Comment on "Fluctuation-induced first-order transition p-wave superconductors" by Qi Li, D. Belitz, and J. Toner [Phys. Rev. B79, 054514 (2009)]

0.11 MB·English

by  D. I. Uzunov

Comment on “Fluctuation-induced first-order transition p-wave superconductors”
by Qi Li, D. Belitz, and J. Toner [Phys. Rev. B 79, 054514 (2009)]
Dimo I. Uzunov∗
Collective Phenomena Laboratory, Institute of Solid State Physics,
Bulgarian Academy of Sciences, BG–1784, Sofia, Bulgaria.
(Dated: 2nd July 2006)
InthisComment,weshowthatthepaperbyQiLi,D.BelitzandJ.Toner,publishedinPhys. Rev.
B 79, 054514 (2009), contains an incomplete mean-field analysis of a simple model of Ginzburg-
2 Landau type. The latter contains a stable non-unitary phase, which has is not been found in
1 this study and is missing in the outlined picture of possible stable phases. Both mean-field and
0 renormalization-groupstudiesinthispublicationoverlookpreviouspapers,includingthoseinPhys.
2 Rev. B. Asit turns,the authors re-deriveand publish known results.

l
u PACSnumbers: 64.60.ae,64.60.De, 74.20.De
J

5
A paper by Qi Li et al. [1] is intended to present new The authors of Ref. [1] correctly claim that f(ψ) is
] results about global and local gauge effects on the phase minimized by nˆ = mˆ and ψ2 = −t/2u, provided v > 0,
0
n transitions in unconventional (p-wave) superconductors. u>0,andt<0. Theminimumisf(t,u)=−t2/4uand,
o
However, this article contains errors, misleading texts, as mentioned in Ref. [1], describes the unitary phase –
c
∗
- anddemonstratesaboldneglectingofpreviouspapers[2– the phase, characterized by the property ψ ×ψ = 0.
r
p 5], where the same results have already been published. According to these authors, in the case of v > 0, this is
u The study in Ref. [1] is based on the same methods as the only available stable order (ψ 6=0).
s thoseappliedinRef.[2–5](seealsoRef.[6–8]): (i)mean- This outline of stable phases in superconductors with
.
t field(MF)approximation[6],(ii)MF-likeapproximation v >0anddescribedbymodel(1)iscompletelywrong,as
a
[6, 7], and (iii) a renormalization group (RG) investiga- weshallseeintheremainderofthisComment. First,the
m
tion within the one-loop approximation combined with description of the unitary phase is incomplete. Second,
-
ǫ-expansion to first order in ǫ = (4−d); d is the spa- there exists another phase - a non-unitary phase, which
d
n tial dimensionality [6]. As in previous studies [2–4], the is stable when v >0.
o main tasks in Ref. [1] are MF derivation of the possi- Thecarefulanalysisoftheequationsofstate∂f/∂ψ0 =
c ble phases and RG study of the so-called “fluctuation- ∂f/∂θ=∂f/∂φ=0inevitablyleadstotwotopologically
[ ∗
induced weakly-first-orderphase transition”[7] in p-wave different domains of the unitary phase (ψ×ψ =0), as

1 superconductors. given by the actual minimizing condition nˆ = ±mˆ, i.e.,
v Here we will present compelling arguments in favor of θ =πl, where l =0,±1,... (in Ref. [1], this condition is
1 ourpointofviewforessentialdisadvantagesofthepaper. presented in a wrong way: nˆ = mˆ, i.e., θ = 2πl). The
6 Besides, we will correct some essential errors in Ref. [1]. twodomainsofthe sameunitaryphasearedistinguished
2
The authors of Ref. [1] perform a MF analysis of the by the parallel and antiparallel mutual orientations of nˆ
1
. free energy density and mˆ.
