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Complex Frequency Fingerprint: Basic Concept and Theory

T0 review · 4 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read A single real-frequency drive exposes the entire complex Green's function after transients decay, including non-Hermitian spectra, skin eigenstates, and many-body quasiparticles.

desk verdict The CFF protocol for non-Hermitian systems is a genuinely useful steady-state trick for accessing complex-frequency Green's functions from a real-frequency drive; the many-body extension overreaches because freezing the self-energy at ω0 replaces the actual quasiparticle poles with eigenvalues of H_eff(ω0). read the letter →

arxiv 2411.12577 v2 pith:H7EOOZZI submitted 2024-11-19 cond-mat.mes-hall physics.opticsquant-ph

Juntao Huang , Kun Ding , Jiangping Hu , Zhesen Yang This is my paper · ORCID
classification cond-mat.mes-hallphysics.opticsquant-ph
keywords complexfrequencyfingerprintGreen'sfunctionnon-Hermitianskineffectnon-Blochresponsespectralwindingnumberdriven-dissipativesystemsquasiparticleresolutiondouble-frequency
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper proposes the complex frequency fingerprint (CFF): drive the system harmonically at one real frequency $\omega_0$, site by site, record the full response matrix $\chi_{\omega_0}(t)$, and form $G_{\omega_0}(\omega_c;t)=1/((\omega_c-\omega_0)+[\chi_{\omega_0}(t)]^{-1})$. The central claim is that as $t\to\infty$ this object converges to the complex-frequency Green's function $G(\omega_c)=1/(\omega_c-H_{\mathrm{nH}})$, because the transient term in $\chi_{\omega_0}(t)$ decays whenever all eigenvalues of $H_{\mathrm{nH}}$ have negative imaginary parts. This matters because it makes the non-Bloch response experimentally accessible: inside regions of nonzero spectral winding the open-boundary Green's function grows exponentially in one spatial direction and is boundary-sensitive, and the paper claims this is the unique steady-state signature that separates the non-Hermitian skin effect from ordinary nonreciprocity. The same construction extends to quantum many-body systems, where the double-frequency Green's function $1/(\omega_c-H_S^{\mathrm{eff}}(\omega_0))$ is claimed to be a measurable object carrying complete single-particle quasiparticle information across the complex frequency plane.

What carries the argument

The load-bearing object is the complex frequency fingerprint itself, Eq. (7): $G_{\omega_0}(\omega_c\in\mathbb{C};t)=1/((\omega_c-\omega_0)+[\chi_{\omega_0}(t)]^{-1})$, built from the measured response matrix $\chi_{\omega_0}(t)$ obtained by sequential single-site harmonic driving at real frequency $\omega_0$. Its convergence is carried by the identity $\chi_{\omega_0}(t)=G(\omega_0)-G(\omega_0)e^{-i(H_{\mathrm{nH}}-\omega_0)t}$: the transient $e^{-i(H_{\mathrm{nH}}-\omega_0)t}$ decays because the dissipative Hamiltonian's eigenvalues have negative imaginary parts, leaving $G(\omega_0)$ dominant and making the long-time limit independent of $\omega_0$. Two companion notions organize the argument: the spectral winding number $\nu(\omega_c)=\frac{1}{2\pi i}\oint dk\,\partial_k\ln\det[H_{\mathrm{nH}}(k)-\omega_c]$, which partitions the complex plane into Bloch and non-Bloch response regions, and the $\beta$-root scaling of the Green's function, which turns a nonzero winding number into the exponential spatial growth $[G^{\mathrm{OBC}}]_{i\alpha,i_0\beta}\sim\beta_3^{-(i_0-i)}$ that defines non-Bloch response. For many-body systems the same formula is transplanted onto the effective non-Hermitian Hamiltonian $H_S^{\mathrm{eff}}(\omega_0)=H_S-i\eta+\Sigma_S(\omega_0+i\eta)$, so that the double-frequency Green's function $1/(\omega_c-H_S^{\mathrm{eff}}(\omega_0))$ is the complex-frequency resolvent of a frequency-dependent non-Hermitian problem.

