Table Of ContentTopological Phase Transitions in Line-nodal Superconductors
SangEun Han,∗ Gil Young Cho,∗ and Eun-Gook Moon
Department of Physics, Korea Advanced Institute of Science and Technology, Daejeon 305-701, Korea
(Dated: January 7, 2016)
Fathoming interplay between symmetry and topology of many-electron wave-functions has deep-
ened understanding of quantum many body systems, especially after the discovery of topological
insulators. Topology of electron wave-functions enforces and protects emergent gapless excitations,
andsymmetryisintrinsicallytiedtothetopologicalprotectioninacertainclass. Namely,unlessthe
symmetry is broken, the topological nature is intact. We show novel interplay phenomena between
symmetryandtopologyintopologicalphasetransitionsassociatedwithline-nodalsuperconductors.
The interplay may induce an exotic universality class in sharp contrast to that of the phenomeno-
logicalLandau-Ginzburgtheory. Hyper-scalingviolationandemergentrelativisticscalingaremain
6 characteristics,andtheinterplayeveninducesunusuallylargequantumcriticalregion. Wepropose
1
characteristic experimental signatures around the phase transitions in three spatial dimensions, for
0
example, a linear phase boundary in a temperature-tuning parameter phase-diagram.
2
n
a Superconductivity is one of the most intriguing quan- sitions is that of the Higgs-Yukawa theory, the theory
J tum many body effects in condensed matter systems : with relativistic fermions and bosons in 2d.
5 electronsformCooperpairswhoseBose-Einsteinconden- Richer structure exists in three spatial dimensions
sation becomes an impetus of striking characteristics of (3d). Inadditiontopoint-nodes, line-nodesareavailable
]
l superconductors (SCs), for example the Meisner effect in 3d. Effective phase space of line-nodal excitation is
e and zero-resistivity [1]. The pair formation suppresses qualitatively distinct from that of order parameter fluc-
-
r gapless fermionic excitation and only the superconduct- tuation as shown by codimension analysis [5–7]. Thus,
t
s ing order parameter becomes important in conventional concomitantappearanceofsymmetrybreakingandtopo-
t. SCs. But, in the unconventional SCs, fermionic exci- logical unwinding in line-nodal SCs has us expect an ex-
a tation is not fully suppressed generically, so the order otic universality class of the topological transitions.
m
parameterandfermionscoexistandrevealintriguingun- Another motivation of our work is abundant ex-
- conventional nature [2–4]. perimental evidence of line-nodal SCs in various
d
n ThefermionicexcitationinunconventionalSCsisoften strongly correlated systems in heavy fermions [13–17]
o protectedandclassifiedbyitstopologicalnature. Apath and pnictides [18–22], for example, CePtSi , UCoGe,
3
c (orsurface)inmomentumspacearoundnodalexcitation (Ba K )Fe As , Ba(Fe Co ) As , and FeSe in ad-
1−x x 2 2 1−x x 2 2
[ definesatopologicalinvariantintermsoftheBerryphase dition to the 3He polar superfluid phase [23]. Reported
1 (flux) of the Bogoliubov de-Gennes (BdG) Hamiltonian. line-nodal SCs are often adjacent to another supercon-
v Inliterature[5–7],structureoftheBdGHamiltonianhas ducting phase with a different symmetry group. Due to
5 been extensively studied and is applied to weakly corre- thesymmetrydifference,nodalstructuresoftwodifferent
7 lated systems. Proximity effects between topologically SC phases are expected to be different, so they become
9 different phases (or defects) have been investigated and ideal target systems of this work.
0
experimentally tested, focusing on a search for novel ex-
0 Inthiswork,weinvestigatequantumphasetransitions
citation such as Majorana modes [8, 9].
. out of line-nodal SCs where intriguing interplay between
1 AmongunconventionalSCs, wefocusonaclasswhose topologyand symmetryappears. We firstprovidea gen-
0
topological nature is protected by a symmetry. Namely, eral rule to investigate adjacent phases of the line-nodal
6
1 unless the symmetry is broken, topologically-protected SCs. Then, phase transitions are investigated by stan-
: nodal structure is intact. In this class, unwinding of dardmeanfieldanalysis,whichshowsgenericallycontin-
v topological invariant and spontaneous symmetry break- uous phase transitions. A novel universality class of the
i
X ingappearconcomitantlyatquantumcriticalpoints,and continuous phase transitions is discovered and charac-
r thus intriguing interplay between symmetry and topol- terized by hyper-scaling violation and relativistic scaling
a ogy is expected to appear. Therefore, topological phase with wide quantum critical region. Its striking experi-
transitions around the class of the unconventional SCs mental consequences are also discussed at the end.
become a perfect venue to investigate the interplay be-
Topological line-nodal SCs protected by a symmetry
tween topology and symmetry.
