Nonequilibrium Bosonic DMFT

• Nonequilibrium bosonic DMFT is a framework that extends traditional DMFT to lattice bosons by incorporating memory effects and nonperturbative local dynamics.
• It maps the complex lattice problem to an effective impurity action with a self-consistent, time-retarded hybridization function to capture damping and thermalization.
• The theory also integrates open-system extensions that model dissipative processes and finite-connectivity corrections, revealing phenomena like quantum-Zeno behavior and dynamical instabilities.

Searching arXiv for recent and foundational papers on nonequilibrium bosonic dynamical mean-field theory, including open-system extensions. Nonequilibrium bosonic dynamical mean-field theory (BDMFT) is a nonequilibrium extension of bosonic dynamical mean-field theory for strongly interacting lattice bosons, formulated so that local quantum dynamics is treated nonperturbatively while lattice feedback enters through a self-consistent effective field and, in the full DMFT formulation, a dynamical hybridization function (Strand et al., 2014, Scarlatella et al., 2020). In the Bose–Hubbard setting, the construction yields an effective single-site impurity problem on a real-time contour with full Nambu structure, allowing one to address normal and Bose-condensed phases, damping, relaxation, and thermalization beyond Gutzwiller mean-field theory (Strand et al., 2014). In Markovian open quantum systems, the same logic maps a Lindblad lattice problem onto a dissipative impurity embedded in a self-consistent coherent field and a non-Markovian bath, thereby incorporating finite-connectivity corrections beyond Gutzwiller and revealing hopping-induced dissipative processes, finite-frequency instabilities, and stationary quantum-Zeno behavior (Scarlatella et al., 2020). A distinct large-dimensional perspective rigorously justifies the simplest Weiss-field closure, showing that in the high-dimensional limit the one-site reduced dynamics converges to a nonlinear, strongly interacting single-site evolution in which the mean field acts in the hopping channel and not in the interaction (Farhat et al., 9 Jan 2025).

1. Conceptual setting and scope

Nonequilibrium BDMFT is designed for bosonic lattice systems whose dynamics is inaccessible to static mean-field treatments because the relevant physics depends on memory, retardation, and the feedback of local fluctuations onto the lattice environment (Strand et al., 2014). The formalism is especially natural for the Bose–Hubbard model, where the local interaction remains fully nonperturbative while the intersite hopping is encoded through a self-consistent bath or Weiss field (Strand et al., 2014, Farhat et al., 9 Jan 2025).

The nonequilibrium formulation in closed systems is built on the Kadanoff-Baym contour

$$\mathcal{C}=(0\rightarrow t_{\max}\rightarrow 0\rightarrow -i\beta),$$

which permits finite-temperature initial conditions and real-time evolution within a single contour-ordered framework (Strand et al., 2014). In open systems, the formalism is recast in Lindbladian Keldysh language, because dissipation is local and Markovian at the microscopic level while non-Markovian memory emerges self-consistently through the DMFT bath (Scarlatella et al., 2020).

A central organizing distinction is between two levels of theory. The full nonequilibrium BDMFT construction uses a dynamical hybridization function $\mathbf{\Delta}(t,t')$, typically related self-consistently to the local Green’s function, and thereby captures damping and thermalization beyond site-decoupled dynamics (Strand et al., 2014, Scarlatella et al., 2020). By contrast, the high-dimensional rigorous result establishes the validity of the simpler Weiss-field equation

$$i \partial_t \varphi(t) = h^{\varphi(t)} \varphi(t),$$

with a self-consistent order parameter $\alpha_\varphi=\langle \varphi,a\varphi\rangle$, but without a dynamical hybridization bath (Farhat et al., 9 Jan 2025). This implies that “nonequilibrium bosonic DMFT” may refer either to the full Keldysh impurity theory with hybridization or to its large-dimensional Weiss-field limit, depending on context.

2. Lattice models and nonequilibrium formulations

For closed bosonic lattices, the canonical model is the Bose–Hubbard Hamiltonian

$$H = -J \sum_{\langle i,j \rangle} \big(b_i^\dagger b_j + b_j^\dagger b_i\big) + \frac{U}{2}\sum_i n_i(n_i-1) - \mu \sum_i n_i,$$

where $b_i^\dagger$ and $b_i$ are bosonic creation and annihilation operators, $n_i=b_i^\dagger b_i$, $J$ is the hopping amplitude, $U$ the on-site repulsion, and $\mathbf{\Delta}(t,t')$0 the chemical potential (Strand et al., 2014). Nonequilibrium protocols are implemented through real-time evolution after parameter changes, most prominently interaction quenches $\mathbf{\Delta}(t,t')$1 at fixed $\mathbf{\Delta}(t,t')$2 and $\mathbf{\Delta}(t,t')$3 (Strand et al., 2014).