What would settle it

Set up the driven-dissipative lattice model of Eq. (2) with $\gamma_1\neq\gamma_2$ (parameters of Fig. 1: $N=100$, $t_1=1.5$, $t_2=1$, $\mu=0.3$, $\lambda=-1$, $\gamma_1=2$, $\gamma_2=1$): drive only site $x_0=60$, assemble $\chi_{\omega_0}(t)$ at $\omega_0=0$, and evaluate the CFF at the in-gap frequency $\omega_2=1.3-1.35i$ at times $t=1,5,10$. If the matrix elements $[G_{\omega_0}(\omega_2;t)]_{xx_0}$ fail to approach the directly computed resolvent $1/(\omega_2-H_{\mathrm{nH}})$ as $t$ grows, or if the converged result shows no exponential spatial growth and no bulk difference from the periodic-boundary Green's function despite $\nu(\omega_2)\neq 0$, the central convergence claim fails. Symmetrically, the same protocol on the $\gamma_1=\gamma_2$ nonreciprocal-without-skin-effect model must show decaying, boundary-insensitive CFF at every complex frequency; an apparent skin signature there would refute the diagnostic.

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Extended reading notes

Core claim

The paper's thesis is that complex-frequency Green's functions, not real-frequency ones, carry the unique physics of the non-Hermitian skin effect, and that these functions can be measured in steady state. Concretely, the CFF obeys $\lim_{t\to\infty} G_{\omega_0}(\omega_c;t)=G(\omega_c)=1/(\omega_c-H_{\mathrm{nH}})$, independent of the driving frequency $\omega_0$, so one real-frequency experiment maps the resolvent over the entire complex plane. The paper proves a no-go complement: in any purely dissipative system all poles of the open-boundary Green's function lie below the real axis, so the spectral winding number $\nu(\omega_r)$ vanishes for real $\omega_r$, and real-frequency Green's functions always belong to the Bloch response region — meaning ordinary nonreciprocal correlations, seen at real frequencies, cannot certify the skin effect. By contrast, at complex frequencies inside the point gap the open-boundary Green's function scales as $\beta_3^{-(i_0-i)}$ for $i<i_0$ with $|\beta_3|<1$, an exponential growth in real space that encodes the non-Bloch response and the skin eigenstates. For quantum many-body systems the analogous object, $G^{\mathrm{CFF}}_S(\omega_c,\omega_0)=1/(\omega_c-H_S^{\mathrm{eff}}(\omega_0))$ with $H_S^{\mathrm{eff}}(\omega_0)=H_S-i\eta+\Sigma_S(\omega_0+i\eta)$, is argued to be an observable containing all single-particle excitation information; in the paper's two-level fermion example it reveals two quasiparticle peaks in the complex plane where real-frequency spectroscopy shows one.

Load-bearing premise

The central premise is that the full $N\times N$ response matrix can be measured by driving each site one at a time, that the system obeys the linear dissipative equation (1), and that $\chi_{\omega_0}(t)$ remains invertible at the readout times — and for the many-body extension, that every matrix element of the retarded Green's function is a directly measurable observable, for which the paper gives a protocol in classical-wave settings but not for interacting quantum systems.

