Simple Lehmann Model Overview

• The Simple Lehmann Model is a compact spectral representation framework that replaces continuous integrals with finite sums of exponentials and poles in various many-body and field theory applications.
• It leverages low-rank decompositions and discrete Lehmann representations to reconstruct imaginary-time Green’s functions and efficiently evaluate Matsubara frequency data.
• The model’s versatility is demonstrated by its implementation across multiple programming languages and its adaptation to contexts ranging from quantum quench dynamics to unitarized chiral approaches in hadronic physics.

The expression “Simple Lehmann Model” is used in several technically distinct but structurally related ways across recent literature. In numerical many-body theory, it commonly denotes a compact discrete Lehmann representation of imaginary-time Green’s functions, in which a spectral integral is replaced by a small sum of exponentials in imaginary time and simple poles in Matsubara frequency (Kaye et al., 2021, Kaye et al., 2021, Kaye et al., 2024). In quantum field theory, the same phrase also appears in connection with standard or generalized Källén–Lehmann decompositions of two-point functions in local, nonlocal, curved, lattice, and Lorentz-violating settings (Briscese et al., 2024, Loparco et al., 2023). In nonequilibrium integrable systems, it further refers to explicit Lehmann sums for post-quench observables and their Quench Action reductions (Senese et al., 19 May 2026). This suggests a common conceptual core: a correlator is represented as a superposition over spectral data, poles, or intermediate states, with the model becoming “simple” when that superposition is reduced to a compact, computationally tractable form.

1. Common structure and scope

The usages appearing in the literature share the same formal motif: an interacting object is expressed as a sum or integral over simpler building blocks.

Context	Object	Representative form
Imaginary-time numerics	Green’s function	$G(\tau)\approx \sum_{l=1}^r K(\tau,\omega_l)\,g_l$
Spectral QFT	Two-point function	$\int d\mu^2\,\rho(\mu^2)/(p^2-\mu^2+i\epsilon)$
Quench dynamics	One-point function	$$\sum_{m,n} c_m^* c_n\,e^{i(E_m-E_n)t}\,\langle m|O|n\rangle$$

In the DLR setting, the “simple” model is explicitly discrete: the continuous spectral density is replaced by an effective spectral density that is a sum of $\delta$ functions,

$$A_{\mathrm{eff}}(\omega)=\sum_{k=1}^r w_k\,\delta(\omega-s_k),$$

so that both the imaginary-time and Matsubara-frequency representations become finite sums (Kaye et al., 2021). In standard Källén–Lehmann theory, the same simplifying idea appears as a decomposition over invariant masses with a positive spectral density, often split into a one-particle pole plus continuum. In nonequilibrium quench problems, the simplification is not necessarily a pole model, but rather a reduction of the full double Lehmann sum to a more manageable sampling problem or to a Quench Action single-sum expansion (Senese et al., 19 May 2026). A further terminological complication is historical: in low-energy hadronic physics, “Lehmann model” can mean H. Lehmann’s 1972 unitarized chiral approach, which is explicitly distinguished from the Lehmann spectral representation and from LSZ (Truong, 2010).

2. Discrete Lehmann representation in imaginary time

In the DLR formulation, imaginary-time Green’s functions are built from the finite-temperature Lehmann representation. In scaled variables, with $\tau\in[0,1]$ and $\omega\in[-\Lambda,\Lambda]$,

$$G(\tau) = -\int_{-\Lambda}^{\Lambda} K(\tau,\omega)\,\rho(\omega)\,d\omega, \qquad K(\tau,\omega)=\frac{e^{-\tau\omega}}{1+e^{-\omega}}.$$

For numerical stability at large negative $\omega$, the equivalent form

$$K(\tau,\omega)=\frac{e^{(1-\tau)\omega}}{1+e^{\omega}}$$

is also used (Kaye et al., 2021).

