Quasiclassical expressions for the free energy of superconducting systems

Artjom Vargunin; Mikhail Silaev; Pauli Virtanen

REVIEW 3 major objections 4 minor 79 references

Quasiclassical expressions for the free energy of superconducting systems

T0 review · 3 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read The paper derives a closed-form quasiclassical free-energy functional covering spin-singlet and spin-triplet superconducting and superfluid states, avoiding coupling-constant integration.

desk verdict A genuinely useful closed-form free energy for quasiclassical superconductors with general spin structure; the singularity handling is a real caveat but not a fatal one. read the letter →

arxiv 1909.00992 v1 pith:WPMS2PA2 submitted 2019-09-03 cond-mat.supr-con

Pauli Virtanen , Artjom Vargunin , Mikhail Silaev This is my paper · ORCID

classification cond-mat.supr-con

keywords quasiclassicaltheoryEilenbergerequationLuttinger-Wardfunctionalspin-tripletpairingsuperfluidhelium-3diffusivelimitUsadelfreeenergy

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

Superconducting and superfluid states are described by quasiclassical Green's functions, but computing their free energy has required a coupling-constant ($\lambda$) integration over auxiliary propagators. The paper removes that integration by constructing a closed functional of the full quasiclassical propagator whose saddle point gives the Eilenberger equations and whose value at that saddle point is the free energy. The construction holds for arbitrary spin structure, including spin-triplet pairing generated by exchange fields or spin-orbit coupling, generalizing the 1968 Eilenberger expression. In the dirty limit the functional reduces to a simple expression in momentum-averaged propagators that yields the Usadel equation. This closes a gap between the fundamental Luttinger-Ward free energy and practical quasiclassical functionals.

What carries the argument

The load-bearing object is the gradient functional $E[\hat g] = \frac12\mathrm{Tr}\bigl(\hat g[\hat\tau_t,\hat g]\, v_F\cdot\check\nabla[\hat\tau_t,\hat g]^{-1}\bigr)$, defined on the quasiclassical propagator $\hat g$ (a $4\times4$ matrix in spin and Nambu space) and an auxiliary matrix $\hat\tau_t$ with $\hat\tau_t^2=1$. It turns a $\lambda$-integration over auxiliary propagators into a surface term, so the free energy becomes a direct functional of $\hat g$; choosing $\hat\tau$ appropriately also avoids most singularities in numerical work. The same object, written in Riccati amplitudes, provides the explicit variational functional whose saddle point reproduces the Riccati form of the Eilenberger equations.

What would settle it

Compute a nonuniform spin-triplet or spin-singlet configuration where $[\hat\tau,\hat g_\lambda]$ has a zero for some $\lambda\in[0,1]$, evaluate the real part of (19), and compare it with the exact $\lambda$-integrated Luttinger-Ward expression (3) at the same saddle point: if the real parts differ by a nonzero winding contribution, the real-part prescription fails.

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Extended reading notes

Core claim

Starting from the Luttinger-Ward formulation, the paper evaluates the $\lambda$-integral analytically and obtains the free-energy functional $\Omega[\hat g,\hat\Sigma] = \frac12 E[\hat g_1[\hat\Sigma]] + \Phi[\hat g] + \frac12\mathrm{Tr}[\hat\Lambda(\hat g_n - \hat g_1[\hat\Sigma])]$ before minimization, with the saddle-point value $\Omega = \frac12 E[\hat g_*] + \Phi[\hat g_*] + \frac12\mathrm{Tr}[\hat\Lambda(\hat g_n - \hat g_*)]$. The central ingredient is the gradient functional $E[\hat g] = \frac12\mathrm{Tr}\bigl(\hat g[\hat\tau_t,\hat g]\, v_F\cdot\check\nabla[\hat\tau_t,\hat g]^{-1}\bigr)$, where $\hat\tau_t$ is any matrix field with $\hat\tau_t^2=1$. This functional is constructed so that its variation reproduces the gradient term of the Eilenberger equation, making the saddle point of the full functional equivalent to the quasiclassical equations plus the self-consistency relation. In the spin-singlet, spin-diagonal case it reduces to Eilenberger's original expression, while the same formula covers spin-triplet correlations; the paper also derives a Riccati-amplitude form and the diffusive-limit free energy in terms of momentum-averaged propagators. Where $[\hat\tau_t,\hat g]$ is not invertible, the functional produces imaginary winding contributions, and the paper identifies the physical free energy with the real part of the expression.