For Markovian open bosonic lattices, the microscopic evolution is given by the Lindblad master equation

$\mathbf{\Delta}(t,t')$4

with local jump operators $\mathbf{\Delta}(t,t')$5 and a Bose–Hubbard Hamiltonian scaled as

$\mathbf{\Delta}(t,t')$6

$\mathbf{\Delta}(t,t')$7

where $\mathbf{\Delta}(t,t')$8 is the coordination number and $\mathbf{\Delta}(t,t')$9 (Scarlatella et al., 2020). The specific driven-dissipative model studied in the open-system extension uses two-body loss and incoherent pump,

$$i \partial_t \varphi(t) = h^{\varphi(t)} \varphi(t),$$0

with $$i \partial_t \varphi(t) = h^{\varphi(t)} \varphi(t),$$1 the two-body loss rate and $$i \partial_t \varphi(t) = h^{\varphi(t)} \varphi(t),$$2 the dimensionless pump-to-loss ratio (Scarlatella et al., 2020).

In the rigorous high-dimensional setting, the Bose–Hubbard Hamiltonian is written with explicit $$i \partial_t \varphi(t) = h^{\varphi(t)} \varphi(t),$$3 hopping scaling: $$i \partial_t \varphi(t) = h^{\varphi(t)} \varphi(t),$$4 on the $$i \partial_t \varphi(t) = h^{\varphi(t)} \varphi(t),$$5-dimensional periodic square lattice $$i \partial_t \varphi(t) = h^{\varphi(t)} \varphi(t),$$6, with coordination $$i \partial_t \varphi(t) = h^{\varphi(t)} \varphi(t),$$7 (Farhat et al., 9 Jan 2025). This scaling is the basis for the large-dimensional limit in which the local mean-field dynamics becomes exact.

3. Impurity mapping, Keldysh structure, and self-consistency

In nonequilibrium BDMFT for closed systems, the lattice problem is mapped to an effective local impurity action on the contour $$i \partial_t \varphi(t) = h^{\varphi(t)} \varphi(t),$$8. In Nambu notation,

$$i \partial_t \varphi(t) = h^{\varphi(t)} \varphi(t),$$9

and the impurity action takes the form

$\alpha_\varphi=\langle \varphi,a\varphi\rangle$0

where $\alpha_\varphi=\langle \varphi,a\varphi\rangle$1 is a $\alpha_\varphi=\langle \varphi,a\varphi\rangle$2 Nambu hybridization matrix and $\alpha_\varphi=\langle \varphi,a\varphi\rangle$3 is a self-consistent symmetry-breaking source field (Strand et al., 2014). The source is

$\alpha_\varphi=\langle \varphi,a\varphi\rangle$4

with condensate $\alpha_\varphi=\langle \varphi,a\varphi\rangle$5 and $\alpha_\varphi=\langle \varphi,a\varphi\rangle$6 (Strand et al., 2014).

The corresponding contour-ordered Nambu Green’s function is

$\alpha_\varphi=\langle \varphi,a\varphi\rangle$7

with normal and anomalous entries treated on equal footing (Strand et al., 2014). The impurity Dyson equation is

$\alpha_\varphi=\langle \varphi,a\varphi\rangle$8

$\alpha_\varphi=\langle \varphi,a\varphi\rangle$9

where $$H = -J \sum_{\langle i,j \rangle} \big(b_i^\dagger b_j + b_j^\dagger b_i\big) + \frac{U}{2}\sum_i n_i(n_i-1) - \mu \sum_i n_i,$$0 is the local self-energy (Strand et al., 2014).

