Dissipative Anderson Impurity Model

• The dissipative Anderson impurity model is a family of open quantum systems where nonunitary dynamics—such as two-body loss, dephasing, or one-body decay—are introduced into the standard Anderson framework.
• It employs various formulations, including reduced system approaches, Lindblad equations, and non-Hermitian effective Hamiltonians, to capture the interplay between impurity correlations and dissipative processes.
• Experimental and numerical studies reveal non-monotonic behavior in observables like loss current and Kondo screening, offering insights for ultracold-atom setups and quantum dot systems.

Searching arXiv for recent and relevant papers on dissipative Anderson impurity models and closely related impurity solvers. The dissipative Anderson impurity model denotes a family of open quantum impurity problems built on the Anderson impurity model and supplemented by relaxation, decoherence, loss, or non-Hermitian decay. In current usage, the term covers at least three closely related settings: the standard Anderson impurity model viewed as an open reduced system after integrating out fermionic reservoirs; Lindblad formulations in which the impurity is exposed to local Markovian channels such as dephasing or two-body loss; and non-Hermitian effective descriptions of postselected impurity dynamics with one-body loss. Across these variants, the common structure is a correlated local orbital with Coulomb repulsion $U$, hybridization to itinerant fermions, and a reduced impurity dynamics that is no longer purely unitary in the impurity sector (Sun et al., 13 Oct 2025, Vanhoecke et al., 27 Jun 2025, Yamamoto et al., 2024).

1. Scope and model classes

The term is not tied to a single microscopic definition. In the narrowest recent sense, it refers to a standard single-impurity Anderson Hamiltonian with an explicit local dissipator, as in the model with impurity two-body loss $L=\sqrt{\gamma}\,d_\uparrow d_\downarrow$. In a broader reduced-dynamics sense, it also includes the ordinary fermionic Anderson impurity model once the bath is integrated out exactly and retained as a temporally nonlocal influence functional. A third usage arises in non-Hermitian treatments of one-body impurity loss, where the Lindblad problem is replaced by a no-jump effective Hamiltonian $$H_{\rm eff}=H-\frac{i\gamma}{2}\sum_\sigma n_{d\sigma}$$. These usages are related, but they are not interchangeable (Vanhoecke et al., 27 Jun 2025, Sun et al., 13 Oct 2025, Yamamoto et al., 2024).

A useful summary of the main constructions is given below.

Variant	Dissipative mechanism	Representative paper
Reduced-dynamics AIM	Bath traced out into hybridization kernel $\Delta(t,t')$	(Sun et al., 13 Oct 2025)
Lindblad dephasing AIM	$L_\sigma=\sqrt{\gamma_\sigma}n_\sigma$	(Vanhoecke et al., 2023)
Lindblad two-body-loss AIM	$L=\sqrt{\gamma}\,d_\uparrow d_\downarrow$	(Vanhoecke et al., 27 Jun 2025)
Non-Hermitian AIM	$$H_{\rm eff}=H-\frac{i\gamma}{2}\sum_\sigma n_{d\sigma}$$	(Yamamoto et al., 2024)
Zeno-engineered infinite-$U$ AIM	Strong localized pair loss projects to dark subspace	(Stefanini et al., 2024)

The sharpest microscopic realization of a dissipative Anderson impurity model is the single lossy dot site coupled to noninteracting leads and subjected to strong localized two-body loss. In that construction, the dark subspace is $$\mathcal H_{\rm dark}=\operatorname{span}\{|0\rangle,|\uparrow\rangle,|\downarrow\rangle\}$$, so the coherent sector is exactly the infinite-$U$ Anderson impurity model, while finite loss leaves residual dissipative corrections that compete with Kondo screening (Stefanini et al., 2024).

A common source of confusion is terminological. The phrase “Anderson model” is also used for Anderson localization in disordered lattices. The randomized-gradient dephasing simulator studies a single-particle disordered tight-binding lattice with Markovian pure dephasing and does not contain an impurity orbital, a fermionic bath, a hybridization function, or a local interaction $L=\sqrt{\gamma}\,d_\uparrow d_\downarrow$0; it is therefore not a dissipative Anderson impurity model (Hunter-Gordon et al., 2019).