Editorial extensions

If this is right

  • The non-Hermitian skin effect becomes certifiable by a steady-state measurement: at complex frequencies with nonzero spectral winding the open-boundary CFF shows exponential growth in one spatial direction and obeys $|G^{\mathrm{OBC}}|\neq|G^{\mathrm{PBC}}|$ in the bulk, while systems without the skin effect show decaying, boundary-insensitive responses at all complex frequencies.
  • Non-Hermitian spectra are read out directly from the complex-frequency density of states $D_{\omega_0}(\omega_c;t)=-\frac{1}{N\pi}\mathrm{Im}\,\mathrm{Tr}\,G_{\omega_0}(\omega_c;t)$, which diverges as $\omega_c$ approaches any eigenvalue $E_n$, including eigenvalues far from the real axis.
  • Near an eigenvalue, the CFF matrix elements reconstruct the right and left eigenstates ($G_{ij_0}\propto\langle i|\psi^R_n\rangle$, $G_{j_0 i}\propto\langle\psi^L_n|i\rangle$), so skin-mode profiles — including the geometry-dependent 2D skin effect — are experimentally recoverable.
  • In quantum many-body systems, the CFF resolves quasiparticle peaks across the complex frequency plane, remains meaningful even when quasiparticle weight $Z\to 0$, and in the two-level fermion model shows an intuitive repulsion between quasiparticles as the chemical potential rises.
  • Because the linear dissipative equation (1) also governs classical wave systems, the protocol transfers directly to photonic, acoustic, electric-circuit, and mechanical platforms where amplitude and phase of the field are measurable.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Editorial inference: the method's practical reach is set by the convergence time of the transient $e^{-i(H_{\mathrm{nH}}-\omega_0)t}$; in systems with eigenvalues whose imaginary parts are near zero, the readout time needed for the key limit to hold may exceed the dephasing or coherence time, so the scheme's range is tied to the dissipative gap of $H_{\mathrm{nH}}$.
  • Editorial inference: because $G^{\mathrm{CFF}}_S(\omega_c,\omega_0)$ is a double-frequency object, sweeping $\omega_0$ amounts to a two-dimensional spectroscopy that traces the frequency dependence of the self-energy $\Sigma_S(\omega_0+i\eta)$; this suggests a concrete program — measuring the CFF at many real driving frequencies — that goes beyond the single-$\omega_0$ examples shown.
  • Editorial inference: the many-body claim would be fully realized by an explicit experimental protocol for measuring all matrix elements of the retarded Green's function of an interacting quantum system; the paper provides that protocol for driven-dissipative and classical-wave systems, but for the interacting case it states the measurability without giving the step-by-step procedure.
  • Editorial inference: the CFF should also detect spectral degeneracies such as exceptional points and point-gap bound states without extra apparatus, since the complex-frequency DOS divergence and the eigenstate reconstruction in Supplementary Section IV are generic; testing this on a two-band model with an exceptional point would be a direct extension.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 5 minor

Summary. The paper introduces the complex frequency fingerprint (CFF), a protocol in which the full time-dependent response matrix of a driven-dissipative system is measured under sequential harmonic drives and then inverted to form G_{ω0}(ω_c; t) = 1 / ((ω_c − ω0) + [χ_{ω0}(t)]^{-1}). For non-Hermitian systems governed by Eq. (1), the paper shows that this object converges to 1/(ω_c − H_nH) as t → ∞, and uses this to access complex-frequency Green's functions, complex spectra, and non-Hermitian skin eigenstates, including a distinction between Bloch and non-Bloch responses. The paper then generalizes the construction to quantum many-body systems by defining G_CFF_S(ω_c, ω0) = 1/(ω_c − H_eff(ω0)), where the self-energy is evaluated at a fixed real frequency ω0, and illustrates the idea on a mean-field two-level model coupled to a thermal bath, claiming that peaks of this object in the complex plane constitute complete quasiparticle information.

Significance. The non-Hermitian part of the paper is a clean and potentially useful contribution: the derivation of the convergence in Eq. (8) from Eq. (S16) is correct under the stated dissipative assumption, and the numerical demonstrations in Figs. 1, 2, and 6 support the claim that the CFF provides a steady-state route to complex-frequency Green's functions, spectra, and eigenstates, including the non-Hermitian skin effect. The CFF is explicitly a measurement reconstruction rather than a new theoretical object, which is a strength from an experimental perspective. However, the many-body extension, which is a central advertised novelty, currently overclaims. The object in Eq. (13) freezes the self-energy at a real frequency and therefore its poles are not generally the quasiparticle poles of the interacting Green's function. Because this issue affects the core interpretation of Fig. 3 and the 'complete quasiparticle information' claim, the manuscript requires substantial revision before its main claims can be accepted.