maintaintheirnodalstructureunlesstheprotectingsym-
In 2d, Sachdev and coworkers have investigated this
metry is broken. Therefore, adjacent symmetry-broken
class in the context of d-wave SCs [10–12]. They found
phases may be described by representations of the sym-
theuniversalityclassofpoint-nodevanishingphasetran-
metry. For example, the polar phase with line-nodes,
A-phase with point-nodes, and nodeless B-phase in liq-
uid 3He are described by investigating symmetry repre-
sentations of SO(3) × SO(3) × U(1) . Below, we take
L S φ
∗ Theseauthorscontributedequallytothiswork. the group G = C × T × P, one of the common lat-
4v
2
T Tc Rep. Lattice (Fs(k)Ms) Continuum #
(a) kz
A τy τy 0
1
A sin(k )sin(k )(cos(k )−cos(k ))τy sin(4θ)τy 16
Tco Quantum Critical 0 B21 x (cos(kyx)−cosx(ky))τy y cos(2θ)τy 8
Region B2 sin(kx)sin(ky)τy sin(2θ)τy 8
E sin(k )sin(k )τy, cos(θ)τyµz, 4
x z
sin(k )sin(k )τy sin(θ)τyµz
TQC (b) kz y z
TRS broken Nodal line
SC SC TABLE I. C representations for topological phase transi-
4v
tions. For simplicity, T broken and spin-singlet representa-
0
r tions are only illustrated. The first column is for represen-
tations. The second column is the matrix structure in the
g Gaussian Nambu space. The third column is for continuum represen-
(g=0)
tations near nodal lines. The last column is for the numbers
of the nodal points in each representation.
FIG. 1. Phase Diagram and RG flow. Three axes are for
temperature (T), the tuning parameter (r), and the coupling
between order parameter and line-node fermions (g). In r-
It is obvious that T-symmetry breaking superconduc-
T plane, critical region is parametrically wider than conven-
tional φ4 theory’s. In r-g plane, the RG flow is illustrated by tivity (the term with τy) changes nodal structure, so or-
arrows. The “Gaussian” fixed point has Laudau MFT’s crit- derparameterrepresentationsfortopologicalphasetran-
ical exponents due to the upper critical dimension. Once the sitionscanbeillustratedasinTableI.Grouptheoryanal-
couplinggturnson,theGaussianfixedpointbecomesdesta- ysisguaranteescouplingtermsbetweenorderparameters
bilized and RG flows go into ‘TQC’. At T=0, the left (red) and fermionic excitation,
sphere is for the ordered phase, and the right (blue) sphere
is for the disordered phase. Tc plays the high energy cutoff, H =(cid:88)φ (cid:88)Ψ†F (k)MsΨ ,
and T is for critical temperature of the symmetry breaking ψ−φ s k s k
co s k
order parameter. (a) Nodal lines in momentum space in the
symmetric phase are illustrated at kz = ±kz∗ in addition to where s is for representations (and multiplicity) and
the zero point k = 0 (black dot). (b) nodal points in mo- F Ms are illustrated in Table I. For detail of this clas-
s
mentum space in a symmetry broken phase (8 nodal points).
sification, see the supplemental material (SM) A. Note
that s = E is a two dimensional representation, so the
corresponding order parameter (φ ) has two compo-
s=E
nents.
tice groups in line-nodal SC experiments (here P and T Two topologically different cases exist. First, mo-
are for particle-hole and time-reversal symmetries), as a mentum independence of A1 representation makes the
prototype. Its generalization to other groups is straight- fermion spectrum gapped completely, so-called is pair-
forward. ing. In the 3He context, this phase corresponds to the
weakly T-broken analogue of the B-phase. Second, the
Itiswellunderstoodinliterature[5]thattheSCmodel
order parameters in A , B , B , and E-representations
with the symmetry group G 2 1 2
leaves point nodes due to angular dependence. Nodal
H =(cid:88)Ψ†(cid:16)h(k)τz+∆(k)τx(cid:17)Ψ , (1) points appear when Fs(k) has zeros on line nodes and
0 k k is in fact Weyl nodes. This phase corresponds to the
k A-phase in 3He.
Armed with understanding of adjacent symmetry bro-
hasline-nodesprotectedbyT-symmetry. Afourcompo-
ken phases, we consider topological phase transitions.
nentspinorΨ† =(ψ†,iσyψT )whereψ† =(c∗ ,c∗ )is
k k −k k k,↑ k,↓ Standard mean field theory (MFT) with on-site inter-
introduced, and the particle-hole (spin) space Pauli ma- action −u(Ψ†τyΨ)2 gives a mean field free energy den-
trices τx,y,z (σx,y,z) are used. The τz term describes a
sity of ‘isotropic’ A representation order parameter (i-s
normalstatespectrumh(k)=(cid:15)(k)−µ+α(cid:126)l(k)·(cid:126)σ,andthe pairing), 1
τx term describes a pairing term ∆(k)=(∆ +∆ d(cid:126)(k)·
s t 1 1
(cid:126)σ). Energy dispersion (cid:15)(k) = −2t(cos(kx)+cos(ky)+ FMF(φ)=(u − u +T)φ2+kf|φ|3+··· . (2)
cos(k )) is introduced with spin-orbit coupling strength c
z
α. Theorbitalaxisofthepairingandspin-orbittermsare Coefficients of each term are scaled to be one and ···
identicald(cid:126)(k)=(cid:126)l(k)=(sin(k ),sin(k ),0)whichusually is for higher order terms. Notice that the unusual |φ|3
x y
maximizes T [24, 25]. The pairing amplitudes {∆ ,∆ } term appears whose presence is solely from line-nodal
c s t
are chosen to be real and positive without losing gener- fermions manifested by k . It also guarantees the phase
f
ality because of the T-symmetry. As illustrated in Fig.1 transition is continuous and makes the usual φ4 term
(a), the system exhibits two topological line nodes sepa- irrelevant. Furthermore, the order parameter critical ex-
ratedinmomentumspaceprotectedbytheT-symmetry. ponent becomes significantly different from one of the
3
Landau MFT (which only contains bosonic degrees of The coupling term is also written in terms of the low-
freedom), (cid:104)φ(cid:105) ∼ (u − u ) giving β = 1 which already energy degrees of freedoms
c
suggests a novel universality class.