The core approximation is a low-rank decomposition of the kernel,

$\int d\mu^2\,\rho(\mu^2)/(p^2-\mu^2+i\epsilon)$0

which yields the DLR expansion

$\int d\mu^2\,\rho(\mu^2)/(p^2-\mu^2+i\epsilon)$1

The same coefficients reconstruct the Matsubara function through

$\int d\mu^2\,\rho(\mu^2)/(p^2-\mu^2+i\epsilon)$2

For fermions,

$\int d\mu^2\,\rho(\mu^2)/(p^2-\mu^2+i\epsilon)$3

and for bosons,

$\int d\mu^2\,\rho(\mu^2)/(p^2-\mu^2+i\epsilon)$4

The basis therefore consists of exponentials in imaginary time and simple poles in Matsubara frequency (Kaye et al., 2021, Kaye et al., 2024).

The numerical efficiency of the representation follows from the low numerical rank of the analytic continuation kernel. The observed scaling is

$\int d\mu^2\,\rho(\mu^2)/(p^2-\mu^2+i\epsilon)$5

with $\int d\mu^2\,\rho(\mu^2)/(p^2-\mu^2+i\epsilon)$6 and user-controlled tolerance $\int d\mu^2\,\rho(\mu^2)/(p^2-\mu^2+i\epsilon)$7 (Kaye et al., 2021, Kaye et al., 2021). The literature contrasts this with other imaginary-time parameterizations: uniform grid/Fourier series require $\int d\mu^2\,\rho(\mu^2)/(p^2-\mu^2+i\epsilon)$8 degrees of freedom, orthogonal polynomials such as Legendre and Chebyshev require $\int d\mu^2\,\rho(\mu^2)/(p^2-\mu^2+i\epsilon)$9, while IR and DLR both exhibit $$\sum_{m,n} c_m^* c_n\,e^{i(E_m-E_n)t}\,\langle m|O|n\rangle$$0 scaling (Kaye et al., 2021). This is the sense in which the DLR-based simple Lehmann model is both compact and explicit.

3. Basis construction, interpolation grids, and software realizations

The DLR basis is constructed by discretizing the kernel on a fine adaptive grid and applying interpolative decomposition through pivoted QR. The procedure first forms

$$\sum_{m,n} c_m^* c_n\,e^{i(E_m-E_n)t}\,\langle m|O|n\rangle$$1

then applies a rank-revealing column-pivoted QR to select the DLR frequencies $$\sum_{m,n} c_m^* c_n\,e^{i(E_m-E_n)t}\,\langle m|O|n\rangle$$2, and finally a rank-revealing row-pivoted QR to $$\sum_{m,n} c_m^* c_n\,e^{i(E_m-E_n)t}\,\langle m|O|n\rangle$$3 to select the DLR imaginary-time nodes $$\sum_{m,n} c_m^* c_n\,e^{i(E_m-E_n)t}\,\langle m|O|n\rangle$$4 (Kaye et al., 2021). The selected time nodes are precisely those from which the coefficients $$\sum_{m,n} c_m^* c_n\,e^{i(E_m-E_n)t}\,\langle m|O|n\rangle$$5 can be stably recovered by solving a small square linear system. A corresponding DLR Matsubara grid $$\sum_{m,n} c_m^* c_n\,e^{i(E_m-E_n)t}\,\langle m|O|n\rangle$$6 is built by increasing a cutoff $$\sum_{m,n} c_m^* c_n\,e^{i(E_m-E_n)t}\,\langle m|O|n\rangle$$7, practically set to $$\sum_{m,n} c_m^* c_n\,e^{i(E_m-E_n)t}\,\langle m|O|n\rangle$$8, until the selected nodes stabilize (Kaye et al., 2021).