major comments (4)
  1. [Eq. (13) and Supplementary Eq. (S21)] The definition G_CFF_S(ω_c, ω0) = 1/(ω_c + iη − H_S − Σ_S(ω0 + iη)) evaluates the self-energy at a fixed real frequency ω0, so that ω_c appears only in the resolvent denominator. As a result, the poles of G_CFF_S are the eigenvalues of the fixed matrix H_eff(ω0), while the poles of the true retarded Green's function in Eq. (11) solve det[ω_c − H_S − Σ_S(ω_c + iη)] = 0. These two sets of poles coincide only when the self-energy is frequency-independent or when ω0 happens to satisfy a self-consistent pole condition. In the example of Eq. (15), the self-energy Σ_↓↓(ω) is strongly frequency dependent, as shown explicitly in Eq. (S27). Therefore the claim that Eq. (13) contains 'complete quasi-particle information' across the complex frequency plane is not supported by the stated equations and requires either a proof of the pole correspondence under well-defined conditions or a substantial reframing of what G_CFF_S actually measures.
  2. [Fig. 3 and surrounding text] Because of the issue in Eq. (13), the identification of the two peaks in Fig. 3 as 'quasiparticle peaks' of the interacting system is not justified. The peaks are poles of the fixed effective Hamiltonian H_eff(ω0 = 0), not necessarily poles of the interacting Green's function G^R_S(ω_c). In a model with strong frequency dependence of the self-energy, a pole of the fixed matrix can be located arbitrarily far from the true quasiparticle pole on the complex plane. The statement that the two peaks 'reveal two distinct quasi-particle peaks' and the conclusion that their repulsion provides an intuitive picture of many-body interactions are therefore unsupported without an additional argument establishing the physical meaning of the fixed-ω0 poles.
  3. [Supplementary Section V and Eq. (S20)] The claimed experimental accessibility of the many-body CFF is not established. Supplementary Section V states only that one should 'measure all matrix elements of the Green's function' and then defines G_CFF_ω0(ω_c) by inverting G^R_S(ω0). For an interacting quantum many-body system, it is not shown how the full single-particle Green's function matrix in a fixed real-space basis can be measured with the required completeness, including off-diagonal elements and phase information. Without a concrete protocol analogous to Steps 1–3 of the non-Hermitian case, the assertion that the double-frequency Green's function is 'a physical observable' remains a postulate rather than a demonstrated property.
  4. [Eqs. (7) and (8)] The construction of the CFF requires the inverse of the measured response matrix χ_{ω0}(t). The paper does not discuss when this inverse exists. At finite times, χ_{ω0}(t) may be singular for certain parameter values and times, and in the long-time limit invertibility requires that ω0 is not an eigenvalue of H_nH. If ω0 lies in the spectrum, G(ω0) is not invertible and Eq. (8) is not well defined without a regularization or generalized inverse. This condition should be stated and its practical consequences, such as which driving frequencies are admissible, should be addressed.
minor comments (5)
  1. [Eq. (5)] The notation [χ_{ω0}(t)]_{j1} is slightly ambiguous because the second index labels the drive site while the first labels the measured site; a more explicit notation such as [χ_{ω0}(t)]_{j1} defined with a sentence identifying the column drive would improve readability.
  2. [Fig. 1] The panels (c) and (d) lack axis labels and color-bar labels; adding these would make the convergence demonstration easier to evaluate quantitatively.
  3. [Eq. (20)] Substituting ω_c exactly equal to an eigenvalue E_n into the resolvent is formally singular; the numerical detection of eigenstates necessarily uses a finite-time or finite-η regularization. The text should state the regularization procedure used in the numerical examples.
  4. [Supplementary Eq. (S27)] The notation 'Sign[x ≥ 0] = 1 and Sign[x < 0] = 0' is unconventional; this is the Heaviside step function, and using θ(x) instead would avoid confusion with the usual sign function.
  5. [Introduction and Table I] The terms 'Bloch response' and 'non-Bloch response' are defined only for the Green's function example; since the paper later argues about responses more broadly, a more general definition of what counts as a physical response would make the table more precise.