(cid:90) (cid:90)
We investigate quantum criticality around the contin- g H ≈g φ F(θ )Ψ† MΨ ,
uous phase transitions. For simplicity, we omit the sub- ψ−φ q,Ω k k+q,ω+Ω k,ω
x k,ω,q,Ω
script r and introduce one real scalar field φ to describe
the order parameter. Its generalization to the E repre- so-called the Yukawa coupling. We use Shankar’s de-
sentation with two scalar fields is straightforward. composition of fermion operators around the line node,
In the phenomenological Landau-Ginzbug theory, or- Ψk ≈Ψ(δkz,δk⊥,θk;µz).
der parameter fluctuation near quantum phase transi- The standard large-Nf analysis is performed by intro-
tions may be described by ducingNf-copiesoffermionflavorscoupledtotheboson
φ. The lowest order boson self-energy Σ (Ω,q) can be
b
(cid:90) 1 1 r λ obtained by the usual bubble diagram. For A represen-
S = (∂ φ)2+ (∇φ)2+ φ2+ φ4. (3) 1
φ 2 τ 2 2 4! tation, the boson self-energy is
x,τ
Ofcourse, thisactionisnotcompleteinoursystemsand (cid:90) (cid:104) (cid:105)
Σ (Ω,q)=N g2 Tr τyG (ω,k)τyG (ω+Ω,q+k) ,
necessary to be supplemented by the corrections from b f f,0 f,0
k,ω
fermions. Without the coupling between the order pa-
risamweetllerunanddersfetormodio,nsso,-ctahlelecdritthicealφt4htehoeryorSyφ:winith3dr,=itrics waghaetroer.GN−f,10o(tωic,ekt)h=at−thiωe+inHte0g(rka)tiiosnthiseobvaerreffeerrmmiioonnicprmopo--
at the upper-critical dimension. Thus, the Landau MFT mentum and frequency, thus main contribution comes
workswelluptologarithmiccorrectionandhyper-scaling from the line-nodal fermions. Basically, the momentum
is satisfied. Below, we show that the coupling to the integration can be replaced with energy integration with
fermionssignificantlychangeslowenergyphysicsandin- D ((cid:15)) ∼ k |(cid:15)|. The integration is straightforward (see
f f
duces a novel universality class. SM C.1) and we find
The total action with fermions is
(cid:113)
(cid:90) (cid:90) δΣ (Ω,q)=C(k N ) Ω2+v2q2+v2q2 el[ρ(Ω,q)],
S =S +S , S = Ψ†(∂ +H )Ψ+g H . b f f z z ⊥ ⊥
c φ ψ ψ τ 0 ψ−φ
x,τ τ with δΣ ≡ Σ (Ω,q)−Σ (0,0) and C = g2 . The
A coupling constant g characterizes strength of the cou- completeb elliptbic integralbel[x] and variabl4eπvρ⊥(vΩz,q) =
ptolinniganbdetewnesietnytHhe ifserimntiroondsuacnedd(bHoso=ns(cid:82), aΨnd†HtheΨ)H.amil- 1/(1+ Ω2v+2vqz22qz2) are used. The elliptic integral is well-
0 0 x 0 defined in⊥all⊥range of momentum andfrequency, thus as
Near the phase transitions, low- energy and momen-
the lowest order approximation, one can treat the inte-
tum degrees of freedom become important, so we only
gral as a constant since 1≤el[x]<2.
needthelow-energycontinuumtheoryoftheBdGHamil-
Two remarks follow. First, the linear dependence in
tonian Eq.(1) near nodes and obtain
momentum and frequency can be understood by power-
H (k)≈v δk µzτz+v δk τx, (4) counting with the linear fermionic density of states. Sec-
0 z z ⊥ ⊥
ond, the boson propagator contains the factor N k .
f f
Thus, one can understand the large-N analysis as an
where the momentum is k = ((k +δk )cos(θ ),(k + f
δk⊥)sin(θk),kz∗µz + δkz). Herefµz =⊥ ±1 rekpresefnts expansion with Nf1kf factor. The presence of kf al-
“which-line-node” index and the effective parameters ready suggests suppression of infrared divergences in
{v ,v }arethefunctionsofthemicroscopicparameters. loop-calculations (see below).
