Quasi-Local Redfield Equation

• Quasi-local Redfield equation is a family of Born–Markov quantum master equations that modify the microscopic Redfield construction to maintain spatial or orbital locality and include level broadening without relying solely on heuristic local Lindblad forms.
• It is applied across various settings—from quadratic lattice systems and molecular transport to many-body simulations—balancing computational tractability with accurate capture of nonsecular dynamics and thermodynamic consistency.
• The approach bridges local and global master equation regimes by incorporating self-consistent time-local generators, partial secular coarse-graining, and short-memory expansions to enforce approximate complete positivity and improve transient and steady-state predictions.

Searching arXiv for the specified papers and closely related work on quasi-local Redfield equations. Quasi-local Redfield equation denotes a family of Born–Markov quantum master equations that retain the microscopic Redfield construction while modifying the closure, operator representation, or truncation so that dissipation remains site-local or quasi-local, level broadening is incorporated, or complete positivity and thermodynamic consistency are addressed without reverting to a fully heuristic local Lindblad ansatz. In the literature considered here, the term refers to several related constructions: a self-consistent time-local generalized quantum master equation based on backward Redfield evolution; a site-local Redfield dissipator for quadratic systems that exactly reduces to a Davies generator under identical independent baths; a short-bath-correlation expansion that converts global Redfield jump operators into sparse local series; a partially secular completely positive Redfield generator interpolating between local and global limits; and a localized-orbital Redfield formalism for electronically open molecules with phenomenological broadening (Esposito et al., 2010, Shiraishi et al., 14 Jul 2025, Schnell, 2023, Farina et al., 2020, Sannes et al., 2024).

1. Foundational structure and the meaning of quasi-locality

The common starting point is the Redfield equation obtained from a total Hamiltonian of the form

$H_{\rm tot}=H_S+H_B+H_{SB},$

with weak system–bath coupling and a Born–Markov closure. In one standard representation,

$$\frac{d}{dt}\rho = -\,i[H_S,\rho]+\mathcal R[\rho],$$

where

$$\mathcal R[\rho]=-\sum_{\alpha\beta}\int_0^\infty d\tau\,\bigl[A_\alpha,\,A_\beta(-\tau)\rho\bigr]\,C_{\alpha\beta}(\tau)+\mathrm{h.c.}$$

and $A_\alpha(-\tau)=e^{iH_S\tau}A_\alpha e^{-iH_S\tau}$, while $C_{\alpha\beta}(\tau)$ are bath correlation functions. In its fully global form, this equation resolves transitions in the eigenbasis of $H_S$, which generally requires diagonalization of the full interacting system Hamiltonian (Schnell, 2023).

Within this framework, “quasi-locality” is not a single formal property. In quadratic lattice systems, the dissipator is called quasi-local because the integrand involves only site-$j$ operators, and for short bath correlation time the dissipative influence is confined by a Lieb–Robinson bound (Shiraishi et al., 14 Jul 2025). In short-memory expansions for many-body systems, quasi-locality means that a jump operator generated from a site-local coupling remains supported only on a finite neighborhood after truncating nested commutators with $H_S$ (Schnell, 2023). In molecular formulations, quasi-locality refers to expressing the Hamiltonian and system–environment transfer terms in a local spin-orbital basis (Sannes et al., 2024). This suggests that the phrase identifies a class of Redfield-type constructions that preserve spatial or orbital locality as far as the chosen approximation permits, rather than a unique canonical equation.

A recurrent source of confusion is the relation between quasi-local, local, and global master equations. The local approach typically keeps bath couplings in local operators but may ignore interaction-induced spectral structure; the global approach resolves exact normal modes or many-body eigenstates and is thermodynamically well behaved when its assumptions hold; quasi-local Redfield schemes occupy an intermediate position by retaining nonsecular or partially nonlocal information while trying to avoid the full nonlocality or computational cost of global diagonalization (Farina et al., 2020, Schnell, 2023).

