Mixed order parameters, accidental nodes and broken time reversal symmetry in organic superconductors: a group theoretical analysis

B. J. Powell

REVIEW

Mixed order parameters, accidental nodes and broken time reversal symmetry in organic superconductors: a group theoretical analysis

B. J. Powell

Superconductivity · Submitted 3 Mar 2006 · 31 references

Pith's verdict: A clean group-theoretic argument about accidental nodes in monoclinic κ-NCS is undermined by a sign error that flips the paper's central d+id prediction.

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arXiv IDcond-mat/0603057 (version 1)

Pith IDJOIPCMXO, Pith's permanent record of this paper and its review

CitationB. J. Powell. "Mixed order parameters, accidental nodes and broken time reversal symmetry in organic superconductors: a group theoretical analysis." arXiv:cond-mat/0603057 (2006). https://pith.science/paper/JOIPCMXO

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Claim paper

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Editor's note

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Colleague,

The thing to know up front: the central Ginzburg-Landau analysis contains a sign error. In Eq. (1), the anisotropic quartic term is β2(η1*η2 − η1η2*)^2. Writing η1 = a e^{iθ1}, η2 = b e^{iθ2}, this equals −4β2 a²b² sin²(θ2 − θ1). The chiral (1, i) state has relative phase π/2 and minimizes the energy only when β2 > 0. The paper argues the opposite: it states that β2 < 0 gives (1, i). So its central prediction — d+id and broken TRS in κ-CN3 and β′-dmit — does not follow from its own equations. That is not a material-specific uncertainty; it is internal algebra. Even if the cited Hubbard-model papers do give a negative β2, the mapping in this paper is inverted. The fix is straightforward, but as written the headline result is unsupported.

That said, there is real value here, mostly in the parts that do not depend on β2. The argument about monoclinic κ-NCS is sound: because C2h has a single one-dimensional singlet irrep (Bg), the order parameter can mix in other Bg components, so nodes along the b-axis are not symmetry-protected. That is a clean, useful point, and it applies more broadly than this one material. The proposal that quarter-filled salts like θ-I3 have dxy + s is also a legitimate group-theoretic observation: in D2h, dxy belongs to the trivial A1g irrep, so s-wave admixture is not forbidden. I would credit both of these.

The softer spots, beyond the sign error, are what you expect from a Letter: ε is introduced as a small symmetry-breaking field but never estimated, and the two-transition scenario depends on the monoclinic distortion being a weak perturbation. Those would be reasonable referee questions, but they are minor compared with the algebra.

Who should read this? Anyone working on the pairing symmetry of organic superconductors, but they should read the κ-NCS and quarter-filled sections and treat the TRS-breaking prediction as unproven in this version. It deserves a serious referee, not a desk reject, because the error is fixable and the non-robustness arguments are worth publishing once cleaned up. I would not cite the d+id claim in its current form.

Best.

Referee report

Major objections (2)

[TRS breaking in κ-CN3 and β′-dmit, Eq. (1)] The sign assignment in Eq. (1) is inverted relative to the ground-state classification stated in the text. Writing η1=a e^{iθ1}, η2=b e^{iθ2}, the anisotropic quartic term equals β2(η1*η2 − η1η2*)² = −4β2 a²b² sin²(θ2−θ1). This term is minimized at relative phase π/2 when β2>0 and at relative phase 0 when β2<0. The manuscript states the opposite: it assigns (1,0) or (0,1) to β2>0 and (1,i) to β2<0. The conclusion that the Hubbard-model value β2<0 implies a d+id state therefore does not follow from Eq. (1) as written. The same issue enters Eq. (2). Please correct either the sign in the free energy or the ground-state analysis and re-derive the phase diagram.
[TRS breaking in κ-CN3 and β′-dmit, Eqs. (1)-(2), Fig. 2] The prediction that κ-CN3 and β′-dmit undergo two superconducting transitions is conditional on two assumptions that are not derived in the paper: the sign of β2, taken from independent Hubbard-model studies rather than from this analysis, and the smallness of the monoclinic symmetry-breaking field ε. The abstract states the two-transition scenario as a definitive prediction, but the paper itself notes that for large ε the model returns to C2v symmetry with a single transition. Please qualify the conclusions in the abstract and summary so that the conditional nature of the central claim is explicit.