2. Reduced open-system formulation of the standard Anderson impurity model

The standard single-impurity Anderson model has Hamiltonian

$L=\sqrt{\gamma}\,d_\uparrow d_\downarrow$1

with

$L=\sqrt{\gamma}\,d_\uparrow d_\downarrow$2

$L=\sqrt{\gamma}\,d_\uparrow d_\downarrow$3

$L=\sqrt{\gamma}\,d_\uparrow d_\downarrow$4

The bath is characterized by the bath spectral function

$L=\sqrt{\gamma}\,d_\uparrow d_\downarrow$5

In this formulation, dissipation is not added phenomenologically. It appears after the noninteracting fermionic bath is integrated out exactly in the Grassmann coherent-state path integral, leaving the impurity with a nonlocal self-interaction in time (Sun et al., 13 Oct 2025).

For imaginary time, the impurity partition function becomes

$L=\sqrt{\gamma}\,d_\uparrow d_\downarrow$6

with influence functional

$L=\sqrt{\gamma}\,d_\uparrow d_\downarrow$7

The hybridization function is

$L=\sqrt{\gamma}\,d_\uparrow d_\downarrow$8

where

$L=\sqrt{\gamma}\,d_\uparrow d_\downarrow$9

After this reduction, the impurity action contains a bilocal term $$H_{\rm eff}=H-\frac{i\gamma}{2}\sum_\sigma n_{d\sigma}$$0, so the impurity dynamics depends on its history. In open-system language, the fermionic reservoir already generates relaxation, decoherence, broadening, finite-temperature damping, and non-Markovian memory. Real-time nonequilibrium dynamics follows by replacing the imaginary contour with the Keldysh contour and the Matsubara bath Green’s function with the contour-ordered bath Green’s function. This formulation therefore treats dissipation in the standard bath-induced sense even without introducing a bosonic environment or a Lindblad term (Sun et al., 13 Oct 2025).

This reduced description is important for the topic because it fixes a broad baseline meaning of “dissipative Anderson impurity model”: an impurity exchanging particles and energy with fermionic continua, with the dissipative back-action encoded in $$H_{\rm eff}=H-\frac{i\gamma}{2}\sum_\sigma n_{d\sigma}$$1. A plausible implication is that some papers use the phrase to emphasize reduced nonunitary impurity dynamics, while others reserve it for models with an additional explicit dissipative channel.

3. Explicit Lindblad and non-Hermitian formulations

The most direct open-system construction adds a local dissipator to the standard Anderson Hamiltonian

$$H_{\rm eff}=H-\frac{i\gamma}{2}\sum_\sigma n_{d\sigma}$$2

and evolves the full impurity-plus-bath density matrix by

$$H_{\rm eff}=H-\frac{i\gamma}{2}\sum_\sigma n_{d\sigma}$$3

This jump removes a pair only when the impurity is doubly occupied. Total particle number is not conserved, and its evolution is

$$H_{\rm eff}=H-\frac{i\gamma}{2}\sum_\sigma n_{d\sigma}$$4

so the loss current is

$$H_{\rm eff}=H-\frac{i\gamma}{2}\sum_\sigma n_{d\sigma}$$5

Because the jump acts only on doublons, the microscopic dark sector is the singly occupied local-moment manifold rather than the empty or fully mixed impurity sector (Vanhoecke et al., 27 Jun 2025).

A second explicit Lindblad variant is the dephasing Anderson model, where the local impurity Hamiltonian is

$$H_{\rm eff}=H-\frac{i\gamma}{2}\sum_\sigma n_{d\sigma}$$6

the bath and hybridization retain the standard form, and the jump operators are

$$H_{\rm eff}=H-\frac{i\gamma}{2}\sum_\sigma n_{d\sigma}$$7

Since $$H_{\rm eff}=H-\frac{i\gamma}{2}\sum_\sigma n_{d\sigma}$$8, the jumps are Hermitian and diagonal in the impurity occupation basis. They do not directly remove particles from the impurity; instead they dephase sectors that differ in occupancy. The local occupation sectors are strong symmetries of the local Lindbladian, so for $$H_{\rm eff}=H-\frac{i\gamma}{2}\sum_\sigma n_{d\sigma}$$9 the occupation is constant even though coherences decay with a finite lifetime set by $\Delta(t,t')$0 (Vanhoecke et al., 2023).

