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Imaginary Time Formalism for Causal Nonlinear Response Functions

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Pith's one-line read This paper proves that, for any Hamiltonian, the $n$-th order causal nonlinear response function is the analytic continuation of the time-ordered Matsubara function, with sign $(-1)^n$, at every order.

desk verdict A clean all-order proof of the Matsubara-causal connection for finite systems, undercut by an unproven continuum-limit extension and a missing advertised example. read the letter →

arxiv 2506.21428 v2 pith:KF26DDHR submitted 2025-06-26 cond-mat.mes-hall cond-mat.stat-mechquant-ph

Sounak Sinha , Barry Bradlyn This is my paper
classification cond-mat.mes-hallcond-mat.stat-mechquant-ph
keywords nonlinearresponseMatsubarafunctionsanalyticcontinuationLehmannrepresentationequationofmotionspectraldensitysumrulesimaginarytime
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

This paper proves that causal nonlinear response functions of any order $n$ can be obtained by analytically continuing the corresponding imaginary-time (Matsubara) correlation function, generalizing the textbook linear-response result. The proof rests on recursion relations for the Lehmann representations of both objects: the $n$-th order causal response and the $n$-point Matsubara function obey the same recursion, with imaginary frequencies $i\nu_i$ replacing $\omega_i^+$ and an overall factor $(-1)^n$. Because the Matsubara function is time-ordered, it can be computed by standard diagrammatic or numerical methods for interacting and disordered Hamiltonians, which is what makes the identity practical for nonlinear spectroscopy and nonlinear optical response. The paper also derives an explicit Lehmann representation and a spectral-density representation of $n$-th order response functions, and uses the large-frequency limit to obtain generalized sum rules including the $n$-th harmonic generation asymptotics.

What carries the argument

The load-bearing mechanism is a pair of recursion relations for the Lehmann coefficients of the response functions. For the causal response, Eq. (4) (or its coefficient form Eq. (6)) relates the $n$-th order function to the $(n-1)$-th order function through commutators $[A,H_0]$ and $[A,B]$; the Matsubara function obeys the same recursion in imaginary frequencies (Eqs. (10) and (12)). Because both recursions produce the same rational functions of the energy denominators $\omega_t^+-\epsilon_{i_1 i_{n+1}}$ and $i\nu_t-\epsilon_{i_1 i_{n+1}}$, induction from the Kubo formula forces the two families to be the same object continued to different frequency domains. A closed-form solution of the recursion is the sum $f^{(n)}_{i_1,\ldots,i_{n+1}}(\{\omega_i\}) = \frac{1}{n!}\sum_{\sigma}\sum_{a=1}^{n+1} \Delta_a \prod_{i\ne a}\frac{1}{q_a-q_i}$ with $q_a=\epsilon_{i_a}+\sum_{k=0}^a \omega_k^+$, which yields the Lehmann representation and, in the equal-frequency limit, the generalized $f$-sum rule.

What would settle it

Take a finite interacting chain where exact diagonalization yields both the direct causal $n=3$ response and the Matsubara four-point function; if analytic continuation of the latter using the recursion disagrees with the direct causal calculation at frequencies where the continuation is unambiguous, the induction's base case or its continuation assumption fails.

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Extended reading notes

Core claim

The paper's central claim is that causal nonlinear response functions are the analytic continuation of imaginary-time ordered correlation functions at every order: $\chi^{(n)}_{AB}(\{\omega_i^+\}) = (-1)^n \bar\chi^{(n)}_{AB}(\{i\nu_i = \omega_i^+\})$ (Eq. 14). Here $\chi^{(n)}_{AB}$ is the $n$-th order causal response of an observable $A$ to $n$ powers of a perturbation $B$, and $\bar\chi^{(n)}_{AB}$ is the time-ordered Matsubara $n$-point function. The equality is established by induction: the equations of motion for the density matrix give a recursion that the Lehmann coefficients $f^{(n)}$ of the causal response satisfy, and identical reasoning from imaginary-time translation invariance gives the same recursion for the coefficients $g^{(n)}$ of the Matsubara function. Since the $n=1$ case is the Kubo formula and the two recursions map onto each other under $\omega_i^+\to i\nu_i$ with a factor $(-1)^n$, the identity holds for all $n$. The result is stated for arbitrary interacting or disordered Hamiltonians and for multiple simultaneous perturbations, and the same construction yields an explicit closed-form Lehmann representation, a spectral-density representation valid for continuous spectra, and a family of generalized sum rules.