The implementation ecosystem is unusually explicit. libdlr provides dlr_it_build to build the basis and imaginary-time grid, dlr_mf to construct the Matsubara grid, dlr_it2cf_init and dlr_it2cf to recover coefficients from imaginary-time grid samples, dlr_it_fit for least-squares fitting from noisy data, dlr_it_eval and dlr_mf_eval for evaluation, eqpts_rel, abs2rel, and rel2abs for relative and absolute time-point conversion, and dlr_it2itr_init for the reflection mapping used by the SYK self-energy (Kaye et al., 2021). Standard operations such as convolution and integration are available in the explicit exponential basis, and the Dyson equation can be solved either in Matsubara frequency through

$$\sum_{m,n} c_m^* c_n\,e^{i(E_m-E_n)t}\,\langle m|O|n\rangle$$9

or in imaginary time via DLR discretization of convolutions (Kaye et al., 2021).

The software landscape spans multiple languages. libdlr is written in Fortran, provides a C header interface, contains the Python module pydlr, and is accompanied by the stand-alone Julia implementation Lehmann.jl (Kaye et al., 2021). cppdlr provides a C++ interface implementing the same DLR logic for functions in imaginary time and Matsubara frequency, with explicit exponentials in time and simple poles in Matsubara frequency, and has been integrated into TRIQS and used in DMFT solvers, Keldysh mixing Green’s function calculations, and diagrammatic evaluations (Kaye et al., 2024).

The practical workflow is standardized. One chooses $\delta$0, estimates $\delta$1, sets $\delta$2, chooses $\delta$3, builds the basis and grids, samples or fits the function, and evaluates the resulting representation in either domain. The reported examples are concrete: for $\delta$4 and $\delta$5, there are $\delta$6 nodes in each domain; for noisy data with $\delta$7, $\delta$8, noise magnitude $\delta$9, and $$A_{\mathrm{eff}}(\omega)=\sum_{k=1}^r w_k\,\delta(\omega-s_k),$$0, the DLR rank is $$A_{\mathrm{eff}}(\omega)=\sum_{k=1}^r w_k\,\delta(\omega-s_k),$$1 and the fit agrees well with $$A_{\mathrm{eff}}(\omega)=\sum_{k=1}^r w_k\,\delta(\omega-s_k),$$2; and for an SYK solver with $$A_{\mathrm{eff}}(\omega)=\sum_{k=1}^r w_k\,\delta(\omega-s_k),$$3, $$A_{\mathrm{eff}}(\omega)=\sum_{k=1}^r w_k\,\delta(\omega-s_k),$$4, and $$A_{\mathrm{eff}}(\omega)=\sum_{k=1}^r w_k\,\delta(\omega-s_k),$$5, there are $$A_{\mathrm{eff}}(\omega)=\sum_{k=1}^r w_k\,\delta(\omega-s_k),$$6 DLR nodes in each domain (Kaye et al., 2021).

4. Sampled, fitted, and learned Lehmann models in many-body dynamics

Outside equilibrium imaginary-time calculations, the simple Lehmann model becomes a tool for explicit state sums. For a homogeneous quantum quench from an initial pure state $$A_{\mathrm{eff}}(\omega)=\sum_{k=1}^r w_k\,\delta(\omega-s_k),$$7, the time-dependent expectation value of a local operator is

$$A_{\mathrm{eff}}(\omega)=\sum_{k=1}^r w_k\,\delta(\omega-s_k),$$8

This direct-sum representation is exact but exponentially hard to evaluate in interacting systems because the number of relevant eigenstates grows exponentially with system size, while overlaps and form factors can be exponentially small and strongly fluctuating (Senese et al., 19 May 2026).

The Quench Action formalism reduces the double sum to a single sum over excitations of a representative saddle-point state,

$$A_{\mathrm{eff}}(\omega)=\sum_{k=1}^r w_k\,\delta(\omega-s_k),$$9

with

$\tau\in[0,1]$0

The Monte Carlo scheme samples configurations with probability

$\tau\in[0,1]$1

using a Metropolis–Hastings acceptance step. The method is benchmarked in the transverse-field Ising chain and the Lieb–Liniger model, and the paper identifies a sign problem for more general dynamical correlators and generic initial states, while integrable initial states with pair structures avoid it (Senese et al., 19 May 2026).

