UNEPT: Unifying Nonequilibrium Perturbative Theory

• UNEPT is a unified framework that integrates weak-coupling perturbative expansions, Keldysh–Matsubara formalism, and cluster methods to model nonequilibrium dynamics.
• It computes time-dependent Green’s functions along complex contours to simultaneously capture transient evolution, steady-state behavior, and relaxation processes.
• UNEPT offers a computationally efficient approach for analyzing strongly correlated electrons, initial-state correlations, and dissipation in open quantum systems.

Unifying Nonequilibrium Perturbative Theory (UNEPT) refers to a systematic framework that integrates diverse perturbative techniques for describing nonequilibrium phenomena in quantum many-body systems, with special emphasis on strongly correlated electrons, transport, and quantum open system dynamics. UNEPT unifies weak-coupling perturbative expansions, time-contour Green's function methods, and cluster-based approaches to address arbitrary initial states, time-dependent evolution, and environmental coupling. Its foundational structure enables the treatment of transient, steady-state, and relaxation processes on equal footing, facilitating analysis of observables, correlations, and dissipation under nonequilibrium conditions.

1. Theoretical Foundations and Reformulations

The starting point of UNEPT is the reformulation of equilibrium perturbation theory in a language that naturally generalizes to nonequilibrium situations. Standard Cluster Perturbation Theory (CPT)—where a lattice is partitioned into clusters, intra-cluster dynamics are solved exactly, and inter-cluster hopping is addressed perturbatively—is recast as a subset (neglecting vertex corrections) of the weak-coupling expansion. The central CPT Dyson equation can be written as

$$G_{\text{CPT}} = G' + G' \cdot (V_{\mathcal{K}}\otimes 1) \cdot G_{\text{CPT}},$$

where $G'$ is the cluster Green’s function and $V_{\mathcal{K}}$ is the inter-cluster potential on the contour.

The exact self-energy would include vertex corrections (inter-cluster, interaction-modified diagrams), but UNEPT takes the re-summation without those, resulting in a computationally manageable yet still highly nontrivial approximation. This diagrammatic reorganization clarifies that CPT is an “all-order” perturbative sum over a restricted diagram class, making the extension to general time contours and out-of-equilibrium scenarios transparent (Balzer et al., 2011).

2. Extension to Nonequilibrium and Time-Ordered Green's Functions

A fundamental aspect of UNEPT is its generalization to the nonequilibrium domain. All diagrammatics and operator expressions are elevated to the Keldysh–Matsubara time contour, which incorporates both real-time evolution (the Keldysh branches) and imaginary time (the Matsubara branch) for proper treatment of initial correlations. The contour-ordered Green's function,

$$iG_{\alpha \alpha'}(t, t') = \langle T_\gamma\, c_{K\alpha}(t) c^\dagger_{K\alpha'}(t') \rangle$$

(with $t$, $t'$ on a contour $\gamma$ consisting of two real-time plus one Matsubara branch), is the primary object. All perturbative integrals are performed along $\gamma$, and the Green's function matrices have indices in both orbital and (discretized) time variables.

Solving the CPT equation on this contour allows direct calculation of transient, nonequilibrium evolution starting from a correlated pure or mixed state. The methodology accommodates arbitrary system preparations, including those with inter-cluster coupling in the initial density matrix, by requiring the Matsubara branch (Balzer et al., 2011).

3. Transient Dynamics and Time-Dependent Observables

UNEPT provides explicit access to the time-dependent dynamics of local and nonlocal observables following a disturbance, quench, or other nonequilibrium protocol. Once the Green’s function $G_{\text{CPT}}(t, t')$ is computed, the expectation value of any one-particle observable,

$$A(t) = \sum_{\alpha\beta} a_{\alpha\beta}(t) c^\dagger_\alpha c_\beta,$$

is obtained via

$$\langle A \rangle_t = -i\sum_{\alpha\beta} a_{\alpha\beta}(t) G_{\beta\alpha}(t, t + 0^+).$$

This formulation captures the relaxation of excitations, propagation of patterns (such as Néel order or local perturbations), and the approach to steady-state, within an approximation neglecting correlations beyond the cluster size (Balzer et al., 2011).