Load-bearing premise

The derivation assumes that the physical free energy equals the real part of the constructed functional, discarding imaginary winding contributions that arise when $[\hat\tau_t,\hat g_\lambda]$ fails to be invertible inside the $\lambda$-integration domain.

Editorial extensions

If this is right

For spin-singlet, spin-diagonal systems the new functional reproduces Eilenberger's original free energy, settling the earlier expression's origin as the saddle-point reduction of the Luttinger-Ward functional.
For spin-triplet states, including superfluid $^3$He and systems with spatially inhomogeneous exchange fields or spin-orbit coupling, the same closed form supplies a workable thermodynamic potential at the same level of rigor as singlet calculations.
In the diffusive limit, the derived expression (21) has the Usadel equation and the gap self-consistency relation as its saddle point, giving a microscopically derived free energy for dirty superconductors.
Previous variational functionals from nonlinear sigma-model approaches and the Serene-Rainer construction coincide with the Eilenberger-type expression derived here, unifying these formulations.
Because no $\lambda$-integration remains, numerical minimization no longer requires solving the auxiliary propagator for many coupling constants, which simplifies phase-competition studies such as $0$-$\pi$ junctions, FFLO states, and vortex structures.

Reading between the lines

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

The freedom in choosing $\hat\tau_t$ suggests a gauge-like tool: one can pick $\hat\tau_t$ locally so that $[\hat\tau_t,\hat g]$ stays invertible, and a natural choice is to rotate $\hat\tau_1$ by the local transformation diagonalizing $\hat g$, which the paper only sketches in the normal-state limit.
The Berry/Wess-Zumino reading of the gradient term hints that free-energy differences between quasiclassical configurations might be computable as geometric phases, which could simplify topological-state comparisons beyond the saddle-point regime.
A direct stress test would evaluate the real part of expression (19) for a nonuniform spin-triplet texture and compare it with the exact $\lambda$-integrated expression (3) at the same saddle point, checking the real-part prescription exactly where $[\hat\tau,\hat g]$ nearly vanishes.
The dirty-limit functional could be applied to Majorana nanowire models to see whether vortex-formation energetics change when using this microscopically derived free energy instead of the commonly used approximate forms.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

This paper delivers what it promises: a λ-integration-free free energy functional for quasiclassical superconductors and superfluids with arbitrary spin structure, derived systematically from the Luttinger–Ward functional. The central result, Eqs. (11)–(12), is new and fills a gap that has been open since Eilenberger's 1968 paper. The spin-singlet limit reduces to the known expression, and the triplet generalization is explicit and usable. The diffusive-limit expression, Eq. (21), will be handy for the Usadel community. The derivation is coherent; the supplement actually shows the main variational calculation, and the Riccati parametrization provides a concrete check. This is not a hand-waving paper.

The stress-test note flags the real-part subtraction at singularities of [τ_t, g_λ]. I read the supplement carefully, and the authors do address this: they compute the singularity contributions and find they are winding numbers of the form i π m, hence purely imaginary. That is more than most papers would do. The remaining gap is narrower than the stress-test suggests: they assume U stays nonsingular at the poles of [τ,g]^{-1}, and they do not prove that taking Re E always selects the correct branch for arbitrary textures. That is a legitimate technical caveat, but it is not a load-bearing flaw. For most practical applications, including the advertised triplet and textured cases, one can choose τ to avoid the worst singularities, and the real-part prescription is physically sensible. I would want the authors to state the conditions more carefully in a final version, but I would not block publication on it.

The diffusive-limit derivation uses a leading-order spherical-harmonic expansion with the standard normalization argument; that is fine. The citation pattern is appropriate, including the authors' own prior work, which is directly relevant. No hidden parameters, no fitting, nothing circular.

Who is this for? Anyone working on superfluid 3He phases, FFLO states, vortex physics, or superconductor/ferromagnet hybrids. This will become a standard citation for free-energy calculations in those areas. I would bring it to our reading group and would cite it. It deserves serious peer review and, in my view, acceptance after minor revisions that clarify the singularity conditions and the branch choice.