z ⊥
Thelowenergyfermionspectrum(say,µz =+1)without The modified boson action is
the fermion-boson coupling is
(cid:90) |φ |2(cid:16) (cid:17)
Seff = q,Ω r˜+q2+Ω2+δΣ (Ω,q)) +··· ,
(cid:15)0(δkz,δk⊥,θk)=±(cid:112)(vzδkz)2+(v⊥δk⊥)2. (5) φ q,Ω 2 b
with r˜= r+Σ (0,0). The self-energy manifestly dom-
Oneparameter,theangle0≤θ ≤2π,characterizeszero b
k
inates over the bare terms at long wavelength, thus
energy states, so a nodal line exists in momentum space.
the bare terms may be ignored near the critical point
Density of states near zero energy vanishes linearly in
(r˜= 0) and the boson propagator becomes G (Ω,q) →
(cid:15), D ((cid:15))∼k |(cid:15)| in a sharp contrast to ones of Fermi sur- b
facesf (∼ (cid:15)0),fnodal points (∼ (cid:15)2), and order parameters δΣb(Ω,q)−1.
(∼ (cid:15)2). It is clear that the phase space of the nodal-line The back-reaction of the bosons to the fermions is ob-
tained by the fermion self-energy,
fermion excitation is different from that of fluctuation of
the order parameter. Such difference in the phase spaces (cid:90)
of the bosons and fermions is a consequence of the codi- Σ (ω,k)=g2 τyG (ω+Ω,k+q)τyG (Ω,q).
f f b
mension mismatch. Ω,q
4
Straightforward calculation shows the corrections to the QCP in 3d z ν β γ η HS
parametersofthebarefermionactionEq.(4)hasthefol-
φ4 theory[26] 1 1 1 1 0 O
lowing structure, 2 2
Higgs-Yukawa[26, 27] 1 1 1 1 0 O
2 2
δΣf(ω,k) ∝ 1 ×(Λ−µ), QBT-QCP[28, 29] 2 1 2 1 1 O
δ(cid:15)a Nfkf Hertz-Millis[31, 32] 2 or 3 1 1 1 0 X
2 2
Nodal line QCP 1 1 1 1 1 X
where (cid:15)a =(ω,v δk ,v k ), and Λ and µ are the ultra-
⊥ ⊥ z z
violet (UV) and infrared (IR) cutoffs. k is the largest
f TABLE II. Critical theories of QCP in three spatial dimen-
momentum scale, k (cid:29) Λ (cid:29) µ in this work. The same
f sions (d = 3). The first raw includes critical exponents
cutoffdependenceinthevertexcorrectionisfound(omit- (Ω∼qz,ξ−1 ∼|r−r |ν,χ ∼|r−r |−γ,and[φ]= d+z−2+η ).
ted here and see SM C for details). ‘HS’ is for hyper-scacling. φBoth Higcgs-Yukawa and φ42theory
Two remarks follow. First, the momentum integra-
areattheuppercriticaldimension,sotheexponentsareones
tion captures order parameter fluctuation, so it may be oftheLandauMFT.Bothquadraticbandtouchingquantum
replaced with energy integration with ∼ (cid:15)2 density of critical point (QBT-QCP) and Nodal line QCP have wider
states. Next, the cutoff dependence is a result of the quantum critical region ν = 1 with large anomalous dimen-
large-Nf expansion with kf as discussed before. The ab- sion η=1 obtained by large-Nf analysis.
sence of the infrared divergence indicates perturbation
theory works well. Thus, fermions and bosons become
basically decoupled at low energy. In renormalization nodal fermions, and the second term is from order pa-
group sense, this indicates the vertex operator is irrele- rameter fluctuations with d/z =3 (see SM D).
vant at low energy. For other representations, the cor- The hyper-scaling is violated even in 3d. If not, one
responding angle dependent functions F(k)M appear in wouldgettheorderparametercriticalexponent,β((cid:104)φ(cid:105)∼
the integrands (see SM C.1 for details) which does not (r − r)β) by the scaling relation, β = (d+z−2+η)ν =
c 2
modify divergence structure. 3. But, we already observe β = 1 in our MFT in Eqn.
2
The critical theory associated with topological line- (2), and also the perturbative calculation in our critical
nodal SCs is theory gives (see SM E)
(cid:90) (cid:113) |φ|2
Sφc = Nfkf Ω2+vz2qz2+v⊥2q⊥2 R(ρ(Ω,q)) 2 (,6) r˜+Σb(0,0;T)−Σb(0,0;T =0)∼r˜+T,
Ω,q
givingthecriticaltemperaturescaling, T ∼|r −r|=r˜
co c
setting r˜= 0. R(ρ(Ω,q)) is an order one non-zero posi- which gives qualitatively wider quantum critical region
√
tive well-defined function to characterize representations than one of the Landau MFT, T ∼ r −r. The
co,L c
(see SM C.1). Therefore, critical exponents do not de- hyper-scaling violation indicates the Yukawa coupling is
pend on R(x). We omit the φ4 term which is justified dangerously irrelevant. In Table II, we compare our crit-
below. ical theory with other critical theories in 3d [26–32] in
Let us list striking characteristics of our critical the- terms of critical exponents and hyper-scaling applicabil-
ory. First,thedampingterm,kf|Ω|,atq =0exists. The ity.