A critical observation is that short-time (transient) dynamics are typically predicted accurately, as the nonequilibrium CPT satisfies a number of initial moment sum rules. The accuracy at longer times depends on system properties, with limitations arising especially from neglected nonlocal (inter-cluster) vertex corrections and finite-size effects.

4. Treatment of Initial-State Correlations and Matsubara Branch

The correct encoding of correlated, thermal, or otherwise nontrivial initial conditions is a central challenge in nonequilibrium quantum theory. Within UNEPT, the initial density matrix $\rho \propto \exp(-\beta B)$, with $B$ possibly including nontrivial inter-cluster coupling terms $V_B$, is incorporated via the Matsubara (imaginary time) branch of the contour. Neglecting this branch leads to errors if initial inter-cluster couplings are significant (e.g., for finite staggered fields or partial decoupling).

Numerically, the retention of the Matsubara branch is demonstrated to be essential for capturing actual post-quench dynamics in cases where initial correlations are not localized to single clusters or sites (Balzer et al., 2011). This contour treatment thus forms a crucial bridge between initial-state physics and subsequent quantum evolution.

5. Dissipation, Baths, and Embedding Schemes

UNEPT is particularly effective for modeling open systems composed of a correlated subsystem coupled to large, uncorrelated baths. The CPT framework accommodates embedding by splitting the reference Green's function

$$G' = \begin{pmatrix} G'_S & 0 \ 0 & G'_B \end{pmatrix}$$

and introducing the hybridization function

$\Delta = V G'_B V^\dagger.$

The system’s Green’s function then obeys

$$G_{\text{CPT}, S} = \left[(G'_S)^{-1} - \Delta\right]^{-1},$$

parallel to embedding theory in equilibrium many-body physics. This allows tracing of spin or charge dissipation from the interacting subsystem into the bath, provided that numerical artifacts (such as reflections in finite baths) are controlled. The later-time dynamics reveal relaxation toward equilibrium, and the formalism applies for arbitrary bath size and coupling (Balzer et al., 2011).

6. Systematic Control, Strengths, and Limitations

UNEPT, in the CPT setting, is controlled by the inverse cluster size $L_c^{-1}$. In the two extremes—noninteracting ($U = 0$) and decoupled cluster ($V = 0$)—the theory recovers the exact results. By increasing $L_c$, accuracy can be systematically improved, albeit at increased computational cost due to the growth in cluster Hilbert space and the scaling of time-grid dimensions.

Strengths:

• Transparent interpolation between atomic and band limits.
• Access to both transient and steady-state regimes, including arbitrary initial states.
• Moderate computational requirements due to the avoidance of self-consistency and full spatial self-energy calculations.

Shortcomings:

• Vertex corrections (nonlocal inter-cluster interactions) are neglected, impacting accuracy in regimes dominated by nonlocal quantum correlations or for extended times.
• Cluster-size limitations may manifest as mean-field–like artifacts, including spurious symmetry breaking.
• The computational effort grows rapidly with finer time-grid resolution or increased maximal time, restricting practical simulations to short/intermediate time windows (Balzer et al., 2011).

7. Contribution to the Unified Nonequilibrium Perturbative Framework

The CPT reformulation and its extension via the Keldysh–Matsubara time contour establish the architecture of UNEPT for quantum lattice models:

• Recasting cluster methods as a summation over explicit diagram classes within weak-coupling perturbation theory enables a uniform treatment of both intra-cluster interactions and inter-cluster kinetic terms.
• The contour-ordered Green's function approach, with proper treatment of initial correlations and bath couplings, places virtually all nonequilibrium scenarios, including open-system dynamics, on common theoretical ground.
• The framework, while approximate due to the neglected vertex corrections, allows systematic approaches to dynamical observables and paves the way for more sophisticated generalizations (e.g., variational cluster approximations, dynamical mean-field extensions), each recoverable as limiting cases within the general UNEPT philosophy.

In summary, UNEPT as developed in this context synthesizes perturbative expansions, cluster-based embeddings, and time-contour Green's function techniques to provide a computationally efficient and conceptually transparent method for studying nonequilibrium quantum dynamics, accommodating transient behavior, initial correlations, and dissipation within a unifying architecture (Balzer et al., 2011).