2. Self-consistent time-local generators and backward Redfield evolution

One influential use of the term arises in molecular transport, where conventional Redfield theory fails to capture level broadening. The exact second-order equation of motion for the reduced density matrix in the interaction picture,

$$\frac{d}{dt}\rho_S^I(t)= -\int_{t_0}^t d\tau\,\mathrm{Tr}_B\{[V^I(t),[V^I(\tau),\rho_S^I(\tau)\otimes \rho_B^{eq}]]\},$$

is usually closed by the Born–Markov replacement $\rho_S^I(\tau)\to \rho_S^I(t)$ together with free-system propagation. The quasi-local modification replaces the free propagator $$\frac{d}{dt}\rho = -\,i[H_S,\rho]+\mathcal R[\rho],$$0 inside the kernel by propagation with the Redfield generator itself, $$\frac{d}{dt}\rho = -\,i[H_S,\rho]+\mathcal R[\rho],$$1 or $$\frac{d}{dt}\rho = -\,i[H_S,\rho]+\mathcal R[\rho],$$2, where

$$\frac{d}{dt}\rho = -\,i[H_S,\rho]+\mathcal R[\rho],$$3

Esposito and Galperin then formulate a more symmetric closure by evolving from the later time backward to the earlier one through “anti-contour” Green’s functions $$\frac{d}{dt}\rho = -\,i[H_S,\rho]+\mathcal R[\rho],$$4, yielding a time-local generalized quantum master equation with generator $$\frac{d}{dt}\rho = -\,i[H_S,\rho]+\mathcal R[\rho],$$5 (Esposito et al., 2010).

The resulting equation has the form

$$\frac{d}{dt}\rho = -\,i[H_S,\rho]+\mathcal R[\rho],$$6

with

$$\frac{d}{dt}\rho = -\,i[H_S,\rho]+\mathcal R[\rho],$$7

Because $$\frac{d}{dt}\rho = -\,i[H_S,\rho]+\mathcal R[\rho],$$8 are themselves generated by $$\frac{d}{dt}\rho = -\,i[H_S,\rho]+\mathcal R[\rho],$$9, the construction becomes a fixed-point problem,

$$\mathcal R[\rho]=-\sum_{\alpha\beta}\int_0^\infty d\tau\,\bigl[A_\alpha,\,A_\beta(-\tau)\rho\bigr]\,C_{\alpha\beta}(\tau)+\mathrm{h.c.}$$0

In practice, one starts from the ordinary Redfield generator and iterates until the difference between successive generators falls below tolerance; the derivation reports that typically only $$\mathcal R[\rho]=-\sum_{\alpha\beta}\int_0^\infty d\tau\,\bigl[A_\alpha,\,A_\beta(-\tau)\rho\bigr]\,C_{\alpha\beta}(\tau)+\mathrm{h.c.}$$1–$$\mathcal R[\rho]=-\sum_{\alpha\beta}\int_0^\infty d\tau\,\bigl[A_\alpha,\,A_\beta(-\tau)\rho\bigr]\,C_{\alpha\beta}(\tau)+\mathrm{h.c.}$$2 iterations are needed (Esposito et al., 2010).

In Schrödinger picture, the quasi-local Redfield equation can be written in a rate form,

$$\mathcal R[\rho]=-\sum_{\alpha\beta}\int_0^\infty d\tau\,\bigl[A_\alpha,\,A_\beta(-\tau)\rho\bigr]\,C_{\alpha\beta}(\tau)+\mathrm{h.c.}$$3

where the generalized rates $$\mathcal R[\rho]=-\sum_{\alpha\beta}\int_0^\infty d\tau\,\bigl[A_\alpha,\,A_\beta(-\tau)\rho\bigr]\,C_{\alpha\beta}(\tau)+\mathrm{h.c.}$$4 collect time integrals of bath correlators together with the broadening encoded in $$\mathcal R[\rho]=-\sum_{\alpha\beta}\int_0^\infty d\tau\,\bigl[A_\alpha,\,A_\beta(-\tau)\rho\bigr]\,C_{\alpha\beta}(\tau)+\mathrm{h.c.}$$5. In a molecular-junction setting, these rates are expressed through the contact self-energies $$\mathcal R[\rho]=-\sum_{\alpha\beta}\int_0^\infty d\tau\,\bigl[A_\alpha,\,A_\beta(-\tau)\rho\bigr]\,C_{\alpha\beta}(\tau)+\mathrm{h.c.}$$6 and the anti-contour propagators (Esposito et al., 2010).