presenceofthedampingtermappearsduetotheabsence
Remark that our low energy theory has a larger sym-
of the Ward identity in our systems in a sharp contrast
metry than one of the original system, namely U(1) ro-
to line-nodal normal semimetal with the Coulomb inter-
tational symmetry not the original C . Thus, k is in-
4v f
action. Its form is the same as the Hertz-Millis theory of
dependent of the angle θ . This is an artifact of the lin-
k
antiferromagnetictransitions,butmomentumdepedence
earizationapproximation,butitisnotdifficulttoseethe
is also linear, so the dynamic critical exponent is rela-
universality class is not modified by inclusion of symme-
tivistic (z =1).
try breaking terms down to C unless singular fermion
4v
Moreover, the anomalous dimension of the order pa-
spectrum such as nesting appears.
rameterislarge(η =1), sothescalingdimensionofthe
φ Thisisbecausethecodimensionmismatchisthekeyof
order parameter is [φ] = d+z−2+ηφ = 3. This is com- lineardependenceofmomentumandfrequencyinthebo-
2 2
pletely different from one of the Landau theory (φ4 the- sonself-energywiththepresenceofk andtheabsenceof
f
ory) at the upper critical dimension (d = 3 with z = 1). IRdivergenceinthefermionself-energy. Thus,allcritical
Due to the large anomalous dimension, the correlation exponents are the same as ones of Eqn.(6). This is also
lengthbehavesξ−1 ∼|r−r |,soν =1. Also,theanoma- consistentwithpreviousliteratureonquantumcriticality
c
lousdimensionmakestheφ4 couplingirrelevant,[λ]<0. [33, 34]. We also explicitly show the linear dependence
So our critical theory is stable which becomes a sanity- without the linearized fermion dispersion approximation
check of the MFT in Eqn.(2). in supplementary information.
The susceptibility exponent is γ = 1, and the Fisher We now discuss experimental implication of our re-
equality is satisfied (2−η )ν = γ. Basically decoupled sults. First, our results provide additional smoking gun
φ
fermions and bosons contribute to specific heat indepen- signature of line-nodal SCs. Namely, the linear phase
dently, C ∼ a T2 +a T3. The first term is from line- boundary T ∼ (r − r), from hyper-scaling viola-
v f b co c
5
tion, between two different SCs identifies the presence itatively different from that of the Landau MFT result
√
of line nodes. Interestingly, some experiments in heavy δσ (r ,T = 0) ∝ r −r, which manifestly shows con-
L c c
fermions, for example UCoGe, suggested that a phase sequences of the hyperscaling violation.
boundary between two different SCs is linear [16, 17]and In conclusion, we have described topological phase
one of SCs at least has line-nodes though further thor- transitions associated with line-nodal SCs where topol-
ough investigation is necessary. ogy and symmetry reveal intriguing interplay phenom-
Ouranalysisindicatesthatcontinuousquantumphase ena. We find quantum criticality naturally appears and
transitions associated with line-nodal SCs have a linear its universality class of the transitions shows novel char-
phase boundary. We argue its converse statement also acteristics such as emergent relativistic scaling, hyper-
works. Quantum criticality without line-nodal SCs have scaling violation, and unusually wide quantum critical
at most point-nodal fermion excitation. Then, codimen- region. Our results can also be applied to topological
sions of order parameter fluctuation and fermion excita- phase transitions out of normal nodal ring semi-metals
tion are the same. Therefore, the Yukawa term and φ4 naturally if chemical potential is fixed to be zero. Fu-
terms would be (marginally) irrelevant as usual. Thus, ture theoretical studies should include more comprehen-
we expect quantum criticality without line-nodes would sive treatment of perturbations of critical points such as
have Landau MFT critical exponents with logarithmic finite temperature and magnetic field effects. Concrete
corrections. Detaileddiscussiononthispointwillappear connection with experiments especially in heavy fermion
in another place. systems would be also desirable.
Furthermore,directmeasurementofcriticalexponents
is possible. In particular, the discussed fluctuation of
the T-breaking order parameters has been extensively
ACKNOWLEDGMENTS
studied in a context of p-wave SCs in high tempera-
ture SCs[35–40]. Following the literature [40–42], one
can investigate a concrete way to measure the fluctua- It is great pleasure to acknowledge valuable discussion
tion, namely, the spin polarized muon scattering. From with H. Choi, Y. Huh, and Y. B. Kim. E.-G. Moon
our critical exponents, we obtain the change in the dis- especially thanks S.-S. Lee for discussion about UV/IR
tribution δσ of internal magnetic fields relative to the mixing and Y. Huh and Y. B. Kim for previous collabo-
T-symmetric phase is δσ(r,T) ∝ (cid:104)φ(r,T)(cid:105). Then, our ration. This work was supported by the Brain Korea 21
scaling analysis gives δσ(r,T)∝(r −r)F( T ) with a PLUSProjectofKoreaGovernmentandKAISTstart-up
c r−rc
scaling function F. Thus, the T-breaking signal is qual- funding.