Referee Report

3 major / 4 minor

Summary. The paper derives a free-energy functional for quasiclassical superconductors and superfluids starting from the Luttinger–Ward formulation. The main result is Eq. (11), with the gradient functional E[g] given by Eq. (12), which avoids the λ-integration of the earlier Serene–Rainer expression (3). The functional is shown to reduce to Eilenberger's free energy in the spin-diagonal case and is generalized to spin-triplet correlations and spin-dependent fields by allowing an arbitrary matrix τt in Eq. (12). The paper also derives a diffusive-limit (Usadel-type) free energy in terms of momentum-averaged propagators, Eqs. (20)–(21), and provides a supplement containing the variation calculation, a Riccati parametrization, and a discussion of singularities of [τ, g].

Significance. If the central claim is correct, the paper fills a genuine gap by providing a closed-form, λ-integration-free free energy for nonuniform superconducting and superfluid states with arbitrary spin structure, including spin-triplet pairing. The construction of E[g] is explicit, the supplement contains a self-contained derivation of its variation, and the reduction to the known Eilenberger and Usadel results in appropriate limits is demonstrated. These are concrete, checkable contributions that would be useful for analyzing competing phases, vortices, and topological systems. The main weakness is that the equality between the real part of the proposed functional and the Luttinger–Ward functional at the saddle point relies on an unproven treatment of singularities of [τ, g], which is especially relevant for the advertised spin-triplet applications.

major comments (3)

[Supplement, Eq. (S10); main text below Eq. (12)] The central claim that Re Ω_{11} equals the Luttinger–Ward functional depends on the assertion that singularities of [τ, g_λ] produce only pure-imaginary winding contributions that are removed by taking the real part. The supporting calculation, Eq. (S10), assumes that U in the representation g = U τ3 U^{-1} is nonsingular at the zeros of [τ, g]. The manuscript does not prove this assumption, nor does it state a condition on g or τ that would guarantee it. For the advertised spin-triplet and textured applications, where the order parameter has zeros, it is not evident that a single global τ exists for which U is nonsingular at all singular points of [τ, g_λ] along the λ-sweep. Without such a proof, the equality Re Ω_{11} = Eq. (3) is not established; at best the paper shows that Eq. (12) has the correct variation away from singularities. Please provide a proof or a precise, physically motivated condition under which the real-part subtraction is exact, or weaken the claim accordingly.
[Eq. (12) and the paragraph following Eq. (17)] The paper states that one should choose τ to avoid singularities, for instance τ1 near the normal state or τ = U0^{-1}τ1 U0 for a reference g0 ≈ g. For spin-triplet pairing, the anomalous component f can vanish on the Fermi surface, and for inhomogeneous textures with nontrivial topology a global smooth U0(x) may not exist. The manuscript does not show that these recommended choices make [τ, g_λ] invertible for all λ ∈ [0,1] and all x, nor does it quantify the error if they do not. Since the spin-triplet generalization is the paper's main advertised advance, this gap is load-bearing rather than cosmetic.
[Discussion of non-uniqueness of E[g] after Eq. (12)] The manuscript correctly notes that E[g] is defined by Eq. (8) only up to topological terms and that the presence of singularities depends on τ. It then asserts that the free energy is real-valued and takes the real part. However, the real part of E[g] itself could depend on the choice of τ if the discarded winding contributions are accompanied by real branch-cut terms, or if different τ choices shift singularities across the λ-path. The paper should either prove that Re E[g] is independent of τ up to total derivatives, or state the conditions under which it is; otherwise the proposed free energy is not manifestly unique.

minor comments (4)

[Reference [20]] The reference to 'Supplementary information at ???' is a placeholder and must be filled with the actual supplementary material identifier.
[Text after Eq. (16)] There is a typo: 'The the general form' should read 'The general form'.
[Title] The title contains a spacing artifact: 'supercon ducting' should be 'superconducting'.
[Eq. (17)] For spin-triplet states, the anomalous component f may fail to be invertible; the text should note explicitly that Eq. (17) is only valid when the chosen τ (here τ3) avoids such zeros, and that Eq. (12) with a different τ should be used otherwise.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: E[g] is constructed and verified, not extracted from the target free energy; self-citations are not load-bearing.

full rationale

The derivation chain from the Luttinger-Ward functional, Eq. (3), to the Eilenberger-type functional, Eqs. (11)-(12), is not circular. The λ-integral is evaluated by introducing a functional E[g] with the variation property (8); this is an existence assumption, not an input that already contains Eq. (12). The paper then constructs E explicitly in Eq. (12), verifies the variation in the appendix (Eqs. S1-S6), and shows it reduces to Eilenberger's singlet expression, Eq. (17). The Riccati-parametrization ansatz, Eq. (16), is presented explicitly and then independently checked by the variation calculation, so the derivation is self-contained rather than relying on an unverified imported theorem. Although refs. [19] and [24] have overlapping authors and suggest the route, the relevant algebra is reproduced in the paper and supplement. The singularities of [τ_t,g_λ] produce winding contributions that are argued to be imaginary and removed by taking the real part; this is an unproven topological assumption, but it is not circular because it does not presuppose the target free energy. No fitted parameters are renamed as predictions, and no known result is merely relabeled. Therefore no circular step is exhibited.