For the single-resonant-level benchmark $$\mathcal R[\rho]=-\sum_{\alpha\beta}\int_0^\infty d\tau\,\bigl[A_\alpha,\,A_\beta(-\tau)\rho\bigr]\,C_{\alpha\beta}(\tau)+\mathrm{h.c.}$$7 with wide-band couplings $$\mathcal R[\rho]=-\sum_{\alpha\beta}\int_0^\infty d\tau\,\bigl[A_\alpha,\,A_\beta(-\tau)\rho\bigr]\,C_{\alpha\beta}(\tau)+\mathrm{h.c.}$$8, the exact nonequilibrium steady-state density matrix is diagonal with

$$\mathcal R[\rho]=-\sum_{\alpha\beta}\int_0^\infty d\tau\,\bigl[A_\alpha,\,A_\beta(-\tau)\rho\bigr]\,C_{\alpha\beta}(\tau)+\mathrm{h.c.}$$9

and the exact current is

$A_\alpha(-\tau)=e^{iH_S\tau}A_\alpha e^{-iH_S\tau}$0

The self-consistent GQME reproduces $A_\alpha(-\tau)=e^{iH_S\tau}A_\alpha e^{-iH_S\tau}$1 and $A_\alpha(-\tau)=e^{iH_S\tau}A_\alpha e^{-iH_S\tau}$2 to within $A_\alpha(-\tau)=e^{iH_S\tau}A_\alpha e^{-iH_S\tau}$3, and exactly in the wide-band limit, whereas plain Redfield fails to capture level broadening. The same derivation also states that average current is reproduced very well, but noise and higher cumulants can deviate significantly unless higher-order or non-Markovian corrections are included (Esposito et al., 2010).

3. Independent baths, coherence cancellation, and equivalence to the Davies equation

A distinct and more recent formulation considers an $A_\alpha(-\tau)=e^{iH_S\tau}A_\alpha e^{-iH_S\tau}$4-site quadratic fermionic or bosonic system,

$A_\alpha(-\tau)=e^{iH_S\tau}A_\alpha e^{-iH_S\tau}$5

coupled locally to identical, independent baths at each site $A_\alpha(-\tau)=e^{iH_S\tau}A_\alpha e^{-iH_S\tau}$6. The system–bath coupling is

$A_\alpha(-\tau)=e^{iH_S\tau}A_\alpha e^{-iH_S\tau}$7

and the bath correlation functions satisfy the KMS condition. Under the usual Born–Markov approximation, but explicitly without the secular approximation, the Redfield dissipator remains written in site-$A_\alpha(-\tau)=e^{iH_S\tau}A_\alpha e^{-iH_S\tau}$8 operators and therefore preserves quasi-locality in the sense described above (Shiraishi et al., 14 Jul 2025).

After transforming to the energy basis,

$A_\alpha(-\tau)=e^{iH_S\tau}A_\alpha e^{-iH_S\tau}$9

the dissipator contains terms indexed by $C_{\alpha\beta}(\tau)$0. No secular truncation of the $C_{\alpha\beta}(\tau)$1 contributions is made at this stage. The central result is that when all baths are identical and coupled homogeneously, the sum over sites collapses by unitarity,

$C_{\alpha\beta}(\tau)$2

so all off-diagonal dissipative terms vanish exactly. The quasi-local Redfield equation therefore coincides exactly with the Davies equation, even though no secular time-coarse-graining has been invoked (Shiraishi et al., 14 Jul 2025).

The resulting master equation is

$C_{\alpha\beta}(\tau)$3

with

$C_{\alpha\beta}(\tau)$4

and a Lamb shift diagonal in the $C_{\alpha\beta}(\tau)$5 basis. The KMS relation implies detailed balance, and the jump operators satisfy quantum-microreversibility relations with the Gibbs state

$C_{\alpha\beta}(\tau)$6

The derivation states that relaxation to Gibbs, nonnegative entropy production, and fluctuation theorems then follow (Shiraishi et al., 14 Jul 2025).