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1
Supplemental Material for “Topological Phase Transitions in Line-nodal
Superconductors”
SangEun Han, Gil Young Cho, and Eun-Gook Moon
Department of Physics, Korea Advanced Institute of Science and Technology, Daejeon 305-701, Korea
Appendix A: Order Parameters and Nodal Structure of T-broken Phases
Inthissupplementalmaterial,wepresentthedetailedderivationoftheorderparameters(inTable1.) ofthelattice
model Eq.(1) whose symmetry group is C ×T ×P symmetry (here T and P represent the time-reversal symmetry
4v
and the particle-hole symmetry) and the nodal structures of T-broken phases. We also discuss the polar phase of
He3, i.e., the nodal line phase experimentally found in He3, and its proximate phases.
1. Derivation of Order Parameters
Here we start with the lattice Hamiltonian Eq. (1) of the maintext,
(cid:88) (cid:16) (cid:17)
H = Ψ† h(k)τ˜z+∆(k)τ˜x Ψ , (A1)
0 k k
k
with h(k)=(cid:15)(k)−µ+α(cid:126)l(k)·(cid:126)σ, where (cid:15)(k)=−2t(cos(k )+cos(k )+cos(k )) and(cid:126)l(k)=(sin(k ),sin(k ),0). The
x y z x y
pairing is given by ∆(k) = ∆ +∆(cid:126)l(k)·(cid:126)σ. The full cubic lattice symmetry is broken by(cid:126)l(k) = (sin(k ),sin(k ),0)
s t x y
down to C with the C rotation in xy-plane.
4v 4
We first demonstrate the existence of the line nodes by employeeing the basis which diagonalizes(cid:126)l(k)·(cid:126)σ =±|(cid:126)l(k)|,
i.e., the helicity basis. In this basis, the Hamiltonian can be block-diagonalized, depending on the signs of the
eigenvalues of(cid:126)l(k)·(cid:126)σ, i.e.,(cid:126)l(k)·(cid:126)σ →σ|(cid:126)l(k)|,σ =±1, to write
(cid:88)(cid:16) (cid:17)
H = Ψ† H (k)Ψ +Ψ† H (k)Ψ , (A2)
0 +,k + +,k −,k − −,k
k
where
H (k)=(cid:32)(cid:15)(k)−µ+σα|(cid:126)l(k)| (cid:16) ∆s+σ∆t|(cid:126)l(k)| (cid:17)(cid:33)=(cid:16)(cid:15)(k)−µ+σα|(cid:126)l(k)|(cid:17)τz+(cid:16)∆ +σ∆ |(cid:126)l(k)|(cid:17)τx, (A3)
σ ∆ +σ∆ |(cid:126)l(k)| − (cid:15)(k)−µ+σα|(cid:126)l(k)| s t
s t
where we have introduced the Pauli matrix τx,y,z acting on the two-component Nambu spinor χ . On writing the
σ,k
Hamiltonian into this form, we can easily calculate the BdG spectrum
(cid:114)
(cid:16) (cid:17)2 (cid:16) (cid:17)2
E (k)=± (cid:15)(k)−µ+σα|(cid:126)l(k)| + ∆ +σ∆ |(cid:126)l(k)| . (A4)
σ s t
Because α > 0, ∆ > 0 and ∆ > 0, σ = +1 is fully gapped and σ = −1 is nodal if 2∆ > ∆ (see below). The
s t t s
position of the zero-energy manifold of the BdG fermion, i.e., line nodes, are identified by
∆ =∆ |(cid:126)l(k)|,
s t
(cid:15)(k)−µ−α|(cid:126)l(k)|=0. (A5)
From the above discussion, it is apparent that the terms ∝τ˜y →τy will gap out the nodes. To see this clearly, we
first imagine to add δH =φ(cid:80) F(k)Ψ†τ˜yΨ to H to find
k k k 0
(cid:88) (cid:16) (cid:17)
H +δH = Ψ† h(k)τ˜z+∆(k)τ˜x+φF(k)τ˜y Ψ . (A6)
0 k k
k
By proceeding to the helicity basis again, we find that
(cid:16) (cid:17) (cid:16) (cid:17)
H (k)→ (cid:15)(k)−µ+σα|(cid:126)l(k)| τz+ ∆ +σ∆ |(cid:126)l(k)| τx+φF(k)τy, (A7)
σ s t
2
whose BdG spectrum is given by
(cid:114)
(cid:16) (cid:17)2 (cid:16) (cid:17)2
E (k)=± (cid:15)(k)−µ+σα|(cid:126)l(k)| + ∆ +σ∆ |(cid:126)l(k)| +φ2F2(k). (A8)
σ s t
We are particularly interested in σ =−1 which is of the lowest energy, and E (k) can be zero if
−
∆ =∆ |(cid:126)l(k)|,
s t
(cid:15)(k)−µ−α|(cid:126)l(k)|=0,
F(k)=0, (A9)
which are more stringent conditions than Eq.(A5). Hence the term ∝τ˜y lifts the line nodes to the full gap for F(k)
being nonzero constant on the line node, or the point nodes for F(k) having the zeros on the line node.