For a Bethe lattice or semi-circular density of states, self-consistency is local: $$H = -J \sum_{\langle i,j \rangle} \big(b_i^\dagger b_j + b_j^\dagger b_i\big) + \frac{U}{2}\sum_i n_i(n_i-1) - \mu \sum_i n_i,$$1 and in the implementation used for a 3D cubic lattice with bandwidth $$H = -J \sum_{\langle i,j \rangle} \big(b_i^\dagger b_j + b_j^\dagger b_i\big) + \frac{U}{2}\sum_i n_i(n_i-1) - \mu \sum_i n_i,$$2 and $$H = -J \sum_{\langle i,j \rangle} \big(b_i^\dagger b_j + b_j^\dagger b_i\big) + \frac{U}{2}\sum_i n_i(n_i-1) - \mu \sum_i n_i,$$3,

$$H = -J \sum_{\langle i,j \rangle} \big(b_i^\dagger b_j + b_j^\dagger b_i\big) + \frac{U}{2}\sum_i n_i(n_i-1) - \mu \sum_i n_i,$$4

This closure feeds normal and anomalous components back into the impurity problem (Strand et al., 2014).

The open-system extension preserves the same basic structure but formulates it directly in Keldysh language for a Lindblad problem. The effective impurity action is

$$H = -J \sum_{\langle i,j \rangle} \big(b_i^\dagger b_j + b_j^\dagger b_i\big) + \frac{U}{2}\sum_i n_i(n_i-1) - \mu \sum_i n_i,$$5

with $$H = -J \sum_{\langle i,j \rangle} \big(b_i^\dagger b_j + b_j^\dagger b_i\big) + \frac{U}{2}\sum_i n_i(n_i-1) - \mu \sum_i n_i,$$6 in Nambu notation (Scarlatella et al., 2020). The effective field includes a finite-$$H = -J \sum_{\langle i,j \rangle} \big(b_i^\dagger b_j + b_j^\dagger b_i\big) + \frac{U}{2}\sum_i n_i(n_i-1) - \mu \sum_i n_i,$$7 memory term,

$$H = -J \sum_{\langle i,j \rangle} \big(b_i^\dagger b_j + b_j^\dagger b_i\big) + \frac{U}{2}\sum_i n_i(n_i-1) - \mu \sum_i n_i,$$8

and Bethe-lattice self-consistency reads

$$H = -J \sum_{\langle i,j \rangle} \big(b_i^\dagger b_j + b_j^\dagger b_i\big) + \frac{U}{2}\sum_i n_i(n_i-1) - \mu \sum_i n_i,$$9

The hybridization scales as $b_i^\dagger$0, so the bath vanishes as $b_i^\dagger$1 and the theory reduces to Gutzwiller mean field (Scarlatella et al., 2020).

In the rigorous large-$b_i^\dagger$2 treatment, the effective single-site dynamics is instead governed by the nonlinear operator

$b_i^\dagger$3

with self-consistency through $b_i^\dagger$4 (Farhat et al., 9 Jan 2025). This is explicitly “a mean-field approximation in the hopping amplitude and not in the interaction,” and the interaction remains fully local and nonperturbative (Farhat et al., 9 Jan 2025).

4. Impurity solvers and computational structure

The practical implementation of nonequilibrium BDMFT requires an impurity solver capable of treating strong local interactions and a retarded, self-consistent bath. In the closed-system formulation, the impurity action is solved by a Nambu real-time strong-coupling perturbative approach based on the non-crossing approximation (NCA) (Strand et al., 2014). Introducing pseudo-particle operators $b_i^\dagger$5 for local many-body states $b_i^\dagger$6, the NCA pseudo-particle self-energy is

$b_i^\dagger$7

and the pseudo-particle propagators satisfy a contour Dyson equation with cyclic convolution on $b_i^\dagger$8 (Strand et al., 2014). The connected impurity Green’s function is computed from the pseudo-particle bubble,

$b_i^\dagger$9

where the second term subtracts the disconnected symmetry-broken contribution (Strand et al., 2014).

The open-system impurity solver is also an NCA, but implemented at the super-operator level so that the local Markovian Lindbladian dynamics is treated exactly while the non-Markovian bath is resummed in non-crossing approximation (Scarlatella et al., 2020). The impurity density matrix evolves as

$b_i$0

and the reduced propagator obeys

$b_i$1

with $b_i$2 the bare local Markovian propagator and $b_i$3 the one-particle irreducible self-energy (Scarlatella et al., 2020). At NCA level,

$b_i$4

where $b_i$5 act by left or right multiplication on the density matrix (Scarlatella et al., 2020). In stationary problems, time-translation invariance simplifies the solver because $b_i$6 and $b_i$7 (Scarlatella et al., 2020).