The assumptions are sharply delimited: the system is quadratic, the baths are identical and mutually independent, each bath is in a Gibbs state at $C_{\alpha\beta}(\tau)$7, the local coupling $C_{\alpha\beta}(\tau)$8 is homogeneous, and only the Born–Markov approximation is used. The result is emphasized to hold even when energy-level spacings $C_{\alpha\beta}(\tau)$9, a regime where the secular approximation fails. Two extensions are sketched: slowly driven quadratic systems, where one obtains an instantaneous Davies equation for $H_S$0, and generic many-body systems, where subextensive scaling of overlap sums can wash out non-energy-conserving terms in the thermodynamic limit (Shiraishi et al., 14 Jul 2025).

4. Partial secular coarse-graining, complete positivity, and the local–global debate

In a two-oscillator benchmark, the system consists of harmonic modes $H_S$1 and $H_S$2 with Hamiltonian

$H_S$3

while only mode $H_S$4 is coupled to a thermal bosonic bath. Standard Born–Markov treatment yields the usual Redfield equation, which in this setting is not completely positive. Passing to normal modes,

$H_S$5

and performing a coarse-graining over $H_S$6 introduces the factors

$H_S$7

This produces a completely positive version of the Redfield equation, denoted CP-Redfield, that retains some nonsecular terms while generating a semigroup (Farina et al., 2020).

The global limit corresponds to $H_S$8, for which $H_S$9, yielding the fully secular GKSL equation. The local equation instead follows from the approximation $j$0 inside the Redfield kernel. The comparison sharpens the local–global tension: the global master equation gives the thermodynamically consistent steady state $j$1 but fails to capture short-time coherent exchange between $j$2 and $j$3; the local equation captures short-time Rabi oscillations but drives the system to $j$4, which violates thermodynamic consistency when $j$5 (Farina et al., 2020).

Complete positivity for CP-Redfield is controlled by a bound on the off-diagonal coarse-grained block: $j$6 Choosing $j$7 to saturate this condition yields a positive semigroup that still contains part of the nonsecular structure. A related construction uses a time-dependent convex mixture

$j$8

or equivalently $j$9. This generator is completely positive at each time, although it is not a strict semigroup (Farina et al., 2020).

The benchmark solves the exact dynamics of the full $H_S$0bath system with $H_S$1 modes using Gaussian methods and compares approximate reduced states by trace distance, Uhlmann fidelity, and an uncertainty-violation measure $H_S$2. In that model, global and local master equations each fail in complementary regimes; CP-Redfield captures both transient and steady-state behavior nearly exactly with $H_S$3, and the convex mixture also achieves $H_S$4 at all $H_S$5, with $H_S$6 roughly $H_S$7 (Farina et al., 2020). In this usage, the term “quasi-local Redfield” denotes a partially secular, positivity-preserving interpolation between local coherence retention and global thermal consistency.

5. Short-memory expansions and local Lindblad embeddings for many-body systems

For interacting many-body systems, the main computational obstacle of the global Redfield equation is the need to diagonalize $H_S$8. A controlled alternative is to expand the global jump operator in the short bath-correlation time $H_S$9. Starting from

$$\frac{d}{dt}\rho_S^I(t)= -\int_{t_0}^t d\tau\,\mathrm{Tr}_B\{[V^I(t),[V^I(\tau),\rho_S^I(\tau)\otimes \rho_B^{eq}]]\},$$0

one rewrites the integral as a Taylor-like series around an energy $$\frac{d}{dt}\rho_S^I(t)= -\int_{t_0}^t d\tau\,\mathrm{Tr}_B\{[V^I(t),[V^I(\tau),\rho_S^I(\tau)\otimes \rho_B^{eq}]]\},$$1,

$$\frac{d}{dt}\rho_S^I(t)= -\int_{t_0}^t d\tau\,\mathrm{Tr}_B\{[V^I(t),[V^I(\tau),\rho_S^I(\tau)\otimes \rho_B^{eq}]]\},$$2

where $$\frac{d}{dt}\rho_S^I(t)= -\int_{t_0}^t d\tau\,\mathrm{Tr}_B\{[V^I(t),[V^I(\tau),\rho_S^I(\tau)\otimes \rho_B^{eq}]]\},$$3. Truncation at order $$\frac{d}{dt}\rho_S^I(t)= -\int_{t_0}^t d\tau\,\mathrm{Tr}_B\{[V^I(t),[V^I(\tau),\rho_S^I(\tau)\otimes \rho_B^{eq}]]\},$$4 avoids diagonalization and replaces dense spectral operators by nested commutators of local operators with the Hamiltonian (Schnell, 2023).