A different computational development is the Lehmann-representation-based PINN for Anderson impurity models. There the self-energy is parameterized by a nonnegative auxiliary spectral function,

$\tau\in[0,1]$2

which is implemented as a finite pole sum,

$\tau\in[0,1]$3

This hard-wires causality, analyticity, Kramers–Kronig consistency, and the correct $\tau\in[0,1]$4 tail into the model, while sparse IR/DLR grids stabilize the representation (Kakizawa et al., 2024). In the reported single-orbital AIM study, the model is trained for $\tau\in[0,1]$5, $\tau\in[0,1]$6, and $\tau\in[0,1]$7, and the Lehmann constraints reduce the maximum test error in electron filling by a factor of about $\tau\in[0,1]$8 relative to a comparable network without the Lehmann projection (Kakizawa et al., 2024). A plausible implication is that the “simple Lehmann model” increasingly functions not only as a compressed representation, but also as an inductive bias in machine-learned many-body surrogates.

5. Källén–Lehmann representations in local, nonlocal, curved, lattice, and Lorentz-violating field theory

In local quantum field theory, the canonical scalar Källén–Lehmann representation is

$\tau\in[0,1]$9

with $\omega\in[-\Lambda,\Lambda]$0, and typically

$\omega\in[-\Lambda,\Lambda]$1

The time-ordered two-point function is correspondingly a superposition of free propagators weighted by the same positive spectral density (Briscese et al., 2024). This basic structure extends, with important modifications, to several nonstandard settings.

In nonlocal quantum gravity with entire form factors,

$\omega\in[-\Lambda,\Lambda]$2

the time-ordered two-point function still admits a generalized Källén–Lehmann representation with the standard momentum dependence

$\omega\in[-\Lambda,\Lambda]$3

while nonlocality enters through the spectral density. The free spectral density coincides with the local one because $\omega\in[-\Lambda,\Lambda]$4, the physical spectrum is unchanged, no extra poles or ghosts appear, and the local limit is recovered smoothly as $\omega\in[-\Lambda,\Lambda]$5 (Briscese et al., 2024).

In de Sitter spacetime, the KL decomposition is organized by unitary irreducible representations of $\omega\in[-\Lambda,\Lambda]$6. For scalars,

$\omega\in[-\Lambda,\Lambda]$7

with nonnegative spectral densities in the Bunch–Davies vacuum; for spinning traceless symmetric operators, the principal-series part sums over spins $\omega\in[-\Lambda,\Lambda]$8 and is supplemented by complementary-series contributions, and in $\omega\in[-\Lambda,\Lambda]$9 also by discrete-series terms (Loparco et al., 2023). A related development gives compact formulas for spinor–scalar and spinor–spinor loop densities in AdS and dS, derives corresponding Källén–Lehmann decompositions of Wightman products, and writes chain-approximation spectral equations for scalar and spinor one-loop self-energies in Yukawa theory in de Sitter space (Altshuler, 10 Aug 2025).

In Lorentz-violating field theory with a constant symmetric observer-Lorentz tensor $$G(\tau) = -\int_{-\Lambda}^{\Lambda} K(\tau,\omega)\,\rho(\omega)\,d\omega, \qquad K(\tau,\omega)=\frac{e^{-\tau\omega}}{1+e^{-\omega}}.$$0, the exact propagators still admit generalized Källén–Lehmann representations, but the spectral densities depend not only on $$G(\tau) = -\int_{-\Lambda}^{\Lambda} K(\tau,\omega)\,\rho(\omega)\,d\omega, \qquad K(\tau,\omega)=\frac{e^{-\tau\omega}}{1+e^{-\omega}}.$$1 but also on the observer scalars

$$G(\tau) = -\int_{-\Lambda}^{\Lambda} K(\tau,\omega)\,\rho(\omega)\,d\omega, \qquad K(\tau,\omega)=\frac{e^{-\tau\omega}}{1+e^{-\omega}}.$$2