Minor objections (4)

[Eq. (2)] The notation Tc± = Tc ± ε/α assumes that α is the coefficient introduced in Eq. (1); please state this explicitly so that the units of ε are clear.
[Table II and Fig. 1] The text says the Bg irrep of C2h has nodes only along the c-axis, while Table II lists a line kc = kx − ky = 0; please harmonize the wording with the table so the reader can see that these statements agree.
[Fig. 2] The phase labels 'd-wave' and 'd+id' in Fig. 2 are not defined in the caption; please specify which basis functions of the E2 representation correspond to each phase.
[Reference [25]] Reference [25] aggregates several Hubbard-model studies with different methods and parameter regimes; please indicate which of these studies obtain β2<0 and over what parameter range, since that sign is load-bearing for the κ-CN3 and β′-dmit predictions.

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The paper

The paper claims

Organic superconductors on frustrated lattices are predicted to enter a time-reversal-breaking d+id state through two phase transitions.

The reading

This paper uses group theory to classify possible superconducting order parameters in several families of layered organic superconductors. Its central claim is that highly frustrated half-filled materials, notably κ-(ET)2Cu2(CN)3 and β'-[Pd(dmit)2]2X, should undergo two superconducting transitions: first into a d-wave state, then into a d+id state that is fully gapped and breaks time reversal symmetry. The paper also argues that the monoclinic distortion in κ-(ET)2Cu(NCS)2 shifts the nodes of the d-wave gap away from high-symmetry directions, and that quarter-filled compounds like θ-(ET)2I3 should have dxy+s pairing. If correct, these predictions would settle a long-standing debate about whether phonons or electronic correlations drive superconductivity in these salts.

What carries the argument

The central object is the Ginzburg-Landau free energy for the two two-dimensional irreducible representations of the hexagonal point group C6v, written as Fs-Fn = α(T-Tc)(|η1|2+|η2|2) + β1(|η1|2+|η2|2)2 + β2(η1*η2-η1η2*)2. The sign of β2 determines whether the order parameter chooses a single real basis function (pure d-wave) or the complex combination η=(1,i), which corresponds to dx2-y2+idxy and breaks time reversal symmetry. A symmetry-breaking field ε, representing the monoclinic distortion, is added as ε(|η1|2-|η2|2), which splits the single transition into two. This free-energy machinery carries the argument from lattice symmetry to observable transitions.

Extended reading notes

Core claim

On the nearly hexagonal lattice of the highly frustrated half-filled organic superconductors, the superconducting order parameter belongs to a two-dimensional irreducible representation of the point group C6v. The Ginzburg-Landau free energy then contains a quartic term with coefficient β2 whose sign selects the ground state: β2>0 gives a pure d-wave state (dx2-y2 or dxy), while β2<0 gives the combination dx2-y2+idxy, which is fully gapped and breaks time reversal symmetry. Because the real crystals are monoclinic rather than hexagonal, a symmetry-breaking field ε splits the transition into two: normal → d-wave at Tc+ and d-wave → d+id at Tc-. The paper argues, on the basis of Hubbard model calculations on the hexagonal lattice, that β2<0 for these materials, and therefore predicts the double transition and broken time reversal symmetry in κ-(ET)2Cu2(CN)3 and β'-[Pd(dmit)2]2X. It further shows that the same reasoning implies the monoclinic κ-(ET)2Cu(NCS)2 has a d-wave gap with nodes only along the c-axis plus an accidental node that can shift or disappear, and that quarter-filled systems such as θ-(ET)2I3 should have dxy+s pairing.