7 The stability matrix Sˆ = {S } = (∂2f/∂µ ∂µ ),
ij i j
0
where µ = µi = (ψ0,θ,φ), describes the stability of the
2
1 f(ψ)=t|ψ|2+u|ψ|4+v|ψ×ψ∗|2, (1) possible phases (equilibria) with respect to variations of
: the order parameters (ψ0,θ,φ) around their equilibrium
v values. For the unitary phase (ψ×ψ∗ =0), this matrix
(cf. Eq. (3.1) in Ref. [1]), where t, u, and v are vertex
i
X (Landau) parameters,and ψ =(ψ1,ψ2,ψ3) is a complex has the form
r vectorfield(the orderparameterofp-wavesuperconduc- −4t, 0, 0
a tors). They use the suitable representation Sˆ= 0, vt2/2u2 sin22φ, 0. (3)
(cid:0) (cid:1)
 0, 0, 0
ψ =ψ0(nˆcosφ+imˆ sinφ) (2)
UsingEq.(3)andtheexistenceconditions(t<0,u>0),
by the modulus ψ0 ≡ |ψ|, the auxiliary (“phase”) angle one obtains the stability conditions u > 0, v > 0, t > 0,
φ, and the unit vectors nˆ and mˆ (nˆ.mˆ = cosθ). The as indicated in Ref. [1].
unit vectors nˆ and mˆ point out the directions of the real Besides, there are additional circumstances, which
and imaginary vector parts of the field ψ = (ψ′+iψ′′): should be emphasized. It is readily seen from Eq. (3)
nˆ kψ′, and mˆ kψ′′. that the unitary phase is marginally stable with respect
to θ-fluctuations for angles φ = πk/2 (k = 0,±1,...),
whereas the same phase has a marginal stability to-
wards φ-fluctuations for any θ (in both cases, t < 0,
∗Electronicaddress: [email protected] u > 0, and v > 0). As shown in previous works,
2
see [2, 5] and references therein, these marginal sta- (in slightly different notations; see Sec. II and especially,
bilities towards the angles θ and φ reflect the global Sec. II.D.4 in Ref. [2]). In Refs. [2, 5], the possible u−v
gauge invariance of the model (1) and the phenomenon phasediagramsofthe free energy(1) havebeen outlined
ofglobalsymmetrybreakinginthesix-dimensionalspace in details as a particular case of a quite more general
(ℜψ1,ℑψ1,...,ℜψ3,ℑψ3) of the order parameter vector description and more realistic variants of such diagrams.
ψ. Thus the zeros of the matrix elements S22 and S33, Thus the statement in the end of the Introduction of ar-
produced by some values of θ and φ, exhibit the fun- ticle[1]thattheauthors“derivetheMFphasediagram”,
damental global symmetry of model (1) rather than an withoutmakinganystipulationforalackoforiginalcon-
uncertainty in the stability degree of the unitary phase tribution, is incorrect and unacceptable.
(the same is valid for the non-unitary phases, discussed In Sec. III.B of Ref. [1], the Eq. (3.5) for the MF-
below) [2, 5]. like renormalization of the free energy f(ψ) contains an
Ref. [1] reports only for a particular type of non- essentialerror. In this equation,the parameterw should
unitary phase (ψ × ψ∗ 6= 0), where nˆ ⊥ mˆ, i.e., θ = be related to the charge q by w ∼ q3 rather than by
π(l+1/2);existenceandstabilityconditions: (u+v)>0, the wrong relationship w ∼ q, given on the line below
v <0, and t <0. The result for θ follows as a direct so- the equation. In order to persuade himself in this, one
lution of the equations of state. Obviously, this phase shouldhavealookontheoriginalarticle[7]orsomeother
′ ′′
has the property ψ ⊥ ψ . In Sec. I-III as well as any- sources, for example, Ref. [6].
whereinthepaper[1],therearenoreferencestoprevious Besides, within the MF-like approximation, in which
papers with MF results for the phases of model (1), al- the spatial fluctuations of the field ψ are neglected but
thoughthismodelisoftenused(see[2–6],andreferences the fluctuations of the magnetic induction in the entire
therein). Moreover, the authors do not make any stip- Ginzburg-Landaufluctuation Hamiltonian are preserved
ulation for some purposive incompleteness of their con- [6,7],theparameterstanduacquireq-renormalizations,
sideration; neither in Sec. III.A nor elsewhere. Thus the too. The latter are important for the description of es-
reader remains with the wrong conclusion that the non- sential properties of the weakly-first-order phase transi-
∗
unitary phase of type ψ ⊥ ψ is the only non-unitary tion[6]. Note,thatinaccordwithEq.(2e)inRef.[4],the
∗
phase (ψ×ψ 6=0) described by the model (1). parameter v does not possess q-renormalizations within
In fact, the careful MF analysis [2] shows that the the model (1), neither inthe one-loopRG treatmentnor
model (1) contains a stable non-unitary phase in more. within the MF-like approximation.