A third formulation begins from one-body impurity loss through Lindblad jump operators

$\Delta(t,t')$1

so that

$\Delta(t,t')$2

The analysis then focuses on the no-jump or postselected short-time dynamics generated by

$\Delta(t,t')$3

Here dissipation first appears as a complex onsite energy rather than as a complex hybridization. The ensuing many-body renormalization is a central part of the non-Hermitian theory (Yamamoto et al., 2024).

The Zeno-engineered realization occupies a distinct position. Its microscopic Lindblad equation uses the jump

$\Delta(t,t')$4

on a single dot site with Hamiltonian

$\Delta(t,t')$5

$\Delta(t,t')$6

$\Delta(t,t')$7

In the strong-loss regime $\Delta(t,t')$8, adiabatic elimination projects out the rapidly decaying doublon and yields an effective coherent Hamiltonian that is exactly the infinite-$\Delta(t,t')$9 Anderson impurity model in the constrained Hilbert space, plus residual dissipative corrections of order $L_\sigma=\sqrt{\gamma_\sigma}n_\sigma$0 (Stefanini et al., 2024).

By contrast, the out-of-equilibrium single-impurity Anderson model attached to two biased metallic leads is an open electronic environment problem in which hybridization broadening, current flow, and bias-induced decoherence arise from the leads themselves. It is relevant to dissipative impurity physics, but it does not add an explicit Lindblad or bosonic dissipator (Yan et al., 2021).

4. Many-body physics under dissipation: Kondo screening, Zeno protection, and decay

The central low-energy question is whether dissipation destroys the local moment before it can be screened, or whether correlations and dissipation reorganize into a new effective Kondo problem. In the two-body-loss Anderson model, the answer is not monotonic. The impurity spin resides mainly in the singly occupied sector, while loss requires a virtual process in which a bath electron first creates a doublon and the jump then removes the pair. Because both strong repulsion $L_\sigma=\sqrt{\gamma_\sigma}n_\sigma$1 and strong monitoring $L_\sigma=\sqrt{\gamma_\sigma}n_\sigma$2 suppress access to the doublon sector, the effective low-energy dissipation is non-monotonic. The steady-state loss current $L_\sigma=\sqrt{\gamma_\sigma}n_\sigma$3 grows at small $L_\sigma=\sqrt{\gamma_\sigma}n_\sigma$4, peaks around $L_\sigma=\sqrt{\gamma_\sigma}n_\sigma$5, and decreases again at large $L_\sigma=\sqrt{\gamma_\sigma}n_\sigma$6. The spin relaxation rate $L_\sigma=\sqrt{\gamma_\sigma}n_\sigma$7, extracted from $L_\sigma=\sqrt{\gamma_\sigma}n_\sigma$8, shows the same structure: it first increases, is maximal near $L_\sigma=\sqrt{\gamma_\sigma}n_\sigma$9, and then decreases in the strong-loss Kondo-Zeno regime. The spectral function reflects this crossover: the doublon band is rapidly destroyed, the Kondo peak survives at weak loss, disappears at intermediate loss, and re-emerges for $L=\sqrt{\gamma}\,d_\uparrow d_\downarrow$0 (Vanhoecke et al., 27 Jun 2025).