A compact summary of the principal computational ingredients is useful.

Formulation	Effective local object	Solver structure
Closed-system nonequilibrium BDMFT	Contour impurity action with $b_i$8 and $b_i$9	Nambu real-time strong-coupling NCA (Strand et al., 2014)
Open-system nonequilibrium BDMFT	Lindbladian impurity with coherent field and non-Markovian bath	Super-operator hybridization expansion with NCA (Scarlatella et al., 2020)
High-dimensional Weiss-field limit	Nonlinear single-site Schrödinger equation $n_i=b_i^\dagger b_i$0	Direct integration after Hilbert-space truncation (Farhat et al., 9 Jan 2025)

In numerical practice, the closed-system implementation uses a uniform time grid on $n_i=b_i^\dagger b_i$1 and a fifth-order multi-step method; the open-system stationary solver uses an implicit second-order Runge–Kutta scheme with typical parameters $n_i=b_i^\dagger b_i$2, $n_i=b_i^\dagger b_i$3, and local Hilbert-space truncation $n_i=b_i^\dagger b_i$4–14 (Strand et al., 2014, Scarlatella et al., 2020). The rigorous high-dimensional work proves global well-posedness of the truncated single-site problem and then controls the $n_i=b_i^\dagger b_i$5 limit (Farhat et al., 9 Jan 2025).

5. Dynamical regimes in closed bosonic lattices

Applied to interaction quenches in the Bose–Hubbard model, nonequilibrium BDMFT captures several dynamical regimes that are absent or frozen in Gutzwiller mean-field theory because the latter lacks the retarded hybridization $n_i=b_i^\dagger b_i$6 and therefore cannot describe hopping-induced relaxation, damping, or thermalization at finite temperature (Strand et al., 2014).

For quenches starting from a normal or Mott state with $n_i=b_i^\dagger b_i$7, the theory identifies rapid thermalization near intermediate $n_i=b_i^\dagger b_i$8 and prethermalization plateaus at both weak and strong $n_i=b_i^\dagger b_i$9 (Strand et al., 2014). The prethermal behavior is attributed to proximity to the integrable limits $J$0 and $J$1, while a qualitative non-equilibrium phase diagram in the $J$2 plane delineates the intermediate region of fast thermalization (Strand et al., 2014). Thermalization in the symmetric phase is monitored by

$J$3

with $J$4 and $J$5 indicating thermalization (Strand et al., 2014).

For quenches starting from a superfluid initial state, five regimes are reported: deep-Mott collapse-and-revival oscillations with frequency $J$6 and exponential damping; intermediate-coupling relaxation to the normal phase after a few oscillations; trapping in a nonthermal superfluid near the phase boundary on the normal side; excitation of a coherent amplitude mode with roughly constant frequency and damping; and rapid self-amplified growth of the condensate for small quenches that remain within the equilibrium superfluid region (Strand et al., 2014). Near the dynamical transition $J$7, the theory finds a kink in the phase frequency $J$8, a maximal relaxation rate for double occupancy, and maximal damping of the amplitude mode (Strand et al., 2014).

Energy conversion is an explicit diagnostic. The kinetic, condensate, and interaction contributions are

$J$9

and nonequilibrium BDMFT shows nontrivial exchange between these channels during relaxation, in contrast to Gutzwiller mean-field in symmetric states (Strand et al., 2014).

Finite temperature plays a systematic role: increasing the initial temperature $U$0 enhances damping, accelerates relaxation, suppresses the plateau values of $U$1, and reduces the trapped superfluid regions (Strand et al., 2014). This establishes nonequilibrium BDMFT as a finite-temperature dynamical theory rather than merely a zero-temperature strong-coupling approximation.

6. Open-system extension: dissipation, finite-frequency instability, and synchronization

In driven-dissipative bosonic lattices, nonequilibrium BDMFT extends beyond coherent relaxation and addresses steady states of Lindbladian many-body systems (Scarlatella et al., 2020). The open-system theory reveals qualitative phenomena that are completely missed by Gutzwiller mean-field because, in the normal phase with $U$2, Gutzwiller predicts uncoupled sites and therefore no dependence of local stationary properties on the hopping $U$3 (Scarlatella et al., 2020). DMFT restores this dependence through the non-Markovian bath $U$4, which encodes virtual hopping processes and finite-connectivity memory.

