times $G(\omega)$8 (Hofmann et al., 2015). Taken together, these works establish a precise meaning of “dynamical Hubbard functional” as a real-time variational self-energy functional for Hubbard dynamics.

3. Green’s-function spectral functionals for correlated solids

A second major meaning is the Green’s-function-based dynamical Hubbard functional for realistic solids. In this formulation, the starting point is a Klein functional of the interacting Green’s function,

$G(\omega)$9

where $\Phi_{\text{dynH}}[\mathbf{G}]$0 is the projection of $\Phi_{\text{dynH}}[\mathbf{G}]$1 onto a localized correlated subspace. The dynamical exchange–correlation term is taken in a localized $\Phi_{\text{dynH}}[\mathbf{G}]$2-like form, so that the self-energy is local in the correlated manifold and explicitly frequency dependent. In the site- and spin-resolved form used for transition-metal monoxides,

$\Phi_{\text{dynH}}[\mathbf{G}]$3

with $\Phi_{\text{dynH}}[\mathbf{G}]$4 computed from RPA screening and $\Phi_{\text{dynH}}[\mathbf{G}]$5 providing a fully localized limit double-counting correction (Chiarotti et al., 2023, Caserta et al., 13 Mar 2025).

A distinctive computational feature is the sum-over-poles representation and algorithmic inversion method. If

$\Phi_{\text{dynH}}[\mathbf{G}]$6

the Dyson equation can be mapped exactly to a noninteracting augmented Hamiltonian $\Phi_{\text{dynH}}[\mathbf{G}]$7, whose diagonalization yields the poles and residues of $\Phi_{\text{dynH}}[\mathbf{G}]$8. This makes addition, multiplication, convolution, and inversion closed operations within the sum-over-poles algebra, avoids analytic continuation, and permits real-axis evaluation of both spectra and total energies (Chiarotti et al., 2023).

The one-shot implementation for SrVO$\Phi_{\text{dynH}}[\mathbf{G}]$9 produced an occupied $v_{xc}(n)$0 bandwidth of $v_{xc}(n)$1 eV, a mass enhancement $v_{xc}(n)$2, a lower satellite at $v_{xc}(n)$3 eV, and upper satellites at $v_{xc}(n)$4 eV and $v_{xc}(n)$5 eV. The fully self-consistent extension preserved the bandwidth at $v_{xc}(n)$6 eV, yielded $v_{xc}(n)$7, placed the lower satellite at $v_{xc}(n)$8 eV and the upper satellites at $v_{xc}(n)$9 eV, and improved equilibrium properties such as lattice parameter and bulk modulus (Chiarotti et al., 2023, Chiarotti et al., 25 Aug 2025).

The same dynH formalism was extended to spin polarization and multiple correlated sites in MnO, FeO, CoO, and NiO. It reproduced, among other features, a small FeO gap of $f_U$0 eV, a NiO main addition peak near $f_U$1 eV, and an overall NiO bandwidth of $f_U$2 eV, while using a scalar orbital-averaged $f_U$3 rather than an explicit Kanamori tensor. In NiO the reported zero-frequency interaction is $f_U$4 eV (Caserta et al., 13 Mar 2025). Within this strand of work, “dynamical Hubbard functional” denotes a variational spectral functional of $f_U$5 that generalizes DFT+$f_U$6 by replacing a static $f_U$7 with a frequency-dependent screened interaction.

4. Density-functional and response-theory realizations

In lattice DFT for the 3D Hubbard model, the dynamical Hubbard functional appears as an exchange–correlation potential extracted from DMFT thermodynamics of the homogeneous reference system. The xc energy density is defined by

$f_U$8

and the xc potential is

$f_U$9

For the simple cubic lattice, the DMFT-derived $U[Q]$0 develops a discontinuity at half filling only for $U[Q]$1. In the time-dependent setting, the potential is used within an adiabatic local density approximation,

$U[Q]$2

which was benchmarked on cluster dynamics and applied to Bloch oscillations in a $U[Q]$3 trapped cluster (Karlsson et al., 2010).