The dissipative Schrieffer-Wolff analysis of the same model makes this mechanism explicit. To second order in the hybridization, the effective low-energy theory contains both a Kondo exchange

$L=\sqrt{\gamma}\,d_\uparrow d_\downarrow$1

and residual nonlocal impurity-bath two-body loss with effective jump operator

$L=\sqrt{\gamma}\,d_\uparrow d_\downarrow$2

At the Fermi energy, the residual effective loss rate is

$L=\sqrt{\gamma}\,d_\uparrow d_\downarrow$3

while the low-energy exchange is

$L=\sqrt{\gamma}\,d_\uparrow d_\downarrow$4

Thus $L=\sqrt{\gamma}\,d_\uparrow d_\downarrow$5 is maximal around $L=\sqrt{\gamma}\,d_\uparrow d_\downarrow$6 but is suppressed in both the correlated limit $L=\sqrt{\gamma}\,d_\uparrow d_\downarrow$7 and the Zeno limit $L=\sqrt{\gamma}\,d_\uparrow d_\downarrow$8, whereas the real part of the Kondo exchange remains finite even for strong loss. This is the organizing principle behind the Kondo-Zeno crossover (Vanhoecke et al., 27 Jun 2025).

The non-Hermitian one-body-loss problem yields a different, but equally nontrivial, outcome. In the infinite-$L=\sqrt{\gamma}\,d_\uparrow d_\downarrow$9 slave-boson treatment, the impurity operator is written as

$$H_{\rm eff}=H-\frac{i\gamma}{2}\sum_\sigma n_{d\sigma}$$0

and the saddle-point variables become genuinely complex. The renormalized impurity Green functions are

$$H_{\rm eff}=H-\frac{i\gamma}{2}\sum_\sigma n_{d\sigma}$$1

with

$$H_{\rm eff}=H-\frac{i\gamma}{2}\sum_\sigma n_{d\sigma}$$2

In the non-Hermitian Kondo regime, the theory finds $$H_{\rm eff}=H-\frac{i\gamma}{2}\sum_\sigma n_{d\sigma}$$3 and $$H_{\rm eff}=H-\frac{i\gamma}{2}\sum_\sigma n_{d\sigma}$$4, so the effective onsite loss is strongly suppressed. Dissipation is instead transmuted into an emergent many-body process encoded in the complex hybridization $$H_{\rm eff}=H-\frac{i\gamma}{2}\sum_\sigma n_{d\sigma}$$5. The associated complex Kondo scale is

$$H_{\rm eff}=H-\frac{i\gamma}{2}\sum_\sigma n_{d\sigma}$$6

with real part

$$H_{\rm eff}=H-\frac{i\gamma}{2}\sum_\sigma n_{d\sigma}$$7

The screened non-Hermitian Kondo state breaks down when $$H_{\rm eff}=H-\frac{i\gamma}{2}\sum_\sigma n_{d\sigma}$$8, giving the critical condition

$$H_{\rm eff}=H-\frac{i\gamma}{2}\sum_\sigma n_{d\sigma}$$9

A key conclusion is that increasing microscopic loss can enhance the impurity lifetime near the transition because correlations suppress the renormalized one-body decay (Yamamoto et al., 2024).

The Zeno-engineered infinite-$U$0 realization connects these themes directly to an experimentally motivated Lindblad construction. In the ideal $U$1 limit, the coherent dark-sector dynamics is exactly the infinite-$U$2 Anderson model, and the Kondo temperature is

$U$3

Finite dissipation introduces a Zeno-suppressed loss rate $U$4. The reported criterion is that Kondo physics survives when $U$5, is significantly smeared once $U$6 exceeds a few times $U$7, and crosses over from Kondo-controlled magnetization decay $U$8 to loss-controlled decay $U$9 as $$\mathcal H_{\rm dark}=\operatorname{span}\{|0\rangle,|\uparrow\rangle,|\downarrow\rangle\}$$0 is reduced (Stefanini et al., 2024).

Dephasing modifies the impurity differently. In the dephasing Anderson model, symmetric local dephasing strongly slows the charge dynamics and only partially affects the spin dynamics; large dephasing leads to Zeno-like freezing of charge, while asymmetric dephasing can generate a long-lived or metastable impurity magnetization plateau. The same work interprets the $$\mathcal H_{\rm dark}=\operatorname{span}\{|0\rangle,|\uparrow\rangle,|\downarrow\rangle\}$$1-dependence of slow spin relaxation at strong dephasing in continuity with Kondo-related slow spin dynamics of the unitary model (Vanhoecke et al., 2023).