When the coupling operator $$\frac{d}{dt}\rho_S^I(t)= -\int_{t_0}^t d\tau\,\mathrm{Tr}_B\{[V^I(t),[V^I(\tau),\rho_S^I(\tau)\otimes \rho_B^{eq}]]\},$$5 acts only on site $$\frac{d}{dt}\rho_S^I(t)= -\int_{t_0}^t d\tau\,\mathrm{Tr}_B\{[V^I(t),[V^I(\tau),\rho_S^I(\tau)\otimes \rho_B^{eq}]]\},$$6 and $$\frac{d}{dt}\rho_S^I(t)= -\int_{t_0}^t d\tau\,\mathrm{Tr}_B\{[V^I(t),[V^I(\tau),\rho_S^I(\tau)\otimes \rho_B^{eq}]]\},$$7 consists of local terms plus short-range residual couplings, each commutator extends the operator support by at most one site. The truncated operator therefore remains quasi-local. The first two orders are

$$\frac{d}{dt}\rho_S^I(t)= -\int_{t_0}^t d\tau\,\mathrm{Tr}_B\{[V^I(t),[V^I(\tau),\rho_S^I(\tau)\otimes \rho_B^{eq}]]\},$$8

This construction provides a non-heuristic route from global Redfield dynamics to local operator expressions (Schnell, 2023).

A further step maps the truncated Redfield generator to an approximate local Lindblad form. Rewriting the exact Redfield superoperator as a difference of two dissipators with operators

$$\frac{d}{dt}\rho_S^I(t)= -\int_{t_0}^t d\tau\,\mathrm{Tr}_B\{[V^I(t),[V^I(\tau),\rho_S^I(\tau)\otimes \rho_B^{eq}]]\},$$9

and choosing $\rho_S^I(\tau)\to \rho_S^I(t)$0, one finds that $\rho_S^I(\tau)\to \rho_S^I(t)$1 vanishes to leading order in the small parameter $\rho_S^I(\tau)\to \rho_S^I(t)$2. Dropping that term gives

$\rho_S^I(\tau)\to \rho_S^I(t)$3

with

$\rho_S^I(\tau)\to \rho_S^I(t)$4

and rates $\rho_S^I(\tau)\to \rho_S^I(t)$5 (Schnell, 2023).

The regime of validity is set by $\rho_S^I(\tau)\to \rho_S^I(t)$6, where $\rho_S^I(\tau)\to \rho_S^I(t)$7 is a characteristic system time scale. The truncation error obeys

$\rho_S^I(\tau)\to \rho_S^I(t)$8

High-temperature baths, for which $\rho_S^I(\tau)\to \rho_S^I(t)$9 is short, favor rapid convergence. The derivation also notes limitations: the expansion degrades for narrow-band or strongly structured baths, higher orders may be required for long-range interactions, and strict positivity of the truncated Redfield generator is not guaranteed before the approximate Lindblad step (Schnell, 2023). The significance of this approach is that it retains microscopic Redfield grounding while producing sparse operators compatible with tensor-network or mean-field methods.

6. Electronically open molecules, fractional charging, and transport observables

In molecular settings, the quasi-local Redfield equation is formulated in a local spin-orbital basis $$\frac{d}{dt}\rho = -\,i[H_S,\rho]+\mathcal R[\rho],$$00, where unbarred indices refer to system orbitals and barred indices to environment orbitals. The full electronic Hamiltonian is decomposed as $$\frac{d}{dt}\rho = -\,i[H_S,\rho]+\mathcal R[\rho],$$01, and the particle-breaking part of the interaction is isolated as

$$\frac{d}{dt}\rho = -\,i[H_S,\rho]+\mathcal R[\rho],$$02

A Nakajima–Zwanzig projection is then constructed so that the projected density operator retains the full system indices while tracing out environment coherences. Applying the Born and Markov approximations yields a Redfield equation for matrix elements $$\frac{d}{dt}\rho = -\,i[H_S,\rho]+\mathcal R[\rho],$$03,

$$\frac{d}{dt}\rho = -\,i[H_S,\rho]+\mathcal R[\rho],$$04

where the tensor $$\frac{d}{dt}\rho = -\,i[H_S,\rho]+\mathcal R[\rho],$$05 is built from environment correlation functions $$\frac{d}{dt}\rho = -\,i[H_S,\rho]+\mathcal R[\rho],$$06 and $$\frac{d}{dt}\rho = -\,i[H_S,\rho]+\mathcal R[\rho],$$07 (Sannes et al., 2024).