Load-bearing premise

The sign of the quartic coefficient β2 in the Ginzburg-Landau free energy is negative for the highly frustrated half-filled organic superconductors, so that the d+id state is the stable ground state; this sign is taken from Hubbard model calculations on the hexagonal lattice rather than derived for these specific compounds.

Editorial extensions

If this is right

κ-(ET)2Cu2(CN)3 and β'-[Pd(dmit)2]2X should show two distinct superconducting transitions, resolvable in specific heat or ultrasound experiments.
The low-temperature state in these materials should be fully gapped, so thermodynamic quantities should not show power-law behaviour at the lowest temperatures.
The d+id state should produce a spontaneous magnetic field detectable by muon spin rotation (µSR), signalling broken time reversal symmetry.
κ-(ET)2Cu(NCS)2 should have a nodal gap whose nodes are not pinned to the b-axis, so experiments probing node positions will see a temperature- or kz-dependent nodal structure.
θ-(ET)2I3 and related quarter-filled salts should have a dxy+s order parameter, so their low-energy quasiparticles are governed by nodes that are shifted away from the lines kxky=0.

Reading between the lines

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

If the sign of β2 is set by the degree of frustration, then tuning frustration via pressure or chemical substitution should move the two transition temperatures apart or together in a predictable way.
The same group-theoretic logic could be applied to other highly frustrated layered superconductors beyond organics, where a hexagonal lattice with strong correlations might also host d+id states.
The accidental nodes in monoclinic κ-NCS might be directly imaged by angle-resolved thermal conductivity or quasiparticle interference, providing a map of the order parameter that distinguishes the Bg basis functions.

Original abstract

We present a group theoretic analysis of several classes of organic superconductors. We argue that highly frustrated half-filled layered organic superconductors, such as kappa-(ET)_2Cu2(CN)_3 (where ET is BEDT-TTF) and beta'-[Pd(dmit)_2]_2X, undergo two superconducting phase transitions, the first from the normal state to a d-wave superconductor and the second to a d+id state. We show that the monoclinic distortion of kappa-(ET)_2Cu(NCS)_2 means that the symmetry of its superconducting order parameter is different from that of orthorhombic kappa-(ET)_2Cu[N(CN)_2]Br. We propose that quarter filled layered organic superconductors, e.g., theta-(ET)_2I_3, have d_{xy}+s order parameters.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 3,830 input tokens, $0.0002 reported.

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.06 · human participants 0.02 · controlled comparison 0.06 · randomized assignment 0.01 · states quantitative result 0.08 · compares with prior results 0.14 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

We present a group theoretic analysis of several classes of organic superconductors. We argue that highly frustrated half-filled layered organic superconductors, such as kappa-(ET)_2Cu2(CN)_3 (where ET is BEDT-TTF) and beta'-[Pd(dmit)_2]_2X, undergo two superconducting phase transitions, the first from the normal state to a d-wave superconductor and the second to a d+id state. We show that the monoclinic distortion of kappa-(ET)_2Cu(NCS)_2 means that the symmetry of its superconducting order parameter is different from that of orthorhombic kappa-(ET)_2Cu[N(CN)_2]Br. We propose that quarter filled layered organic superconductors, e.g., theta-(ET)_2I_3, have d_{xy}+s order parameters.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We present a group theoretic analysis of several classes of organic superconductors.	contribution 0.98	0.92	0.04	asserted 0.91
s1	We argue that highly frustrated half-filled layered organic superconductors, such as kappa-(ET)_2Cu2(CN)_3 (where ET is BEDT-TTF) and beta'-[Pd(dmit)_2]_2X, undergo two superconducting phase transitions, the first from the normal state to a d-wave superconductor and the second to a d+id state.	result 0.85	0.71	0.41	asserted 0.51
s2	We show that the monoclinic distortion of kappa-(ET)_2Cu(NCS)_2 means that the symmetry of its superconducting order parameter is different from that of orthorhombic kappa-(ET)_2Cu[N(CN)_2]Br.	result 0.98	0.79	0.11	asserted 1.00
s3	We propose that quarter filled layered organic superconductors, e.g., theta-(ET)_2I_3, have d_{xy}+s order parameters.	result 0.56	0.74	0.13	speculative 0.66

What kind of paper the classifier reports this is: theoretical result 0.98 · evidence from proof or derivation 0.58.