In the normal phase, the retarded and Keldysh hybridizations have distinct roles: $U$5 where $U$6 encodes spectral damping and $U$7 bath occupation or noise (Scarlatella et al., 2020). Because the system is far from equilibrium, $U$8 and $U$9 are independent and are not constrained by an FDT relation (Scarlatella et al., 2020).

The theory identifies hopping-induced dissipative channels in which a boson is pumped on one site, hops to a neighbor, and is lost there through the local dissipators (Scarlatella et al., 2020). These channels suppress local gain, redistribute steady-state populations, and reduce population inversion (Scarlatella et al., 2020). At large two-body loss $\mathbf{\Delta}(t,t')$00, the steady state enters a stationary quantum-Zeno regime controlled by the effective scale

$\mathbf{\Delta}(t,t')$01

with non-monotonic dependence on $\mathbf{\Delta}(t,t')$02 and a collapse of observables as functions of $\mathbf{\Delta}(t,t')$03 (Scarlatella et al., 2020). In this regime, probabilities for $\mathbf{\Delta}(t,t')$04 become exponentially suppressed and the steady-state weight concentrates in the $\mathbf{\Delta}(t,t')$05 sector (Scarlatella et al., 2020).

The open-system phase structure contains a normal phase with $\mathbf{\Delta}(t,t')$06 and a non-equilibrium superfluid with a time-oscillating order parameter

$\mathbf{\Delta}(t,t')$07

which breaks both $\mathbf{\Delta}(t,t')$08 and time-translation invariance (Scarlatella et al., 2020). In the normal phase, the spectral function

$\mathbf{\Delta}(t,t')$09

may develop a negative-density-of-states region $\mathbf{\Delta}(t,t')$10 for $\mathbf{\Delta}(t,t')$11, implying gain and negative absorbed power under a weak coherent probe through

$\mathbf{\Delta}(t,t')$12

The onset of the ordered phase is a finite-frequency instability determined, on the Bethe lattice, by

$\mathbf{\Delta}(t,t')$13

$\mathbf{\Delta}(t,t')$14

This instability has no equilibrium analog because in equilibrium $\mathbf{\Delta}(t,t')$15 follows from FDT, whereas here the unstable mode appears at a nonzero $\mathbf{\Delta}(t,t')$16 (Scarlatella et al., 2020).

Finite-connectivity fluctuations strongly renormalize the phase boundary relative to Gutzwiller mean field. The ordered region shrinks as $\mathbf{\Delta}(t,t')$17 decreases and moves toward larger $\mathbf{\Delta}(t,t')$18 and $\mathbf{\Delta}(t,t')$19 because the DMFT bath-induced dissipation suppresses negative density of states and local gain (Scarlatella et al., 2020). The same transition is connected to many-body synchronization of quantum van der Pol oscillators. In the semiclassical limit, the Keldysh action yields

$\mathbf{\Delta}(t,t')$20

with white noise $\mathbf{\Delta}(t,t')$21, so the finite-frequency superfluid instability becomes the onset of a collective limit cycle (Scarlatella et al., 2020).

7. Large-dimensional limit, rigor, and relation to mean-field theory

A rigorous mathematical foundation for the simplest nonequilibrium bosonic DMFT closure is provided by the high-dimensional analysis of the Bose–Hubbard model with hopping scaled as $\mathbf{\Delta}(t,t')$22 (Farhat et al., 9 Jan 2025). The central result is that the one-site reduced density matrix of the exact Schrödinger evolution converges, in trace norm, to a rank-one projector $\mathbf{\Delta}(t,t')$23 generated by the nonlinear single-site mean-field equation (Farhat et al., 9 Jan 2025).

The exact reduced one-site dynamics satisfies

$\mathbf{\Delta}(t,t')$24

so the deviation from the mean-field closure is controlled by the two-site reduced density matrix $\mathbf{\Delta}(t,t')$25 (Farhat et al., 9 Jan 2025). The large-$\mathbf{\Delta}(t,t')$26 analysis shows that these corrections vanish with increasing dimension.