This second non-unitary phase (2ndNUP) is given by
In Sec. III.C-D of Ref. [1], for the aim of study of
spatial fluctuations of ψ, the approximation of uniform
t π
ψ2 =− , θ 6=πl, φ= k, (4) field [ψ(x) ≈ ψ] is abandoned and the x-variations of
0
2u 2
ψ are taken into account. Within this extended scheme,
(k = 0,±1,...), the free energy minimum f = −t2/4u the critical fluctuations of the field ψ(x) can be stud-
ied with the help of RG. Here we briefly discuss the
– the same as for the unitary phase, and the stability
RG analysis of the “action” (2.3), performed in Ref. [1]
matrix
(Sec.III.C) within the one-loop approximation and first-
−4t, 0, 0 order ǫ-expansion. Note, that the mentioned level of ac-
Sˆ= 0, 0, 0 . (5) curacy in the application of RG is enough to reveal the
 0, 0, 2vt2/u2 sin2θ  main features of the fluctuation effects on the properties
(cid:0) (cid:1) of the phase transition from the normal metal state to
The 2ndNUP has the same modulus ψ0 and energy f thep-wavesuperconductingstateatthephasetransition
as the unitary phase. The difference with the unitary temperature,wherethemeanmagneticinductionisequal
phase is in the restriction φ = (π/2)k as well as in the to zero, and a particular type of fluctuation-induced
equilibrium values of angle θ. The Eq. (5) shows that weakly-first-orderphase transition may appear [6, 7].
the second non-unitary phase is stable for t < 0, u > 0, First, we note, that Eq. (2.3) does not present an ac-
and v > 0 – stability conditions, which are identical to tion, as wrongly noted in Ref. [1]. Rather, Eq. (2.3)
those for the unitary phase. In the limit θ → πl the presentsa“generalizedfreeenergy(functional)”(alias,a
structureandthestabilitypropertiesof2ndNUPcoincide “fluctuationHamiltonian”ratherthanan“action”). The
with those of the unitary phase. Having in mind the same is valid for the quantity SA, given by Eq. (4a) and
samefreeenergiesofthesephases,onemayconcludethat describing the time-independent spatial fluctuations of
the unitary phase and the second non-unitary phase are themagneticinductionfortheparticularcaseofuniform
identical for θ → πl. These results show that the u−v ψ.
phase diagramoutlined in Fig. 1 of Ref. [1] is wrong and Second, the RG analysis within the ǫ-expansion per-
should be corrected along our instructions. formedinRef. [1] is anentire repetitionofthe RGstudy
The authors of Ref. [1] have missed to find the 2nd- in Ref. [3]. Apart of some slight differences in the nota-
NUP, contained in model (1). On the other hand, they tions,the RGEqs.(3.7a)–(3.7c)arethe sameasthe first
have missed to take advantage from paper [2] (and ref- threeequations(3)inRef.[3]. Note,thatowingtoasim-
erences therein), where this phase is described in MF ple misprint in Ref. [3], the number factor 12 in the last
3
term of the equation for du/dl should be 96. This mis- pear in the known superconductors, where n does not
print is without any consequence for the results derived exceed 6.
in Ref. [3]. A second note is needed to point out that in
Thedifferenceintheǫ-analysisperformedinthesetwo
Ref. [3] we use known RG equations for the parameters
papersis inthe widerscope ofRef. [3], wherethe crystal
c, q, and µ (Refs. [7, 8]). These equations are exactly
anisotropyhasbeen takeninto account,andthe effectof
the same as Eqs. (3.7d)–(3.7f) in Ref. [1]. They do not
this anisotropy on the stability of the critical behavior
containtheparameterv andareknownfromtheoriginal
has been investigated in details. The crystal anisotropy
paper [7] as well as from numerous further publications;
enhances the effect of first-order transition. For exam-
see, e.g., Ref. [6, 8].
ple, in the present of crystal anisotropy, the critical be-
Thus our RG equations, presented in Ref. [3] and the
havior appear for m = 2n > 5494 (see Ref.[3]). Cer-
RG equations, re-derived in Ref. [1] are the same. Then
tainly,inrealmetalsandmetalliccompounds,thecrystal
itisnotstrangethattheauthorsofRef.[1]haverediscov-
anisotropyeffectonthep-wavesuperconductivitycannot
ered our RG results about the possible fixed points and
be neglected.