What was examined, and what was not

sources read

arXiv listing abstract v1

not examined

full text
figures and tables
references and citing sentences
relations to other papers

levels run

P abstract

not run

S sections
T body sentences
C citations
R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures

**Figure 1.** FIG. 1: (Color online.) Sketch of the differences between the

**Figure 2.** FIG. 2: Sketch of the proposed phase diagram for supercon

Figures reproduced from arXiv:cond-mat/0603057 by the authors.

References

Works cited (31)

[1]

most notably µSR [27]. Experimental conﬁrmation of a double superconducting transition and broken TRS would be extremely important, not only because of the intrinsic interest in these phenomena, but also because they would be conclusive proof of unconventional su- perconductivity in the layered organic superconductors. The implication is then that phonons...

View on Pith
[2]

See, e.g., M. Sigrist and K. Ueda, Rev. Mod. Phys. 63, 239 (1991) and references therein

View on Pith 1991
[3]

See, e.g., J.F. Annett, Adv. Phys. 39, 83 (1990)

View on Pith 1990
[4]

See, e.g., T. Ishiguro, K. Yamaji, and G. Saito, Organic Superconductors (Springer Verlag, Heidelberg, 1998)

View on Pith 1998
[5]

For a review see R.H. McKenzie, Comments Condens. Matter Phys. 18, 309 (1998)

View on Pith 1998
[6]

K. Kanoda, Physica C 282-287, 299 (1997)

View on Pith 1997
[7]

B.J. Powell and R.H. McKenzie, Phys. Rev. Lett. 94, 047004 (2005)

View on Pith 2005
[8]

F. L. Pratt et al ., Polyhedron 22 2307 (2003); Phys. Rev. Lett. 94, 097006 (2005); B.J. Powell and R.H. McKenzie, J. Phys.: Condens. Matter. 16, L367 (2004)

View on Pith 2003
[9]

F. Kagawa, K. Miyagawa, and K. Kanoda, Nature 436, 534 (2005)

View on Pith 2005
[10]

Y. Shimizu et al ., Phys. Rev. Lett. 91, 107001 (2003)

View on Pith 2003
[11]

M. Tamura and R. Kato, J. Phys.: Condens. Matter 14, L729 (2002)

View on Pith 2002
[12]

For a review see Y. Shimizu et al ., Prog. Theo. Phys. Supp. 159, 52 (2005)

View on Pith 2005
[13]

For a review see K. Miyagawa, K. Kanoda, and A Kawamoto, Chem. Rev. 104, 5635 (2004)

View on Pith 2004
[14]

J. Merino and R.H. McKenzie, Phys. Rev. Lett. 87, 237002 (2001); R.H. McKenzie et al ., Phys. Rev. B 64, 085109 (2001); A. Greco et al ., ibid. 71, 144502 (2005)

View on Pith 2001
[15]

R.H. McKenzie, Science 278, 820 (1997)

View on Pith 1997
[16]

For a review see K. Kuroki, cond-mat/0601060

View on Pith Pith review arXiv
[17]

For a review see B.J. Powell and R.H. McKenzie, Phys. Rev. B 69, 024519 (2004)

View on Pith 2004
[18]

X=Me4Z, Et 2Me2Z; Z=P, Sb; Me=CH 3 and Et=C 2H5

View on Pith
[19]

See, e.g., A. Damascelli, Z. Hussain, and Z.-X. Shen, Re v. Mod. Phys. 75, 473 (2003)