Hence we classify the possible mass term according to the symmetry C ×T ×P for the lattice model. For the
4v
classification, it is instructive to write out the mass term
(cid:88) (cid:88)(cid:16) (cid:17)
δH =φ F(k)Ψ†τ˜yΨ =φ× iF(k)ψ†(iσy)ψ∗ +h.c., , (A10)
k k k −k
k k
which is the imaginary component of the singlet pairing between the electrons (remember ψ =(c ,c )T). Hence
k k,↑ k,↓
we immediately notice that it is time-reversal odd, i.e., δH breaks T-symmetry as expected (otherwise, the line node
is protected and stable). Furthermore, it is part of the pairing and thus, by definition, is particle-hole symmetric.
Secondly, it is the singlet pairing between the electrons. This implies that
(cid:88) (cid:88)
g ∈C : F(k)ψ†(iσy)ψ∗ → F(g−1[k])ψ†(iσy)ψ∗ , (A11)
4v k −k k −k
k k
in which g−1[k] is the map of k under g−1 with g ∈ C (because the symmetry C is a unitary symmetry). Thus
4v 4v
the form factor F(k) solely determines the representation class of the order parameters. Now given this information,
it is straightforward to classify the mass terms (or order parameters).
1.A representation: F(k)=const. (is- pairing)
1
2.A representation: F(k)=sin(k )sin(k )(cos(k )−cos(k )). (ig-pairing)
2 x y x y
3.B representation: F(k)=cos(k )−cos(k ). (id -pairing)
1 x y x2−y2
4.B representation: F(k)=sin(k )sin(k ). (id -pairing)
2 x y xy
5.E representations: F(k)=sin(k )sin(k ), or F(k)=sin(k )sin(k ).(id - and id - pairings)
x z y z xz yz
This is the set of the order parameters present in table 1 of the maintext.
2. Nodal Structure of T-broken Phases
We now present the detailed nodal structure of the T-broken phases. To investigate the nodal structure, it is
beneficial to proceed to the low-energy continuum limit of the lattice Hamiltonian Eq. (1) in the maintext and the
T-breaking order parameters in table 1 of the maintext.
To project to the low-energy limit, we first ignore the σ = +1 band and take only the σ = −1 band in Eq.
(A2) supplemented by the approximations (cid:15)(k) = −2t(cos(k )+cos(k )+cos(k )) → k2 with m = 1 and(cid:126)l(k) =
x y z 2m 2t
(sin(k ),sin(k ),0) ≈ (k ,k ,0) = k . Then it is straightforward to demonstrate that the σ = −1 band of the BdG
x y x y ⊥
Hamiltonian of Eq. (A2) becomes,
H ≈v δk τzµz+δk (v τx+ζτz), (A12)
z z ⊥ ⊥
in which τα is acting on the particle-hole basis (in this σ = (−1)-band), µz is acting on the “valley” index, i.e.,
µz =+1 for the node at k =k∗ and µz =−1 for the node at k =−k∗. Here
z z z z
k∗ k
v = z, v =−∆ , and ζ = f −α. (A13)
z m ⊥ t m
3
Thus the low-energy Hamiltonian is given by
(cid:88) (cid:90) dθ (cid:16) (cid:17)
H ≈ Ψ(δk ,δk ,θ)† (v δk µz+ζδk )τz+v δk τx Ψ(δk ,δk ,θ), (A14)
2π z ⊥ z z ⊥ ⊥ ⊥ z ⊥
δk⊥,δkz
in which the momentum of the quasiparticle is given by
k=(k∗µz+δk ,(k +δk )cos(θ),(k +δk )sin(θ)), (A15)
z z f ⊥ f ⊥
i.e., we have moved from the cartesian coordinate to the polar coordinate.
To investigate the nodal structure of T-broken phases, we next project the coupling between the order parameters
and BdG fermions δH =φ(cid:80) F(k)Ψ†τ˜yΨ to the σ =−1 band. The projection can be effectively done through :
k k k
sin(k )→k ∝cos(θ ),
x x k
sin(k )→k ∝sin(θ ),
y y k
cos(k )−cos(k )→k2−k2 ∝cos(2θ ),
x y x y k
sin(k )→µz. (A16)
z
With this in hand, we can write out the mass terms in terms of the low-energy fermions,
(cid:88) (cid:90) dθ (cid:16) (cid:17)
δH =φ Ψ(δk ,δk ,θ)† F(θ)τy+G(θ)τyµz Ψ(δk ,δk ,θ), (A17)
2π z ⊥ z ⊥
δk⊥,δkz
where we have
1.A representation: F(θ)=1. G =0.
1
2.A representation: F(θ)=sin(4θ). G =0.