If only populations $$\frac{d}{dt}\rho = -\,i[H_S,\rho]+\mathcal R[\rho],$$08 are retained and fast coherences are neglected, one obtains the Pauli master equation

$$\frac{d}{dt}\rho = -\,i[H_S,\rho]+\mathcal R[\rho],$$09

with rates expressed through spectral densities $$\frac{d}{dt}\rho = -\,i[H_S,\rho]+\mathcal R[\rho],$$10 and Fermi–Dirac occupations $$\frac{d}{dt}\rho = -\,i[H_S,\rho]+\mathcal R[\rho],$$11. In the quasi-local limit, system states are often expressed in a common spin-orbital basis of determinants so that only diagonal matrix elements of $$\frac{d}{dt}\rho = -\,i[H_S,\rho]+\mathcal R[\rho],$$12 survive, and the energy-conserving $$\frac{d}{dt}\rho = -\,i[H_S,\rho]+\mathcal R[\rho],$$13-function is replaced by a Gaussian of width $$\frac{d}{dt}\rho = -\,i[H_S,\rho]+\mathcal R[\rho],$$14 to mimic finite lifetime broadening of molecular levels (Sannes et al., 2024).

The physical content is explicit. The unitary commutator with $$\frac{d}{dt}\rho = -\,i[H_S,\rho]+\mathcal R[\rho],$$15 generates coherent intramolecular dynamics, while the Redfield tensor encodes incoherent electron exchange with the environment. Because $$\frac{d}{dt}\rho = -\,i[H_S,\rho]+\mathcal R[\rho],$$16 breaks particle number in each subsystem, the steady state can populate determinants with different electron numbers, so

$$\frac{d}{dt}\rho = -\,i[H_S,\rho]+\mathcal R[\rho],$$17

need not be an integer. The resulting noninteger charge is the fractional charging emphasized in the molecular formalism. The width $$\frac{d}{dt}\rho = -\,i[H_S,\rho]+\mathcal R[\rho],$$18 and the shape of $$\frac{d}{dt}\rho = -\,i[H_S,\rho]+\mathcal R[\rho],$$19 control how sharply molecular levels resonate with the environment’s Fermi sea and therefore how large the fractional charge becomes. The formalism is illustrated for benzene physisorbed on a graphene sheet as a toy model (Sannes et al., 2024).

This molecular usage also clarifies a distinction among quasi-local Redfield schemes. In electronically open molecules, level broadening is introduced phenomenologically and does not arise self-consistently from $$\frac{d}{dt}\rho = -\,i[H_S,\rho]+\mathcal R[\rho],$$20; the authors explicitly list this as a limitation. By contrast, in the self-consistent transport GQME, broadening is folded into the generalized rates through the backward-evolution propagators $$\frac{d}{dt}\rho = -\,i[H_S,\rho]+\mathcal R[\rho],$$21, and the resulting method reproduces the nonequilibrium steady-state density matrix and current of solvable transport models far more accurately than plain Redfield for those observables (Sannes et al., 2024, Esposito et al., 2010).

Overall, the quasi-local Redfield equation is best understood as a research program rather than a single formula. Across transport theory, quantum thermodynamics, many-body open-system simulation, and molecular electronic structure, it denotes Redfield-derived generators that preserve locality in operator support, incorporate broadening or nonsecular effects more faithfully than plain local master equations, and in some regimes recover GKSL or Davies structure without the standard secular approximation. The main unresolved boundary, already visible across these formulations, is that success for average observables, locality, or complete positivity does not by itself guarantee uniform accuracy for fluctuations, higher cumulants, or strongly non-Markovian regimes (Esposito et al., 2010, Shiraishi et al., 14 Jul 2025, Farina et al., 2020).