View on Pith 2003
[20]

J. Liu, J. Schmalian, and N. Trivedi, Phys. Rev. Lett. 94, 127003 (2005); J. Y. Gan et al ., ibid. 94, 067005 (2005)

View on Pith 2005
[21]

A. Carrington et al ., Phys. Rev. Lett. 83, 4172 (1999) measured the penetration depth of κ-Br and κ-NCS down to 0.4 K and observed power law dependences, although other measurements (which did not go to such low tem- peratures) have found exponentially activated behaviours occur in the same materials, e.g., H. Elsinger, et al ., ibid. 84, 6098 (2000); J....

View on Pith 1999
[22]

J.G. Analytis et al ., cond-mat/0512388

View on Pith Pith review arXiv
[23]

See, e.g., J. Schmalian, Phys. Rev. Lett. 81, 4232 (1998)

View on Pith 1998
[24]

T. Komatsu et al ., J. Phys. Soc. Japan 65, 1340 (1996)

View on Pith 1996
[25]

W. Zheng et al ., Phys. Rev. B 71, 134422 (2005)

View on Pith 2005
[26]

M. Ogata, J. Phys. Soc. Japan 72, 1839 (2003); C. Hon- erkamp, Phys. Rev. B 68, 104510 (2003); B. Kumar and B.S. Shastry, ibid. 68, 104508 (2003); G. Baskaran, Phys. Rev. Lett. 91, 097003 (2003); B. Braunecker, P.A. Lee and Z. Wang, ibid. 95, 017004 (2005)

View on Pith 2003
[27]

D. W. Hess, T. A. Tokuyasu, and J. A. Sauls, J. Phys.: Condens. Matter l, 8135 (1989)

View on Pith 1989
[28]

For example, G.M. Luke et al ., Nature 394, 558 (1998) performed just this kind of experiment to detect broken TRS in the superconducting phase of Sr 2RuO4

View on Pith 1998
[29]

R. Prozorov et al ., Phys. Rev. B 63, 052506 (2001)

View on Pith 2001
[30]

M. Sigrist et al ., Phys. Rev. B 53, 2835 (1996)

View on Pith 1996
[31]

K.A. Kouznetsov et al ., Phys. Rev. Lett. 79, 3050 (1997). 4

View on Pith 1997

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

[1] [1]

most notably µSR [27]. Experimental conﬁrmation of a double superconducting transition and broken TRS would be extremely important, not only because of the intrinsic interest in these phenomena, but also because they would be conclusive proof of unconventional su- perconductivity in the layered organic superconductors. The implication is then that phonons...

View on Pith

[2] [2]

See, e.g., M. Sigrist and K. Ueda, Rev. Mod. Phys. 63, 239 (1991) and references therein

View on Pith 1991

[3] [3]

See, e.g., J.F. Annett, Adv. Phys. 39, 83 (1990)

View on Pith 1990

[4] [4]

See, e.g., T. Ishiguro, K. Yamaji, and G. Saito, Organic Superconductors (Springer Verlag, Heidelberg, 1998)

View on Pith 1998

[5] [5]

For a review see R.H. McKenzie, Comments Condens. Matter Phys. 18, 309 (1998)

View on Pith 1998

[6] [6]

K. Kanoda, Physica C 282-287, 299 (1997)

View on Pith 1997

[7] [7]

B.J. Powell and R.H. McKenzie, Phys. Rev. Lett. 94, 047004 (2005)

View on Pith 2005

[8] [8]

F. L. Pratt et al ., Polyhedron 22 2307 (2003); Phys. Rev. Lett. 94, 097006 (2005); B.J. Powell and R.H. McKenzie, J. Phys.: Condens. Matter. 16, L367 (2004)

View on Pith 2003

[9] [9]

F. Kagawa, K. Miyagawa, and K. Kanoda, Nature 436, 534 (2005)

View on Pith 2005

[10] [10]