For the Hubbard–Holstein model, the functional variables are the site occupations $U[Q]$4 and phonon coordinates $U[Q]$5. The Lang–Firsov transformation produces a screened local interaction

$U[Q]$6

and the DFT construction yields $U[Q]$7 for the phonon xc field. For the isolated site, the analytic electronic xc potential depends on $U[Q]$8, and the location of the discontinuity changes with the sign of $U[Q]$9: for $\Omega[\Sigma]$0 it is at $\Omega[\Sigma]$1, whereas for $\Omega[\Sigma]$2 it shifts to $\Omega[\Sigma]$3 and $\Omega[\Sigma]$4. The corresponding ALDA was shown to describe linear conductance and real-time dynamics with good accuracy against exact benchmarks (Boström et al., 2019).

In TDDFPT for magnons, the functional is instead a nonempirical Hubbard augmentation of the response kernel. In the Dudarev form,

$\Omega[\Sigma]$5

which gives the Hubbard potential

$\Omega[\Sigma]$6

Its linear variation defines the kernel contribution $\Omega[\Sigma]$7, and the full susceptibility obeys

$\Omega[\Sigma]$8

with $\Omega[\Sigma]$9. The method was formulated in a general noncollinear setting, implemented through the Liouville–Lanczos approach, and reported to satisfy the Goldstone condition without empirical rescaling of the exchange–correlation kernel (Binci et al., 2024). In this context, the functional is “dynamical” because the Hubbard correction enters the full frequency-dependent response, even though $U[Q]$00 and $U[Q]$01 themselves are static parameters determined nonempirically by linear response.

5. Structural and mode-resolved interaction functionals

A more recent usage makes the Hubbard interaction itself a functional of a lattice coordinate. For monolayer 1T-TaS$U[Q]$02, the star-of-David charge-density-wave amplitude is parameterized by

$U[Q]$03

where $U[Q]$04 is the central–outer Ta distance in the 13-Ta cluster. Constrained RPA gives the partially screened interaction

$U[Q]$05

projected onto the narrow molecular Wannier orbital that forms the half-filled correlated band. Near equilibrium, the static screened interaction is approximately linear in the distortion,

$U[Q]$06

with $U[Q]$07 eV at $U[Q]$08 and $U[Q]$09 eV/\% for $U[Q]$10 (Notter et al., 30 Oct 2025).

The structural dependence is large. The bare on-site interaction decreases from $U[Q]$11 eV at $U[Q]$12 to $U[Q]$13 eV at $U[Q]$14. Over the interval $U[Q]$15 to $U[Q]$16, the Wannier spread grows from $U[Q]$17 $U[Q]$18 to $U[Q]$19 $U[Q]$20, the effective bandwidth $U[Q]$21 grows from $U[Q]$22 meV to $U[Q]$23 meV, and the $U[Q]$24-resolved Mott gap shrinks from $U[Q]$25 eV to $U[Q]$26 eV. At $U[Q]$27, DFT+DMFT yields an insulator-to-correlated-metal transition with a quasiparticle peak at $U[Q]$28 (Notter et al., 30 Oct 2025).

Because the coherent star-of-David amplitude mode has $U[Q]$29 THz, the time-dependent interaction is written as

$U[Q]$30

For $U[Q]$31, the expected modulation is $U[Q]$32 meV; for $U[Q]$33, $U[Q]$34 eV. The same work therefore proposes a minimal driven Hubbard Hamiltonian in which both hopping and interaction are functionals of the instantaneous lattice coordinate,

$U[Q]$35

This use of “dynamical Hubbard functional” is literal: the screened interaction becomes a calculable function of a collective structural mode.

6. Exact, field-theoretic, and renormalization-group constructions

The term also appears in more formal many-body settings. In the Kotliar–Ruckenstein slave-boson representation of the two-site extended Hubbard model, an exact imaginary-time functional integral was constructed in the radial gauge, with fermions, a complex $U[Q]$36-boson, radial slave-boson amplitudes, and time-dependent constraint fields. The discrete-time action includes a regularized time-shifted form of the KR $U[Q]$37-factors and a slice-by-slice projection operator that enforces the constraints exactly. In the continuum limit, the projected functional integral reproduces the exact partition function and exact Matsubara Green’s function of the operator formulation, while resolving the normal-ordering and square-root objections associated with the KR representation (Dao et al., 2024).