Load-bearing premise

The argument assumes every response function can be written as a sum of simple energy poles (a Lehmann representation) and that continuing from imaginary frequencies to upper-half-plane real frequencies has a unique answer; if the continuation is ambiguous, the identity may fail even though the recursion still defines some function.

Editorial extensions

If this is right

  • For any interacting or disordered system, nonlinear optical and spectroscopy responses can be computed from time-ordered Matsubara $n$-point functions, where diagrammatic perturbation theory is standard.
  • The explicit solution of the recursion gives a closed-form Lehmann representation for the $n$-th order response, so the induction also supplies a practical formula rather than an existence statement.
  • The spectral-density representation extends the equivalence to systems with continuous spectra, so the identity is not limited to finite or discrete systems.
  • The high-frequency limit yields generalized sum rules: $\lim_{\omega\to\infty}\omega^n\chi^{(n)}_{AB}(\omega,\ldots,\omega)=\frac{1}{n!}\langle[\cdots[[A,B],B]\cdots,B]\rangle_0$, and a family of higher sum-rule coefficients from spectral-density moments.
  • The generalization to multiple perturbations covers multi-field probes such as tensor components, so the result applies beyond scalar single-probe setups.

Reading between the lines

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

  • A direct corollary the paper hints at but does not develop: because Matsubara $n$-point functions are standard outputs of quantum Monte Carlo and diagrammatic codes, the identity turns nonlinear response into an analytic-continuation problem, and the practical bottleneck becomes the numerical continuation rather than the nested commutators.
  • The generalized sum rules express high-frequency moments of the nonlinear response in terms of nested commutators; one testable extension is to read off interaction and disorder corrections to quantum-geometric quantities, such as shift-current or Berry-curvature dipoles, from those moments when band-geometry language is unavailable.
  • The proof is set up for bosonic observables with even Matsubara frequencies; applying the recursion to fermionic or odd-frequency operators would likely require tracking extra signs from time ordering, a natural check of the formalism's limits.
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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

2 major / 3 minor

Summary. The paper claims to prove, to all orders in perturbation theory, that causal nonlinear response functions of a thermal system can be obtained by analytic continuation of imaginary-time-ordered Matsubara functions, χ_AB^(n)({ω_i^+}) = (-1)^n \barχ_AB^(n)({iν_i = ω_i^+}). The proof is based on a recursion relation for the nth-order response derived from the equation of motion of the density matrix, and a matching recursion for the Matsubara function obtained by differentiating the imaginary-time correlation function. The authors establish the induction in the Lehmann representation for finite systems, provide an explicit closed-form solution for the nth-order Lehmann coefficients, derive a spectral-density representation intended to extend the result to continuous spectra, and apply the formalism to obtain asymptotic sum rules and the high-frequency limit of nth-harmonic generation.

Significance. If fully established, the main result would provide a practical and general all-order bridge between imaginary-time diagrammatic methods and causal nonlinear response, with direct applications to interacting and disordered systems. The paper also gives a useful explicit Lehmann formula for arbitrary n and a family of generalized sum rules. The finite-system proof is self-contained, the second-order check matches the known Lehmann expression, and the explicit solution for f^(n) is verified by induction. These are concrete, non-circular contributions. However, the advertised extension to the thermodynamic limit for arbitrary interacting or disordered Hamiltonians is not fully supported by the manuscript as written.