Y. Shimizu et al ., Phys. Rev. Lett. 91, 107001 (2003)

View on Pith 2003

[11] [11]

M. Tamura and R. Kato, J. Phys.: Condens. Matter 14, L729 (2002)

View on Pith 2002

[12] [12]

For a review see Y. Shimizu et al ., Prog. Theo. Phys. Supp. 159, 52 (2005)

View on Pith 2005

[13] [13]

For a review see K. Miyagawa, K. Kanoda, and A Kawamoto, Chem. Rev. 104, 5635 (2004)

View on Pith 2004

[14] [14]

J. Merino and R.H. McKenzie, Phys. Rev. Lett. 87, 237002 (2001); R.H. McKenzie et al ., Phys. Rev. B 64, 085109 (2001); A. Greco et al ., ibid. 71, 144502 (2005)

View on Pith 2001

[15] [15]

R.H. McKenzie, Science 278, 820 (1997)

View on Pith 1997

[16] [16]

For a review see K. Kuroki, cond-mat/0601060

View on Pith Pith review arXiv

[17] [17]

For a review see B.J. Powell and R.H. McKenzie, Phys. Rev. B 69, 024519 (2004)

View on Pith 2004

[18] [18]

X=Me4Z, Et 2Me2Z; Z=P, Sb; Me=CH 3 and Et=C 2H5

View on Pith

[19] [19]

See, e.g., A. Damascelli, Z. Hussain, and Z.-X. Shen, Re v. Mod. Phys. 75, 473 (2003)

View on Pith 2003

[20] [20]

J. Liu, J. Schmalian, and N. Trivedi, Phys. Rev. Lett. 94, 127003 (2005); J. Y. Gan et al ., ibid. 94, 067005 (2005)

View on Pith 2005

[21] [21]

A. Carrington et al ., Phys. Rev. Lett. 83, 4172 (1999) measured the penetration depth of κ-Br and κ-NCS down to 0.4 K and observed power law dependences, although other measurements (which did not go to such low tem- peratures) have found exponentially activated behaviours occur in the same materials, e.g., H. Elsinger, et al ., ibid. 84, 6098 (2000); J....

View on Pith 1999

[22] [22]

J.G. Analytis et al ., cond-mat/0512388

View on Pith Pith review arXiv

[23] [23]

See, e.g., J. Schmalian, Phys. Rev. Lett. 81, 4232 (1998)

View on Pith 1998

[24] [24]

T. Komatsu et al ., J. Phys. Soc. Japan 65, 1340 (1996)

View on Pith 1996

[25] [25]

W. Zheng et al ., Phys. Rev. B 71, 134422 (2005)

View on Pith 2005

[26] [26]

M. Ogata, J. Phys. Soc. Japan 72, 1839 (2003); C. Hon- erkamp, Phys. Rev. B 68, 104510 (2003); B. Kumar and B.S. Shastry, ibid. 68, 104508 (2003); G. Baskaran, Phys. Rev. Lett. 91, 097003 (2003); B. Braunecker, P.A. Lee and Z. Wang, ibid. 95, 017004 (2005)

View on Pith 2003

[27] [27]

D. W. Hess, T. A. Tokuyasu, and J. A. Sauls, J. Phys.: Condens. Matter l, 8135 (1989)

View on Pith 1989

[28] [28]

For example, G.M. Luke et al ., Nature 394, 558 (1998) performed just this kind of experiment to detect broken TRS in the superconducting phase of Sr 2RuO4

View on Pith 1998

[29] [29]

R. Prozorov et al ., Phys. Rev. B 63, 052506 (2001)

View on Pith 2001

[30] [30]

M. Sigrist et al ., Phys. Rev. B 53, 2835 (1996)

View on Pith 1996

[31] [31]

K.A. Kouznetsov et al ., Phys. Rev. Lett. 79, 3050 (1997). 4

View on Pith 1997