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

The derivation rests on standard Luttinger-Ward and quasiclassical machinery. There are no fitted free parameters; the matrix tau_t is a gauge choice that does not affect the physical free energy. The key technical axiom is the existence and explicit form of the gradient functional E[g], whose validity requires invertibility of [tau_t,g] except for imaginary winding contributions that are subtracted.

assumptions (6)

domain assumption Luttinger-Ward functional (Eq. 2) is the exact free energy for the fermionic system.
Starting point of the derivation; quoted from Refs [16] and [5] without re-derivation.
domain assumption Quasiclassical approximation: off-shell contributions in the xi_p integration are neglected and the propagator is normalized as g^2 = 1.
Standard quasiclassical approximation, introduced in Refs [1,2]; used throughout.
domain assumption There exists a functional E[g] whose variation yields the gradient term (Eq. 8), and Eq. (12) is such a functional.
Key technical assumption. The paper constructs E and verifies the variation property in the supplement, but the construction depends on the choice of tau_t and the invertibility of [tau_t,g].
domain assumption Boundary terms in the lambda-integration vanish because vF(-p)=vF(p), or g_lambda(s=infinity)=g_lambda(s=-infinity), or the system is in the normal state at infinity.
Stated in the supplement (Eq. S8 discussion); needed to equate the line integral to bulk gradient terms.
domain assumption In the diffusive limit, the Born approximation for impurity self-energy (Eq. 22) and the spherical harmonic expansion (Eq. 23) are valid.
Used to derive the Usadel-limit free energy (21); standard dirty limit assumptions.
standard math Riccati parametrization (13) is a valid parametrization of the quasiclassical Green's function and the Riccati equations (14)-(15) hold.
Used to construct an intermediate form of E; from Refs [21-23].

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Pith. "Pith review of Quasiclassical expressions for the free energy of superconducting systems." pith.science (2026). https://pith.science/paper/WPMS2PA2

@misc{pith2026190900992,
  author       = {Pith},
  title        = {Pith review of: Quasiclassical expressions for the free energy of superconducting systems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WPMS2PA2}},
  note         = {Machine review of arXiv:1909.00992}
}

abstract

In the seminal work by G. Eilenberger [Z. Phys. 214, 195 (1968)], the quasiclassical expression for the free energy of spin-singlet superconductor has been suggested. Starting from the Luttinger-Ward formulation we derive the Eilenberger free energy and find its generalization for superconductor or superfluid with spin-triplet correlations. Besides ordinary superconductors with various scattering mechanisms, the obtained free energy functional can be used for systems with spin-triplet pairing such as superfluid $^3$He and superconducting systems with spatially-inhomogeneous exchange field or spin-orbit coupling. Using this general result we derive the simplified expression for the free energy in the diffusive limit in terms of the momentum-averaged propagators.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

79 extracted references · 68 canonical work pages

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is implemented in the vicinity of the Fermi sphere and the oﬀ-shell contribution is neglected resulting in the following expression for the free energy [5] Ω = 1 2 Tr[ˆΣˆg − 1 π ˆ dξp ln(−i ˆΣ − ˆG−1 0 )] + Φ[ˆg] (2) where ˆΣ is the self-energy and the last term is the func- tional generating the self-energy ˆΣ = −2δΦ/δ ˆgT . The generalized trace operato...

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However, the remaining λ-integration necessitates solving Eq

can be used for any weakly-coupled su- perconducting or superﬂuid state with arbitrary pairing interactions and ﬁelds A(r), h(r) and ˆA(r). However, the remaining λ-integration necessitates solving Eq. ( 4) for the auxiliary propagator ˆgλ for many λ. This makes the functional (3) less convenient for numerical work, and hinders analytical calculations in ...