major comments (2)
  1. [Supplementary Material, Sec. V, Eqs. (94)-(96)] The extension to continuous spectra is not established. Equations (94) and (95), namely ∫dω1 ρ^(n) = πρ^(n-1) and ω1 ρ^(n) = ρ^(n)_{[A,H0]}, are asserted to follow from Eq. (89) but no derivation is given. More importantly, Eq. (96) is not a closed recurrence from order n-1 to order n: its right-hand side contains X^(n) evaluated on [A,H0], so it does not determine X^(n) from X^(n-1) without an additional analyticity or uniqueness condition. The statement that (-1)^n X^(n)(iν_i) is the Matsubara n-point function therefore does not follow by induction as written. Consequently, Eq. (14) is rigorously proven only for finite systems with discrete spectrum, where the f/g recursions close because the Lehmann basis diagonalizes H0. Please derive (94)-(95) explicitly and supply a uniqueness argument for the fixed-point equation (96) before claiming the thermodynamic-limit result for arbitrary interacting or disordered systems.
  2. [Main text, Eq. (6)] The recursion relation for f^(n) as printed is inconsistent with the corresponding Matsubara recursion in Eq. (12) and with the correct derivation in SM Eq. (15): the second f^(n-1) term should have frequency arguments {ω_i}_{i=2}^n, not {ω_i}_{i=1}^{n-1}. As written, the term-by-term comparison of Eq. (6) with Eq. (12) that underlies the main-text induction does not hold. The SM version is correct, so this is repairable, but Eq. (6) must be corrected in the main text.
minor comments (3)
  1. [Main text, Eq. (12)] The symbol ω_t in the denominator of Eq. (12) is not defined at that point; since the Matsubara frequencies are iν_i, the intended quantity is i∑_{i=1}^n ω_i, and this should be stated explicitly for clarity.
  2. [Main text, Eq. (21) and surrounding text] The permutation structure in the spectral representation is somewhat opaque: the index σ appears both on the z variables and on the B labels, but the ω integration variables are not permuted. It would improve readability to state explicitly that the sum over σ is over permutations of the pairs (I_i, z_i), as is done in SM Eq. (91).
  3. [General] The notation χ^(n)_AB({ω_i^+}) is used before the definition of ω_i^+ = ω_i + iϵ is given; moving that definition to the first occurrence would avoid confusion.

Circularity Check

0 steps flagged · score 1.0 of 10

No circular reduction in the central induction; the only self-citation (Ref. 12) is a benchmark, and the continuum-limit extension has an asserted rather than derived closure step, which is a proof-completeness gap, not circularity.

full rationale

The central claim, Eq. (14), is obtained by an induction whose base case is the externally established Kubo linear-response formula (Eq. 7, citing Ref. 47) and whose recursive steps (Eqs. 6 and 12) are derived, not postulated: Eq. (6) follows from the density-matrix equation of motion (Eq. 2) and the definition of the causal response (Eq. 3), while Eq. (12) follows from the chain rule and imaginary-time translation invariance applied to the independently defined Matsubara function (Eq. 8). Because [A, H0] acts diagonally in the Lehmann basis, the recursion closes and determines f^(n) explicitly from f^(n-1), so substituting the induction hypothesis f^(n-1) = (-1)^(n-1) g^(n-1)|_{iν=ω^+} into Eq. (12) gives g^(n)| = (-1)^n f^(n) algebraically, with the denominator matching since iν_t = ω^+_t. No parameter is fitted anywhere, and no prediction is an input by construction. The n=2 case is cross-checked against published results (SM Eq. 17 matching Ref. 4), and Eq. (19) is derived from the Fourier transform of the step function; Ref. 12 (one author overlapping) is cited only as a benchmark to be generalized, which is external falsifiable support and does not raise the circularity score. Two caveats are flagged in the paper itself: (i) after Eq. (14), the continuation is only defined 'provided the continuation is carried out in the Lehmann representation,' and (ii) SM Section V admits that the Lehmann argument assumes discrete many-body energies and must be extended to the thermodynamic limit. That extension asserts the spectral identities ∫dω1 ρ^(n) = πρ^(n-1) and ω1ρ^(n) = ρ^(n)_{[A,H0]} (SM Eqs. 94-95) as following from Eq. (89) without showing the derivation, and it closes with 'Thus, by induction' from SM Eq. (96) even though Eq. (96) also contains X^(n)_{[A,H0]} at the same order n, so an unstated uniqueness or analyticity condition is needed to make the continuum induction rigorous. This is an omitted-proof or rigor gap in the advertised thermodynamic-limit scope, not a reduction of the conclusion to its inputs; the finite-system derivation is self-contained. Verdict: no significant circularity.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No free parameters are fitted anywhere, and the derivation introduces no new entities. The central argument rests on standard domain assumptions: thermal initial state, adiabatic switching, existence and uniqueness of the Lehmann or spectral representation, and bosonic Matsubara conventions. The n=1 base case is the externally established Kubo formula.