Circularity Check

2 steps flagged · score 6.0 of 10

Many-body CFF reduces by construction to a resolvent of the fixed self-energy matrix; its 'quasiparticle peaks' are eigenvalues of H_eff(ω0), not poles of the interacting complex-frequency Green's function.

  1. self definitional [Main text, Section 'CFF in the quantum many-body system', Eq. (13) and following paragraph.]
    "GCFF S (ωc ∈ C, ω0) = 1/ωc − H eff S (ω0). Note that for any given ω0 ∈ R, H eff S (ω0) is in general non-Hermitian and ω0-dependent. Therefore, Eq. (13) is nothing but the complex frequency Green's function of the non-Hermitian Hamiltonian H eff S (ω0)."

    G_CFF is defined directly as the resolvent of the fixed matrix H_eff(ω0), so its poles in ω_c are, by construction, the eigenvalues of H_eff(ω0). The actual interacting retarded Green's function, Eq. (11), is G^R_S(ω_c) = 1/(ω_c + iη - H_S - Σ_S(ω_c + iη)), with the self-energy evaluated at ω_c. The paper gives no argument that the poles of the former coincide with the poles of the latter; in the example, Σ_↓↓(ω) is strongly frequency-dependent (Eq. S27), so the 'quasiparticle peaks' in Fig. 3 are not derived to be quasiparticle poles. The claim that CFF contains complete quasiparticle information is thus true only in the sense that it restates the defining eigenvalue problem of H_eff(ω0).

  2. renaming known result [Supplementary Material, Section V, Eq. (S21).]
    "GCFF ω0 (ωc) = 1/ωc − ω0 + [GR S (ω0)]−1 = 1/ωc + iη − HS − ΣS(ω0 + iη) , (S21)"

    This equation shows that the CFF is obtained by inverting the single-frequency Green's function G^R_S(ω0) and inserting it into a resolvent with an auxiliary complex parameter ω_c. Scanning ω_c therefore only sweeps the spectrum of a matrix fixed by data at ω0; it is an algebraic rewriting of G^R_S(ω0), not a measurement of the complex-frequency Green's function G^R_S(ω_c). The complex-ω_c structure ('quasiparticle peaks across the complex plane') is a relabeling of this resolvent, which is a known object once the single-frequency Green's function is known. Presenting the relabeling as a new detection of complex-frequency quasiparticles is renaming rather than derivation.

full rationale

The non-Hermitian part of the paper is self-contained: Eq. (7) defines the CFF from the measured response matrix; Eqs. (S14)-(S16) derive the exact time-dependent response from the linear dissipative Schrödinger equation; the long-time limit then yields Eq. (8) as a theorem, not as a fit or as a tautology. No parameter is fitted to the target Green's function, and the cited literature on spectral winding provides independent support. The many-body generalization, however, is where circularity enters. Eq. (13) and Eq. (S21) define G_CFF as the resolvent of the fixed matrix H_eff(ω0) = H_S - iη + Σ_S(ω0 + iη), i.e., as an algebraic reshuffling of the single-frequency Green's function G^R(ω0). The resonance peaks in Fig. 3 are therefore, by construction, the eigenvalues of that fixed matrix. The actual interacting complex-frequency Green's function given in Eq. (11) has the self-energy evaluated at ω_c, not at ω0; its poles solve det(ω_c - H_eff(ω_c)) = 0. The paper never proves these two pole sets coincide, and in the example Σ_↓↓(ω) is strongly frequency-dependent (Eq. S27), so the identification of CFF peaks with quasiparticle peaks is not a derived consequence. Calling those eigenvalues 'quasiparticle peaks across the complex plane' is a relabeling of the defining matrix's spectrum; the claim of complete quasiparticle information is thereby true by definition rather than by independent derivation. This is partial, not total, circularity: the non-Hermitian skin-effect fingerprint, which is the paper's main technical contribution, stands independently.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The central claim rests on linear-response theory, the resolvent identity, the dissipative decay assumption, and the ability to measure a full response matrix. The many-body generalization additionally assumes that a double-frequency Green's function is experimentally accessible, which is not demonstrated. No free parameters are fitted to data; model parameters are illustrative. The CFF itself is a proposed observable, not a postulated physical entity.