In the two-dimensional Hubbard model, a “dynamical functional” viewpoint arises within functional renormalization group calculations that retain full Matsubara-frequency dependence of effective interactions and gap functions. The scale-dependent effective action $U[Q]$38 is evolved in the symmetric regime, then continued below $U[Q]$39 with dynamical magnetic and pairing gap functions $U[Q]$40 and $U[Q]$41. For $U[Q]$42 and $U[Q]$43, this produced a regime with robust $U[Q]$44-wave pairing coexisting with antiferromagnetism between $U[Q]$45 and $U[Q]$46, magnetic order up to $U[Q]$47, and a Kosterlitz–Thouless superconducting dome centered around $U[Q]$48 hole doping (Vilardi et al., 2020). Here the “dynamical” content is the explicit retention of full frequency dependence in the functional flow.

A geometric and algebraic variant appears in the supergroup approach to the Hubbard model. There, even Hubbard operators are related to generators of rotations in four-dimensional space, translations are obtained by deformation and contraction to an ISO(3)-type algebra, and a supercoherent-state construction yields an effective functional for spinless fermions in the large-$U[Q]$49 regime. The resulting effective Lagrangian,

$U[Q]$50

contains bosonic Cartan-form contributions and fermionic gradient terms on a curved group manifold (Zharkov et al., 2011). This is the most geometrically literal interpretation of a Hubbard functional as an effective action defined on a nontrivial supergroup space.

7. Limitations, comparisons, and prospective directions

Across these formulations, dynamical Hubbard functionals trade locality and controllability against completeness. In nonequilibrium SFT with a single bath site, particle number and spin are conserved, but energy conservation is not exact unless a continuum of variational degrees of freedom is included; the two-site reference also limits long-time thermalization and misses nonlocal correlations (Hofmann et al., 2016). In Green’s-function dynH for solids, the self-energy is strictly local to the correlated manifold, the interaction is typically reduced to an orbital-averaged scalar $U[Q]$51, and nonlocal self-energy effects are absent; the self-consistent SrVO$U[Q]$52 studies therefore identify nonlocal extensions, intersite corrections, and combinations with DFT+$U[Q]$53 as natural continuations (Chiarotti et al., 25 Aug 2025, Caserta et al., 13 Mar 2025).

Density-functional realizations inherit the limitations of local and adiabatic approximations. In the 3D Hubbard TDDFT construction, the ALDA lacks memory and spatial nonlocality, and the derivative discontinuity near half filling can produce sizable deviations under fast driving or in strongly inhomogeneous clusters (Karlsson et al., 2010). In the Hubbard–Holstein DFT, the lattice functional is local and memoryless, and the attractive regime was treated analytically only for the isolated site, while fuller DMFT-based attractive functionals were deferred (Boström et al., 2019). In TDDFPT for magnons, the dynamical response is improved by the Hubbard kernel, but the ground-state correction still uses static $U[Q]$54 and $U[Q]$55, not an explicit $U[Q]$56 (Binci et al., 2024).

The structural $U[Q]$57 formulation for 1T-TaS$U[Q]$58 makes a different approximation: DFT+DMFT uses only the static $U[Q]$59, with no dynamic $U[Q]$60, no nonlocal $U[Q]$61, and no EDMFT corrections. The authors explicitly identify EDMFT or GW+EDMFT as directions for refining both frequency-dependent screening and momentum-dependent self-energy effects (Notter et al., 30 Oct 2025). A plausible synthesis is that future “dynamical Hubbard functionals” will increasingly couple three ingredients that are separated in current implementations: local dynamical self-energies, nonlocal screening or correlations, and explicit coupling to collective coordinates such as phonons, magnons, or structural modes.

In that broader sense, the subject is less a closed formalism than an evolving research program. Its unifying objective is to cast Hubbard correlation physics into a functional language that remains variational, self-consistent, and computationally tractable while retaining genuinely dynamical information—whether in real time, in frequency space, or along a driven structural coordinate.