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can be evaluated using an approach suggested in Ref. 19. Let us note the general relation Tr[ˆΣ(ˆg − ˆgλ )] = ∂λ Tr[ ˆMλ (ˆg − ˆgλ )] + Tr[ ˆMλ∂λ ˆgλ ]. (5) Here the ﬁrst term on the r.h.s. is a full λ-derivative and easily integrated, but further treatment is needed for the second term. To calculate its contribution, we note that the variation of GF pres...

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Such points, if they occur in- side the region swept by the λ-integration, produce imag- inary winding number contributions

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J. M¨ akinen, V. Dmitriev, J. Nissinen, J. Rysti, G. Volovik, A. Yudin, K. Zhang, and V. Eltsov, Nat. Commun. 10, 237 (2019). Derivation of Eq. (12) We now derive Eq. (12). We assume ˆ τ2 = 1, ˆg2 = 1, ∂λ ˆg = [ ˆW, ˆg], and ∂λ ˆτ = 0. Moreover, we denote ∂ ≡ vF vF · ˇ∇ as the...

2019

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

The generalized trace operator in Eq

is implemented in the vicinity of the Fermi sphere and the oﬀ-shell contribution is neglected resulting in the following expression for the free energy [5] Ω = 1 2 Tr[ˆΣˆg − 1 π ˆ dξp ln(−i ˆΣ − ˆG−1 0 )] + Φ[ˆg] (2) where ˆΣ is the self-energy and the last term is the func- tional generating the self-energy ˆΣ = −2δΦ/δ ˆgT . The generalized trace operato...

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[2] [2]

One way to do this is based on the observation (c.f

is rather nontrivial. One way to do this is based on the observation (c.f. [11]) ´ dξp∂λ Tr ln(−iλ ˆΣ − ˆG−1 0 ) = π Tr ˆΣˆgλ resulting in the general expression for the free energy density of a non- uniform superconductor or Fermi superﬂuid [4–6, 11]: Ω[ˆg, ˆΣ] = 1 2 ˆ1 0 dλ Tr[ˆΣ(ˆg − ˆgλ )] + Φ[ˆg], (3) 0 = vF · ˇ∇ˆgλ + [ ˆMλ, ˆgλ ], ˆg2 λ = 1, (4) whe...

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[3] [3]

However, the remaining λ-integration necessitates solving Eq

can be used for any weakly-coupled su- perconducting or superﬂuid state with arbitrary pairing interactions and ﬁelds A(r), h(r) and ˆA(r). However, the remaining λ-integration necessitates solving Eq. ( 4) for the auxiliary propagator ˆgλ for many λ. This makes the functional (3) less convenient for numerical work, and hinders analytical calculations in ...

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[4] [4]

can be evaluated using an approach suggested in Ref. 19. Let us note the general relation Tr[ˆΣ(ˆg − ˆgλ )] = ∂λ Tr[ ˆMλ (ˆg − ˆgλ )] + Tr[ ˆMλ∂λ ˆgλ ]. (5) Here the ﬁrst term on the r.h.s. is a full λ-derivative and easily integrated, but further treatment is needed for the second term. To calculate its contribution, we note that the variation of GF pres...

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J. W. Serene and D. Rainer, Phys. Rep. 101, 221 (1983)

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A. B. Vorontsov and J. A. Sauls, Phys. Rev. B 68, 064508 (2003)

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[7] [7]

we get ˆ d3r Tr[ ˆMλ∂λ ˆgλ ] = d dλ ˆ d3rE [ˆgλ ]. (9) Finally, we can perform the λ integration to obtain the general expression for the free energy functional: Ω[ˆg, ˆΣ] = 1 2E[ˆg1[ ˆΣ]] + Φ[ˆg] + 1 2 Tr [ˆΛ(ˆgn − ˆg) ] (10) + 1 2 Tr[(ˆΛ + ˆΣ)(ˆg − ˆg1[ ˆΣ])] . where ˆgn ≡ ˆgλ =0 = sgn( ω )ˆσ0 ˆτ3 and we have chosen E[ˆgn] = 0. Using Eqs. ( 4),(8), the ...

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[8] [8]

Such points, if they occur in- side the region swept by the λ-integration, produce imag- inary winding number contributions

is not deﬁned at points where [ˆτt, ˆgλ ] is not invertible. Such points, if they occur in- side the region swept by the λ-integration, produce imag- inary winding number contributions. [20] For exam- ple, in the singlet case, E[ˆgn] ˆ =ivF · ∇ψ (excluding the Matsubara sum and angle average), where ψ is the po- lar angle of rotation of the unit vector t ...

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