their stability properties, as well as about the possible
Moreover,theRGanalysisinRef.[1]isalimitedcaseof
typesofcriticalbehaviorandfirst-orderphasetransition.
theRGstudypresentedinRef.[4]. InRef.[4],thesimul-
For example, using Eqs. (3.11a) and (3.11b), the au-
taneous effect of crystal anisotropy and quenched impu-
thors of Ref. [1] rediscover the couple of fixed points of
ritieshasbeenobtainedbyageneralizationofthesimple
typeu± 6=0,u˜± 6=0,v± =0(notationsofRef.[3],where
fluctuation model (2.1)-(2.3), given in Ref. [1]. The sim-
u˜correspondstov inRef.[1];thefixedpointcoordinates
ple check of article contents undoubtedly demonstrates
v± correspond to the crystal anisotropy parameter con-
that the RG equations in Ref. [1] are a limited case of
sidered there but neglected in Ref. [1]). Moreover, the
those in Ref. [4] and describe a simplified picture, where
condition for the existence of this couple of fixed points,
both crystal anisotropy and quenched impurity effects
m > m = n /2 = 211, derived for the first time in
b b
are ignored. Hence, the RG analysis and the respective
Ref. [3] has also been re-derived in Ref. [1] in a slightly
differentnotation: 2m=n>nP ≈420.9(inaminordif- RG results in Ref. [1] exactly follow from the RG study
c
in Ref. [4] when the parameters, describing the crystal
ference with the result in Ref. [3]). Let us remind, that
anisotropy and quenched disorder are neglected.
within the present problem, the symmetry index n is an
integer equal to the total number of component of the Finally, we summarize our main findings. In Ref. [1],
complexvectorψ,whilem=n/2isequaltothe number the MF analysis is incomplete and a stable non-unitary
′ ′′
of components of the real vectors ψ and ψ . Following phase is missed. In result, the domain of stability of this
Ref. [1], in our discussion of MF results for the phases, phase should be indicated in the u−v phase diagram,
we have used n = 2m = 6. But in the RG studies large shown in Fig. (1). Excepting this error, the MF anal-
integer n are considered, too. ysis in Ref. [1] repeats previous studies and re-discover
ThemainconclusionsofRefs.[1]and [3]arethesame: known results (see Ref. [2] and references therein). The
The critical behavior appears for n > n . For n < n , parameter w in Sec.III.B should be corrected, as shown
b b
the phase transition from normal metal state to p-wave in this Comment. The RG study, performed in Ref. [1]
superconductivity is a fluctuation-induced weakly-first- within the ǫ expansion, totally repeats essential parts of
order phase transition. Under suitable physical condi- the previous RG studies (Refs. [3, 4]) and, hence, re-
tions[6],thisweakly-first-orderphasetransitionmayap- derive known results.
[1] Qi Li, D. Belitz, and J. Toner, Phys. Rev. B 79, 054514 1412 (1985) [Sov. Phys.– JETP 61, 843 (1985)].
(2009). [6] D. I. Uzunov, Introduction to the Theory of Critical Phe-
[2] E.J.Blagoeva, G.Busiello, L.DeCesare, Y.T.Millev,I. nomena(WorldScientific,Singapore–NewJersey–Lon-
Rabuffo,andD.I.Uzunov,Phys.Rev.B42,6124(1990); don, 1993); Second extended edition (World Scientific,
See Sec. Sec. II and, in particular, Sec. II.D.4; the nota- Singapore – New Jersey – London,2010).
tions are slightly different from those in Ref. [1]. [7] B.I.Halperin,T.C.Lubensky,andS.-K.Ma,Phys.Rev.
[3] Y. T. Millev and D. I. Uzunov, Phys. Lett. A 145, 287 Lett. 32, 292 (1974).
(1990). [8] G. Busiello, L. De Cesare, and D. I. Uzunov, Phys. Rev.
[4] G. Busiello, L. De Cesare, Y. T. Millev, I. Rabuffo, and B 34, 4932 (1986).
D. I. Uzunov,Phys. Rev.B 43, 1150 (1991).
[5] G.E.VolovikandL.P.Gor,kov,Zh.Eksp.Teor.Fiz. 88,