assumptions (6)
  • domain assumption The driven dissipative system obeys the linear inhomogeneous non-Hermitian Schrödinger equation i d⟨a⟩/dt = HnH⟨a⟩ + F(t), derived from a Lindblad master equation for bosonic modes.
    Used for Eq (1) and all subsequent response-function derivations. This excludes fermionic systems and nonlinear or non-Markovian dynamics.
  • domain assumption All eigenvalues of HnH have negative imaginary parts so the transient term e^{-i(HnH-ω0)t} vanishes as t→∞.
    Needed for convergence of Eq (8); excludes PT-symmetric or gain-loss regimes with purely real or growing eigenvalues.
  • domain assumption The full N×N response matrix χ(t) is measurable by sequential single-site driving and is invertible at the measurement times used.
    Required by Eq (7); the paper does not prove invertibility away from the t→∞ limit and does not discuss singular configurations.
  • standard math Standard linear-response algebra gives χ(t)=G(ω0)-G(ω0)e^{-i(H-ω0)t}.
    Derived in Supplementary Section III from the general solution of Eq (S13) and the bosonic commutation relations.
  • domain assumption For quantum many-body systems, the exact retarded Green's function has the resolvent form G^R_S(ω)=1/(ω+iη-H_S-Σ_S(ω+iη)), and the double-frequency CFF object is an observable.
    Used for Eqs (11)-(13). The experimental accessibility of the double-frequency object is only sketched in Supplementary Section V, not proven.
  • ad hoc to paper Mean-field approximation for the two-level many-body example: D approximates diag(⟨n↓⟩,⟨n↑⟩) times the retarded Green's function.
    Used in Eq (S25); the illustrative quasiparticle peaks in Fig. 3 depend on this approximation, which is not exact.

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Pith. "Pith review of Complex Frequency Fingerprint: Basic Concept and Theory." pith.science (2026). https://pith.science/paper/H7EOOZZI

@misc{pith2026241112577,
  author       = {Pith},
  title        = {Pith review of: Complex Frequency Fingerprint: Basic Concept and Theory},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/H7EOOZZI}},
  note         = {Machine review of arXiv:2411.12577}
}
abstract

We introduce the complex frequency fingerprint (CFF), an experimentally accessible method for detecting the complex frequency Green's function (GF). Unlike the real frequency GF, where $\omega$ is real, this complex frequency GF is shown to play a necessary role in both non-Hermitian and quantum many-body systems. For non-Hermitian systems, we will prove that our method detects complex energy spectra, eigenstates, and complex frequency GFs throughout the complex plane, providing necessary identification of the non-Hermitian skin effect. For quantum many-body systems, our method reveals quasiparticle peaks across the complex plane and intuitively illustrates interaction effects. This information is difficult to obtain with real frequency detection. Our method paves the way for exploring exotic phenomena in both non-Hermitian and quantum many-body systems, bridging theory and experiment across diverse physical areas.

Figures

Figures reproduced from arXiv: 2411.12577 by the authors.

Figure 1
Figure 1. FIG. 1. (a) Illustration of the region possessing the non-Bloch [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. The OBC (a) and PBC (b) complex frequency DOS of [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. (a)-(d) Comparison of real-frequency spectral function [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4: FIG. 4. (a) Spectra for [PITH_FULL_IMAGE:figures/full_fig_p005_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. (a) The Green’s function for [PITH_FULL_IMAGE:figures/full_fig_p006_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6. (a) [PITH_FULL_IMAGE:figures/full_fig_p007_6.png]

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Reference graph

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