5. Methods of analysis and impurity solvers

Dissipative impurity problems are numerically demanding because they combine strong local correlations, long memory from the fermionic bath, and local nonunitary dynamics. Method development has therefore become a central part of the subject.

For the reduced standard Anderson model, the influence-functional tensor-network method compresses the exact bath back-action as a temporal Grassmann matrix product state. When the hybridization function can be approximated by a sum of $$\mathcal H_{\rm dark}=\operatorname{span}\{|0\rangle,|\uparrow\rangle,|\downarrow\rangle\}$$2 exponentials, the influence functional can be built from $$\mathcal H_{\rm dark}=\operatorname{span}\{|0\rangle,|\uparrow\rangle,|\downarrow\rangle\}$$3 small bond-dimension-2 Grassmann MPS blocks. The reported worst-case bond-dimension scaling is

$$\mathcal H_{\rm dark}=\operatorname{span}\{|0\rangle,|\uparrow\rangle,|\downarrow\rangle\}$$4

and the computational cost is reduced to $$\mathcal H_{\rm dark}=\operatorname{span}\{|0\rangle,|\uparrow\rangle,|\downarrow\rangle\}$$5. The formal result is that, for the Grassmann objects appearing here, the WII exponentiation step is exact rather than approximate. This directly targets long-time real-time and imaginary-time impurity dynamics with non-Markovian fermionic baths (Sun et al., 13 Oct 2025).

For the Lindblad two-body-loss Anderson model, the principal solver is a self-consistent hybridization expansion based on the non-crossing approximation in a vectorized or superfermion representation. After vectorizing the density matrix into $$\mathcal H_{\rm dark}=\operatorname{span}\{|0\rangle,|\uparrow\rangle,|\downarrow\rangle\}$$6, the evolution is generated by a doubled-space Lindbladian $$\mathcal H_{\rm dark}=\operatorname{span}\{|0\rangle,|\uparrow\rangle,|\downarrow\rangle\}$$7, and tracing out the noninteracting bath yields an exact integro-differential equation for the reduced impurity dynamical map. The self-energy is then approximated by a non-crossing resummation. This framework is used to compute real-time impurity observables, the steady-state reduced density matrix, two-time Green functions, and the steady-state spectral function. Exact quantum-trajectory simulations on finite chains serve as a qualitative benchmark (Vanhoecke et al., 27 Jun 2025).

For the dephasing Anderson model, diagrammatic Monte Carlo is formulated directly for the vectorized Lindblad problem on a single real-time contour rather than the conventional double Keldysh contour. The key reorganization is that the doubled Hilbert-space index replaces the upper and lower Keldysh branches. For diagonal jump operators such as $$\mathcal H_{\rm dark}=\operatorname{span}\{|0\rangle,|\uparrow\rangle,|\downarrow\rangle\}$$8, the impurity trace admits a generalized segment representation analogous to equilibrium hybridization-expansion continuous-time Monte Carlo. The paper’s central algorithmic conclusion is that local Markovian dissipation generally helps convergence by reducing the sign problem, because dissipative sectors with negative real Lindbladian eigenvalues are exponentially suppressed in time (Vanhoecke et al., 2023).

A related nonequilibrium, though not explicitly Lindbladian, approach is the two-particle semi-analytic reduced-parquet treatment of the biased single-impurity Anderson model. There the open-system character arises from two metallic leads at different chemical potentials, and two-particle vertex renormalization is used to avoid spurious magnetic transitions and unphysical hysteresis in the current-voltage characteristic. This framework is useful for interpreting how electronic reservoirs alone produce decoherence and suppress Kondo correlations when the bias becomes comparable to the Kondo temperature (Yan et al., 2021).

Closed-system impurity solver architectures remain relevant because they provide starting points for dissipative generalizations. The hybrid classical/quantum algorithm that combines tensor-network ground-state preparation with quantum subspace expansion for Green’s functions is formulated for equilibrium DMFT Anderson impurity models and contains no Lindblad, Keldysh, or non-Hermitian dynamics. Nevertheless, it supplies a solver template for large bath discretizations and dynamical correlators that could, in principle, be adapted to open-system impurity settings (Jamet et al., 2023).