2
3.B representation: F(θ)=cos(2θ). G =0.
1
4.B representation: F(θ)=sin(2θ). G =0.
2
5.E representations: F(θ)=0. G(θ)=cos(θ) or G(θ)=sin(θ) (two-dimensional representation).
With these in hand, we can now investigate the nodal structures of each phase. Specifically, we will show
the existence of the Weyl nodes for A , B , B , and E representations, and that of the full gap for A representation.
2 1 2 1
We will mainly consider the Hamiltonian for the one-dimensional representation cases
(cid:90) (cid:16) (cid:17)
H ≈ Ψ† (v δk µz+ζδk )τz+v δk τx+φF(θ )τy Ψ , (A18)
k z z ⊥ ⊥ ⊥ k k
k
but it is straightforward to generalize to the two-dimensional representation E. To see the structure clearly, we first
transform v δk µz+ζδk →v δk µz by translating k →k −ζδk µz/v . Then we have
z z ⊥ z z z z ⊥ z
(cid:90) (cid:16) (cid:17)
H ≈ Ψ† v δk µzτz+v δk τx+φF(θ )τy Ψ . (A19)
k z z ⊥ ⊥ k k
k
Below we consider only A and B representations but the consideration below can be easily generalized to the other
1 2
representations.
a. A -representation
1
We show the full gap of A representation, we simply need to diagonalize the Hamiltonian
1
(cid:90) (cid:16) (cid:17)
H ≈ Ψ† v δk µzτz+v δk τx+φτy Ψ , (A20)
k z z ⊥ ⊥ k
k
and find
(cid:113)
E(k)=± v2δk2+v2δk2 +φ2. (A21)
z z ⊥ ⊥
It is clear that as far as φ(cid:54)=0, the spectrum is fully gapped.
4
b. B -representation and others
2
We now keep the dependence on F(θ) here. By diagonalizing
(cid:90) (cid:16) (cid:17)
H ≈ Ψ† v δk µzτz+v δk τx+φF(θ )τy Ψ , (A22)
k z z ⊥ ⊥ k k
k
we have
(cid:113)
E(k)=± v2δk2+v2δk2 +φ2F2(θ ). (A23)
z z ⊥ ⊥ k
The spectrum is gapless where the form factor F(θ) vanishes. Other points on the ring such that F (cid:54)= 0 will be
gapped out. The point nodes are in fact Weyl point nodes which corresponds to the hedgehogs in momentum space.
To demonstrate this explicitly, we choose B -representation as an example and expand the mean-field Hamiltonian
2
near the point node at θ = 0 and µz = 1 for simplicity. Near this point, the fermionic BdG Hamiltonian can be
expanded
(cid:90) (cid:16) 2φk (cid:17)
H = Ψ† v k τz+v k τx+ yτy Ψ, (A24)
z z ⊥ x k
k f
inwhichδk =k andθ = ky atthevicinityofθ =0. ThisistheHamiltonianforthetopologicalWeylfermionswith
⊥ x kf
the winding number is +1. The analysis can be generalized to the other point nodes in the other representations.
3. Line-nodal p -paired Phase and Proximate T-broken Phases
z
Here we discuss the line nodal p -paired phase which may arise from the liquid He3. This is so-called the polar
z
phase. The normal state is described by ξ(k)= k2 −µ. Note the absence of the spin-orbit coupling. We concentrate
2m
onaparticularline-nodalpairedstatehere, butitcanbeeasilygeneralizedtoanyline-nodalp-wavesuperconducting
state. The pairing state that we are interested in is given by the pairing
∆ =(cid:104)c† [d(cid:126)(k)·(cid:126)σ(iσy)]α,βc† (cid:105), (A25)
t k,α −k,β
in which the orbital axis of the triplet pairing is given by d(cid:126)(k)=(0,0,k ).
z
For this paired state, we can use the Nambu basis Ψ =(c ,c† ) to write out the BdG Hamiltonian
k k,↑ k,↓
(cid:32) (cid:33)
(cid:88) ξ(k) ∆ k
H = Ψ†(k) t z Ψ(k). (A26)
∆ k −ξ(k)
k t z
It is easy to confirm that this paired state has the symmetry group C ×T ×P.
4v
There is a line node at
(cid:112)
k =0, and k =|(k ,k ,0)|= 2mµ, (A27)
z f x y
which is protected by T-symmetry. Furthermore, by expanding the Hamiltonian near the node, we obtain the low-
energy theory
k
H = fδk τz−∆ δk τx (A28)
k m ⊥ t z
renaming the variables kf →v and −∆ →v , we arrive at the low-energy Hamiltonian
m ⊥ t z
H =v δk τz+v δk τx. (A29)
k ⊥ ⊥ z z
As in the noncentrosymmetric SC case, we now need to classify the mass terms. To investiate the mass terms, we
first identify the symmetry actions on the low-energy BdG fermions.
1. Time-reversal symmetry
T :Ψ(δk ,δk ,θ)→iτyΨ∗(δk ,δk ,θ) (A30)
z ⊥ z ⊥