Two explicit error bounds are established. Under exponential-tail assumptions on the initial occupation distribution, Theorem 1 gives a trace-norm estimate whose $\mathbf{\Delta}(t,t')$27-dependent prefactor tends to zero as $\mathbf{\Delta}(t,t')$28 (Farhat et al., 9 Jan 2025). Under the assumptions $\mathbf{\Delta}(t,t')$29 and $\mathbf{\Delta}(t,t')$30, Theorem 2 yields

$\mathbf{\Delta}(t,t')$31

and for initial Gutzwiller product states this reduces to

$\mathbf{\Delta}(t,t')$32

The same work proves convergence of the order parameter,

$\mathbf{\Delta}(t,t')$33

This rigorously justifies the time-dependent Weiss field as the large-dimensional limit of the microscopic condensate (Farhat et al., 9 Jan 2025).

The rigorous results also clarify the relation between BDMFT and Gutzwiller theory. The proven equation corresponds to the time-dependent Gutzwiller or Weiss-field dynamics, not to the full hybridization-based Keldysh impurity theory (Farhat et al., 9 Jan 2025). A plausible implication is that the rigorous large-$\mathbf{\Delta}(t,t')$34 limit establishes the controlled starting point from which full nonequilibrium BDMFT incorporates finite-connectivity corrections through the dynamical bath $\mathbf{\Delta}(t,t')$35, as made explicit in the closed- and open-system DMFT constructions (Strand et al., 2014, Scarlatella et al., 2020).

8. Comparison with related approaches, limitations, and open directions

Compared with Gutzwiller mean-field theory, nonequilibrium BDMFT retains local interactions nonperturbatively while adding a retarded hybridization term that captures memory, damping, and thermalization (Strand et al., 2014). In the open-system context, this same retarded bath produces hopping-induced dissipative channels and finite-$\mathbf{\Delta}(t,t')$36 memory feedback that are absent in Gutzwiller mean field, especially in the normal phase where Gutzwiller predicts completely uncoupled sites (Scarlatella et al., 2020). The distinction is therefore not merely quantitative.

Compared with equilibrium or low-frequency strong-coupling approaches, nonequilibrium BDMFT is formulated directly in real time and at finite temperature, with full Nambu structure and explicit treatment of normal and anomalous components (Strand et al., 2014). Compared with fermionic DMFT, the bosonic problem requires explicit handling of condensation and anomalous propagators; in the open-system case it additionally requires a Lindbladian Keldysh formalism in which $\mathbf{\Delta}(t,t')$37 and $\mathbf{\Delta}(t,t')$38 are independent and no FDT constraint exists (Scarlatella et al., 2020).

The theory also has clear limitations. DMFT neglects nonlocal spatial correlations and is controlled in the limit of large coordination or high dimension (Strand et al., 2014, Scarlatella et al., 2020, Farhat et al., 9 Jan 2025). The NCA impurity solvers are approximate: in the closed-system case spectral sum-rule checks show small errors deep in the Mott regime but larger ones near the superfluid boundary; in the open-system case higher-order crossing diagrams may be needed for very small $\mathbf{\Delta}(t,t')$39 or deep inside symmetry-broken phases (Strand et al., 2014, Scarlatella et al., 2020). Numerical cost is substantial because real-time two-time objects scale as $\mathbf{\Delta}(t,t')$40 in memory and bosonic local Hilbert spaces grow rapidly with occupancy (Strand et al., 2014). In the open-system steady-state solver, accessible parameter ranges are also limited by Hilbert-space truncation, particularly at high pump $\mathbf{\Delta}(t,t')$41 (Scarlatella et al., 2020).

Several extensions are explicitly identified. For driven-dissipative bosons, possible developments include cluster DMFT for short-range correlations and improved impurity solvers such as full hybridization expansion via diagrammatic Monte Carlo, auxiliary master equations with a discretized bath, MPO-based non-Markovian impurity solvers, and renormalized Lindblad NRG (Scarlatella et al., 2020). From the rigorous side, extending the analysis from the Weiss-field closure to full nonequilibrium bosonic DMFT with anomalous Nambu components and a dynamic hybridization bath remains an open problem (Farhat et al., 9 Jan 2025).

Taken together, these results define nonequilibrium bosonic dynamical mean-field theory as a hierarchy of local self-consistent descriptions for bosonic lattice dynamics: at the simplest level, a rigorously justified Weiss-field evolution in large dimension; at the full DMFT level, a Keldysh impurity theory with retarded hybridization that captures relaxation, damping, thermalization, dissipative renormalization, finite-frequency instabilities, and collective synchronization in both closed and Markovian open quantum many-body systems (Strand et al., 2014, Scarlatella et al., 2020, Farhat et al., 9 Jan 2025).