assumptions (4)
  • domain assumption The initial state is the thermal density matrix ρ0 = e^{-βH0}/Z, and the external perturbation is switched on adiabatically with e^{ϵt}, taking ϵ→0 after all calculations.
    Used in Eqs. 1-3 to define the n-th order response and throughout the recursion derivations in SM Section I.
  • domain assumption The causal response function admits a Lehmann representation with spectral density written as a sum of delta functions; the proof of Eq. 14 is carried out in this representation.
    Stated in the paragraph after Eq. 5; the continuous-spectrum case is handled later through the spectral density representation of Eqs. 19-20.
  • domain assumption The analytic continuation from the discrete Matsubara frequencies iν_i to the real-frequency upper-half-plane values ω_i^+ is unique.
    The text says 'provided the continuation is carried out in the Lehmann representation' after Eq. 14, but no uniqueness theorem such as Carlson's theorem is invoked.
  • domain assumption A and B are bosonic observables, so the Matsubara frequencies are ν_i = 2πm/β and the time-ordered function is normalized with 1/n! in Eq. 8.
    Required for the Fourier conventions and the symmetry factor in Eq. 8, as stated around Eq. 10.

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Cite this review

Pith. "Pith review of Imaginary Time Formalism for Causal Nonlinear Response Functions." pith.science (2026). https://pith.science/paper/KF26DDHR

@misc{pith2026250621428,
  author       = {Pith},
  title        = {Pith review of: Imaginary Time Formalism for Causal Nonlinear Response Functions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KF26DDHR}},
  note         = {Machine review of arXiv:2506.21428}
}
abstract

It is well established that causal linear response functions can be found by computing the much simpler imaginary time-ordered Matsubara functions and performing an analytic continuation. This principle is the basis for much of our understanding of linear response for interacting and disordered systems, via diagrammatic perturbation theory. Similar imaginary-time approaches have recently been introduced for computing nonlinear response functions as well, for example in [Annalen der Physik 536, 2300504 (2024); Physical Review X 11, 041006 (2021)], where the authors analytically continue the Matsubara functions to obtain the Keldysh response functions. In this work, we provide a proof of this connection to all orders in perturbation theory using an equation of motion based approach. We show by induction that causal nonlinear response functions at every order can be obtained from an analytic continuation of an appropriate time-ordered Matsubara function. We demonstrate this connection explicitly for second order response functions in the Lehmann representation. As a byproduct of our approach, we derive an explicit expression for the Lehmann representation of $n$-th order response functions by solving the equations of motion. We also use our result to find an analytic spectral density representation for both causal response functions and Matsubara functions. As an example, we apply our method to derive the non-linear $A^3$ term in the $SU(2)$ spin Hall response of an insulator with spin rotation symmetry. Finally, we show how our results lead to a family of generalized sum rules, focusing explicitly on the asymptotic expression for $n$-th harmonic generation rate. Our work opens the door to using imaginary time approaches to study nonlinear response functions in general condensed matter systems.

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