For fermions, the spectral matrix involves the four Dirac structures $$G(\tau) = -\int_{-\Lambda}^{\Lambda} K(\tau,\omega)\,\rho(\omega)\,d\omega, \qquad K(\tau,\omega)=\frac{e^{-\tau\omega}}{1+e^{-\omega}}.$$3 and $$G(\tau) = -\int_{-\Lambda}^{\Lambda} K(\tau,\omega)\,\rho(\omega)\,d\omega, \qquad K(\tau,\omega)=\frac{e^{-\tau\omega}}{1+e^{-\omega}}.$$4 for $$G(\tau) = -\int_{-\Lambda}^{\Lambda} K(\tau,\omega)\,\rho(\omega)\,d\omega, \qquad K(\tau,\omega)=\frac{e^{-\tau\omega}}{1+e^{-\omega}}.$$5, and equal-time commutator and anticommutator sum rules generalize the usual normalization conditions (Potting, 2011).

In lattice field theory, the Umezawa–Kamefuchi–Källén–Lehmann representation follows from Hermiticity, translational invariance, reflection positivity, and polynomial boundedness. In momentum space one obtains

$$G(\tau) = -\int_{-\Lambda}^{\Lambda} K(\tau,\omega)\,\rho(\omega)\,d\omega, \qquad K(\tau,\omega)=\frac{e^{-\tau\omega}}{1+e^{-\omega}}.$$6

with $$G(\tau) = -\int_{-\Lambda}^{\Lambda} K(\tau,\omega)\,\rho(\omega)\,d\omega, \qquad K(\tau,\omega)=\frac{e^{-\tau\omega}}{1+e^{-\omega}}.$$7. The positivity of the spectral density is thus necessary under the stated axioms, and the overlap scalar boson provides a counterexample: its continuum density changes sign, so the model violates reflection positivity (Usui, 2012).

Across these settings, the common point is not a fixed formula but a conserved architecture: two-point functions are decomposed into simpler free or harmonic building blocks, with positivity, analyticity, and pole structure encoded in the spectral weights.

6. Historical usage and terminological ambiguity

A final usage is historically important because it changes the meaning of “Lehmann model” altogether. In the review of low-energy soft hadronic physics, the term refers to H. Lehmann’s 1972 unitarized chiral approach to low-energy $$G(\tau) = -\int_{-\Lambda}^{\Lambda} K(\tau,\omega)\,\rho(\omega)\,d\omega, \qquad K(\tau,\omega)=\frac{e^{-\tau\omega}}{1+e^{-\omega}}.$$8 scattering. There, the one-loop chiral amplitude in the chiral limit is combined with exact elastic unitarity and analyticity through an effective-range expansion for $$G(\tau) = -\int_{-\Lambda}^{\Lambda} K(\tau,\omega)\,\rho(\omega)\,d\omega, \qquad K(\tau,\omega)=\frac{e^{-\tau\omega}}{1+e^{-\omega}}.$$9, the inverse-amplitude method, or Padé approximants. The relevant formulas are

$\omega$0

together with the effective-range form

$\omega$1

and the Padé unitarization

$\omega$2

The review states explicitly that this is not the Lehmann spectral representation and not the LSZ formalism (Truong, 2010).

This terminological divergence matters because the same phrase can denote either a spectral representation of correlators or a unitarized low-energy scattering model. The literature therefore supports a careful, context-dependent reading. In numerical many-body work, “simple Lehmann model” usually means a low-rank discrete spectral ansatz. In field theory, it more often points to Källén–Lehmann spectral decompositions and their generalizations. In older hadronic applications, it denotes a unitarized chiral partial-wave construction. The unifying thread is the use of Lehmann-type structure to compress, organize, or reconstruct nontrivial physics from a minimal spectral or state-sum description.