6. Observables, realizations, and conceptual boundaries

The standard observables of the dissipative Anderson impurity model combine impurity spectroscopy, transport, and open-system decay diagnostics. In the two-body-loss model, the principal one-time observables are impurity density $$\mathcal H_{\rm dark}=\operatorname{span}\{|0\rangle,|\uparrow\rangle,|\downarrow\rangle\}$$9, double occupancy $U$0, loss current $U$1, impurity magnetization

$U$2

and the spin relaxation time extracted from $U$3. Two-time observables are built from the retarded Green function

$U$4

and the spectral function

$U$5

Finite-chain benchmarks additionally use the nearest-neighbor spin correlation $U$6 between impurity and bath site. In the Zeno-engineered realization, the spectral function, differential conductance, and impurity magnetization decay are emphasized, with the late-time linear conductance tracking the Kondo resonance height through $U$7 (Vanhoecke et al., 27 Jun 2025, Stefanini et al., 2024).

The principal experimental settings are ultracold-atom transport geometries and quantum-dot-like impurity platforms. The dissipative realization of the infinite-$U$8 Anderson model is designed for ultracold fermions with localized two-body loss on selected impurity sites; the required regime is strong localized loss, weak enough lead-dot coupling for dark-subspace dynamics, broad leads, and temperatures low compared with $U$9 (Stefanini et al., 2024). The non-Hermitian one-body-loss theory is proposed for semiconductor quantum dots coupled to leads and for ultracold Fermi gases or quantum point contacts, where tightly focused beams can induce local one-body loss and postselection can realize non-Hermitian dynamics (Yamamoto et al., 2024).

The conceptual boundaries of the subject are as important as the models themselves. First, a dissipative Anderson impurity model is not the same as a dissipative Anderson localization model. The latter is a disordered lattice problem with Markovian pure dephasing in the site basis, and it lacks the defining ingredients of the impurity problem: a distinguished correlated local orbital, a hybridization function, and Kondo-scale physics (Hunter-Gordon et al., 2019). Second, a Hamiltonian equilibrium impurity solver for DMFT is not by itself a dissipative impurity theory; it becomes relevant only indirectly, as a computational architecture for standard AIM dynamics (Jamet et al., 2023). Third, the reduced-dynamics standard Anderson model with its temporally nonlocal fermionic influence functional already captures bath-induced dissipation, relaxation, and non-Markovian memory, but it does not automatically include additional bosonic baths, mixed fermion-boson environments, or explicit Lindblad channels (Sun et al., 13 Oct 2025).

Several limitations recur across the literature. The effective infinite-$L=\sqrt{\gamma}\,d_\uparrow d_\downarrow$00 dark-subspace description is exact only in the ideal strong-loss limit, and the full lossy microscopic model may admit true stationary states very different from the local steady state around the impurity. The noncrossing approximation used in open Anderson problems is nonperturbative and suitable for strong-coupling impurity dynamics, but remains approximate for the finest low-temperature features. Non-Hermitian formulations describe no-jump or postselected dynamics and therefore omit the full quantum-jump structure of a Lindblad steady state. Finally, many exact factorization results are specific to quadratic fermionic baths and do not straightforwardly extend to bosonic dissipators or non-Gaussian environments (Stefanini et al., 2024, Vanhoecke et al., 27 Jun 2025, Yamamoto et al., 2024, Sun et al., 13 Oct 2025).

Taken together, these results show that the dissipative Anderson impurity model is best understood as a class of open correlated impurity theories rather than a single canonical Hamiltonian. In some formulations dissipation is simply the reduced effect of fermionic reservoirs; in others it is an explicit local Markovian or non-Hermitian channel. The most robust recent lesson is that dissipation does not have a uniform effect on impurity correlations: one-body loss, dephasing, and correlated two-body loss act on different sectors of the impurity Hilbert space and therefore reshape Kondo physics in qualitatively different ways.